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Unit 1 Why Study Science?
Discussion 1.1
Discussion 1.2
Unit 1 Study Questions
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STUDY GU IDE
Unit 1 Why Study Science?
This unit introduces the idea that a better public understanding of science is
essential if policy decisions involving science, or the technological products of
science, are to serve productive social goals. As you work through the unit, you will
be introduced to the scientific point of view, and asked to think about the role of
science in society.
Objectives
When you have completed Unit 1, you should be able to
1. describe the scientific point of view in a general way.
2. describe the inductive attitude, and state the moral qualities associated with it.
3. provide at least three reasons why it is important to understand science, at least
in a general way.
4. discuss, in your own words, some general aspects of the relation between
science and society, and between science and religion.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 1.1, “What Is the Importance of Point of View?” in this Study
Guide.
2. Read Chapter 9, “Ultimate Questions: Science and Religion,” pages 125-132
of What Science Is and How It Works, by Gregory N. Derry.
3. Read Chapter 10, “More Practical Questions: Science and Society,” pages 133144 of What Science Is, by Gregory N. Derry.
4. Answer Unit 1 Study Questions 1-4 in this Study Guide.
5. Read Discussion 1.2, “Why Do We Need to Understand Science?” in this Study
Guide.
6. Answer Unit 1 Study Questions 5-7 in this Study Guide.
Note: in the remaining units, Discussions and Study Questions will not be identified as being located “in
this Study Guide.”
Discussion 1.1 What Is the
Importance of Point of View?
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No man ever looked at the world with pristine eyes. He sees it edited by a definite set of
customs and institutions and ways of thinking.
—Ruth Benedict (1887-1948)[1]
What we see of the world, of other people, and of our place in the world is
determined by where we stand. Culturally, everybody is conditioned into a particular
point of view by parents and peers, and many people never question it, never step
beyond it, and never recognize that their perceptions of the world are
preprogrammed. They assume that the world really is the way that they have been
taught to see it, and live their lives in a state that can be likened to sleep, or hypnosis.
If we are instructed to consider a particular thing, event, situation or idea from a
point of view different from our own, it may require much effort and almost certainly
will feel unnatural. But in a pluralistic world, the ability to see from different points
of view is essential.[2]
This course attempts to present the scientific point of view. In science, the goal is to
obtain as accurate a description of nature as is possible. This means that the
scientific point of view seeks to strip away anything that might introduce bias or
misperception into its methods and modes of analysis, to the extent that this is
possible. Far more difficult than learning facts, learning a new point of view can be
requires that we learn to step back from ordinary, apparently obvious assumptions
and ways of thinking, to break old habits of mind. For this reason, as you work
through this course, it is important to realize that simply learning the material
presented, in a mechanical way, is not sufficient. You must also learn a way of
thinking about this material. To learn to “do” scientific reasoning, you need
something other than memory—the lights have to go on, as it were. The phrase
“learning how to learn” captures an essential quality of what is required. The
geophysicist Robert M. Hazen and the physicist James Trefil remark that
[T]he logical structure of science is analogous to a spider’s web. Start anywhere on the web
and work inward, and eventually you come to the same core. Understanding this core of
knowledge, then, is what science is all about.[3]
But, surprisingly, this knowledge is the knowledge of a point of view—of how to go
about doing science—rather than any specific knowledge of the nature of the world
itself, other than the knowledge that the world is such that science is possible. A
person who knows how to approach nature in a scientific way knows how to
construct theories; a person who knows a scientific theory knows something that
may be replaced in the future.
In this course, you will be exposed to some of the forms of reasoning used by
scientists as they go about their work. On the one hand, you will find that many of
these forms of reasoning are nothing more than ordinary reason made more precise.
According to the biologist Thomas Henry Huxley (1825-1895), “The method of
scientific investigation is nothing but the expression of the necessary mode of
working of the human mind.”[4]
On the other hand, at first exposure you may find that much of the language of
science appears dense and incomprehensible, and wonder how it could have
anything to do with the ordinary workings of the mind. This is an effect of “point of
view.” Common-sense thinking is carried out in terms of everyday images and
analogies, which also provide a basis for much of our scientific thought. Our minds
are adapted to think in terms of the ordinary world in which we live, and we use
these common images to try to understand scientific concepts that apply to worlds
vastly different from ours. The actual entities science deals with often do not behave
in ways that our everyday images would lead us to expect.
In everyday life, the length of measuring rods or the rate of clocks do not depend on
their state of motion relative to an observer. No thing can be in two places at once, or
jump from one place to another without passing continuously through the space
separating the two places. Things also have a unique identity independent of
whether or not they are observed.
But none of these common-sense ideas is true in the exotic realms of relativity and
quantum mechanics. The language of science is subtle, not because it uses a
technical vocabulary—easy enough to learn—but because we necessarily think in
terms of everyday images, and use them to attempt to represent entities that do not
behave in everyday ways.
As the scientist and humanist Jacob Bronowski (1908-1974) stated,
The analysis of our total impression of the outside world into coherent things with specific
properties is evidently a human trait, and we may presume that it has been crucial in giving
us the power to master and manipulate our environment as no other animal can. But it does
not follow that the invisible units for which we search in science to explain the visible events
have the character of things at all. And if we give them this character, it does not follow that
the properties with which we then have to endow them are consistent with the way in which
predicates work in sentences.[5]
Some authors have pointed out the great difficulty that people have in learning to
reason accurately and logically, both in science and in everyday life. This difficulty is
used to support the idea that logical, rational thought is unnatural.[6]
What we will find, however, is that for the most part Huxley is correct: not only
scientific thinking, but all rational thinking, is simply ordinary thinking made
precise. What is difficult is breaking the habits of everyday thinking, learning to pay
attention to the processes of thought as they go on, and learning how to use the tools
required to make these processes precise. As we will see, the conditions necessary for
thinking and speaking about experience are part of our everyday mental processes,
but without the tools of critical, logical and scientific thought, our mental processes
can easily lead to error.
The proper use of a tool requires that we be trained in its use, but it also requires
that we have a correct understanding of its capacities, including an understanding of
the conditions under which the tool will fail or produce undesirable results. A
surgeon would not want to use a dull scalpel, or operate in a non-sterile
environment. Likewise, the proper use of the tools of science requires that they be
kept “sharp,” and that they be employed from the proper point of view. Otherwise,
the conclusions reached may be “infected” by elements that compromise their
scientific validity.[7] As the German philosopher Immanuel Kant (1724-1804) stated
in his monumental work, The Critique of Pure Reason,
[T]he critique of reason leads . . . naturally and necessarily to science; and, on the other
hand, the dogmatical use of reason without criticism leads to groundless assertions, against
which others equally specious can always be set, thus ending unavoidably in skepticism.[8]
At bottom, the scientific point of view is based on the faith that the human mind can
understand the world, and on what the mathematician George Pólya (1887-1985)
called the “inductive attitude.”[9] Pólya points out that in our personal life there may
be certain beliefs we do not dare question, because doing so would prove too
upsetting to our emotional balance. In science, however, we must be willing to
question everything. In his words, the inductive attitude “aims at adapting our
beliefs to our experience as efficiently as possible. It requires a certain preference for
what is matter of fact.” He goes on to list three moral qualities associated with this
attitude:
1. Intellectual Courage: To be ready to revise any one of our beliefs.
2. Intellectual Honesty: To be willing to change a belief when there is a compelling
reason to change it.
3. Wise Restraint: Not to change beliefs capriciously, without good reason.
Pólya sums up these qualities in the injunction “Do not believe anything, but
question only what is worth questioning.” It is difficult to find a better summation of
the point of view required for science.
Discussion 1.2 Why Do We Need
to Understand Science?
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Science has become a form of magic practised by an elite priesthood whose members have
been subjected to a long and arduous apprenticeship in secret arts and rites from which the
layman is firmly excluded.
—Robin Dunbar[10]
[I]n order to discover where the professors of any branch of knowledge have erred, one
must make a profound study [and] must equal and even surpass those who know most of it.
—Abu Hamid al Ghazali (1058-1111)[11]
It is clear that science can lead to the development of technological products that
enhance human life and for this reason, if no other, society has an interest in
encouraging scientific studies. But what value could there be in studying science
itself? Isn’t it a question of “If it ain’t broke, don’t fix it”?
It is the case, however, that negative consequences arising from the improper use of
technology, or as unexpected side effects of technology, have led some people to take
an anti-science view. Also, some of the social effects of science, arising through ideas
diffused into the general social world, have been a cause of concern for some
individuals. Many believe that these issues should be addressed by a public debate,
intended to reach an eventual social consensus on ways to limit scientific theorizing
and technological applications.
The prospect of such restriction is a grave threat to science. Science claims unlimited
freedom of inquiry, both in the topics chosen for research, and in the dissemination
of ideas and results. Other groups in society, however, would challenge this freedom.
Laws have been passed against experiments in human cloning, some religious
groups oppose all research on methods for the control of human fertility, and other
groups seek to control what is taught in high school biology classes about human
origins. Politicians routinely deny scientific consensus on topics, such as climate
change, that seem to go against their ideological commitments. Genetic engineering,
a technological spin-off of basic research in genetics, is a subject of heated public
debate, and research carried out for military or commercial purposes is kept secret.
Ethical questions of what ought and what ought not to be permissible in the search
for knowledge of the world are debated daily. In Western society we have strong
limitations on what sort of experiments can be done with animal and human
subjects, and require informed consent for dangerous or potentially dangerous
experiments. By contrast, in ancient Alexandria the debate was whether or not
vivisection of prisoners condemned to die was morally permissible.[12]
How is it possible to make judgements about the role of science in society until we
are clear not only about the nature of science, but also about the way that science
interacts with the rest of human society? And can we ever hope to gain this necessary
clarity without applying the methods of science itself?
But we do not have to be trained scientists to understand at least the basics of
science any more than we have to be a trained musician to appreciate a symphony,
or an architect to judge the practical and aesthetic qualities of a building. All that is
required is a sufficient background in the basic goals, methods, and principles of
science so that we can recognize the difference between good science writing and
biased distortions.
The success of science over the past several hundred years has led to the slow
diffusion of scientific ideas into popular culture. The time lag in this process,
however, means that many popular images of science are out of date. In addition,
there is little popular understanding of the scientific point of view. As a result,
scientific concepts and results are often misinterpreted. For these reasons, it is
important to understand science and the scientific point of view, even if one has no
plans to engage in scientific activity.
On the one hand, there is great public interest in science, as demonstrated by the
explosion of popular books written by scientists or science writers to describe some
currently hot aspect of scientific thought. On the other hand, the actual nature of
scientific ideas and the activities carried out by scientists have become more obscure.
In the public eye, as indicated in the Dunbar quote at the beginning of this
discussion, scientists are too often seen as an occult priesthood, guarding hidden and
incomprehensible secrets. And this is not helped by the popular press, which often
publishes inflated claims about a scientific discovery; or, in some egregious cases,
false claims intended to spread fear in support of an agenda.
Public stereotypes of scientists are the stuff of comic book stories and B-grade
movies. Scientists are viewed as somewhat comical, absent-minded types with bad
hair, thick glasses and white lab coats. Perhaps they are knowledgeable in their
narrow specialty, but hopelessly inept otherwise. When not seen as bumbling nerds,
scientists may be thought of as Vulcans—cousins of the coldly logical Mr. Spock
of Star Trek fame. Or, they may be seen in two paired images out of B-grade horror
movies: the wild-eyed mad scientist bent on destroying the world, or the steadfast
adventure hero, also a scientist, who foils each and every evil plot. Sometimes
scientists are portrayed as idealistic altruists working to save humanity. This image
of scientific dedication is especially prominent in worshipful biographies, although
some admit focus on more mundane motives, such as personal ambition and desire
for the glory of a Nobel Prize.
In actual fact, the public perception of the everyday process of science is badly
skewed. Most scientists are relatively normal people, and the actual practice of
science is not at all as pictured in the popular media. The ordinary daily routines of
work in the lab, or the days of just thinking in an attempt to resolve a difficult
theoretical question, are generally overlooked. It is much more exciting to focus on
the moments of great discovery. The idea that everything is preprogrammed also
shows up in the public view of science. Scientists are seen as following preset
programs of work in a linear fashion until they reach a result. This is far from the
reality, better described by Bronowski,
[T]he B-movies . . . always represent the scientist at the moment of triumph, with the notion
that there it is, here is the discovery. Well, that is not how the discovery is made. The
discovery is made with tears and sweat (at any rate, with a good deal of bad language) by
people who are constantly getting the wrong answer.[13]
The physicist Sunny Auyang describes the actual practice of science in the following
illuminating terms:
[S]cientists maneuver their positions, shift their viewpoints, idealize judiciously, postulate
creatively, discard irrelevant details . . . , introduce novel concepts to represent organized
wholes, and repeatedly reformulate their problems to make them more manageable. In the
process they use logical reasoning and mathematical deduction, but they also think
realistically and intuitively, trading off between detail and generality, authenticity and
tractability. They rely on robust common sense, familiarity with the subject matter, active
observation, and experimentation on the objective world. This informal and creative
thinking marks theoretical reason from mere instrumental reason.[14]
The effects of public confusion about science can be far more serious than mere
academic dispute. Because people in general have only hazy ideas about the nature
of science, while science and its attendant technology plays such a central role in the
modern world, the word “science” is open to all sorts of abuse. Companies are quick
to advertise “scientific” support for their products, or to imply such support by
covert and not so covert techniques. Actors in commercials are dressed up as doctors
or laboratory workers, and supposedly “scientific” statistics are quoted.
Advertisers, political consultants, and propagandists of all sorts are quick to use the
language of science as a form of magical incantation, and the public response—if it
sounds scientific it must be scientific—is totally predictable. Furthermore, the
methods that these opinion manipulators use to sell us their particular brand of
beer, automobile, prescription medication, political candidate or whatever, are
firmly grounded in scientific research on conditioning, hypnosis and brainwashing.
Even in the academic world there is a great deal of misinformation and ignorance
about science, once one moves outside of the science faculty. Academics in the
humanities tend to look down on scientists as almost illiterate technicians, lacking in
cultural sophistication. Robin Dunbar addresses this point when he writes,
The claim that scientists as a group are cultural philistines simply isn’t true. The average
scientist is probably as literate and artistic as the average graduate in the humanities.
Instead, we ought to be asking this question: is the average professional in the humanities
as scientifically literate as the average scientist is culturally literate?[15]
Unfortunately, the answer to Dunbar’s question is—No! Isaac Asimov (1920-1992), a
chemistry professor better known for his voluminous output of popular science
articles, science fiction short stories and novels stressed this point. In an article
lamenting the woeful state of science education, Asimov remarked that whereas a
chemistry professor would be ashamed to admit that he only read comic books (even
if it were true), an English professor would boast of his ignorance of calculus.[16]
Ignorance of this sort contributes to a belief that science is static. It is thought of as a
17th century discovery with fixed methods and truth criteria, rather than a dynamic,
knowledge-generating process that continues to evolve. People with a static view of
science are far behind the times. Jacob Bronowski writes
Most scientists felt in 1930 that philosophers had just caught up with nineteenth century
physics, and were trying to make it the model for all knowledge, at the very moment when
physicists had painfully discovered its shortcomings.[17]
The division between science and the humanities is particularly distressing in light of
the historical fact that the beginnings of modern science (in ancient Greece and then
in the Scientific Revolution) coincide with two of the greatest periods of literary and
artistic production in the history of the world. Surely this cannot have been
coincidental! The general ignorance of so pervasive an influence in the modern
world is a cause for great concern. “The world today is made, it is powered by
science; and for [anyone] to abdicate an interest in science is to walk with open eyes
toward slavery.”[18]
Unit 1 Study Questions
TOP
Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. State three significant differences between the scientific and religious
approaches to the world.
2. Write a short essay (300-500 words) contrasting the scientific and religious
senses of the word “meaning,” as described in Chapter 9 of What Science Is.
3. Write a short essay (300-500 words) discussing a question of public concern
involving some aspect of science or technology.
4. Give examples from your own life of times when you have acted in accordance
with the three moral qualities that are involved in the inductive attitude.
5. Find at least six cases from newspapers, magazines or television in which an
unsubstantiated overt or covert appeal to science is used to sell a product.
6. Consider one of your own beliefs about the world (for example, a belief about an
important social issue).
a. How does this belief function in allowing you to make judgements about
your own behaviour? How does it help you in fitting together a view of
appropriate actions in the world?
b. How does this belief fit into your larger model of what the world is like?
c. What evidence would cause you to change this belief?
FOOTNOTES
[1]
Benedict, Ruth. Patterns of Culture, p. 18. New York: Mentor Books, 1959.
[2]
At a lecture given at the Edmonton Public Library (c. 1986), the writer Robert Anton
Wilson recommended spending an hour each day reading the literature of groups with
whose ideas and viewpoint we completely disagree (the stronger the disagreement, the
better), attempting to understand their point of view. A study of points of view was initiated
by John C. Lilly in his book Simulations of God: The Science of Belief. New York: Simon and
Schuster, 1976.
[3]
Hazen, Robert M., and James Trefil. Science Matters: Achieving Scientific Literacy, pp.
xvii-xviii. New York: Anchor, 1991.
[4]
Huxley, T. H. “Six Lectures to Working Men ‘On Our Knowledge of the Causes of the
Phenomena of Organic Nature’ [1863],” p. 359. Retrieved July 25, 2002, from
http://aleph0.clarku.edu/huxley/CE2/
Bronowski, Jacob. Nature and Knowledge, the Condon Lectures, pp. 42-43. Eugene, OR:
[5]
Oregon State System of Higher Education, 1969.
See, for example, Cromer, Alan. Uncommon Sense: The Heretical Nature of Science. New
[6]
York: Oxford University Press, 1993.
[7]
This is a major failing in pseudosciences, such as creation science.
[8]
Kant, Immanuel. (1781). Preface to the First Edition. The Critique of Pure Reason, trans.
J. M. D. Meiklejohn. Retrieved July 3, 2002, from
http://philosophy.eserver.org/kant/critique-of-pure-reason.txt/
[9]
Pólya, George. Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy
in Mathematics, pp. 7-8. Princeton, NJ: Princeton University Press, 1990.
[10]
Dunbar, Robin. The Trouble with Science, p. 7. Cambridge, MA: Harvard University
Press, 1996.
[11]
Ghazali, Abu Hamid al. Deliverance from Error (c. 1100 CE). Retrieved July 3, 2002,
from http://www.fordham.edu/halsall/basis/1100ghazali-truth.html/
[12]
The pro argument was that they were going to be executed in any case, and in being a
subject in a vivisection experiment they would at least be making a contribution to medical
knowledge.
[13]
Bronowski, Jacob. The Origins of Knowledge and Imagination, p. 111. New York: Yale
University Press, 1978.
[14]
Auyang, Sunny. Foundations of Complex Systems Theory, p. 7. Cambridge: Cambridge
University Press, 1998.
[15]
Dunbar, Robin. The Trouble with Science, p. 152.
[16]
Asimov, Isaac. Column in The Magazine of Fantasy and Science Fiction, c. 1958-60.
[17]
Bronowski, Jacob. A Sense of the Future, p. 75. Cambridge, MA: MIT Press, 1977.
[18]
Bronowski, Jacob. Science and Human Values, p. 6. New York: Harper and Row, 1956.
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Unit 2 Can Science Be Defined?
Discussion 2.1
Discussion 2.2
Unit 2 Study Questions
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STUDY GU IDE
Unit 2 Can Science Be Defined?
Science: the observation, identification, description, experimental investigation, and
theoretical explanation of natural phenomena.
—American Heritage Dictionary of the English Language[19]
The range of activities under the heading of science is tremendous. In addition to the
traditional sciences—astronomy, physics, chemistry, medicine, geology and
biology—we have a host of newer human sciences, such as psychology and
anthropology; social sciences, such as sociology and political science; and hybrid
fields, including management science, economics (called “the dismal science”) and
library science.
Such diverse subjects can all claim to be sciences only if they have something in
common. But the unifying principle cannot be found in their subject matter. We
cannot even say that all of the disciplines laying claim to the title “science” share the
same methods. Each has its own version of something called “the scientific method,”
but comparisons between fields such as physics, biology, sociology and anthropology
show that “the” scientific method is more aptly thought of as a bundle of disciplinedependent methods sharing only basic similarities of form. What does show up in
common is the goal of understanding, and the basic inductive attitude and point of
view. Without these core characteristics, we do not have science.
Objectives
When you have completed Unit 2, you should be able to
1. formulate a general, if incomplete, definition of science.
2. describe, briefly, the history of several cases of scientific discovery: Roentgen’s
discovery of X-rays, Kekulé’s discovery of the structure of benzene, the discovery
of energy bands in solids, Humbolt’s biogeographical studies and Jenner’s
discovery of the smallpox vaccine.
3. show how each of these cases of scientific discovery exemplifies some aspect of
your definition of science.
Indications
1. Read Discussion 2.1, “Is Science Possible?”
2. Read the Prologue, “What Is Science?” pages 1-8 of What Science Is and How It
Works.
3. Answer Unit 2 Study Questions 1 and 2.
4. Read Discussion 2.2, “Is Science Unique?”
5. Read Chapter 1, “A Bird’s Eye View: The Many Routes to Scientific Discovery,”
pages 11-25 of What Science Is.
6. Answer Unit 2 Study Questions 3-5.
Discussion 2.1 Is Science
Possible?
TOP
Now, my suspicion is that the universe is not only queerer than we suppose, but queerer
than we can suppose. . . . I suspect that there are more things in heaven and earth than are
dreamed of in any philosophy.
—J. B. S. Haldane (1892-1964)[20]
This is a course in scientific reasoning; it is designed to enable students to study
some of the forms of reasoning that are used in the practice of science, together with
some of the deeper philosophical issues relating to the nature and possibility of
science.
It may seem strange to question the possibility of science when science is so
evidently present as a force in the modern world. Certainly, we cannot question the
fact that there are people who call themselves scientists, who engage in a variety of
activities that they refer to as science. Does this not prove that science is possible?
The question, however, goes deeper than just asking about appearances. Does the
fact that for more than 2000 years there were people who called themselves
alchemists who engaged in alchemical activities prove the possibility of alchemy? If
we assume that the goal of alchemy was to change lead into gold by supernatural
means it would seem that alchemy is not really possible, that the tens of thousands
of highly intelligent and educated individuals who engaged in it were simply
deluded.[21] On the other hand, the psychologist Carl Jung (1875-1961) suggested that
the goal of alchemy was to produce a transformation in the practitioner—changing
the dross of the lower self into the gold of personal growth and individuation.[22] And
the historian Titus Burkhardt claimed that the actual goal of alchemy was neither
material wealth nor psychological integration, but spiritual illumination.[23] So
perhaps alchemy is possible after all, it depends on our point of view, and what we
take as its purpose.
In the same way, our view on the possibility of science depends on what we take to
be the goals of science, and on our acceptance or rejection of the basic metaphysical
assumptions underlying it. In the remainder of this discussion, however, we bracket
the question of purpose, and instead sketch out a few preliminary thoughts about
science.
To date, science has proved to be the most accurate method known for gaining
knowledge of the world. Most people will agree with this claim, without realizing that
it leaves many fundamental questions unanswered. Or, to be more precise, it
assumes certain answers to these questions without providing any justification. Is
there a material world? What is knowledge? How is knowledge gained, and how is it
communicated to others? How does one go about verifying a knowledge claim? If
knowledge is something that is purely personal, for example, to be acquired only in
ecstatic trance, then our claims for science do not hold water.
These questions are considered in more detail in subsequent units. They are not the
kinds of questions that are open to final answers, but it is important for every
scientifically or philosophically oriented person to be aware that they exist, and to
devote some careful thought to developing a position on their possible answers. In
addition, if one is to engage in scientific work, it is a good idea to exclude certain
potential answers to these questions, simply from a practical point of view. It does
little good, for example, for a physicist to deny the existence of the material world, or
to believe that it is impossible to gain and share knowledge of this world, if for no
other reason than that he or she would have little motivation for doing physics. Who,
after all, will spend time and energy attempting to acquire impossible knowledge of a
non-existent world?
To develop the view taken in this course, it will be necessary to
o
o
o
o
distinguish between two different kinds of knowledge—absolute and relative.
consider two different kinds of truths—absolute (or certain) and relative.
explore two different criteria of truth—coherence and correspondence.
View science as an enterprise that generates relative knowledge by the use of
methods which, although they do not provide certainty, do yield bodies of theory
that can be accepted as relatively true in the context of currently available
methods for exploring the nature of the world.
The philosophical issues surrounding these dichotomies are highly controversial,
with a variety of different positions being strongly defended. We refer to such issues,
but our main interest is in the actual enterprise of science. Thus, we leave the
philosophical arguments to more qualified philosophers, and focus on ideas and
techniques that are of practical use in the doing of science. We will attempt to do the
best we can without chasing the wild goose of perfection too far.
Discussion 2.2 Is Science
Uniquely Human?
TOP
Science is not an inhuman or superhuman activity. It’s something that humans invented,
and it speaks to one of our great needs—to understand the world around us.
—Maxine Singer[24]
One basic fact about science is that it is a human activity. To the best of our
knowledge, no other species on this planet engages in the systematic questioning of
nature characterizes science. No other species has separated itself from nature to the
extent that such an enterprise becomes possible—and necessary. These facts suggest
that we look for the roots of science in that which is fundamentally human.
But wait! Does that assumption make sense? To find out, we need to make the
argument for this claim in a more accurate way.
Assumption 1: Human beings have not changed in any essential way in, say, the past
10,000 years.
Assumption 2: Based on all current observations it seems that only human beings
practice science.
Conclusion 1: The development of science is something that is essentially human.
Conclusion 2: Since, so far as we know, science was not practised 10,000 years ago,
and is now; and since science is a human activity, it must have evolved out of some
more basic human characteristic.
Although plausible, neither the assumptions nor the conclusions can be irrefutably
supported. Why, then, are they justified?
In support of Assumption 1, we can suggest that records left by different peoples
over the past 10,000 years tend to show the same concerns as modern people, and at
least understandable forms of thinking. Furthermore, experience with extant
“primitive” societies seems to indicate that their members are, with the exception of
their type of cultural and technical development, very much like us.
A skeptic might attack Assumption 2 by claiming that many species could be
practising science. Apes and other primates have been known to experiment when
facing a problem, and some scientists have claimed that whales and dolphins have at
least as much intellectual capacity as humans. For all we know, whales and dolphins
may have developed extremely elaborate scientific theories of their watery
environment, but just haven’t developed a technology based on these theories.
The response to this contention is that such an argument lacks empirical evidence.
At present nothing indicates that whales or dolphins have anything like science. If
such evidence were to appear, of course, we would need to make some choices. On
the one hand, we could revise the assumption that science is uniquely human. On the
other, we could revise our definition of human so that it included whales and
dolphins. In the first case, we might decide to assume, for example, that humans,
whales, and dolphins were all examples of a higher category called “intelligent life,”
and that the development of science was characteristic of it. In the second case we
might decide that for theoretical or other reasons we wanted to keep “the
development of science” as a characteristic of humans.[25] It would then be necessary
to classify whales and dolphins as human on the basis of their similarity to us with
regard to the development of science, and as a result, to make a distinction between
being a human being and being a member of the species Homo sapiens.[26]
The skeptic is not done, however, and points out that both conclusions can be
criticized by saying that science is just something that happened accidentally. After
all, it is only found in late developments of human culture. Maybe it was introduced
by aliens—perhaps as a subtle means of getting us to destroy ourselves. Again, we
can answer that there is no evidence of alien intervention. But it could be that
science was just a chance discovery. Somebody said “Eureka!” and it began.
This final concern and the previous criticism of Assumption 2 are answered most
effectively by presenting a theory that describes the basic human qualities which are
claimed to underlie the activity we call science, and explains the way in which the
development of science might have occurred. Such a theory would not provide a
certain answer to the skeptical challenge, but it would provide a response that might
succeed in being tentatively accepted by most people with the background and
training needed to understand the theory and the questions that it claimed to
answer.
Let’s review: we began by deciding to study science as a human activity.
We observed that it appears to be a fact that science has not always existed, and that
it is only practised by human beings. This observation directed attention to the
question of the origins of science, and we hypothesized that these must lie in the
development and elaboration of some essentially human characteristic.
Skeptical criticism of this hypothesis indicated the need to develop a theory that
would support the conclusion by providing an acceptable explanation.
This line of reasoning is an example of scientific thinking. It can be compared to
some of the examples of scientific reasoning that you will encounter in your later
readings. A problem or question is recognized; some possible answers are proposed
and hypotheses are generated; these hypotheses are criticized, resulting in
recognition of the need for a theoretical explanation to provide sufficient reason for
their acceptance or modification.
What is absent from this example is the condition of empirical testing. Indeed, it is
difficult to imagine how such a hypothesis could be tested; thus, the hypothesis itself
is relegated to the borderlands of science. However, if you are able to follow this and
later examples with clear understanding, you are well on your way to understanding
scientific reasoning.
Unit 2 Study Questions
TOP
Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. Write a short essay (300-500 words) discussing the similarities and differences
between research in the sciences and in the humanities.
2. Write out summary definitions of science from the readings presented in
this Study Guide and in the textbooks. Then, comparing your summaries, write
a definition of science that is consistent with all of them.
3. Robin Dunbar[27], an animal behaviourist and primatologist, had claimed that
science is nothing other than learning. On this basis, he has argued that every
creature with any capacity to learn, however limited, is engaging in a form of
science. Write a short essay (300-500 words) discussing whether or not this
claim is consistent with the definitions of science that you have encountered in
this unit.
4. Choose one of the examples of a scientific discovery given in Chapter 1 of What
Science Is, by Gregory N. Derry, and list the ways in which this example fits with
the definition of science that you developed in study Question 2, above.
FOOTNOTES
[19]
Pickett, Joseph, ed. American Heritage Dictionary of the English Language, 4th ed.
Boston: Houghton Mifflin, 2000.
[20]
Haldane, J. B. S. Possible Worlds and Other Essays, p. 286. London: Chatto and Windus,
1927.
[21]
Even Isaac Newton carried out extensive alchemical experiments!
[22]
Jung, Carl. Psychology and Alchemy. Collected Works of C. G. Jung, Volume 12, trans. R.
F. C. Hull. Princeton, NJ: Princeton University Press, 1953.
[23]
Burkhardt, Titus. Alchemy: Science of the Cosmos, Science of the Soul, trans. William
Stoddart. Baltimore: Penguin, 1971.
[24]
Quoted in Jones, Shirley, ed. The Mind of God and Other Musings: The Wisdom of
Science, p. 40. San Rafael, CA: New World Library, 1994.
[25]
Suppose that we encounter an alien species, for example. Serious ethical and moral issues
would be raised if we were to decide that they were “not human.”
[26]
The idea of changing the definition of human beings in order to preserve the
characteristic of having developed science is rather far-fetched. In other cases, however, this
kind of change has actually been made to preserve a scientific principle. The definition of
energy, for example, has undergone a number of such changes to preserve the principle of
the conservation of energy.
[27]
Dunbar, Robin. The Trouble with Science. New Haven, CT: Harvard University Press,
1995.
[28]
See, for example, Wolpert, Lewis. The Unnatural Nature of Science. New Haven, CT:
Harvard University Press, 1992; Cromer, Alan. Uncommon Sense: The Heretical Nature of
Science. New York: Oxford University Press, 1993.
[29]
Wilder, Raymond. Mathematics as a Cultural System. Oxford: Pergamon Press, 1981.
[30]
Diamond, Jared. Guns, Germs and Steel. New York: Norton, 1997.
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Unit 3 How Does Science Proceed?
Discussion 3.1
Discussion 3.2
Discussion 3.3
Discussion 3.4
Unit 3 Study Questions
TOP
STUDY GU IDE
Unit 3 How Does Science
Proceed?
New organs of perception arise as a result of need. Therefore, increase your need.
—Jalaluddin Rumi (1207-1273)[31]
We open this unit with a brief consideration of why people engage in science; then
introduce the view of science as a consciously directed process of questions and
answers. As a method of transmitting technical skills from master to apprentice, this
process probably extends back to the dawn of humanity. The master poses a
question, and it is up to the apprentice to find an answer; or the apprentice asks a
question and the master waits for the correct circumstances to provide the answer.
The story of the Polynesian healer, her apprentice and the man who needed coconut
milk is a good illustration of this teaching technique.[32]
A healer was seated in front of her hut in conversation with her apprentice on the
importance of noticing detail. She saw a man approaching and said, “Here comes somebody
with a minor ailment that can easily be cured by coconut milk. Can you handle this case?”
The apprentice, an alert young fellow, jumped at this opportunity to practice at least a part
of the healer’s craft. “Of course!” he replied with alacrity.
When the man arrived at the hut, the apprentice leaped to his feet and said, “You are mildly
ill, take coconut milk.”
“That’s crazy!” the man said, “What sort of healer are you? Can’t you see how I am
suffering? Nuts to you and your coconuts!” And he stalked away muttering about the quack
healers in these parts.
Crestfallen, the apprentice turned to the healer. “What did I do wrong?” he asked, “I told
him what he needed but he didn’t believe me.”
“I will answer your question when I can,” the healer replied.
Sometime afterward, the healer and apprentice were again seated in front of her hut as a
man approached. “Do you recall your question about the man who needed coconut milk?”
she said, “Now I can provide an answer. Here is a similar case.”
The man approaching reached the hut and the healer asked him his business. In response,
he gave a longwinded description of his symptoms.
“Ah, yes,” the healer said, “I can see that this is an important case, one requiring serious
thought. Perhaps I should consult with the oracle.” She went into her hut and emerged a
moment later with a bowl made from half a coconut shell, containing some small pieces of
bone and some seashells. Muttering to herself, she shook the bowl and cast its contents onto
the ground. Looking carefully at the scattered bones and shells she began to speak,
apparently to herself.
“The condition can be cured . . . let’s see, what is required? Ah, something from a tree I
think . . . . Yes . . . something about the size and shape of a person’s head, round and hard
. . . but wait . . ., the cure itself is a liquid, whitish and translucent . . . how can something
round and hard yield that sort of liquid?” She paused and scratched her chin in thought.
“Of course!” she exclaimed, “How stupid of me, the cure is coconut milk.”
The man thanked the healer and promised her a fine reward for her services. After he had
gone, she turned to the puzzled apprentice. “You see,” she said, “he needed coconut milk,
but he also needed time to digest the idea.”
As far as is known, credit for developing this empirical teaching method into an
abstract method of discovery belongs to Socrates (469-399 BCE). Although Socrates
left no written works, his method of questioning was preserved by his students, most
notably Plato (427-347 BCE).
In Discussion 3.2, we consider the question-and-answer process as it is found in
present day science; describe some of the kinds of questions that are commonly
asked in science; and finish with a brief description of different stages that can be
identified in the process of scientific development.
You will then read a number of brief case histories of scientific discoveries. The
examples given in Chapters 2 to 4 of What Science Is illustrate various ways in which
new science can originate, while Chapter 5 of the same book gives a history of the
development of our theory of planetary motion, illustrating the various factors
contributing to this development at its different stages. Derry emphasizes the
importance of asking the right questions at the right time. The unit ends with a
discussion of John Snow’s discovery of the mode of transmission of cholera.
Objectives
When you have completed Unit 3, you should be able to
1. describe some of the factors that motivate people to engage in scientific activity.
2. discuss science as a guided process of questions and answers.
3. state the forms of some of the common questions that are asked in science.
4. list some of the basic criteria for valid answers to scientific questions.
5. describe the stages in the development of a scientific idea, or an entire field of
science.
6. describe some of the different factors that can lead to new scientific discoveries.
7. describe some case histories of scientific discovery in terms of the question-andanswer process.
Indications
1. Read Discussion 3.1, “What Are the Roots of Science?”
2. Answer Unit 3 Study Questions 1 and 2.
3. Read Discussion 3.2, “How Does Science Proceed?”
4. Answer Unit 3 Study Questions 3-6.
5. Read the chapters listed below from What Science Is.
a. Chapter 2, “Nature’s Jigsaw: Looking for Patterns as a Key to Discovery,”
pages 26-34
b. Chapter 3, “New Vistas: Expanding Our World with Instrumentation,”
pages 35-41
c. Chapter 4, “Close, But No Cigar: Discrepancies as a Trigger to Discovery,”
pages 42-51
d. Chapter 5, “Ingredients for a Revolution: Thematic Imagination, Precise
Measurements, and the Motions of the Planets,” pages 52-65.
6. Read Discussion 3.3, “Tycho and Kepler: The Odd Couple.”
7. Read Discussion 3.4, “Snow on Cholera.”
8. Answer Unit 3 Study Questions 7-10.
Discussion 3.1 What Are the Roots
of Science?
TOP
Tiger gotta hunt. Bird gotta fly
Man gotta sit and wonder why, why, why.
Tiger gotta sleep. Bird gotta land.
Man gotta tell himself he understand.
—Kurt Vonnegut, Jr. (1922-2007)[33]
The scientist does not study nature because it is useful. . . . He studies it because he delights
in it, and he delights in it because it is beautiful.
—Henri Poincaré (1854-1912)[34]
The most exciting phrase to hear in science, the one that heralds new discoveries, is not
“Eureka! I found it!” but “That’s funny . . . .”
—Isaac Asimov (1920-1992)[35]
Wonder and Curiosity
In Unit 2, we suggested that science is a uniquely human activity, and that this idea
can best be supported by the development of a theory that explains how science
evolved from more basic modes of human thought. We will not go further in
developing such a theory, other than to suggest that it has to do with the human
desire to understand the world—to know “the truth” about it—and our capacity to
wonder about the world and our purpose in it. Curiosity may have killed the cat, but
it is a driving force behind science. Aristotle (384-322 BCE), for example, states,
“For it is owing to their wonder that men . . . begin to philosophize . . . they were
pursuing science in order to know, and not for any utilitarian end.”[36]
This sense of wonder and curiosity, and the questioning it evokes, is beautifully
illustrated in the following quotation taken from Isaac Newton’s Opticks:
What is there in places almost empty of Matter, and whence is it that the Sun and Planets
gravitate towards one another, without dense Matter between them? Whence is it that
Nature doth nothing in vain; and whence arises all that Order and Beauty which we see in
the world? To what end are Comets, and whence is it that Planets move all one and the same
way in Orbs concentrick, while Comets move all manner of ways in Orbs very excentrick,
and what hinders the fix’d Stars from falling upon one another? How came the Bodies of
Animals to be contrived with so much Art, and for what ends were their several Parts? Was
the Eye contrived without Skill in Optics, or the Ear without Knowledge of Sounds? How do
the Motions of the Body follow from the Will, and whence is the Instinct in Animals?[37]
This same sense of wonder is found in modern scientists. The physician and
endocrinologist Hans Selye (1907-1982), for example, was quoted in the popular
press as saying “The true scientist never loses the faculty of amazement. It is the
essence of his being.”[38]
The capacity to wonder, the desire to understand, and the desire for truth are at least
good candidates for the psychological roots of science. Out of wonder we ask
questions, in order to understand we seek answers, and to know the truth, we test
whatever answers we obtain: against experience, for internal consistency, and
against the scrutiny of other scientists.
From this perspective, all of the activities of science can be seen as originating from a
set of fundamental questions: Who are we? Where did we come from? Where are we
going? Of course it could be pointed out that these fundamental questions are
behind much of human activity in addition to science. What distinguishes science is
that it seeks knowledge of the world, and our place in the world, subject to a rigorous
set of constraints which provide publicly available knowledge that is “as accurate as
we can get” at any given time in history. Science is, therefore, a particular form of
guided questioning. But the motive is wonder, and the satisfaction that comes with
understanding.
We are an intelligent species and the use of our intelligence quite properly gives us pleasure.
In this respect the brain is like a muscle. When it is in use we feel very good. Understanding
is a kind of ecstasy.[39]
The Scientific Tradition
One aspect of science that is not generally appreciated is the extent to which it is a
social activity. The public image of a scientist is often that of an isolated individual
working with highly abstruse ideas that, if fortune smiles, may result in a great
discovery. On the basis of this sort of image, we sometimes encounter speculation
about what would have happened if, for example, Archimedes had gone just a bit
further and discovered calculus, or if quantum mechanics had been developed in the
18th century. We are led to wonder why great scientists of the past failed to note
some idea or other that in hindsight appears glaringly obvious, or why they believed
in an obviously false theory. Why, for example, did the great astronomer Isaac
Newton (1546-1601) reject the heliocentric Copernican theory? Why did the physicist
Ernst Mach (1836-1916) reject the existence of atoms? Why did geologists, for over
40 years, universally reject the theory of continental drift?
In viewing the history of science, however, it is important to avoid projecting our
own understanding and point of view onto the past. The social historian
Giambattista Vico (1668-1744) observed that there are two common mistakes made
when viewing the past. The first is to look back to a golden age, populated by
individuals of incomparable intellectual and moral genius. Vico called this mistake
the “conceit [error] of nations,” referring to the deification of “founding fathers.” The
second sort of mistake, called the “conceit of scholars,” is the opposite. It is to view
people of the past as ignorant and primitive, having nothing to teach the present.[40]
We can avoid these errors by recognizing the historical continuity of humanity, but
to do so requires inclusion of the social and cultural dimensions. This continuity is
particularly evident in science. We sometimes hear of scientific revolutions, in which
old theories and ways of thinking are suddenly cast into history’s dustbin. Such
stories tempt us to direct our attention toward the future and discount the past, to
think that we’ve “been there, done that,” and wonder why earlier people were so
ignorant.[41] We may admit that they produced some great art and literature, and
built some magnificent edifices, while thinking that scientifically—which today is
often taken as the only thing that counts—they were backward and uninformed.
What we must clearly understand is the falseness of this belief. Despite some
postmodern claims to the contrary, science is a progressive enterprise. The science of
the past is not just theories that have been chucked out because we have found
others that are more fashionable, it is the very foundation on which our present-day
theories rest. None of the ancient Greek theories of nature survive, but many of their
basic conceptual forms are essential aspects of modern science. The Greek idea of
atoms is very different from the atoms of modern physics, but the idea of seeking for
the elementary components of matter remains. We no longer accept the Stoic
conception of pneuma, an ultra-refined form of living fire filling the cosmos, but the
underlying thought reappears in the modern concepts of electromagnetic, quantum
and gravitational fields.
Moving closer to the present, the Scientific Revolution of the 17th century did not
blossom from nothing. Galileo was able to carry out his experiments on the motion
of bodies only because medieval mathematicians had clarified the ideas of velocity
and acceleration. René Descartes’ philosophy was firmly rooted in medieval
scholasticism, and the logical work of Leibniz is based on the medieval concept of the
knowledge of God (scientia dei).
Throughout history scientists and seekers after truth have used the material and
conceptual tools available to address issues of current need and, on rare occasions,
to develop new and more refined tools. If many of their theories have been found
wanting, this in itself is a gain. When Thomas Edison (1847-1931) was told that his
thousands of failed experiments to make an electric light bulb had produced no
result, he shot back, “Results! Why, man, I have gotten a lot of results. I know several
thousand things that won’t work.”[42]
Far from being an isolated individual, a scientist is firmly embedded in a historical
movement having its own traditions, values and developmental dynamics.
Individuals are drawn to science by a sense of wonder and a desire to understand the
world. When they become scientists, however, their desire to understand must be
disciplined. It is no longer the case of a single individual constructing a personal
worldview; rather, the individual contributes to the social construction of a cultural
worldview whose richness, the result of the effort of many individuals past and
present, more than makes up for the loss of unconstrained imaginative speculation.
O man! If you only knew how many of the false fantasies of the imagination were nearer to
the Truth than the careful conclusions of the cautious. And how these truths are of no
service until the imaginer, having done his work with the imagination, has become less
imaginative.[43]
Discussion 3.2 How Does Science
Proceed?
TOP
A good question is one that grabs you right in the gut, and it doesn’t let go until you have an
answer.
—Bryn Beorse (Shamcher)[44]
[H]aving a good question, a fundamental question, and having some tools of inquiry that
allow you to take the first step toward an answer—those are the conditions that make for
exciting science.
—Herbert A. Simon (1916-2001)[45]
Those who do not stop asking silly questions become scientists.
—Leon Lederman[46]
Science Begins with Questions
In the philosophy of science, there are many different views of what science is and
how it is to be characterized. It is generally agreed, however, that the process of
science involves questions and answers. Indeed, for a working scientist, scientific
reasoning is a question-and-answer process that involves asking the right questions
at the right time and in the right way, and paying attention to the answers. The
philosopher Karl Popper (1902-1994) labeled this process as one of “conjectures and
refutations.”[47] A conjecture is made, and the scientific community attempts to refute
it, leading to a progressive sharpening and refinement of the original conjecture. But
there is more to it than this. Before a scientist can make a conjecture, he or she must
have a question that the conjecture is intended to answer. Likewise, our interest,
once there is a conjecture, is not directly in its refutation. Rather, we have a new set
of questions: How well does the conjecture answer the initial question? Is it
consistent with currently accepted theory? Does it correspond to the facts?
In addition, questions can only be asked in context. Simply asking “Why?” or “How?”
without a context within which the question “makes sense” will not produce a useful
answer.
When a scientific theory is undergoing development, there are always questions that
arise relating to the logical consistency of the theory, the need to explain
experimental data and observations, and possible applications and consequences of
the theory. In attempting to answer these questions, scientists generate further
conjectures (hypotheses). These conjectures are criticized by other scientists, within
the context of the developing theory. If a hypothesis manages to survive this
criticism it may eventually be accepted as a part of the theory.[48]
Another useful view of this question-and-answer process is given by the historian of
science Thomas Kuhn, who refers to it as “puzzle solving.”[49] In Kuhn’s view, an
existing scientific “paradigm” (i.e., the basic set of assumptions and beliefs guiding
development in a scientific discipline, hence providing the context for asking
significant questions) raises questions of consistency and empirical fit, and it is
assumed that it is possible to construct answers to these questions (puzzles) within
the assumptions that define the paradigm. Only in extreme cases, where a set of
central questions have gone unanswered for a long period of time, is the paradigm
itself questioned. When this occurs, the field of potential questions and answers
widens and a scientific revolution begins, eventually resulting in the emergence of a
new paradigm within which the previously unanswered questions of the old
paradigm are either answered, or seen to be irrelevant.[50]
The philosopher Stephen Toulmin, in his book Foresight and Understanding, makes
a similar point in describing what he calls “Ideals of Natural Order.” These are
assumptions about what is natural, about how nature would behave if there were no
interference. All that is required is explanation of deviations from this ideal
behaviour. One example is the ancient Greek assumption that planetary orbits were
perfect circles. The baroque detail of the Ptolemaic theory of planetary motions
consists of adjustments and rationalizations to explain apparent deviations from
“perfect” motion. Likewise, Newton’s first law of motion (that natural motion is
motion in a straight line at constant speed) is an Ideal of Natural Order. Newton’s
formula F=ma can be turned around to give a=F/m, indicating that any acceleration
(producing a deviation from straight line motion at constant speed) must be
explained in terms of the application of an external force. Thus, in Toulmin’s view,
the context for questions in science involves attempts to explain deviations from the
natural order: why did this planet move in a way that appears to indicate a noncircular orbit; why did this motion follow the path that it did?
What is central is the importance of good questions. This point was strongly
emphasized by the cybernetician Heinz von Foerster, who suggests that it is not the
failure to answer long-standing questions that eventually kills off an old paradigm,
but its failure to generate new questions of interest.[51] When there are no more
significant questions in a field of study, the field stagnates and dies, much as a gold
mine is abandoned once all of its gold has been mined out. Of course, the alternative
is that the theories involved simply become standard—nobody is going to alter the
laws governing levers discovered by Archimedes, for example.
Scientists are interested in good questions—ones that grab their attention and
demand to be answered. In his autobiography, Albert Einstein (1879-1955) tells us
that the reason he chose to specialize in physics rather than mathematics was that,
although he found mathematics interesting, he was not able to distinguish the
important questions from the trivial ones, whereas in physics he had a very good
instinct for those questions, and the resulting lines of research that were
significant.[52]
One of the most interesting views on the question-and-answer process of science is
found in an analogy suggested by the physicist John Archibald Wheeler (19112008).[53] Wheeler likened scientific research to a modified game of Twenty
Questions. In this game one person (the “scientist”) is “it.” This player leaves the
room while the other players (“nature”) ostensibly think up a target object to be
discovered. What they really agree on, however, is how questions will be answered.
No predetermined object is chosen. Rather, the rules for answering questions are as
follows:
o
o
when a person responds to a question they must have some definite object in
mind for which the response is true.
each response must be consistent with all previous answers.
Now the person who is “it” re-enters the room and begins to ask questions, under the
impression that there is some predetermined object to be discovered. And in fact, the
game will generally end with the identification of some definite object. But this
object has been constructed in the interactive question-and-answer process. A
different sequence of questions might well have led to a different “answer.”
Wheeler’s analogy has sometimes been used to argue that there is no reality other
than that which we create in our own minds, and that there is no possible standard
for comparison of different belief systems. This view is known as an “anti-realist”
position, and in one form or another, it has supporters in both science and
philosophy. In this course, however, we adopt a “realist” view; that is, we assume
that at least some of the theoretical entities we talk about—electrons, hydrogen
atoms, natural selection, economic forces and so on—actually exist in a reality that is
independent of human thought. Realism of this mild sort is the dominant view in
science.
The realist response to the anti-realist argument based on Wheeler’s analogy is to
point out that the analogy itself is only partial, and the anti-realist conclusion pushes
it too far. It could be, for example, that in science any honestly followed questionand-answer sequence will eventually lead to the same place, in the sense that two
different sequences yield different formulations of the same theory (something that
has actually happened in science, as with the different formulations of quantum
mechanics given by Erwin Schrödinger and Werner Heisenberg). It is also the case
that in science, such question-and-answer sequences often reach a point at which the
next question, although obvious, is anomalous, in that it cannot be answered in a
way that maintains consistency with all previously obtained answers. In such cases,
new paradigmatic assumptions are required.
Questions in Science
Ask questions. Don’t be afraid to appear stupid. The stupid questions are usually the best
and the hardest to answer. They force the speaker to think about the basic problem.
—Paul Ehrenfest (1880-1933)[54]
There are many ways to begin asking questions in science. A few of them are listed
below.
1. We may ask whether or not a particular hypothesis is
a. “true.”
b. consistent with a given theory.
c. consistent with experiment.
For example, Snow’s questioning of various hypotheses about the way in which
cholera was spread led to his acceptance of the hypothesis that it was spread
through the water supply.
2. Assuming that a given theory or hypothesis is “true,” we ask what consequences
its truth implies
a. for further theory development.
b. for possible predictions that can be made.
For example, Einstein’s questioning the consequences of assuming the speed of
light was constant led to the development of his special theory of relativity.
3. We may ask what must be assumed as true in order for a particular result to
follow; for example, asking what must be assumed to explain the motion of the
planets.[55]
4. We may ask about the nature of the relation between elements of some set or
collection. Linnaeus, for example, asked about the relations between different
kinds of plants and animals, and answered in terms of a species concept based
on the possibility of interbreeding: plants or animals of the same species could
mate and produce fertile offspring, those of different species could not.
5. We may ask about the composition or structure of some entity. For example,
Dalton asked how chemical compounds might be composed, and developed the
atomic theory of matter.
6. We may ask why a thing is a particular way, and not otherwise. For example, the
astronomer Wilhelm Olbers (1758-1840) asked the question: Why is the night
sky dark? The query was proposed as a paradox: if an infinite number of stars
exist, then any line of sight should eventually encounter a star, and the entire
sky should be bright. Why is this not so?
7. We may ask what data must be obtained, and how one might go about obtaining
it, to test a particular hypothesis or theory. For example, in 1919 an expedition
to Paraguay was undertaken to measure the bending of starlight by the sun, an
effect predicted by Einstein’s general theory of relativity.
There are also questions that must be asked before any experimental investigation
can be undertaken. Some of the most important of these questions are given below.
1. Will the experimental procedure interfere with the natural behaviour of the
system being studied to the extent that the results obtained are not
characteristic of this system? For example, keeping a bird in a small cage might
completely disrupt its normal behaviour, but inserting a small thermometer into
a large beaker of liquid will not significantly alter the temperature of the liquid
(although it will alter it very slightly).
2. Are the systems to be studied capable of exhibiting the expected behaviour? It
would be useless, for example, to carry out a test of the ability to spell of infants
who had not yet learned how to talk.
3. Is the available equipment capable of detecting the results expected? For
example, the Copernican theory predicted that observations of star positions
made 6 months apart would show slight shifts because the Earth would be at
different points in its orbit. But it was well over 100 years before astronomical
instruments were developed that were sensitive enough to measure this effect
(called stellar parallax).
4. Is the initial state of the system being studied properly prepared? For example,
if rats have been trained to perform a complex task in order to obtain food, then
in studies of how well they perform this task, it is important to make sure that
they are hungry.
5. Will the experimental procedure destroy or disable the system being studied,
and if so does this fact invalidate the experiment? For example, it would be
useless to study the electrical behaviour of a nerve cell by inserting an electrode
so large that it kills the cell. On the other hand, physicists regularly study the
structure of atomic nuclei by bombarding them with high energy particles and
analysing the sorts of fragments that result.
There are also a variety of criteria for judging a good question. The most significant
one was given at the beginning of this discussion. A good question must grab a
scientist’s imagination and demand an answer. It should be something that she or he
will lie awake at night wondering about. Some of the characteristics of such a
question are listed here.
1. It has depth—it goes to the heart of a situation.
2. It is tantalizing—it seems to hold out the possibility of an answer.
3. It is suggestive—it almost tells one how to look for an answer.
4. It is fruitful—the answer, when obtained, will have significant meaning for
several areas of science.
There are also criteria for judging answers that are to be considered scientific.
1. No answer can imply the violation of an established empirical fact, unless it also
offers an explanation that shows why the cases in which this violation will occur
are ones that have as yet not been observed. In this case, it offers predictions
that can be tested.
2. Every answer must either be consistent with existing theories, or provide a good
reason for changing these theories.
3. No answer can involve a logical contradiction.
It is the criteria for acceptable answers that give us science. Answers such as “it is
obvious,” “it just feels right,” “it is the will of God” or “it is morally and ethically
correct,” may be satisfactory in other areas of human concern, but as far as science
goes, they are inadmissible.
Depending on the circumstances, it is not always necessary for a question to be
precise. Often, in the initial phases of research, it is enough to ask rather vague
questions. Indeed, asking questions that are too precise at the beginning may
eliminate consideration of lines of thought that later turn out to be important. At 16,
years before he asked about the consequences of assuming a constant speed of light,
Einstein questioned what one would see if one were moving along with a light beam
at the speed of light. It was only after much study that he was ready to rephrase this
query as a precise question that could be given a precise answer.
The beginnings of a research project are a time for great care—questions and
procedures that are formulated will tend to persist and determine the future
development of the project. Often, in science, attempts to answer the interesting
questions must be deferred until answers are obtained for many narrower questions.
If we liken scientific theorizing to constructing a building, then a comment of
Johann Wolfgang von Goethe (1749-1832) is most appropriate: “Three things are to
be looked to in a building: that it stand on the right spot; that it be securely founded;
that it be successfully executed.”[56] Before we can attempt to answer a question it
must be suitably located within a context; there must be a firm foundation of
empirical and theoretical work on which to build; and we must have the tools
necessary for finding an answer, or at least the means to develop these tools.
Mistakes at the beginning can lead to wasted effort, and being overly precise too
soon is often a serious mistake.
Stages of Scientific Development
It takes little talent to see clearly what lies under one’s nose, a good deal of it to know in
what direction to point that organ.
—W. H. Auden (1907-1973)[57]
I have a very good nose.
—Albert Einstein
While science involves asking questions, not just any question will do. Furthermore,
different questions will be relevant at different stages of development in a science. In
mechanics, the nature of the relationship between the concepts of acceleration,
velocity, momentum and kinetic energy was an important question in the 17th
century. Today, nobody considers it—the answers were worked out long ago. In a
different way, no chemist today asks about the caloric content of materials; the
demise of the caloric theory of heat makes such questions not only irrelevant but
scientifically incomprehensible, somewhat like asking about the weight a unicorn
could carry or the crystalline structure of the celestial spheres.
Consideration of historical examples shows that as a science evolves, it goes through
a sequence of developmental stages. This does not mean that every science has gone
through every stage in a way that can be easily recognized, or that work in a scientific
field at a given time will only involve a single stage. Nor do all people who study the
development of a science identify the same set of stages, although there are parallels.
Rather, there are interconnections between the different stages, and their
description is intended only as a means of giving a general classification of the goals,
procedures and types of questions that predominate in each stage.
The French physicist Roland Omnès describes a four-stage model that he attributes
to Pierre Duhem (1861-1916).[58]
1. Exploratory Stage: This initial stage “consists in the observation of facts, the
performing of experiments ‘to see what happens,’ the compiling of a catalogue of
data, and, eventually, the discovery of empirical rules.”
2. Conceptual Stage: This stage “consists in the development and the selection of
appropriate concepts permitting a representation of Reality, the invention of the
principle, or principles, that might govern this representation.”
3. Developmental Stage: This stage “consists in examining all possible
consequences of the principles. . . . In most cases, only certain consequences are
considered, primarily those concerning known facts.”
4. Verification Stage: In this stage, “each prediction is systematically subjected to
the test of experience.” Successful predictions lead to tentative acceptance,
unsuccessful ones mean that the theory is false, or at least incomplete, and that
further clarification is required.
Breaking this model down into eight stages allows clearer insight into the
connections between stages and the cyclic nature of the overall process.
1. Initial Interest: Every line of research in science begins when some
phenomenon attracts somebody’s attention. Something unusual happens, or
somebody thinks about something in a new way and thinks, “That’s funny,
what’s going on here?” The object or event of interest is roughly described along
with its more obvious properties and peculiarities.
2. Delineation of the Phenomenon: By familiarizing themselves with the objects
and events of interest, researchers begin to build a more comprehensive
description of the phenomenon, and to develop an empathetic “feel” for the
subject. On this basis, they choose or develop appropriate tools for a more indepth study.
3. Empirical Database: Through carefully designed observations and experiments
targeting specific aspects of the phenomenon, researchers develop an empirical
database.
4. Classification and Conceptualization: Regularities in the empirical data are
grouped into classes, and concepts are developed to characterize these classes. A
taxonomy or typology may be constructed as a conceptual net to capture
patterns appearing in the data. This is the stage where Plato’s injunction holds,
“First the survey of scattered particulars, leading to their comprehension in one
idea [then dividing the idea into parts] where the joint is, not breaking any part
as a bad carver might.”[59]
5. Generalization and Theory Construction: Relationships between the conceptual
entities that have been defined are studied. Causal regularities are discovered
and/or posited, and are either incorporated into existing theory or used as a
basis for new theory.
6. Hypothesis Generation and Testing: Hypotheses are put forward and used to
generate new predictions that are tested in newly devised experiments or
observational situations. In general different research groups may produce a
variety of hypotheses but empirical tests alone may be insufficient to definitively
eliminate all but one of the hypotheses.
7. Analysis and Debate: There is a general discussion within the concerned
scientific community, comparing the merits and demerits of hypotheses that
have not been eliminated by empirical tests. Modifications of various
hypotheses are proposed and tested as the scientific community works toward
consensus.
8. Assimilation into Existing Theory: At some point, the preponderance of
evidence will lead to the acceptance of a hypothesis or theoretical construct as
most likely, or if no definitive empirical selection emerges, as most elegant. That
hypothesis is then assimilated as a part of the existing theoretical structure of
the science, and is incorporated into undergraduate textbooks.
When we compare this formulation to the four stages described by Omnès, we see
that his Exploration Stage involves our Stages 1, 2 and 3; his Conceptual Stage is our
Stage 4; his Developmental Stage is our Stage 5; and his Verification Stage covers
our Stages 6, 7 and 8.
When presented in this form, the process described seems linear, moving from
initial interest to accepted theory. This impression is just an artifact of the form of
presentation. In the actual practice of science, work is going on in each of these eight
stages simultaneously, although one stage may receive more attention and effort at
any given time. In addition, there are feedback and feedforward links among stages,
and the overall process itself is cyclic—the accepted theory that appears at Stage 8 is
also part of the background against which new phenomena of interest will emerge.
The discussions and debates going on in Stage 7 are carried out within the context of
this background of accepted theory, and they are the source of many of the new
questions and new phenomena that initially excite scientific interest in Stage 1.
Likewise, the move from Stage 1 to Stage 2 requires consideration of Stage 4, since
the tools developed, and the way that the data are to be worked over in Stage 2, both
require consideration of the sort of conceptual structures that are available for
representation. If we want a taxonomy that carves nature with a scalpel we don’t
want to carry out our experiments using equipment that carves with an axe.
The accumulation of empirical data moves the process from Stage 2 to Stage 4, but it
also requires us to think in terms of the currently accepted theories (Stage 8). That
is, the way that we develop concepts and classify entities in Stage 4 is determined by
the condition that it be possible eventually to incorporate these concepts into
existing theory, or at least to show how they are consistent with this theory. If it
appears impossible to do so, then we will need to consider changes to the existing
theory. In either case, the work at Stage 4 is carried out in a conceptual framework in
which current theory provides the paradigm. The work of harmonizing the new work
with existing theory is carried out in Stage 5, and will automatically lead to the
generation of new hypotheses to be tested in Stage 6. Again, this process is
automatically followed by the discussions and debates of Stage 7, which prepare the
new work for general acceptance, and also initiate a new cycle of activity by raising
new questions of interest. You will see an example of this process in action later in
the course in the development of the Bohr model of the atom.
Discussion 3.3 Tycho and Kepler:
The Odd Couple
TOP
Perhaps no more unlikely pair of individuals is found in the history of science than
the wealthy Danish nobleman Tycho Brahe (1546-1601) and the poor son of a
German soldier, Johannes Kepler (1571-1630). Fittingly, both men observed
supernova explosions which bear their names.
Tycho became fascinated with astronomy as a youth in 1560 when he saw an eclipse
of the sun that occurred at the predicted time. Later, he observed a conjunction of
Jupiter and Saturn whose predicted occurrence was in error by several days.
Realizing the need for more accurate astronomical measurement, he devoted his life
to measuring the positions of the moon, sun, stars and planets. Tycho gained initial
fame with his interpretation of his observation of the 1572 supernova, which was
radical. At the time, Church dogma held that the heavens, being perfect, could not
change. Thus any observed change must be an atmospheric phenomenon. Tycho,
however, insisted that the bright light that appeared and slowly faded was a new
star. If it were an atmospheric phenomenon, he argued, it would display a parallax
relative to the fixed stars, but according to his best measurements, it did not. Hence,
it could not be an atmospheric phenomenon. This was a tremendous shock to the
16th century conception of the world—if the heavens could change, what could
remain stable? What else might change?
King Frederick II of Denmark gave Tycho the island of Ven to build an observatory
that soon became the centre of European astronomy. Tycho kept detailed records of
all of his observations, carried out continuously, in contrast to other astronomers
who were satisfied to record the position of a planet only a few times in its orbit. It
has been estimated that during its operation his observatory consumed as much as
1.5% of the entire Danish national budget.
Tycho himself was immensely wealthy, reputed to control as much as one percent of
the entire wealth of Denmark. He was also vain and wildly eccentric. He had lost his
nose in a duel with a fellow student (over a mathematical equation) and wore, as the
occasion warranted, artificial noses of copper, silver or gold. He lived in a large castle
surrounded by an assortment of entertainers, including a dwarf named Jepp whom
Tycho believed to have psychic powers. Jepp acted as court jester and spent most of
his time at banquets under the table. Tycho also had a tame elk, which died falling
down a flight of stairs after drinking large quantities of beer.
While Tycho’s relations with Frederick were apparently cordial, they were decidedly
less so with his successor, Christian IV (Tycho was rumoured to have had an affair
with the monarch’s mother), so he moved to the court of the Emperor Rudolf II in
Prague. He died in 1601 of a bladder infection following a court banquet. Some said
he was actually poisoned by a dose of mercury and in a recent study, traces of
mercury were found in some of his preserved hairs. One theory is that Christian IV
ordered the murder, while another theory is that it was done by none other than
Kepler as a means of gaining access to Tycho’s accumulation of astronomical data. A
less dramatic possibility is that Tycho had poisoned himself, over time, through
alchemical experiments involving the ingestion of small quantities of mercury. The
jury is still out.
While Tycho is best known for the quality of his observations, he also attempted to
resolve the difference between the Ptolemaic and Copernican systems. He proposed
a hybrid theory in which all planets orbited in circular orbits around the sun while
the sun followed a circular orbit around the earth. While Tycho did not believe the
Ptolemaic theory, he could not accept the Copernican theory, and for a very good
reason—if the Copernican theory were correct, it ought to be possible to observe
stellar parallax, the slight shifting of stellar positions in the sky at different points in
the Earth’s orbit. But given the accuracy of instruments available at the time, no such
effect was seen. Indeed, it was more than 100 years before instruments sufficiently
powerful to detect this effect were developed. Tycho was an observational
astronomer and, based on the best available observations, rejected Copernican
theory.
Johannes Kepler became interested in mathematics at 18. After graduation he had
obtained a position teaching mathematics and astronomy at the Protestant
University in Graz, Austria. In 1598, however, the Catholic archduke of Graz closed
this university, throwing Kepler out of work. He and his wife were exiled and
travelled to Prague. Here he met Tycho, who had also recently arrived. Tycho offered
him a position as his assistant, with a good salary. He was given the job of working
out the orbital motions of the planets. Tycho’s hope was that this would prove his
cosmological theory. Kepler, on the other hand, was a committed Copernican. He
had also studied philosophy and theology and was prone to making abstract mystical
speculations about the cosmic order.
Kepler worked for Tycho for only a year before Tycho died. Kepler needed Tycho’s
observational data on planetary positions to work out a mathematical theory of their
orbits, but Tycho refused to share more than fragments of his data. Perhaps he
feared that Kepler, whose talent he recognized, would replace him as Europe’s
greatest astronomer. When Tycho died, however, his last words to Kepler were
prophetic: “Do not let me have lived in vain.” On Tycho’s death Kepler appropriated
(i.e., made off with) his papers and data. In Kepler’s words: “I confess that when
Tycho died, I quickly took advantage of the absence, or lack of circumspection, of the
heirs, by taking the observations under my care, or perhaps usurping them.”
Kepler was given Tycho’s position as court astronomer and mathematician, but this
paid little and did not last long. The religious wars raging across Europe reached
Prague and again Kepler was forced into exile. He lost his wife and son to plague,
and spent 5 years defending his mother against charges of witchcraft. He lived in
poverty, earning his living by casting horoscopes for members of the nobility.
Even with these distractions, Kepler remained busy analyzing Tycho’s exhaustive
measurements of the positions of the planet Mars. He tried every possible way to fit
these observations to a circular orbit (as Copernicus assumed) without success.
Finally, with no other possibility, he found that the orbit could be described as an
ellipse with the Sun at one of the foci. He was forced to accept what the data was
telling him, and developed two of his three laws of planetary motion:
1. Planetary orbits are ellipses with the sun at one focus.
2. The line joining a planet to the Sun sweeps out equal areas in equal times as the
planet moves on its orbit.
About a decade later, Kepler found his third law of planetary motion:
3. The ratio of the squares of the revolutionary period of two planets is equal to the
ratio of the cubes of the semi-major axes of the ellipses that are their orbits.
Some 50 years later, these laws were essential in Newton’s development of his theory
of universal gravitation.
Tycho Brahe was a brilliant and meticulous observer; Johannes Kepler was a
speculative genius. The two only interacted directly for one contentious year. But
Tycho’s measurements would have counted for little without Kepler’s brilliance in
calculation and theorizing. And Kepler would have had nothing to rein in his wilder
metaphysical speculations without Tycho’s painstakingly accurate data. Individually,
they were brilliant scientists. In combination, they changed the world.
Discussion 3.4 Snow and Cholera
TOP
In 1850, sanitation in the city of London was next to non-existent. The Sanitation
Reform Movement had constructed a sewer system to replace cesspools, but a result
of this well-intentioned movement was that raw sewage was dumped into the river
Thames, while drinking water for the city was either drawn from the Thames or from
shallow wells. Disease was rampant, with repeated epidemics of cholera. The onset
of cholera is rapid and it causes massive diarrhea. If not treated, it leads to death
from dehydration and electrolyte imbalance, often within one or two days.
Before the discovery of bacteria, made possible by the invention of the microscope,
the cause of cholera (and many other diseases) was not known. Cholera was first
recognized in the 18th century in India, but as travel between Asia and the West
increased and urbanization resulted in crowded cities with poor sanitation,
epidemics appeared in many areas of the world.
A textbook example of scientific investigation is the work of Dr. John Snow (18131858) on the transmission and prevention of cholera, carried out during an outbreak
of the disease in London in 1853-54. England had suffered cholera epidemics in
1831-1832, 1848-1849, and again in 1853-1854. Medical speculation on
communicable diseases centered on the idea of transmission by microscopic living
organisms. The competing theory was that such diseases were caused by “effluvia,”
something given off in the exhalations of patients or by the bodies of the dead. At the
time the germ theory was considered highly speculative and was not widely
accepted.
The question of how cholera spread was complicated: evidence indicated that it
could be transmitted by close personal contact, yet some people (especially
physicians) who had close contact with cholera victims did not get the disease. In
addition, the disease could appear far away from existing outbreaks. A number of
people, both physicians and others, blamed the water supply. Snow adopted this
theory, refining it by specifically implicating the excretions of victims. He was able to
provide convincing empirical proof in support of this theory. He did this by
recognizing that in a particular district of London where an outbreak had occurred,
some houses got their water from one water company and other houses got theirs
from a different water company. Further, that these two companies drew their water
from different parts of the River Thames.
Snow’s monographs on the transmission of cholera are models of scientific analysis.
While his studies did not demonstrate that his theory was true, they provided strong
support; in later language, Snow was able to offer “sufficient reasons” that his claims
regarding the spread of cholera through contaminated water were correct. He began
by offering a number of examples in support of the claim that cholera was a
communicable disease. He then made two strong arguments against the effluvia
theory: (1) even though everybody who is in close contact with a victim of the disease
will breathe in the effluvia, not everybody in such contact gets sick; and (2) in some
cases, cholera breaks out in new areas where nobody has been exposed to previous
victims. He pointed out that because there are other known ways for diseases to
spread, there was no need to focus on the effluvia theory. Further, the inability to
identify microscopic agents is no argument against the hypothesis that cholera is
caused by the ingestion of some form of “morbid material.” In Snow’s language “It is
no objection to this view that the structure of the cholera poison cannot be
recognized by the microscope, for the matter of small-pox and of chancre can only be
recognized by their effects, and not by their physical properties.”[60]
Another case in which a theory was adopted even though predicted empirical
evidence could not be observed is the Copernican theory, which predicted stellar
parallax, an effect that could only be observed with the development of telescopes of
greater power than was available in the 17th century.
Snow continued by laying out arguments as to why not everybody attendant on a
cholera patient would get sick—those involved with caring for and washing the
clothing and bedclothes of a sick person had a high chance of getting sick themselves
while physicians and others who were careful to wash their hands and/or who did
not eat food prepared in the same room as a victim had a very low probability of
sickness.
In addition, there was substantial evidence that contamination could be spread
through the water supply. During the 1853-1854 outbreak in Soho, Snow noticed
that many victims came from households that obtained their water from a pump on
Broad Street. He convinced the parish Board of Guardians to remove the pump
handle, and the incidence of cholera in the area dropped significantly. Further
evidence supported the idea that contaminated water from this pump had been a
source of the spread of cholera: two groups living near the Broad street pump had
very few cases. These were the inhabitants of a workhouse and the employees of a
brewery, neither of whom drank from the pump. The workhouse had its own water
supply, and the employees of the brewery were allowed a certain quantity of beer
each day, so did not drink water.
Another line of Snow’s argument involved analysis of death rates between customers
of the Lambeth Waterworks Company (which had moved its water intake upstream
to a place not contaminated by London sewage) and the Southwark and Vauxhall
Water Company (which drew its water from contaminated regions of the river). This
provided an excellent comparison in support of his theory. In Snow’s words:
The pipes of each Company go down all the streets, and into nearly all the courts and alleys.
A few houses are supplied by one Company and a few by the other, according to the decision
of the owner or occupier at that time when the Water Companies were in active competition.
In many cases a single house has a supply different from that on either side. Each Company
supplies both rich and poor, both large houses and small; there is no difference either in the
condition or occupation of the persons receiving the water of the different Companies.[61]
Thus, there were no other circumstances distinguishing the two sample populations,
allowing for an unbiased comparison:
No fewer than 300,000 people of both sexes, of every age and occupation, and of every rank
and station, from gentlefolks down to the very poor, were divided into two groups without
their choice, and, in most cases, without their knowledge; one group being supplied with
water containing the sewage of London, and amongst it, whatever might have come from
the cholera patients, the other group having water quite free from such impurity.[62]
Through painstaking survey work during the cholera epidemic of 1853-54 Snow
accumulated data on cholera deaths in the populations supplied by each water
company. He represented his data in terms of deaths per 10,000 households and
compared the results to the rest of London.
Number of
Houses
Deaths from
Cholera
Deaths in each
10,000 houses
Southwark & Vauxhall Co.
40,046
1,263
315
Lambeth Company
26,107
98
37
Rest of London
256,423
1,422
59
Table 3: Death Rates from Cholera in Houses Supplied by the Two Water
Companies. Source: The Experience of Science: An Interdisciplinary Approach.
Goldstein & Goldstein, p. 47. New York: Plenum Press, 1984.
Based on his work, Snow was able to propose a number of public health measures,
many of which are in effect today:
1. Those attending a sick person ought to be extremely careful to wash their hands.
2. Soiled bed linen, etc., should be immediately washed, or immersed in water
until they can be washed. Things that cannot be washed should be boiled or
otherwise exposed to temperatures of 212 degrees or higher.
3. Care must be taken that water used for drinking or food preparation is not
contaminated.
4. When cholera is present in a neighborhood, all provisions brought into a house
should be washed with clean water and/or exposed to high temperature.
5. When cholera appears among those living in crowded conditions, the healthy
ought to be removed to another location.
6. Mine workers ought to have 4- rather than 8- hour shifts so that they can eat at
home rather than taking food into the mine.
7. People ought to be educated as to how cholera (and other such diseases) spread.
8. Good drainage is needed.
9. A water supply free of contamination is required.
10. Model lodging houses for vagrants and the poor should be provided.
11. People ought to be taught habits of personal cleanliness.
12. Persons and especially ships arriving from places where an epidemic is in
progress need to be quarantined until the sick can be separated from the
healthy.
Snow’s work is an exemplary model of scientific inquiry. He collected extensive data,
gives interpretations based on a theoretical perspective, and offers refutations of
alternative explanations. It was another 30 years, however, before the bacteria that
caused cholera, Vibrio cholera, was identified, providing the definitive final touch to
Snow’s investigation.
Unit 3 Study Questions
TOP
Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. What are the two sorts of error Giambattista Vico says can be made in thinking
about the past? Can you identify times when you have made these errors? Do
you find that you tend to make one kind rather than the other, or that you make
one kind in some areas of your life and the other in other areas?
2. Write a short essay (300-500 words) linking the sense of wonder as a motive for
science, the search for understanding and the goal of generality.
3. What are the four characteristics of a good question?
4. Write a short essay (300-500 words) comparing the criteria for a valid answer
in science to criteria for a valid answer in some other area of human activity
such as law, literature, art or religion.
5. Identify at least five sorts of questions that can be asked in science.
6. Describe Roland Omnès’s four stages of scientific activity as presented in
Discussion 3.2.
7. Describe three different ways that scientific discoveries have been made, and
give an example of each.
8. What were the major pieces of evidence that led Snow to his discovery of how
cholera is transmitted?
9. Identify the main questions that Snow asked, and describe how he answered
them as he carried out his research.
10. Chapter 5 of What Science Is describes the history of our theories of planetary
motion. Apply the eight-stage model of scientific development given in
Discussion 3.2 to map out the corresponding stages in this history. Note that
you may need to consider more than a single cycle of this process.
FOOTNOTES
[31]
Rumi, J. Teachings of Rumi (The Masnavi): The Spiritual Couplets of Jalaluddin Rumi,
trans. E. H. Whinfield. London: Octagon, 1994.
[32]
This tale is adapted from the story “Pomegranates,” which appears in Shah, Idries. The
Dermis Probe, p. 92. London: Octagon, 1980.
[33]
Vonnegut, Kurt, Jr. Cat’s Cradle, p. 150. New York: Delta/Seymour Laurence, 1986 ©
1963.
[34]
Quoted in Jones, Shirley, ed. The Mind of God and Other Musings, pp. 64-65.
[35]
Quoted in Skeptic Magazine, 9, 2 (2002): p. 2.
[36]
Aristotle. Metaphysics (I, 2 982b), p. 258. In A New Aristotle Reader, pp. 255-360. J. L.
Ackrill, ed. Princeton, NJ: Princeton University Press, 1987.
[37]
Newton, Sir Isaac. Opticks, or a Treatise of the Reflections, Refractions, Inflections and
Colours of Light (based on the 4th edition London 1730), pp. 369-370. New York: Dover,
1952.
[38]
Newsweek, March 31, 1958.
[39]
Sagan, Carl. Broca’s Brain, p. 17. New York: Ballantine, 1993.
[40]
Vico, G. The New Science of Giambattista Vico, trans. from 3rd ed. by T. G. Bergin and
M. H. Fisch, rev. and abr. ed., pp. 18-19. Ithaca, NY: Cornell, 1970.
Young children often make the first sort of error when they regard their parents as the
source of all wisdom and truth. Adolescents are prone to the second kind of error when they
assume that their parents are hopelessly ignorant and out of touch. Mark Twain referred to
this second sort when he remarked that as a youth he found his father ignorant, but when he
was older he was amazed at how much the old man had learned.
[41]
In an interesting contrast, people during the Renaissance looked to the past glories of
Greece and Rome, and saw themselves as living in an impoverished age.
[42]
Quoted in Jones, Shirley, ed. The Mind of God, p. 74.
[43]
Shab-Parak, quoted in Shah, Idries. Wisdom of the Idiots, p. 153. London: Octagon, 1991.
[44]
Shamcher. Personal communication. 1976.
[45]
Quoted in Harte, John. Consider a Spherical Cow: A Course in Environmental Problem
Solving. Sausalito, CA: University Science Books, 1988.
[46]
Quoted in Jones, Shirley, ed. The Mind of God, p. 40.
[47]
Popper, Sir Karl. Conjectures and Refutations: The Growth of Scientific Knowledge, 3rd
ed. London: Routledge, 1969.
[48]
There is an evolutionary ring to this view, in that a variety of initial conjectures are acted
on by a selective process which leads to the survival of those best able to fit with current
theorizing and experimental evidence.
[49]
Kuhn, Thomas. The Structure of Scientific Revolutions, 2nd ed. Chicago: University of
Chicago Press, 1970.
[50]
For example, when the caloric theory of heat (based on the paradigm that heat is a
material fluid substance) was replaced by the kinetic theory (based on the paradigm that
heat is a measure of the internal motion of molecules), the question of how much caloric
fluid a body contained ceased to have meaning.
[51]
Foerster, Heinz von. Ludwig von Bertalanffy Memorial Lecture, delivered at the meeting
of the Society for General Systems Research held in Washington, DC, 1982.
[52]
Einstein, Albert. “Autobiographical Note,” pp. 15, 17. In Albert Einstein: Philosopher-
Scientist, Volume 1, Paul A. Schilpp, ed. New York: Harper and Row, 1959.
[53]
Wheeler, John A. “Information, Physics, Quantum: The Search for Links,” pp. 3-28.
In Complexity, Entropy and Information, Wojciech H. Zurek, ed. Reading, MA: AddisonWesley.
[54]
Quoted in Jones, Shirley, ed. The Mind of God, p. 40.
[55]
Here is an example of two different theories emerging in question-and-answer sequences
that begin at the same point; that is, both the Ptolemaic and the Copernican theories are
attempts to explain planetary motion, but they posit very different explanatory hypotheses.
[56]
Goethe, Johann Wolfgang von. Elective Affinities. Quoted in Bartlett, John. Familiar
Quotations, 14th ed. Emily Morison Beck, ed., p. 477. Boston: Little, Brown, 1968.
[57]
Auden, W. H. “Writing,” p. 21. In The Dyer’s Hand and Other Essays, pp. 13-27. New
York: Vintage, 1968.
[58]
Omnès, Roland. Quantum Philosophy: Understanding and Interpreting Contemporary
Science, trans. Arturo Sangalli, pp. 257-260. Princeton, NJ: Princeton University Press,
1999.
[59]
Plato. “Phaedrus,” 265d, e. In The Dialogues of Plato, Volume III, 4th ed., trans.
Benjamin Jowett, pp. 107-189. Oxford: Clarendon Press, 1953.
[60]
Snow, John. “Snow on Cholera.” In Goldstein, Martin, and Inga Goldstein. The
Experience of Science: An Interdisciplinary Approach, pp. 29-71. New York: Plenum Press,
1984.
[61]
Snow, in Goldstein and Goldstein, p. 45.
[62]
Snow, in Goldstein and Goldstein, p. 46.
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Unit 4 What Is Understanding?
Discussion 4.1
Discussion 4.2
Unit 4 Study Questions
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STUDY GU IDE
Unit 4 What Is Understanding?
In Unit 1, we considered the importance of understanding science and its relation to
the rest of society, and noted that the inductive attitude is a major aspect of science.
In Unit 2, we discussed science as an activity that seeks to understand the world, and
pointed out that the methods used in this search have evolved in human cultures
over time. Unit 3 introduced the method of directed questions and answers, and
briefly described the modern version of this method as it is used in science, including
some of the criteria to be satisfied by acceptable answers. Thus, as we begin Unit 4,
we have the idea that science seeks understanding of the world through a process of
interrogation, carried out from a particular point of view, in which criteria for
formulating questions and for accepting provisional answers have evolved over the
course of human history.
The word “understand,” however, can have a number of meanings. When we say that
we understand an instruction, we are acknowledging that we know what action is
required. A statement of this kind is different from a claim that we understand
Euclid’s proof that there is no largest prime number, and both are different from our
understanding of another person’s feelings. Thus, to add more substance to the idea
that science seeks understanding, it is necessary to say what sort of understanding it
is that science seeks.
Unit 4 begins with a more detailed study of the purpose of science. We find that
while “understanding the world” is acceptable as an overall, generic statement of
purpose, there are more issues to consider. We also introduce an important
distinction that will reappear throughout the remainder of the course: the distinction
between the top-down and bottom-up approaches to the study of systems.
In Discussion 4.2, we take up the question of understanding. We introduce the idea
of a conceptual framework, and define understanding in science as being able to see,
both analytically and intuitively, how something fits within a conceptual framework.
This is related directly to the notion of ideals of natural order as introduced in Unit 3
as basic elements of a conceptual framework that ground theoretical work by telling
us what does not need to be explained. Only deviations from these ideals require
explanation.
Thus, having a “scientific understanding” means being able to offer an explanation in
terms of a conceptual framework that is based on certain ideals of natural order, and
that is constrained by the conditions of logical coherence and empirical agreement.
Objectives
When you have completed Unit 4, you should be able to
1. describe the top-down and bottom-up approaches to science, including some of
the ways each approach is used, and some of the difficulties with them.
2. state the difference between “formal purpose” and “functional purpose,” and
identify at least three different functional purposes of science.
3. provide a general definition of “conceptual framework,” and describe how
conceptual frameworks are related to points of view.
4. define understanding in science in terms of conceptual frameworks.
5. state the difference between “prediction” and “explanation,” or “forecasting”
and “understanding.”
6. describe the Aristotelian ideals of natural order in dynamics, planetary motion
and the theory of matter, and contrast them to the ideals of natural order
introduced by Galileo (in dynamics), Newton (in dynamics and planetary
motion), and 18th century chemists (in the theory of matter).
7. describe the distinction between “physiology” and “pathology” in terms of ideals
of natural order.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 4.1, “What is the Purpose of Science?”
2. Answer Unit 4 Study Questions 1-4.
3. Read Discussion 4.2, “What Does It Mean to Understand?”
4. Answer Unit 4 Study Questions 5-8.
Discussion 4.1 What is the
Purpose of Science?
TOP
Science, like art, religion, commerce, warfare, and even sleep, is based on presuppositions.
It differs, however, from most other branches of human activity in that not only are the
pathways of scientific thought determined by the presuppositions of the scientists but their
goals are the testing and revision, even abandonment of old presuppositions and the
creation of new ones.
In this latter activity, it is clearly desirable for a scientist to know consciously and be able to
state his or her own presuppositions.
—Gregory Bateson (1904-1980)[63]
Before you continue reading, take a few minutes to answer each of the questions
given below.
1. What is the purpose of science?
2. Why should society support scientific research?
3. What are your reasons for an interest in science?
Science and Philosophy
Science is the systematic enterprise of gathering knowledge about the world and organizing
and condensing that knowledge into testable laws and theories.
—E. O. Wilson[64]
When considering a complex human activity such as science, we can take a variety of
approaches. We can, for example, seek to understand what it is like to be involved in
scientific research, to get the inside view. Or, we can look at the multitude of
activities in which scientists engage, and attempt to discover some overall scheme
that will allow us to view these activities as part of a coherent whole. We can make
the attempt to compare science with other forms of human activity with which we
believe we are more familiar. We can suppose that we already know the intended
purpose of science, and construct criteria for the satisfaction of that purpose. Each of
these approaches could be taken alone, or they could be pursued in various
combinations.
Philosophy of science attempts to study the general nature of science, together with
its conceptual apparatus and methodology. This course is not a course in philosophy
of science. Nevertheless, it is important that you be familiar with some of the main
ideas that are current in philosophy of science.
In science, the idea is to discover a unifying theory that all scientists in a given field
can accept. In philosophy of science, this is not the case. Philosophers of science
frequently disagree, even on the most basic points, and argue these disagreements
with other philosophers. In this and other units, you will encounter a variety of
divergent philosophical opinions.
From the point of view of a scientist, the approach to these different ideas is to learn
a little about each, asking whether or not it is useful for actually doing science. It is
not a particularly good idea to become a dogmatic adherent of any one of the
different positions put forward by philosophers of science, because situations will
arise in scientific work that require greater philosophical flexibility than any
particular one of these positions will allow.
Albert Einstein, who called himself an “epistemic opportunist,” emphasized this
point when he wrote
Epistemology without contact with science becomes an empty scheme. Science without
epistemology is . . . primitive and muddled. However, no sooner has the epistemologist, who
is seeking a clear system, fought his way through to such a system, than he is inclined to
interpret the thought-content of science in the sense of his system, and to reject whatever
does not fit. The scientist cannot afford to carry his striving for epistemological systematic
that far. External conditions, which are set for him by the facts of experience, do not permit
him to let himself be too much restricted by the adherence to an epistemological system.[65]
Again we encounter the question of point of view, and the importance of not
allowing thinking to be confined within any particular formalized system. Such
systems are tools for thought, not its masters and, as with all tools, what is most
important is learning how to use them correctly.
Science Studies Systems
Science is the search for truth—it is not a game in which one tries to beat his opponent, to
do harm to others.
—Linus Pauling (1901-1994)[66]
The subjects of scientific study are almost always “systems”: complex units made up
of diverse parts that exchange something—molecules, energy, information—and that
are more or less segregated from their surroundings. One characteristic of a system
is that changing one of its parts can have consequences, often unforeseen, for other
parts and for the system as a whole.
In the study of complex systems, two complementary views can be taken—“topdown” and “bottom-up.” In the bottom-up view, we begin with the most basic
components of the system, and the way in which these elementary components
interact and fit together to make up the system. The idea is that it may be possible to
understand properties of the whole system by knowing how these properties are
determined by the interactions of its elementary components. This approach has
been useful in chemistry, for instance, where the properties of chemical compounds
are explained in terms of their molecular and atomic composition. It has also proved
useful in elementary particle physics, and in medicine, where many diseases are
understood and treated in terms of the malfunction of a particular organ, group of
cells or gene.
In the top-down approach, it is assumed that a system must satisfy certain
constraints, either on the basis of general laws of nature, or in order to maintain its
integrity as a system. We then ask what conditions these constraints impose on the
system’s components and their interactions. This approach often shows up, for
example, in applications of the various symmetry, optimality and conservation
principles that are used in science. Human physiology, for example, contains many
feedback mechanisms to insure that things such as body temperature, heart rate,
blood sugar and blood pressure remain within acceptable ranges. Thus, it is
reasonable to assume that the conditions arising from the necessity of organismic
survival also impose certain anatomical and physiological conditions on the human
body.
Neither the bottom-up nor the top-down approach alone can provide a complete
picture of how science is done. Rather, the complementary use of bottom-up and
top-down thinking is perhaps the fundamental process involved in scientific
reasoning. You will encounter it often in the remainder of this course.
A difficulty with the bottom-up approach is that it is often unclear how overall
systemic properties emerge from the interaction of system components. There is
nothing obvious in the atomic nature of hydrogen and oxygen, for example, telling us
that we can expect to experience water as being wet. In the top-down approach, on
the other hand, it is often unclear what general principle, or set of principles is
appropriate to characterize a given system. A lack of clarity can lead to the
imposition of inappropriate conditions and requirements. In medicine, for example,
it is easy to say that the overall principle is health. But then it is necessary to define
health. Is it a lack of pain? Then we might focus attention on painkillers. Is it a
healthy appearance? Then we might focus on cosmetic surgery.
The same kinds of problems arise in attempts to define science. Taking a bottom-up
approach, we can study the various activities of individual scientists, the politics of
journal publication, the ways in which scientists interact at conferences and in
scientific communities, the instruments used in experimental work, and the uses
that are made of the products of scientific research. But we are left with the problem
of trying to generate a unified picture of science from this mass of data. How do the
pieces fit together?
Taking a top-down approach, we can begin with an ideal of the purpose of science,
and develop a philosophical view from it—asking what is necessary in order for
science to serve this purpose. This strategy generates a unified system, but it may
bear little resemblance to the actual reality that we want to understand.
Purposes of Science
In his book Foresight and Understanding, the philosopher Stephen Toulmin
presents a bottom-up analysis, leading to the eventual conclusion that science has no
single purpose; individual scientists carry out their work for a variety of different
reasons and purposes. Yet the reasons that individual scientists have for their choice
of profession cannot be considered to be the purpose of science itself, and by the end
of the book, Toulmin presents a viable candidate for a “purpose of science”—to
understand the world. He advocates this over a contrasting “purpose” of making
accurate predictions; comparing ancient Babylonian and ancient Greek astronomy
as an illustration. The Babylonians could make excellent predictions of planetary
positions, eclipses, and so on. The Greeks, initially, could not make such predictions.
Rather, they thought in terms of an overall system in which the Earth was at the
centre and the planets revolved around it in perfect circles. In this way they could
understand what they saw in the sky, even if their predictions were not as accurate as
those of the Babylonians. In contrast, the Babylonians did not understand what they
saw—they attributed the planetary movements to the gods, who chose to behave as
they did for unknown reasons. The predictivist thesis—that the goal of science is to
make accurate predictions is thus in Toulmin’s view secondary to the goal of
providing a means of understanding the world.
In the context of science itself, understanding can be viewed as a final purpose, the
goal toward which all scientific work is directed. But we might say that from a
broader perspective, understanding is really the aim of science, the motivating
impulse that gives it direction, while the purpose of science is involved in the way in
which our understanding is used. The use of the term purpose in this latter sense
invites us to think teleologically, to ask what purpose science serves in the wider
context of human activity. For example, what is the purpose of science in terms of
human cultural evolution? This context suggests a functional rather than formal
purpose,[67] relating science to the broader context of human development. Taking
this approach to the ultimate extreme, we must think in terms of a final purpose of
human life.
That, of course, is the Big Question.
Without taking a position here on the nature of this ultimate purpose, we note that it
should be clear that, at the level of an individual scientist, the development of a
rational understanding of the world, even if only partial and subject to revision, can
help in the personal search for meaning in life. That is, for individual scientists, the
discipline of their work can provide a vehicle, training them in the mental and moral
qualities that will inform their personal realization of purpose. From that
perspective, any discoveries made are secondary to the personal qualities developed
in the process of discovery. This point of view was clearly stated by Pierre Lecomte
du Nouy (1849-1919) when he wrote, “Our intellectual endeavors, our whole science
will be of no avail if they do not lead man to a better comprehension of himself, of
the meaning of his life, and of the resources buried in his inner self.”[68] This
conception of the purpose of science is one that traces back to Aristotle.
The idea that the pursuit of science can serve the purpose of individual development
is not emphasized in modern society, but it is important to mention it, as it closely
relates both to the moral qualities required of a good scientist and to the motivation
that prompts an individual to go into science.
As an analogy, athletes may train for years to qualify for the Olympics. Although they
aim to win, winning is, in a wider context, secondary. From the viewpoint of
personal development, what has a more lasting importance is the effect of the
training process on the athlete. The improvements in strength and grace, selfdiscipline and sportsmanship have far more significance than the actual competitive
results. From this higher viewpoint, then, the purpose of competitive sports is
human development, as is the purpose of science.
Another approach to the question of purpose asks about the social value of science.
One answer in this line is obtained by pointing to the technological results science
has made available to improve human life.[69] This view bases evaluation on social
utility. It looks at the by-products of science and using them for a definition of
purpose. At one level, this perspective certainly has validity. A corporation will
sponsor scientific research because it expects to profit from the results of that
research. There is, however, a deeper viewpoint.
Remarks of the ancient Greek philosopher Epicurus (341-271 BCE) can be summed
up as asserting that the purpose of science and philosophy is, “To free man from
fear.”[70]
From the perspective of 2300 years of intervening history, we might discount this
comment. The temptation is to think of the material benefits that have come from
science, the technology that has eased our lives. We are distracted by the obvious
and scarcely recognize how well science has carried out the task Epicurus identified.
It takes a major effort of imagination to put ourselves in the place of a citizen of
ancient Greece, but suppose we make that effort.
We find ourselves in a society in which ignorance is all-pervasive. Even if we are a
member of the very small educated elite of this culture, we do not understand the
reason for the tides, the movements of the planets, the source of the sun’s light, the
nature of the seasons, the weather, the causes of almost all diseases . . . the list goes
on and on. And all of this ignorance is a source of fears that we now barely
remember. Wesley C. Salmon (1905-2001) gives a striking example of how science
has helped to eliminate fear in his book Scientific Explanation and the Causal
Structure of the World. Salmon gives two quotes, the first from a letter written
shortly after the appearance of the great comet of 1682 (later known as Halley’s
Comet) by the Jesuit priest Eusebio Francisco Kino:
It appears that this comet, which is so large that I do not know whether or not the world has
ever seen one like it or so vast, promises, signifies, and threatens many fatalities . . . its
influence will not be favorable. And therefore it indicates many calamities for all Europe . . .
and signifies many droughts, hunger, tempests, some earthquakes, great disorders for the
human body, discords, wars, many epidemics, fevers, pests, and the deaths of a great many
people, especially of some very prominent persons. May God our Lord look upon us with
eyes filled with pity.
And because this comet is so large it signifies that its fatalities will be more universal and
involve more peoples, persons, and countries. And since it is lasting so long a time . . . it
indicates that its evil influence will afflict mortals for many years.[71]
The second quote, from Pierre-Simon Laplace (1749-1827), a French mathematician,
refers to the Newtonian theory of universal gravitation as the explanation for the
behaviour of comets:
But as these phenomena occurring and disappearing at long intervals, seemed to oppose the
order of nature, it was supposed that Heaven, irritated by the crimes of earth, had created
them to announce its vengeance. Thus, the long tail of the comet of 1456 spread terror
throughout Europe. . . . This star after four revolutions has excited among us a very different
interest. The knowledge of the laws of the system of the world acquired in the interval had
dissipated the fears begotten by the ignorance of the true relationship of man to the
universe; and Halley, having recognized the identity of this comet with those of 1531, 1607,
and 1682, announced its next return for the end of the year 1758 or the beginning of the year
1759. The learned world awaited with impatience this return which was to confirm one of
the greatest discoveries that have been made in the sciences.[72]
Another, more modern example is smallpox, the disease that killed more people and
caused more human suffering than any other in history. In 1980, the World Health
Organization announced that smallpox had been eradicated. Nobody today lives in
fear of an outbreak of “the Pox.” Beyond the obvious material benefits of science is a
dustheap of ancient fears that no longer haunt humankind.
Of course we could play the cynic and suggest that all science has done is to replace
old fears with new and more realistic ones: environmental pollution, nuclear war,
global warming, depletion of the ozone layer, and other modern sources of stress.
Even the fear of flying didn’t exist before we learned how to fly. But the cynic forgets
the past, and so finds it easy to blame science for the problems and dangers of the
present.
There is another reading of Epicurus’ statement, highlighting both a personal and
social role for science. There is a psychological truism that reads:
Ignorance → Fear → Belief
We fear that of which we are ignorant and to control this fear we create a belief. We
will then attack anybody who questions this belief with a ferocity proportional to the
underlying fear. History is filled with examples of fanaticism acting in support of
some belief, examples of the most atrocious inhumanities, which we read of and
wonder, “Why didn’t they exercise a little common sense?” But when our own deeply
held beliefs are questioned . . . well that’s different. We need to recall a comment
made by Marie Curie (1867-1934): “Nothing in life is to be feared. It is only to be
understood.”[73]
It seems that the only way to deal with the natural psychological reaction— to fear
that which is not understood—is to adopt a cool, detached point of view, suspending
judgement with respect to all belief. With this “inductive” attitude, we can admit
ignorance and begin to reason. Out of this attitude have come both philosophy and
science—the best tools for overcoming ignorance that have been discovered.
Philosophy helps us to acknowledge ignorance and face it without fear, as well as
providing an analysis of knowledge: what it is, how it can be verified, how far it can
be trusted. Science provides a method— the most effective method known—for the
accumulation of publicly verifiable knowledge.
We have seen that what is taken as the purpose of science is relative to the
perspective from which we choose to evaluate science. Some of
the functional purposes of science, according to the context of our viewpoint, are
1. to free people from fear.
2. to free people from drudgery and repetitive labour.
3. to provide leisure and vehicles for personal self-expression.
4. to provide a vehicle of personal development.
5. to attain understanding of ourselves and the world.
6. to improve human health and security.
7. to provide a basis for standards of moral and ethical conduct, and social
regulation.
8. to foster social cohesion and interaction through a basis of commonly accepted
knowledge.
9. to provide means of defining and attaining ideals.
For science itself, however, it would seem that the best statement of formal purpose
that can be given is simply “to understand the world rationally.”
Discussion 4.2 What Does It
Mean to Understand?
TOP
Familiar things happen, and mankind does not bother about them. It requires a very
unusual mind to undertake the analysis of the obvious.
—Alfred North Whitehead (1861-1947)[74]
What do we mean when we say that we understand something? Or, turning this
question around, when we understand a thing or an idea, we believe that we know
what it means, but what does it mean to have meaning?
Sometimes we use sensory metaphors to describe understanding: we get the picture,
grasp the idea, take the point, see what’s really going on, have digested the concept
. . . there are many others. A very common meaning for understanding is that we
have constructed an internal image, something that can be viewed in the mind’s eye.
But meaning is relative to context: the cross means something very different to
Christians than it did to the ancient Romans. In other words, as we noted in
Discussion 1.1, point of view must be taken into account. The very same thing can
have radically different “meanings” when seen from different points of view.
In addition to point of view, however, it is important to include the idea of a
conceptual framework. While a point of view relates to our attitude, and to our
willingness to look without fear, it is our conceptual framework that determines how
we go about looking and how we interpret what we see.
A conceptual framework provides the context within which meaning arises. When
facing the complexity of the world, the number of questions we can ask is unlimited.
Without some means of sorting out what is really important, we can make no real
progress. In the question-and-answer process of science, conceptual frameworks—
what Thomas Kuhn calls paradigms—provide the context that determines which
questions are important, and what sorts of answers are significant.
The philosopher of science N. R. Hanson (1924-1967) describes an imaginary
situation in which Tycho Brahe and Johannes Kepler stand together on a hillside
watching the sunrise.[75] Although both see the same phenomenon, they understand it
in completely different ways. Tycho, who accepted a form of geocentric cosmology,
would say that the sun actually rose in the east as the solar sphere rotated about the
earth from east to west. As a consequence, his questions related to a search for ways
to find kinematic fits for the planetary orbits in terms of perfect circles.
Kepler, on the other hand, a believer in the new heliocentric theory of Copernicus,
would say that the sun did not move, while the earth rotated from west to east. His
questions were dynamic, aimed at explaining planetary orbits in terms of the action
of a central force. Holton quotes Kepler’s own words, as follows:
My aim is to show in this that the celestial machinery is to be likened not to a divine
organism but rather to a clockwork . . . , insofar as nearly all the manifold movements are
carried out by means of a single, quite simple magnetic force.[76]
While Kepler did not succeed in finding the central force law he sought, Isaac
Newton did, and the temptation today is to say that Tycho did not really understand
the motion of the planets and the nature of the solar system. But Tycho was the
greatest astronomer of the 16th century. His observations of planetary motion were
exceptionally accurate for the time, and the Ptolemaic, or later his own geocentric
theory seemed, to him, not only to give a better fit to his data, but also to be in
accord with the way that he believed a scientific explanation in astronomy ought to
be given.
Gerald Holton quotes a letter from Tycho to Kepler written on December 9, 1599:
[E]ven if it should appear to some puzzled and rash fellow that the superposed circular
movements on the heavens yield sometimes angular or other figures, mostly elongated ones,
then it happens accidentally, and reason recoils in horror from this assumption. For one
must compose the revolutions of celestial objects definitely from circular motions; otherwise
they could not come back on the same path eternally in equal manner, and an eternal
duration would be impossible, not to mention that the orbits would be less simple, and
irregular, and unsuitable for scientific treatment.[77]
Ironically, it was Tycho’s observations of the motion of the planets, in particular
Mars, gathered over many years, which provided the essential data Kepler used to
derive his three laws of planetary motion. Nevertheless, it is unfair to say that Tycho
did not understand the motions of the planets.
What can fairly be said is that his understanding was based in a conceptual
framework that later proved to be in error, and that his own observations played an
essential role in the overthrow of the framework in which they were made. This
conclusion brings out the very significant difference between the world itself, and the
conceptual frameworks and theories that are used by scientists to understand the
world. Often the very same experimental or observational data can be explained
within two or more distinct theories or conceptual frameworks. When this happens,
we must search for other ways of determining which theory or framework is to be
accepted. This process may involve further observations or experiments, but it can
also involve aesthetic and other factors.
The success of the Copernican theory was not based on observational results. Both
the geocentric and heliocentric theories could fit the observational data, and
especially at the beginning, the Ptolemaic theory often gave a better fit. In addition,
the Copernican theory predicted that it ought to be possible to observe stellar
parallax, and this was not found. Nor, in its initial form, was the Copernican theory
significantly simpler than that of Ptolemy. But it did open up new directions for
exploration, while in the conceptual framework of geocentrism there were no new or
truly interesting questions to be asked. There was also the important social factor
that the Copernican theory was more in tune with the attitude of discovery and
exploration that gripped Europe following the discovery of the New World.
In science, a conceptual framework includes assumptions about how to reason and
interpret evidence, as well as certain assumptions about what is natural—the “ideals
of natural order.” Phenomena and events that are in accord with these ideals require
no explanation—they are just how we expect the world to behave. All that we are
required to explain are deviations from these ideal behaviours.
In Newtonian physics, for instance, the states of rest and uniform straight-line
motion are the ideals for the natural motion of bodies. Deviations from these natural
motions are explained in terms of the action of external forces. This aspect of the
Newtonian conceptual framework has been incorporated in many other areas of
science, and in popular culture as well, as the belief that if a system is not in an
equilibrium state, some external force must be acting.
This way of interpreting the material world has proved to be extremely productive.
Unfortunately, it is often extended to encompass the entirety of the social and
psychological worlds as well. While the idea does have some validity in these areas,
its over-application can lead to serious error. For example, a basic aspect of the
Newtonian ideal is that no overall systemic change can come from inside of a system
itself. This view can lead to the idea that individuals are not responsible for their
actions, because these actions are controlled by external forces, and to the belief that
social institutions cannot change from within, but must be overthrown by outside
forces. But when systems are able to change their state through their own inner
dynamics, the Newtonian ideal of equilibrium states disturbed by external forces
cannot fully apply. Indeed, the concept of force itself may need to be revised or
abandoned in these cases. In a complex ecosystem, for example, are the species
distributions determined by selective forces; by the mutual interactions of all species
in the ecosystem; or by the relative attraction of different niches that are themselves
determined by overall system dynamics?
The way that we interpret the world tells us as much about ourselves as it does about
the world. As the physicist and philosopher Sunny Auyang remarks,
Scientific theories [do not just] represent the objective world: they represent it in ways
intelligible to us. Thus while their objective content illuminates the world, their conceptual
frameworks also illuminate the general structure of theoretical reasoning, an important
aspect of our mind.[78]
In other words, we understand the world in terms of our conceptual frameworks, but
the sorts of frameworks that we find available to use tell us something about the
nature of mind itself.
For our purposes, then, to understand something means to be able to interpret it
within a conceptual framework, and to experience intuitively the validity of the
interpretation, to see that it is a good fit. Without a conceptual framework to give
meaning to observations and theories, we do not even know what questions need to
be asked. Thus, to explain something is to show how it fits within a conceptual
framework, how it may be understood. Explanation, as well as understanding, is
always relative to a particular conceptual framework.
The ancient Greeks used the word epistemé to indicate this kind of understanding.
From this we get the word epistemology—the study of knowledge. The word itself
derives from epistanai, “to stand upon.” Understanding, in this sense, refers to
knowledge that stands upon a secure conceptual framework. That is, it can be
supported by rational arguments showing its coherence with the basic assumptions
and axioms of a conceptual framework. In science, this framework itself must satisfy
the condition of logical coherence and agreement with empirical observation.
Thus, in saying that the general purpose of science is to understand the world
rationally, we are saying that science aims at the development of conceptual
frameworks within which observed aspects of the world can be rationalized. This
formulation implies that the conceptual frameworks used must be internally
consistent, must allow experimental testing, and must give conclusions that can be
shown to correspond to the world. In philosophical language, they must satisfy both
the coherence and correspondence notions of truth (i.e., must be coherent, or
rationally consistent, and must also correspond to observation and experimental
facts).
As with PG-rated movies, however, a warning is required. There is a false
understanding that relies only on the ability to manipulate language. In
Discussion 14.1, we consider the empathetic and intuitive aspects of reason, and
show that they are essential for real understanding. Not only must we be able to offer
an interpretation within a conceptual framework, we must also have an intuitive
grasp of the coherence of our interpretation with that framework. Without this
intuitive aspect, we are only playing with words, and our understanding will be
superficial at best. This point is well stated by the mathematician Gian-Carlo Rota:
Mathematicians ask the question “What is this good for?” when they are puzzled by some
mathematical assertion, not because they are unable to follow the proof or the applications.
Quite the contrary. What happens is that a mathematician has been able to verify its truth in
the logical sense of the term, but something is still missing. The mathematician who is
baffled and asks the question, “What is this good for?” is missing the sense of the statement
that has been verified to be true. Verification alone does not give us a clue as to the role of a
statement within the theory; it does not explain the relevance of the statement. In short, the
logical truth of a statement does not enlighten us as to the sense of the statement.[79]
In the everyday world there are far too many people who manipulate opinion
through clever arguments that give a superficial impression of understanding, but
lack any real sense. Such individuals tend to avoid science, where their sophistries
would be confronted with demands for hard fact, but they love using scientific words
to make themselves sound knowledgeable, and they can be found in droves in the
pseudosciences that abound.
Unit 4 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. Write a short essay (300-500 words) describing the use of top-down and
bottom-up approaches in the development of the periodic table of the elements,
as described in Chapter 2 of What Science Is.
2. Write a short essay (300-500 words) discussing the way in which the formal
purpose of science relates to and supports one (or more) of its functional
purposes.
3. What difficulties can arise in the top-down and bottom-up approaches to
science?
4. The “predictivist thesis” in philosophy of science is that the goal of science is to
make successful predictions. Do you agree with this? If not, why not? What is
the role of predictions in science?
5. Describe the role of conceptual frameworks in science.
6. Write a short essay (300-500 words) on the importance of intuitive insight for
understanding. Why is being able to provide a verbal or written explanation not
sufficient?
Hint: Think of times that you have listened to an apparently authoritative speaker and thought, “He
really doesn’t know what he is talking about.”
7. What different ideals of natural order are found in the stages of the development
of the theory of planetary motion, as described in Chapter 5 of What Science Is?
Note how the ideals become more abstract, more idealized, as the theory
develops.
8. The alchemists believed that all of nature was in a process of development,
striving toward a state of perfection. The process involved definite stages. A
butterfly, for example, begins as an egg, becomes a caterpillar, enters a cocoon,
and eventually emerges, as a butterfly that lays eggs. It was believed that this
process could be speeded up by an appropriate application of heat under the
appropriate conditions. So, for example, beginning with lead one might produce
tin, copper, silver, and eventually, gold.
a. What is the ideal of natural order behind this theory?
b. What are some possible means of explaining deviations from this order?
c. What is the basic pattern of explanation that is used?
d. Is there any field today, scientific or otherwise, where this form of theory
might apply? If so, what would be the analogue of heat? Of the closed
vessel (or “womb”) in which the transformations are to take place?
FOOTNOTES
[63]
Bateson, Gregory. Mind and Nature: A Necessary Unity, p. 25. New York: Bantam, 1979.
[64]
Wilson, E. O. Consilience: The Unity of Knowledge. New York: Little, Brown, 1998.
Quoted in Skeptic 9, 2 (2002): 28.
[65]
Einstein, Albert. “Reply to Criticisms,” pp. 683-684. In Albert Einstein: Philosopher-
Scientist, Volume 2. Paul A. Schilpp, ed. New York: Harper and Row, 1959.
[66]
Quoted in Jones, Shirley, ed. The Mind of God, p. 86.
[67]
The distinction between formal and functional purposes was worked out by Aristotle. An
example is a bow and arrow: the formal purpose is to project the arrow over some distance
with sufficient force to embed itself in a target. The functional purpose involves our choice
of target and why we want to shoot at it.
[68]
Quoted in Jones, Shirley, ed. The Mind of God, p. 78.
[69]
There are those who claim that all that our technology has done is to bring us to the point
where we are destroying the Earth, and that we would be better off without science.
Although there is not room in this course to deal with the truly serious questions of
technological abuses and scientific ethics, we should be aware that they do exist.
[70]
Epicurus. “Letter to Herodotus.” The Essential Epicurus: Letters, Principal Doctrines,
Vatican Sayings, and Fragments, trans. Eugene O’Connor, pp. 19-43. Buffalo, NY:
Prometheus Books, 1993. The text of this letter is available at the site below. Retrieved
August 4, 2002. http://www.epicurus.net/en/herodotus.html/
[71]
Salmon, Wesley C. Scientific Explanation and the Causal Structure of the World, p. 11.
Princeton, NJ: Princeton University Press, 1984.
[72]
Ibid., p. 12.
[73]
Quoted in Jones, Shirley, ed. The Mind of God, p. 79.
[74]
Whitehead, A. N. Science and the Modern World: Lowell Lectures, 1925, p. 4. New York:
Free Press, 1967.
[75]
Hanson, N. R. “Chapter 13, Observation.” In Introductory Readings in the Philosophy of
Science, rev. ed. E. D. Klemke, Robert Hollinger, and A. David Kline, eds., pp. 184-195.
Buffalo, NY: Prometheus Books, 1988.
[76]
Ibid., p. 56.
[77]
Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein, rev. ed., p.
61. Cambridge, MA: Harvard University Press, 1988. Note the circularity of Tycho’s
argument: he claims that only circular orbits are suitable for scientific treatment because
only circular orbits can be treated scientifically. But, of course, an elliptic orbit also “comes
back on the same path.”
[78]
Auyang, Sunny. Foundations of Complex-Systems Theory, p. ix. Cambridge: Cambridge
University Press, 1999.
[79]
Carlo-Rota, Gian. Indiscrete Thoughts, p. 131. Boston: Birkhauser, 1977.
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Unit 5 What Is Not Science?
Discussion 5.1
Discussion 5.2
Unit 5 Study Questions
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STUDY GU IDE
Unit 5 What Is Not Science?
In Units 1 to 4, we considered various aspects of the nature of science as a human
activity. We discussed the importance of the point of view and the inductive attitude,
the definition of science as the effort to understand the world, the question-andanswer process of scientific inquiry, and the nature of understanding in science.
In this unit, we take a different approach and consider some examples of what is not
science. Of course there are many things that are not science—law, art and religion,
to name only three. But these forms of endeavour do not claim to be science. Here
we consider enterprises that falsely claim scientific status. Seeing how
pseudosciences fail to be scientifically credible will help you to understand the
requirements of real science.
Objectives
When you have completed Unit 5, you should be able to
1. describe the characteristics of “pathological science.”
2. describe the characteristics of “pseudoscience,” “crackpot science” and “cargo
cult science.”
3. explain why astrology and creationism are pseudosciences.
4. provide examples of how advertisers and other opinion manipulators make use
of the cargo cult mentality.
5. state the difference between “factual understanding” and “functional
understanding,” and provide examples of each.
6. state the difference between “open” and “closed” cultures or belief systems.
7. describe the difference between “evidence” and “instances,” and discuss how
each is related to theories and beliefs.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Chapter 12, “Questions of Authenticity: Science, Pseudoscience, and How
to Tell the Difference,” pages 158-173, of What Science Is; then read Chapter 13,
“Contentious Questions: The Shadowy Borderlands of Science,” pages 174-188.
2. Read Discussion 5.1, “Pseudoscience, Crackpots and Cargo Cults.”
3. Answer Unit 5 Study Questions 1-5.
4. Read Discussion 5.2, “Why Witchcraft Is Neither Science Nor Pseudoscience.”
5. Answer Unit 5 Study Questions 6-9.
Discussion 5.1 Pseudoscience,
Crackpots and Cargo Cults
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Man is a credulous animal, and must believe something; in the absence of good grounds for
belief, he will be satisfied with bad ones.
—Bertrand Russell (1872-1970)[80]
Bad science, in the sense of poorly executed experiments or flagrantly incorrect
theorizing, is filtered out by peer review, and by the mechanisms of evaluation and
social consensus established within scientific communities. However, there are other
forms of bad science that are just plain nonsense. They can be pigeonholed under the
headings of pathological science, pseudoscience, cargo cult science, and crackpot
science.
Irving Langmuir (1881-1957), the 1932 Nobel Prize winner in chemistry, described
pathological science as research that exhibited the characteristics listed below.[81]
o
o
o
o
o
o
Experimental effects observed either remain very close to the extreme limits of
detectability, or require massive numbers of measurements because of their
exceptionally low statistical significance.
The magnitude of the observed effect seems to be independent of the postulated
causes, which are barely detectable.
The accuracy of the experiments is claimed to be exceptionally high.
The claimed results seem to imply revolutionary theoretical consequences.
Ad hoc justifications and excuses are provided to counter all criticism of
experimental technique.
The results may initially attract as many supporters as critics, but eventually the
excitement dies down when results cannot be reproduced, or when flaws in the
experimental techniques are discovered.
The October 1989 issue of Physics Today published a transcript of a lecture that
Langmuir gave on the nature of pathological science. After describing several
examples, Langmuir characterized the basic failing as follows:
These are cases where there is no dishonesty involved but where people are tricked into
false results by a lack of understanding about what human beings can do to themselves in
the way of being led astray by subjective effects, wishful thinking, or threshold
interactions.[82]
Nicholas J. Turro offers the following recommendations for doing good science and
avoiding the pitfall of pathological science:[83]
1. Always generate and test several plausible hypotheses in attempts to explain a
result.
2. Use imaginative experimental design to increase objectivity and decrease the
chance that the initial observation contains artifacts.
3. Follow the best available paradigm until you are certain that your results may
require its revision.
4. Be conservative with statistical significance and margin of error, especially when
analyzing phenomena on the threshold between signal and noise.
5. Reproduce your results many times.
6. Discuss surprising results with peers, both formally and informally, and make
constructive use of any criticism offered.
7. When discussing your results with non-scientists be careful not to claim more
for your results than is there. Do not claim world-shaking results. Be modest.
8. If your results are shown wrong, accept this with grace and learn from the
experience.
9. Try your best to think of ways that you could be wrong, try to falsify your results.
Pathological science is most likely to occur when an ambitious researcher pays
insufficient heed to the injunction of the Nobel Prize winning physicist, Richard
Feynman (1918-1988): “The first principle is that you must not fool yourself—and
you are the easiest person to fool.”[84]
Feynman also describes “cargo cult science,” a type of nonsense science in which the
forms of scientific practice are imitated with little understanding of their actual
nature and underlying rationale. Cargo cult science often shows up in fields that are
only beginning to introduce formal scientific methods, where a basis of solid and
reliable practice has yet to be established. As these fields develop, they either become
established sciences or decline into pseudoscience. Parapsychology is the best
current example of an attempt at science that is still seeking to establish itself as
legitimate but remains on the edge of nonsense.
The term “cargo cult” comes from a phenomenon that occurred among some of the
peoples of New Guinea and islands of the South Pacific in the period following the
Second World War. During the war, allied forces had established air bases on these
islands as part of their supply chain. A great many goods, “cargo,” passed through
these bases, and some of it was passed on to the natives, either in payment for
services, to maintain good will, or through theft.
After the war, the bases were dismantled and the distribution of cargo to the natives
stopped. Soon, a cult movement arose, centered on the activities of clearing strips of
jungle and constructing “control towers” from bamboo, in this way seeking to attract
the return of the cargo. The basic belief in these cults was: cargo had been made by
the spirits for the native peoples, who therefore had a right to receive it. By trickery
the white man had stolen it; thus it was necessary to use magical imitation in order
to entice the cargo to return.
This sort of behaviour is not limited to primitive Pacific Islanders. An article from
the Wall Street Journal, reprinted in the January 7, 2001, issue of the Arizona Daily
Star, relates that many computer technology entrepreneurs in South Korea believe
that the route to success includes dressing like Bill Gates. And how many high school
students dress and act like their favorite TV or movie star as a means of establishing
a fashionable identity? The cargo cult phenomenon appears whenever somebody
imitates apparently successful behaviour while lacking the understanding and
capacity that was behind the actual success.
Two other forms of nonsense are pseudoscience and crackpot science. Neither of
these activities is really science at all, but they are associated with science in the
public mind. At the extreme, a single individual promotes a crackpot belief, although
he or she may gain followers. This person claims to have made a revolutionary
discovery that violates all current scientific theory. Descriptions of this earth-shaking
discovery or new theory are generally vague and incomprehensible. The crackpot
may protest that secrecy is necessary because others are trying to steal the ideas, or
that the scientific establishment wants to suppress it as a threat to their power.
Other forms of crackpot science involve assertions blatantly in contradiction with
well-established scientific fact, or at least with all that are currently known to be
scientifically valid. Examples would be the beliefs that the Earth is flat, that the
Great Pyramid of Egypt encodes accurate predictions about the future of the world,
that highly advanced cultures existed on the now sunken continents of Atlantis and
Lemuria, or that UFOs are abducting humans and performing medical experiments
on them.
My favorite example is a book I came across in the University of Texas library in
1970, called Repeal Newton’s Laws. In the preface, the author claimed that he was
presenting a radically new theory that would make all of modern science obsolete.
He went on to say that he had anticipated resistance from the scientific
establishment, but had not expected political persecution as well. But government
agents had abducted him and imprisoned him in a state hospital (the name of which
I recognized as a mental asylum), waking him at all hours of the night to inject
poison into his veins. It was, he said, only by the luckiest of chance that he had been
able to escape. . . .
These sorts of beliefs have their adherents, in what can only be described as the
modern equivalent of fringe religious cults. Most of those involved in such cults are
deluded, and a few are just plain dishonest, using whatever beliefs the cult they
associate with professes as a means to acquire money and power. Not surprisingly,
many New Age theories and practices, while well intentioned, have strong crackpot
leanings.
While crackpot science simply rejects science and asserts its own beliefs, often
coupling its assertions with claims of a conspiracy on the part of vested interests,
pseudoscience attempts to give itself an aura of scientific respectability. If there is
any method to be found in pseudoscience, it can be described as involving imaginary
extrapolations from bad data, and grand, all-encompassing speculations.
A number of examples of pseudoscience are described in the textbooks by Lee and
Derry. The best-known pseudosciences today are astrology, creationism and
intelligent design. Astrology is really the fossilized remnant of the ancient GrecoRoman geocentric worldview, in particular, Stoic cosmology. If this view of the
structure of the cosmos were correct, astrology would be worth investigating. With
the Copernican revolution, however, it lost all credibility.
In contrast to astrology, creationism and its more recent embodiment as intelligent
design is a new pseudoscience that grew out of fundamentalist religious opposition
to the theory of evolution and to 19th century geological discoveries about the age of
the Earth.
“Creation scientists” provide an excellent example of the cargo cult and
pseudoscience phenomenon. They seem to be following scientific practices. They
attempt to show evidence that discredits their opponent’s theories, in this case, the
theory of human evolution; they search for evidence in support of their own theory;
and they develop and elaborate their theory. They claim to be carrying out legitimate
scientific research. On closer inspection, however, this claim turns out to be strongly
dependent on “the argument from ignorance” (argumentum ad ignorantiam), the
logical fallacy of arguing that a proposition must be true because there is no evidence
proving it false (or, conversely, that a claim must be false because there is no
evidence proving it true). This is often called the God of the Gaps argument and,
roughly speaking, takes the form: “Modern science has not explained XX,
therefore XX can never be explained by modern science.” For example, creationists
will argue that science has not yet explained how life could have emerged from nonliving matter, therefore this must imply the existence of a Creator (and if science
were to demonstrate the creation of life in the laboratory tomorrow, creationists
would simply move the goalposts).
While astrology is a pseudoscience because it has no empirical support or legitimate
theoretical foundation to connect planetary motions with human affairs, creationism
is a pseudoscience because it flagrantly abuses the rhetoric and practice of science in
support of a religious and political agenda based on a belief system that is
incompatible with the actual practice of science. There is nothing to prevent a person
from believing in creationism as a religious belief. But science requires doubt, even
of one’s most firmly held beliefs, so the creationists are caught in a dilemma—either
accept skepticism with regard to their basic religious beliefs, or cease to claim that
their beliefs are part of a scientific enterprise.
Discussion 5.2 Why Witchcraft Is
Neither Science Nor Pseudoscience
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The plain consequence is (and it is a general maxim worthy of our attention), “That no
testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its
falsehood would be more miraculous than the fact which it endeavours to establish.”
—David Hume (1711-1776 )[85]
The Azande belief in witchcraft is sometimes presented as an example of “how a
culture determines what kind of science is possible.”[86] The argument goes as
follows:
The Azande believe that all events in the world are controlled by witchcraft. Any
misfortune, for example, is a result of the work of a witch. If a granary collapses, it
was witchcraft. If a woodworker is making a bowl and the wood splits, it was
witchcraft. If a person stubs his toe and it becomes infected, witchcraft again. An
interesting and important feature of the Azande system is the poison oracle. When
seeking to identify the source of a misfortune (e.g., who was the witch that caused it),
or seeking advice about important decisions, this oracle would be consulted. A form
of plant poison would be collected, called benge, and prepared in two equal batches.
The first batch would be fed to a chicken and a question (having a yes or no answer)
posed. The question would be posed in such a way that the yes or no responses were
determined by whether or not the chicken lived or died. After the answer was
received, however, it would be tested by feeding the second batch of poison to a
second chicken, this time posing the question so that a yes answer the first time
corresponded to a no answer the second time, and vice versa. This, according to
some, is a form of scientific hypothesis testing.
The argument then goes on to claim that the Azande belief in witchcraft:
has many of the features of a scientific system. First, it has great explanatory power and is of
great generality in application; it explains more of the events and misfortunes of daily life
than any Western scientific system does. Also, it is supported by experimental evidence: the
stubbed toe that becomes infected, the wooden bowl that splits, the granary that collapses.
Further experimental evidence is provided by the poison oracle.[87]
On this basis, the conclusion is that, in some ways, there is really little difference
between Western science and Azande witchcraft. As have the Azande, we have been
culturally conditioned to believe that our explanations of the world are correct; and
as do the Azande, we hold on to a belief “in spite of awkward contradictory facts,
unless we have a better one to replace it with.”[88]
This argument is an example of a line of thought called cultural relativism, which has
had a strong influence in anthropology and culture studies. Cultural relativism is the
belief that it is not possible to make culture-free value judgements because our
criteria for evaluation are always biased by what our own culture considers right and
proper. In this case, we are not qualified, and could never become qualified, to judge
the relative merits of Western science and Azande witchcraft, nor would a Zande be
able to make such a judgement. All that can be said is that Western science and
Azande witchcraft fill similar roles in their respective cultures.
Further, no argument against witchcraft we might make would convince a Zande
that witches do not exist.[89] Parallel scientific subcultures show the same kind of
resistance to the questioning fundamental tenets of their beliefs as the Azande.
While the attitude of tolerance that cultural relativism seeks to promote is
commendable, the argument as it stands risks giving a false impression of the nature
of science.
Science is indeed a part of Western culture, and the practice of science in the West is
subject to cultural influences. In Discussion 4.1, we saw that the inner purpose of
science is to understand the world, and in Discussion 4.2, we established the idea
that understanding involves interpretation within a conceptual framework. And
there can be no doubt that our culture strongly influences the sorts of explanatory
conceptual frameworks that we use. But this is only half of the story. As we will show
in Discussion 7.2, not only does culture influence science, but science has strongly
influenced culture. The cultural values science promotes have led to an open system
in which, while cultural factors may influence what science studies, and even the way
in which results are described and disseminated, the results obtained still provide
factual understanding of the world.
Before continuing, it is necessary to make clear the distinction between “factual
understanding” and “functional understanding.” In both cases, understanding arises
through explanation in terms of a conceptual framework. In factual understanding,
this explanation must be logically coherent, as well as open to confirmation through
repeatable observations or experiments. In functional understanding, all that is
required is that it leads to appropriate behaviour. Thus, factual understanding is a
highly restricted form of functional understanding.
The film A Beautiful Mind tells the story of John Nash, a brilliant mathematician
who had paranoid schizophrenia.[90] Under the influence of hallucinations produced
by this disease, he came to believe that a Soviet nuclear weapon was being smuggled
into an American city, and that information on the location of this weapon was being
transmitted to Soviet agents in encrypted form through American newspapers and
magazines. This is an example of understanding (i.e., his interpretation of events
within the framework of his delusion) that is neither factual nor functional.
Earlier in his life, while a graduate student at Princeton University, Nash had
believed that he had a roommate who was a graduate student in English. This
roommate encouraged him, gave social advice, and in general acted as a welcome
balance to his extreme shyness and lack of social skills. The only problem was that
the roommate was a hallucination. Nevertheless, this imaginary friend provided
support and assistance through difficult times. Here is an example of a framework
that was functionally effective but not factual.
Finally, after coming to accept that he was sick, Nash was able to come to terms with
his illness. He realized that he saw people who were not there, and projected
meaning into things that were simply random events. Through great personal effort,
he learned to distinguish hallucinations from actual perceptions, to test whether or
not new individuals entering his world were real by confirming their existence with
other people whose existence he trusted, and to ignore the hallucinations and
fantasies, no matter how demanding and insistent they became. This is an example
of factual understanding.
There are, of course, many cases in which the distinction between factual and
functional understanding is irrelevant. If we are walking through the jungle and
encounter a hungry tiger, it does not matter whether we see it as a large hungry
carnivore or as a demon jungle spirit. What is important is that we run. In the long
run, however, factual understanding offers us the best chance of survival. This is the
point of the quotation from Lords of Light, by Roger Zelazny, given at the beginning
of Chapter 13 in the textbook by Derry:
“Then the one called Raltariki is really a demon?” asked Tak.
“Yes—and no,” said Yama. “If by ‘demon’ you mean a malefic, supernatural creature,
possessed of great powers, life span, and the ability to temporarily assume virtually any
shape—then the answer is no. This is the generally accepted definition, but it is untrue in
one respect.”
“Oh? And what may that be?”
“It is not a supernatural creature.”
“But it is all those other things?”
“Yes.”
“Then I fail to see what difference it makes whether it be supernatural or not—so long as it is
malefic, possessed of great powers and life span and has the ability to change its shape at
will.”
“Ah, but it makes a great deal of difference, you see. It is the difference between the
unknown and the unknowable, between science and fantasy—it is a matter of essence.”[91]
What is unknown may, with effort, become known and thus controllable. What is
unknowable can only be feared and avoided.
Returning to our comparison of Western science and Azande witchcraft, we can
restate the argument that it is a form of science:
o
o
both systems offer explanations of the world in terms of culturally specific
conceptual frameworks, thus yielding different understandings of the world.
both conceptual frameworks are internally coherent and yield functionally
effective worldviews. Indeed, in this regard the Azande system is superior, since
it explains more in the Azande world than science explains in ours.
o
o
both systems carry out experimental tests and offer experimental evidence in
support of their beliefs.
therefore, both systems are of equal status in terms of their cultural roles.
Furthermore, because they are completely culturally determined, no argument
made by an adherent of one system can convince an adherent of the other
system that he or she is wrong.
This argument is fine as far as it goes. However, if we are tempted to draw the
conclusion that Azande witchcraft and Western science are in fact equally effective
ways of understanding the world, we would be making a serious error based on a
failure to recognize what is missing from the argument.
In Discussion 1.1, we saw the importance of point of view, and in Discussion 4.2, the
difference between point of view (how we look at the world) and conceptual
frameworks (how we interpret what we see from our point of view) was drawn. What
is missing from the comparison of Azande witchcraft and Western science is the
difference in point of view.
The anthropologist Robin Horton addresses the issue of whether or not it is
reasonable to say that “traditional” cultures (e.g., African tribal cultures) have
developed theoretical science.[92] He points out that theoretical science involves not
only having the right kinds of theories, but also the right sort of attitude. In this
regard, Horton makes a distinction between open and closed cultures: traditional
cultures do not have any real awareness of alternative possibilities to their
established theories and beliefs. In scientific cultures, this awareness is highly
developed.[93] The awareness of different possibilities and the need to test theories is
a basic characteristic of open cultures. In closed cultures, there can be no
questioning of basic assumptions. The hallmark of an open mind, or an open culture,
is the willingness to question basic assumptions, and to alter these assumptions if
there is sufficient reason for doing so.
This willingness to question is directly related to the ability to distinguish the
elements that make up a theory, conceptual framework or belief system from the
system itself. If every element of a belief system is connected to and reinforced by
every other element, it is difficult to focus on any particular element. Therefore, the
entire system must be accepted or rejected as a whole. Closed systems begin to open
up when their component elements can be separated out and considered as distinct
ideas that can be questioned.[94]
One of the most fundamental skills required for scientific reasoning is the ability to
appreciate the interplay of theory and evidence. In closed cultures, these aspects of
understanding are distinguished poorly if at all. When there is no conception of
alternative possibilities, the data of experience cannot be regarded as evidence for or
against existing beliefs. Instead, they become instances that are illustrative of these
beliefs. The examples presented as “experimental evidence” used by the Azande (the
stubbed toe that became infected, the split bowl, the collapsed granary) are neither
experimental (because they are single incidents that are non-repeatable) nor
evidence. Rather, the Azande take them as instances that validate their beliefs.
Neither is the Azande use of the poison oracle analogous to a scientific experiment.
The important difference is found in attitude. While the Azande were well aware that
the oracle sometimes gave incorrect answers, this fact never led them to question its
validity. When the oracle gave accurate answers, they were taken as instances that
confirmed the oracle. Occasions of incorrect answers were explained away by
referring to other elements of the belief system: the poison was bad, or poorly
prepared, or some aspect of the ritual had not been carried out properly, or
somebody had used witchcraft to interfere.
While scientific methods or theories that give incorrect results or predictions may
also be defended by apparently similar arguments (the experimental apparatus was
faulty; the procedures were not performed carefully enough; there must have been
some external factor that was not taken into account), the methods or theories
themselves remain open to question. Scientists will even acknowledge that a theory
is incorrect, yet continue to use it, exploring its limitations in the hope of finding a
better one.[95]
The Azande, on the other hand, did not question the validity of the poison oracle.
The oracle was not consulted with the inductive attitude or from a scientific point of
view. Rather, it was used as a basis for making decisions. If the oracle indicated that
a proposed activity would fail, the activity would not be carried out. The
counterfactual question of confirmation or refutation—of what would have happened
if the activity had been carried out—could never arise. In the words of E. E. EvansPritchard,
In this web of belief every strand depends on every other strand, and a Zande cannot get
outside its meshes because this is the only world he knows. The web is not an external
structure in which he is enclosed, it is the texture of his thought and he cannot think that his
thought is wrong.[96]
While in Western science, we sometimes hold on to our beliefs “in spite of awkward
contradictory facts, unless we have a better one to replace it with,” this fails to
emphasize what is the point: if a better theory comes along, we replace the old. As
Carl Sagan (1934-1996) commented, “In science it often happens that scientists say,
‘You know that’s a really good argument; my position is mistaken,’ and then they
would actually change their minds and you never hear that old view from them
again.”[97] If we were to discover that witchcraft actually worked in a repeatable and
verifiable way, we would change our theories. A Zande would not change his belief
and we could not, within that belief system, convince him that he was wrong.
On the other hand, Azande witchcraft cannot be considered a pseudoscience since no
attempt is made by the Azande to present it as science in the first place. Indeed, the
very idea of claiming that their system of witchcraft was a science would be foreign
to the Azande. Their beliefs form a complete and closed worldview with no need for
external validation. Pseudoscience, to the contrary, desperately seeks to gain
credibility by passing itself off as science.
Thus there is a radical difference between the closed world of the Azande and the
open world of Western science. Lest we feel smug, however, we might consider the
many areas of belief where our own mind and our own culture are closed. Even
somebody as with as strong a scientific orientation as George Pólya notes that in our
personal lives there are certain beliefs that we dare not question because to do so
would prove too upsetting to our emotional stability.[98] If it is the truth that sets us
free, then these areas of mental closure are a measure of our imprisonment.
Unit 5 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. What are the defining characteristics of pathological science, according to Irving
Langmuir?
2. What are the five defining characteristics of pseudoscience given in Chapter 12
of What Science Is?
3. What are the defining characteristics of “crackpot science”? —of “cargo cult”
science?
4. Consider the various examples of pseudoscience given in Chapter 12 of What
Science Is, and classify each as pseudoscience, crackpot science or cargo cult
science. Note that some examples may fit more than one category.
5. Give examples (other than that given in Discussion 5.2) of beliefs that are
neither factual nor functional, beliefs that are functional but not factual, and
beliefs that are both factual and functional.
6. Find six examples from newspapers, magazines or television where advertisers
attempt to use the cargo cult phenomenon to sell their products.
7. Write a short essay (300-500 words) discussing the difference between open
and closed belief systems.
8. What is the importance for scientific theories of the distinction between
evidence and instances?
9. Discussion 5.2 presented a distinction between open and closed cultures. In fact,
however, it is more realistic to speak of cultures that are relatively open and
relatively closed. A similar distinction applies to relatively open-minded and
relatively closed-minded individuals. In Discussion 4.1, we encountered the
connection between ignorance, fear and belief. Choose two cultural beliefs, one
that you accept and the other that you do not, and for each, identify the fear
behind the belief and the ignorance that produces that fear. [Notice how much
easier this exercise is for the belief that you do not accept.]
FOOTNOTES
[80]
Russell, Bertrand. “Outline of Intellectual Rubbish.” In Unpopular Essays, p. 149. New
York: Simon and Schuster, 1950.
[81]
Langmuir, Irving. Physics Today 42: 36-48. A transcription of the original talk is given at
the site below. Retrieved July 15, 2002.
http://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm/
[82]
Ibid.
[83]
http://www.columbia.edu/cu/21stC/issue-3.4/turro.html/
[84]
Feynman, Richard. “Cargo Cult Science,” p. 313. In Surely You’re Joking Mr. Feynman,
pp. 308-317. Retrieved July 15, 2002.
http://www.physics.brocku.ca/etc/cargo_cult_science.html/
[85]
Hume, David. “On Miracles,” p. 115. In An Enquiry Concerning Human Understanding.
L. A. Selby Bigge, ed. Oxford: Clarendon, 1902. For the full text of the essay, see the web site
below. Retrieved July 15, 2002. http://www.fordham.edu/halsall/mod/humemiracles.html/
[86]
Goldstein, Martin, and Inge Goldstein. The Experience of Science: An Interdisciplinary
Approach, p. 378. New York: Plenum, 1984. Note that the fieldwork on which the account of
Azande practices they cite was based had been conducted about 50 years before Goldstein
and Goldstein wrote.
[87]
Ibid., p. 386.
[88]
Ibid., pp. 386-387.
[89]
Ibid., pp. 381-386.
[90]
Howard, Ron, director. A Beautiful Mind. Hollywood, CA: Universal Studios/Vivendi
Universal Entertainment Company, 2001. Note that the hallucinations depicted in the
movie were not actually suffered by Nash. The filmmakers chose to personify the voices he
would hear (and attribute to space aliens) as hallucinated individuals.
[91]
Quoted in Derry, Gregory N. What Science Is and How It Works, p. 174. Princeton, NJ:
Princeton University Press, 1999.
[92]
Horton, Robert. Patterns of Thought in Africa and the West: Essays on Magic, Religion
and Science, pp. 221-223. Cambridge: Cambridge University Press, 1993.
[93]
In this regard there is a difference between fundamentalist cultures and traditional
cultures. Fundamentalists are aware that there are alternative belief systems to that which
they profess, and they react by attempting to suppress these alternatives as evil.
[94]
Applying this principle to societies, it is the individual who “stands out” from the group
who can ask questions about social beliefs and values. This is why tyrannical systems cannot
tolerate individualism, and why all societies view those individuals who stand out with
suspicion.
[95]
You will encounter an example of this use of incorrect theory in Discussion 14.1. The Bohr
model of the atom was published in 1913 and provided the basis for intense theoretical and
experimental research on atomic structure for the next dozen years, even though it was clear
to all involved that this model could not be correct as an actual description. Without the
research carried out on this factually “incorrect” model, however, it is unlikely that quantum
mechanics would have been discovered.
[96]
Quoted in Horton, Patterns of Thought in Africa and the West, p. 222.
[97]
Sagan, Carl. “Keynote Address: The Burden of Skepticism.” Delivered at the Annual
Conference, of the Committee for the Scientific Investigation of Claims of the Paranormal,
Pasadena, CA, April 3-4, 1987.
[98]
Pólya, George. Mathematics and Plausible Reasoning, Volume 1, p. 3. Princeton, NJ:
Princeton University Press, 1990.
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Unit 6 What Does Philosophy Have to Say?
Discussion 6.1
Unit 6 Study Questions
STUDY GU IDE
Unit 6 What Does Philosophy
Have to Say?
Once upon a time an elderly woman lived alone in a small house on the outskirts of a
large city. One day a large eagle landed on her front balcony railing. The woman had
never seen such a magnificent creature; her only experience with birds had been
with the domesticated sort. So she got a pair of scissors and clipped the eagle’s wings
and trimmed its claws and beak. “There,” she said, “Now you look more like a proper
bird.”
On occasion, science has suffered a similar fate when it has fallen into the hands of
philosophers. In their attempts to define science and to give a definitive statement
of the scientific method, some philosophers have established systems that, if
followed rigorously, would effectively cripple science.
On the other hand, scientists do need to be aware of some important philosophical
considerations in order to avoid basic errors. These considerations are generally
built into a science curriculum, to be absorbed by students in the course of their
scientific education. A scientist must also be aware of philosophical issues arising in
his or her field, and as they apply to science in general, but it must be a detached sort
of awareness. As we noted in Unit 4, Albert Einstein commented that “Epistemology
without contact with science becomes an empty scheme. Science without
epistemology is . . . primitive and muddled.”[99]
Or, as Hamlet remarks: “There are more things in heaven and earth, Horatio, than
are dreamt of in your philosophy” (Hamlet, Act I, Scene V).
In Discussion 4.2, we defined “understanding” of a thing or idea in terms of being
able to fit it into a conceptual framework. But every such framework is limited, so
that certain aspects of things, or certain things themselves, may need to be ignored
in order to obtain a satisfactory fit. In science, when there is a significant lack of fit
between theory and data, the theory is changed. Philosophers tend to be more
stubborn in their epistemological commitments.
In this unit, you will be introduced to some of the philosophical considerations that
are important for science. The reading from What Science Is provides a view of the
more significant philosophical perspectives on science and how it ought to be carried
out, introduces two important dichotomies (causality and chance, and reduction and
emergence), discusses some basic questions of epistemology, gives four criteria for
theory selection, and describes how judgements made in science are a result of the
social application of rational and empirical tools. The basic issues of epistemology
raised in the readings are then considered more carefully in Discussion 6.1.
Objectives
When you have completed Unit 6, you should be able to
1. describe the conditions a statement must meet in order to be, at least
potentially, a scientific statement.
2. define the philosophical positions of logical positivism, operationalism,
falseficationism, realism, relativism, rationalism and empiricism as they apply
to science.
3. describe, briefly, the relationship between causality and chance, and that
between reduction and emergence.
4. explain the statement, “Science is ‘theory-laden.’”
5. describe the basic ideas of “normal science” and “revolutionary science,” and
explain how these types of science are related in Thomas Kuhn’s theory of how
science evolves.
6. state four criteria for selecting among competing theories.
7. discuss both the formal and the social aspects of scientific judgements.
8. define the different types of knowledge associated with the
terms epistemé, tekhné and gnosis, and describe some of the constraints on
each.
9. give the truth conditions for both the coherence and correspondence theories of
truth, describe how both theories are used in science, and explain why science
can never claim to have discovered absolute truth.
10. describe the difference between the radical skepticism of the sophist Gorgias
and Pyrrhonian skepticism, and explain why the former is useless for science
while the latter is essential.
11. state the three fundamental metaphysical assumptions of science.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Chapter 14, “Very Abstract Questions: The Philosophy of Science,” pages
189-206 in What Science Is.
2. Answer Unit 6 Study Questions 1-4.
3. Read Discussion 6.1, “Epistemology, Science and the Search for Knowledge.”
4. Answer Unit 6 Study Questions 5-9.
Discussion 6.1 Epistemology,
Science and the Search for
Knowledge
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I believe that there is no philosophical highroad in science, with epistemological signposts.
No, we are in a jungle and find our way by trial and error, building our roads behind us as
we proceed. We do not find signposts at cross-roads, but our own scouts erect them to help
the rest.
—Max Born (1882-1970)[100]
From a practical point of view the most significant thing about science is that it
works; that is, it provides a body of theory and practice that has demonstrated its
usefulness in the real world of everyday human affairs. Science provides the means
to treat and cure many diseases, to fly through space, to turn night into day, to
communicate around the world at the speed of light, and to build computers that can
play chess at the grand master level. We may be horrified at some of the ways in
which people have misused the technological products that science has made
available, but we cannot deny the fact that science has radically changed the
conditions of human life.
For the philosopher, however, a number of questions arise concerning this
enterprise, which has so dramatically altered life on this planet.[101] We suggested, in
Discussion 4.1, that that one of the purposes of science is to free humanity from fear,
and that science works toward this goal by increasing our store of knowledge.
Fundamentally, science is a knowledge-producing enterprise. This definition, of
course, raises the question of just what knowledge might be.
The subdiscipline of philosophy that deals with knowledge is called “epistemology.”
This field deals with questions such as: What is knowledge? How is knowledge
obtained? Is certain knowledge ever possible? And, How can a knowledge claim be
verified? There are also questions of comparison between different kinds of
knowledge. For example, is there any similarity between the mathematician’s
knowledge of the Pythagorean theorem, the chemist’s knowledge that water is
composed of hydrogen and oxygen, the glassblower’s knowledge of how to make a
flask, the historian’s knowledge that William of Normandy won the battle of
Hastings in 1066, the psychoanalyst’s knowledge that a patient’s dream of drowning
symbolizes a fear of being overwhelmed by life, the fundamentalist Christian’s
knowledge of personal salvation, and the mystic’s knowledge of ultimate reality?
The ancient Greek’s had three different words for knowledge: epistemé, from which
we get the word epistemology; tekhné, which is the source for our words technique
and technology; and gnosis, from which come the words Gnostic and gnosiology.
Epistemé meant knowledge in the sense of understanding; that is, fitting within a
conceptual framework. It was derived from the word epistanai, which meant “to
stand upon.” In other words, it referred to knowledge that stood upon or was
supported by a secure and rational conceptual foundation. It could be supported by
rational argument; one could take a stand on it.
Tekhné referred to skill, or know-how; that is, to practical knowledge as found in arts
and crafts. The roots of this word trace back to a complex of ideas that include
weaving, fabrication and construction. In modern terms, it can be thought of as craft
knowledge, as well as technical and engineering knowledge.
Gnosis meant, and still means, the intuitive apprehension of truth, or knowledge
gained through direct insight into reality. In modern usage, it is generally taken to
refer to spiritual truths, and so is often considered irrelevant from the viewpoint of
scientific consideration. This is a mistake. If science ignores the idea of intuition as a
source of valid knowledge, then it has no direct way of understanding human
creativity, which is the real source of both science and culture.[102] The question is not
which ways of knowing are important for science, but how all ways of knowing
contribute to science.
In the study of any form of knowledge, the major issue revolves around the questions
of generation and validity. How, is knowledge obtained, and once obtained, how do
we know that it is true? How, and by what criteria are we to judge the truth of a
knowledge claim? Indeed, what does it even mean to say that a claim is “true”?
In the case of technical skill, these questions are relatively simple to answer.
Technical knowledge is gained primarily through practice, through an
apprenticeship. Claims to possess technical knowledge are stated in terms of a
capacity to do something. Thus, the validation of the claim is found in the actual
performance. An applicant for a data entry position may claim to be able to type so
many words per minute. A professional athlete claims certain physical skills. A
software engineer claims to be able to solve tough programming problems. A
musician claims to have mastered a particular musical instrument. These are all
claims to have the capacity and know-how necessary to do something, and they are
put to the test by demonstrating that that something can, in fact, be accomplished.
The musician demonstrates mastery of an instrument by actually playing it.
With tekhné the proof is in the pudding.
In contrast, philosophers have argued for centuries over validity criteria for
conceptual knowledge. The sticking point is that such knowledge is only given with
respect to a conceptual framework that exists in the human mind, yet knowledge
claims are made about things in the external world. In the final analysis, however,
the only direct information we have about the world outside of our own mind is what
we experience through our senses, and the senses are limited and often deceptive
channels. Thus, the problem of conceptual knowledge is that of how our concepts,
which are based on highly selected forms of sensory input, can be said to provide
accurate knowledge of the world—how, or at least to what extent, is the mind able to
match, fit or reflect reality? Since science is the search for a rational understanding
of the world, the issue here is the extent to which it is actually possible to attain such
an understanding.
This question goes back to the very beginnings of philosophy. Plato devotes an entire
dialogue, the Theaetetus, to an analysis of the conditions for valid knowledge, only to
reach no firm conclusion. Knowledge is not sense perception, nor is it opinion, nor is
it opinion with a rational explanation. In practical terms, however, Plato’s formula
for developing conceptual knowledge has persisted to this day, “First the survey of
scattered particulars, leading to their comprehension in one idea [then dividing the
idea into parts] where the joint is, not breaking any part as a bad carver might.” [103]
That is, we first seek a unifying concept that encompasses a wide class of particular
instances. We seek to discover a unity in the multitude of experiences. This is a
bottom-up, process. Then, having discovered this unity, we seek to determine its
natural articulations, elements or components, and their forms of manifestation. We
seek to determine the multiple possibilities inherent in the unitary concept, asking
which components are essential and which are not in terms of our developing
understanding. In other words, the process is top-down and deductive. In a
paraphrase of Gregory Bateson’s words, “information is the difference that makes a
difference.”[104]
Carving nature at the joints is finding the differences that make a difference, and
discovering these differences—finding out where to carve nature—is a primary goal
in science. And here, science must look to nature. To discover where the joints are
we must, as it were, carefully touch. We have learned through centuries of effort that
we cannot tell nature how it must be. If we could, the earth would be the centre of
the cosmos. We cannot impose our idea of order on nature, we must attempt to tease
out an expression of the natural order—and that can be difficult. As Jacob Bronowski
notes,
[Order] is the most difficult question in science. The notion of order cannot be defined on
any ground except its success. It cannot be put into a science in advance. . . . Order is the
selection of one set of appearances rather than another because it gives a better sense of the
reality behind the appearances.[105]
This selection necessarily leads to an important constraint on conceptual knowledge.
In developing our concepts we always leave something out—we choose to carve
nature in one way rather than another—so our conceptual knowledge of the world
can never lay claim to absolute certainty. The only case where absolute certainty can
be attained is in the pure abstract analysis found in logic and mathematics, which
does not depend on anything other than our own mental construction. There, and
only there, insofar as we are secure in knowing that we have not made a mistake, we
can have complete and perfect certainty (as any mathematician knows, however,
coming to such certainty requires an extensive process of checking and rechecking
every detail).
As far as intuitive knowledge is concerned, validity criteria are even more difficult to
discover. When we have a creative inspiration, the experience itself carries a feeling
of complete conviction. We soon learn, however, to suspend judgement until we have
checked this new idea to make sure that it fits with what we already know to be true.
Intuitive responses are far quicker and more sensitive than reason, but this
sensitivity is purchased at a cost. As with any sensitive instrument, organism or
medium, intuition is strongly vulnerable to perturbing and distorting influences.
Furthermore, since intuition occurs as a mental event, the influences that disrupt its
operation are also aspects of the mind. Anything that occupies attention or fixates
the mind in one way or another also distorts intuition, including emotional states
and such things as envy, pride and greed. Indeed, as a well-known Middle Eastern
proverb has it, “Greed is the mother of incapacity.”
From a scientific point of view, a superficial answer to the question of knowledge is,
“Scientific knowledge is knowledge that is produced by the scientific method, and
this method provides a means of verification through the requirement of
experimental testing.” The problem with this answer becomes apparent, however, if
we consider an example.
For over 200 years, scientists accepted the Newtonian theory of gravity, which
asserted the existence of an attractive force between any two masses, inversely
proportional to the square of the distance between them. During this time, a scientist
could claim that he knew the reason the earth revolved around the sun—there was a
force exerted on the earth by the sun. But in 1916, Albert Einstein published his
general theory of relativity, and by 1920, this theory was accepted as the scientific
theory of gravity. In general relativity theory, there is no gravitational force. Rather,
the presence of matter or energy distorts the metrical properties of space-time. In
general relativity theory, the earth revolves around the sun, not because a force is
exerted on the earth by the sun, but because the space-time around the sun has been
so distorted by the solar mass that the earth is simply following the shortest path
that it can follow.[106] So we must ask: Did the knowledge of those earlier scientists
who had accepted the Newtonian theory of gravity suddenly become ignorance? Was
it ever “true” that such a thing as gravitational force existed?
One way of responding to such questions is to distinguish two kinds of knowledge:
absolute and relative. We can then say that scientific knowledge is relative
knowledge, and cannot be judged on the basis of criteria, such as permanence, that
only apply to absolute knowledge. This distinction, in turn, gives rise to new
epistemological questions: Is there any such thing as absolute knowledge? If so, what
is it? What is the boundary between absolute and relative knowledge? Are there
different kinds of relative knowledge? What degree of certainty can be assigned to a
relative knowledge claim?
Closely tied to the question of knowledge is the question of truth: Is it ever possible
for a person to know something that is false? In order to address this question,
philosophers make a distinction between knowledge and belief, and it is often
asserted that knowledge is true belief. That is, if I believe that something is so, and it
actually is so, then I am justified in claiming to know that it is so. For example, I
know that the sky is blue because I believe that it is blue, and it actually is blue.
But we encounter the same problems with truth as with knowledge, and as before,
we make the distinction between absolute or certain truth, and relative truth. And
the questions still remain: What is certainty? How may it be obtained? What criteria,
if any, can we use to assign degrees of certainty to relative truths?
There are two main theories of truth: the “coherence theory” and the
“correspondence theory.” In the coherence theory, a statement AA is said to be true
with respect to a system of statements SS if it is implied by SS and is consistent with
all of the statements of SS. That is, if it is related to all other statements of SS by ties
of logical implication. In Euclidean geometry, for instance, it is true that the sum of
the angles of any triangle equals two right angles, because this relationship is a
logical consequence of the basic Euclidean definitions and axioms. But in spherical
geometry, this statement is false—it is not coherent or consistent with the axioms
and definitions of spherical geometry.
Thus we need some other criteria for the truth of any system of propositions SS if we
are going to apply it to describe the world. We might hope to find these criteria in the
correspondence theory, which defines a statement as true if it corresponds to the
facts. The catch with this definition, as we will see in Unit 9, is that “the facts” are not
always that obvious. What counts as a “fact” may depend on the conceptual
framework that is used. This limitation raises the danger of a vicious circle in which
a conceptual framework only allows “facts” that support its basic assumptions, or
interprets all events as instances supporting these assumptions. The closed system of
Azande witchcraft described in Discussion 5.2 is an example of such a system, but we
need not look so far away—in Discussion 8.2 we will see that it is a universal human
tendency to interpret experience in ways that support our beliefs.
As a beginning, however, we can say that a system of statements (e.g., a scientific
theory) is true if it is internally coherent and also corresponds in some way to “the
facts” as we know them. But we cannot claim absolute truth. In that regard, we must
suspend judgement. Indeed, it is best not to use the word “true” at all. Instead, we
say that, to the best of our knowledge, the statements provide an accurate
description of nature.
The Sophists of ancient Greece first raised the question of certain knowledge. They
claimed that such a thing did not exist. The Sophist Gorgias (483-378 BCE) gave the
strongest argument for this claim, writing an essay with arguments in support of the
three contentions listed below.
1. Nothing exists.
2. Even if something did exist, nobody could know it.
3. Even if somebody could know about it, they could never communicate their
knowledge.[107]
If one accepts this radical skepticism, then one is forced to the position not only that
all truth is relative, but also that there is no way to compare conflicting knowledge
claims, other than on the basis of non-rational criteria, such as social custom or who
happens to be the most persuasive speaker.[108]
We begin, then, with skepticism in almost its most radical form.[109] Every
philosopher must take some view with respect to the claims of Gorgias, and in doing
so defines a personal position on the possibility of knowledge.
A main philosophical method for doing so is through doubt, called the via
negativa (negative way). This method is best known through the work of the 17th
century philosopher René Descartes (1596-1650), who set himself the task of
doubting everything. He believed that if he could find something that could not
possibly be doubted, this would provide him with a certain basis for knowledge.
Suppose, he began, that I am being deceived by some evil demon, and that none of
my thoughts or perceptions are true. Continuing this line of thought, Descartes was
able to cast doubt on everything—except the fact that he doubted. From this he
derived his famous statement: cogito ergo sum, I think, therefore I am.
From this start, Descartes proceeded to argue for the existence of God, and to assert
that anything that appeared to him as a clear and present idea must be true, since an
all-benevolent God would not deceive.[110]
Without theological assertions, it is still true that there are two areas where it is
generally agreed that certain knowledge is possible. We will term them the truths of
reason and the truths of experience. The truths of reason are those truths that are
necessary by definition. For example, that a true statement cannot be false, or that a
bachelor is an unmarried man. In philosophy, such truths are said to be known a
priori (Latin, “from the former”).
The truths of experience are those truths that a person cannot deny without denying
their own experience. Such truths are said to be known a posteriori (“after the fact”).
There is, however, a subtle point that arises in discussions of experience. I can deny
that the sky is blue. I could suppose that I have been hallucinating; that there is no
sky, and that everything I have ever seen is an illusion. What I cannot deny is that
when I look out my window on a clear day I have the experience of the sky being
blue. That is, I can doubt that my experiences tell me anything about an external
reality, but I cannot doubt that I have experience.
In the case of reason, all “true” statements in a system SS are relative to some set of
initial assumptions, and to the rules of deductive inference. To see that this is so,
suppose that a statement A0A0 is derived from another statement A1A1. This does
not prove that A0A0 is true, since deductive logic is, as you will see in Unit 9, strictly
formal. All that is known is that if A1A1 is true, then A0A0 is true as well. Thus we
must either assume that A1A1 is true, or deduce it from another statement A2A2.
And by the same reasoning, we must either assume that A2A2 is true, or deduce it
from some A3A3. To avoid falling into an infinite regression, we must either stop at
some point with a statement AnAn, which is assumed true as an axiom, or close the
line of deductions by showing that some statement AnAn in this sequence is implied
by A0A0. In this latter case, however, the argument becomes circular. If we want to
say that the statements involved are true, we must assume that some one of them is
true, a priori.
This example can be summarized by saying that what is certain in reason is that
valid deduction from consistent initial premises will yield a valid conclusion, but the
truth of this conclusion is relative to the truth of the initial assumptions.[111] To
investigate the empirical truth of a set of deductively connected statements, we must
have some external, non-deductive check on the truth of at least one of these
statements. Thus a set of statements that satisfies the coherence theory of truth, if it
is to be applied to a description of the world, must also satisfy the correspondence
theory. Unfortunately, this theory is itself problematic.
In considering the truths of experience, it is necessary to distinguish between the
absolute truth of an experience in consciousness, and the relative truth of a
particular individual who is aware of having that experience. In the awareness, “I am
having such and such an experience,” the absolute truth of the experience itself is
made relative to the “I” who has it, and to the kinds of interpretations of meaning
that “I” makes.[112]
On this basis, we conclude that although absolute knowledge does exist, it is not the
sort of knowledge that can be called scientific knowledge. Any scientific knowledge
must be treated as relative. A reasoned statement depends on the assumptions from
which it was derived, and an experiential statement is relative to the individual who
reports on the experience. Thus the question of scientific knowledge becomes that of
determining how and to what extent reason and experience can provide reliable
guides in making judgements about the world.
Furthermore, reason and experience, as discussed here, involve only internal mental
functions. Experience reduces to the apparent having of sensations, and reason
reduces to a process of mental construction according to predefined laws. Neither
requires the existence of an external world. It can always be claimed that the
experience of a sensation is merely an illusion, and there is no guarantee that a
logical or mathematical theorem will apply to the world as we experience it.
This realization brings us face-to-face with the question Einstein called the greatest
mystery of science: How is it that our minds can produce theories that bear any
relation to the real world? Why is it that our reasoning is so effective in fitting
reality?
Gorgias would say that there is no reality. Everything is our own subjective
construction. Reasoning has created it, so there is no surprise that reason fits.
Taking the view that there is a real world, however, we might think to respond to this
question by saying that if our reasoning did not provide a fit to the world our species
would not have survived. Evolution has ensured that our minds are so constructed as
to be able to fit reality. This argument is not, however, an adequate response. All that
is necessary to survive in the world is the ability to deal with everyday problems,
such as avoiding predators, finding food, and reproductive opportunities. Our
evolution to meet these conditions gives no reason to expect that theories of the
interstellar, atomic and subatomic worlds will fit at all.
As is always the case with reason, initial assumptions are necessary, and so we are
led to the fundamental metaphysical assumptions of science.
1. Nature is ordered.
2. This order is comprehensible to human reason.
3. Our understanding of this order can be communicated without subjective bias.
These statements can be seen to be direct negations of the sophisms of Gorgias. But
in denying sophistry, we cannot simply accept a dogmatic belief. Doubt is still a
necessary tool. Rather than the dogmatic skepticism of the sophists, however,
scientific research requires an attitude based on the skeptical position of the
philosopher Pyrrho (c. 365-275 BCE). In Pyrrhonian skepticism, one does not
believe or doubt, one “suspends judgement.”[113] The assumption is that with
suspension of judgement one attains a mental attitude, the inductive attitude, which
is unbiased and most suited to perceiving whatever order may become apparent in
experience. Given the importance of this attitude, it may seem strange that scientists
are often passionate in their convictions about what is true or false. Such a
passionate involvement is necessary, however, for a scientist to work with
conviction. It is the concomitant of having a “good question” to work on. Even so,
however, the acknowledgement remains that, in the final analysis, one must suspend
judgment about all claims of validity.
The concept of suspension of judgement well describes the theoretically preferred
scientific attitude. This non-judgmental attitude provides a framework within which
creative insights can occur. It is the recognition of these insights, together with their
developmental potential and consequences, which is the real source of growth in
science; and the search for, recognition of, and elaboration of such moments of
insight is the core of scientific work.
In the well-known phrase, this kind of work involves a “free play of ideas,” controlled
only by the constraints imposed by the necessary conditions of reason and publicly
repeatable experience. The development of the capacity to engage in this free play of
ideas requires that we learn to suspend judgment. But it also requires the tenacity to
follow out a line of thought to its conclusion, regardless of the difficulties
encountered along the way. Just because we are playing freely with ideas does not
mean that we are playing in a superficial way—a champion chess player plays with
ideas about possible moves, but does so with intensity.
Returning to the question of knowledge, we can say that the working scientist is not
concerned with the attainment of absolute knowledge, or knowledge of reality “as it
really is.” Rather, the concern is with his or her role in the process of theoretical and
experimental construction involved in the doing of science.
In everyday conversation, and also in thinking, scientists generally make the realist
assumption that the entities they talk and think about actually exist. With regard to
the ontological truth of the products of the scientific process, however, there is
suspension of judgement. A good description of the scientific attitude in this regard
is given by the philosopher Frederick Suppe.
When a sophisticated theory is undergoing active development, it is commonplace for
scientists working on it to suppose that the present version of the theory is defective in
various respects, which is to say that it is literally false, at best being only an approximation
to the truth, or a promising candidate; and if one is convinced this is so, it would be
pointless to attempt to either refute or . . . confirm the theory. What is to the point is to use
observation and experiment to discover shortcomings in the theory, to determine how to
improve the theory, and to discover how to eliminate known artificialities, distortions, oversimplifications, and errors in the descriptions, and predictions, of reality that the theory
affords.[114]
Rather than asking about the possibility of knowledge, a scientist will ask about the
criteria for having sufficient reasons to make a knowledge claim. We want to be able
to make statements of the form: If AA, BB and CC are true, then we are justified in
concluding DD; or, Given AA, BB and CC, then I know DD. For example, given the
presuppositions of modern atomic theory, I know that water is composed of
hydrogen and oxygen.
Ignoring the skeptical position, it seems that both reason and experience appear as
possible grounds for relative knowledge claims. All that is needed is a set of accepted
validity criteria that will allow us to make use of these grounds. Thus we define
“rationalism” as the epistemic position asserting that a knowledge claim is justified if
it can be derived from self-evident premises by strictly logical deduction, and
“empiricism” as the position that a knowledge claim is justified if it is possible for
any sufficiently trained person to undergo the experiences stated in the claim. This
last requirement is the basis for the scientific requirement of repeatable
experiments.
The question remains, however, as to how reason and experience can best be
employed in gaining knowledge of the real world.
Unit 6 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. A scientific statement posits a logically consistent explanation for naturally
occurring phenomenon. An important characteristic of the scientific statement
is that it can be used to generate predictions that are testable, allowing the
statement to be supported (not proven) or falsified through experimentation
that produces consistent observations that are both replicable and natural (as
opposed to supernatural). In other words, a scientific statement should
o
o
provide an explanation or causal mechanism of a naturally occurring
phenomenon, and
generate predictions that are testable via observations that are replicable
and natural.
Consider the following statements and discuss whether each fulfills the criteria
for a scientific statement.
c. The universe and everything in it came into existence at 3 p.m., Eastern
Daylight Time, on June 1, 2000.
d. Water is composed of hydrogen and chlorine.
e. The moon is made of green cheese.
f. The speed of light in a vacuum is about 300,000 kilometres per second.
g. Our personalities are influenced by the position of the planets at the time
we were born.
2. Define each of the terms listed below.
a. logical positivism
b. relativism
c. realism
d. operationalism
e. reductionism
f. emergence
3. Outline Thomas Kuhn’s theory of how science evolves.
4. Write a short essay (300-500 words) discussing the way formal criteria are used
by scientific communities in forming judgements.
5. Define the three types of knowledge labeled by the Greek
terms epistemé, tekhné and gnosis.
6. What are the coherence and correspondence theories of truth, and what are the
limitations of each?
7. Write a short essay (300-500 words) discussing how suspension of judgement
fits the inductive attitude as defined by Pólya (see Discussion 1.1).
8. What are the three fundamental metaphysical assumptions of science?
9. Write a short essay (300-500 words) structured as a dialogue between a
rationalist and an empiricist on the topic of the proper way to do science.
FOOTNOTES
[99]
Einstein, Albert. “Reply to Criticisms.” In Albert Einstein: Philosopher-Scientist, pp. 683-
684. P. A. Schilpp, ed. Cambridge: Cambridge University Press, 1949. The text of this article
is available at the web site noted below. Retrieved July 16, 2002.
http://www.marxists.org/reference/subject/philosophy/works/ge/einstein.htm/
[100]
Quoted in Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein,
rev. ed., p. 7. Cambridge, MA: Harvard University Press, 1988.
[101]
We will not, in this course, consider the ethical questions that have arisen as a result of
the technological advances made possible by science, aside from noting that the success of
science-based technologies has brought humanity face-to-face with possible extinction, thus
charging questions of the uses and abuses of science with the intensity associated with any
question of survival.
[102]
This problem is now becoming apparent in the difficulties that are being encountered in
attempts to attain a scientific understanding of human consciousness.
[103]
Plato. “Phaedrus,” 265d, e. In The Dialogues of Plato, Volume III, 4th ed., trans.
Benjamin Jowett, pp. 107-189. Oxford: Clarendon Press, 1953.
[104]
Bateson, Gregory. Mind and Nature, p. 68.
[105]
Bronowski, Jacob. The Common Sense of Science, p. 48. London: Heinemann, 1951.
[106]
In the language to be introduced in Discussion 9.1, we would say that this is to be
expected on the basis of the “principle of sufficient reason.”
[107]
See, for example, Smith, T. V., ed. From Thales to Plato, pp. 66-71. Chicago: University
of Chicago Press, 1960.
[108]
Since the sophists made money by teaching members of the Greek elite to speak well in
public assemblies, they had a vested interest in maintaining this view.
[109]
The most radical skepticism would deny everything and sink into solipsism or nihilism.
The solipsist need not worry about knowing anything, since he or she is all that there is; and
nihilism, too, is self-defeating as a philosophical position, because if nothing exists, then
nobody exists to question, and there is, obviously, nothing to worry about.
[110]
Descartes, René. “Meditations on the First Philosophy, III. Of God: That He Exists.”
In Discourse on Method, Meditations and Principles, trans. John Veitch, pp. 88-102.
Boston: Tuttle, n.d. This argument is stronger than it might appear. We use a version of it
every time we assume that something which is “intuitively obvious” is also true.
[111]
The medieval Islamic theologian al Ghazali (1060-1111 CE) strongly criticized
mathematicians for the tendency to assume that the certainty of their mathematical
arguments extended to the external world; and there is a modern folk saying that the only
thing mathematicians ever argue about is their definitions.
Ghazali, Abu Hamid al. Confessions, or Deliverance from Error, c. 1100 CE. Retrieved July
3, 2002, from http://www.fordham.edu/halsall/basis/1100ghazali-truth.html/
[112]
One of the goals in the philosophical school of phenomenology is to be able to simply
have experiences while bracketing all interpretation so as to be able to investigate the nature
of the experiential phenomenon itself.
[113]
See Inwood, Brad, and L. P. Gerson. Hellenistic Philosophy: Introductory Readings, pp.
173-198. Indianapolis, IN: Hackett, 1988.
[114]
Suppe, F., ed. The Structure of Scientific Theories, 2nd ed., p. 706. Urbana: University of
Illinois Press, 1977.
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Unit 7 Is Science Value Free?
Discussion 7.1
Discussion 7.2
Unit 7 Study Questions
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STUDY GU IDE
Unit 7 Is Science Value Free?
[S]cience can only ascertain what is, but not what should be, and outside of its domain value
judgments of all kinds remain necessary.
—Albert Einstein[115]
Science claims to be objective and to produce knowledge that is value-free. In this
view, the way that scientific knowledge is used, or misused, is thought to be a matter
of public, but not scientific, concern. The origin of this idea can be found in the 17th
century, when science developed an uneasy truce with the Christian churches.
According to the terms of this truce, science was given the study of the material
world, while all questions of social and moral concern were left to governments and
the churches.
More recently, however, as science has grown more and more central to everyday
life, and especially following the development of nuclear weapons, an anti-scientific
view has developed that blames science for the ills of the modern world. According
to this view, scientists are morally responsible for the negative effects of their
discoveries.[116]
Another view has arisen as well, rooted in the Marxist belief that people always act
on the basis of class interest, and extensions of this belief in the radical constructivist
views of culture. In these views, science has become an instrument of political
oppression used by the dominant elite to keep the mass of humanity in bondage.
The natural question that occurs is whether any of these views can be substantiated.
What sort of relation is there between science and values? Clearly there are values
within science; otherwise there would be no way to tell good science from bad. But
how do these scientific values relate to social values, moral values or both? How does
culture influence science, and how does science influence culture? A real attempt to
answer such questions would require a thick book rather than a single unit in this
course. All that can be provided here is a very sparse outline of a particular point of
view that attempts to point the direction in which an answer might lie.
One source of confusion that arises is a lack of clarity in formulating the questions.
How, for example, are we to conceive of culture? What are the natural subunits of
culture so that we can “carve at the joints”? How do these subunits interact? How are
individuals influenced by their culture, and how do they influence it in return? What
are the processes by which a culture persists or changes through time? Without at
least provisional answers to these questions, the actual relationship between science
and the larger culture in which it is embedded remains problematic.
One point that does seem clear, however, is that science is a major source of cultural
change. Changes arising from new technology are obvious, and they appear
continually in everyday life. The scientific ideas behind these technological changes
also diffuse into the general culture, where they interact with already established
cultural ideals and values. In this process, they often lose their scientific precision
and take on a more metaphoric meaning, sometimes even the opposite of the way
they are used in science.[117]
Scientific ideas enter the general culture with the support of successful applications
in science and technology, and they can spread rapidly. In many cases they do not
challenge existing values in any serious way—the computer revolution is a good
example,[118] as is the germ theory of disease. Darwin’s theory of evolution, on the
other hand, has generated controversy since its original publication in 1859.
Misapplications of Darwin’s theory led to the “social Darwinism” of the late 19th
century, and fundamentalist religious opposition to the theory is still strong.
For the past 400 years, science has been an engine of social change, not only through
its immediate influence, but also in the spread of scientific ideas and values. This
effect has met with reactionary opposition that has grown in stridency with the
increasing influence of science in the world. The British writer Brian Appleyard, for
example, writes that
[For science] there is nothing special about the way we happen to see things, nothing special
about the way the universe looks from a human-size perspective. In short, there is nothing
special about us. . . . Science . . . is spiritually corrosive, burning away ancient authorities
and traditions.[119]
Also tied into the way that science and culture are related is the question, considered
in Discussion 2.2, of whether science is unique to Western cultures. In that
discussion, the answer was “yes and no.” Such an answer indicates that the question
itself is not well posed. It is like asking if the ability to sing is unique to canaries, or if
all animals have it. Since the ability to sing is an evolutionary adaptation, it shows up
in some animals and not in others. Furthermore, it shows up differently in the
different species in which it does appear. Whale songs and bird songs are very
different, but both are songs.
In Discussion 5.2, we saw that some cultures do not have science, although they do
have belief systems that fill a similar functional role. All of the world’s more
advanced cultures do have one form of science or another, but only in Western
culture has science developed to the point that it has become the central defining
aspect of the culture.
The evolution of science has passed through several stages, and can be viewed as the
gradual discovery and refinement of tools for making accurate use of our everyday
patterns of thought. Western science has progressed further than the science of other
cultures as far as developing tools for gaining knowledge of the physical world is
concerned. Whether these tools, by themselves and without further development, are
sufficient for studies of the mental and social worlds remains an open question.
In this unit, we consider several related topics concerning questions of science and
values. Discussion 7.1 introduces the ethical values of science, the rules of
professional conduct that all scientists are expected to observe and explores the
deeper underlying values that are necessary for science to flourish. These values are
immediate consequences of the goal of understanding the world, combined with the
social nature of science. That is, they are the values that a social group committed to
the factual understanding of nature must have, at least as ideals, if it is to succeed in
the attainment of its goal.[120]
Discussion 7.2 gives a plausible account of the evolution of science in the West,
including some of the social and cultural factors that may have been involved.
Finally, the reading in What Science Is describes and responds to the postmodern
attack on the ideal of scientific objectivity. The basic point made is that the fact that
science cannot produce certain knowledge does not mean that the knowledge
produced by science is no better than that based on any other belief system. While it
is true that the basic tools of science—logic, mathematics, the use of repeatable
experiments—are to a large extent products of the Greek and European cultures,
they have, as indicated in Discussion 7.2, developed to their present state through an
evolutionary process in which the selection criteria are the production of conclusions
satisfying both the coherence and correspondence theories of truth. While belief
systems that do not satisfy these conditions may be functionally adequate in allowing
their adherents to deal with the world, they are far less likely to turn out to be
factually correct. Science certainly cannot explain everything—many areas of human
activity and concern are beyond its scope—but within the areas where the methods
of science can be applied, it produces the best understanding that can be had. That
is, after all, its job.
Objectives
When you have completed Unit 7, you should be able to
1. discuss both the rules of professional conduct in science, and the more basic
values that underlie these rules.
2. describe the sort of activities that are considered to be misconduct in science.
3. state Gerald Holton’s four principles of integrity in science.
4. describe the “first crisis of science,” and state how Aristotle resolved it.
5. identify some of the cultural factors that contributed to the emergence of
modern science in Europe.
6. describe some of the ways that science has influenced modern Western and
world culture.
7. describe and analyse some of the postmodern claims about science.
8. discuss the interrelation between science and culture.
9. state four ways in which scientific language is abused.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Chapter 11, “Difficult and Important Questions: Science, Values and
Ethics.” Pages 144–156 of What Science Is.
2. Read Discussion 7.1, “Science and Values.”
3. Answer Unit 7 Study Questions 1-3.
4. Read Discussion 7.2, “The Rise of Western Science.”
5. Answer Unit 7 Study Question 4-6.
6. Read Chapter 15, “Questions of Legitimacy: The Postmodern Critique of
Science,” pages 207-213 of What Science Is.
7. Answer Unit 7 Study Questions 7-9.
Discussion 7.1 Science and Value
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The high-minded man must care more for the truth than for what people think.
—Aristotle[121]
The values by which we are to survive are not rules for just and unjust conduct, but are
those deeper illuminations in whose light justice and injustice, good and evil, means and
ends are seen in fearful sharpness of outline.
—Jacob Bronowski[122]
It is often claimed that, although questions of values may arise in the applications of
scientific results, science itself is morally neutral and value free. In this section, we
will see that this claim is false, both in regard to science as a social enterprise, and in
regard to the individual scientists who participate in this enterprise.
Clearly the impact that science has on society can lead to serious questions of values
and morals. These sorts of value questions are inevitable so long as our culture
contains themes that are either no longer functional in a changing world, or that, as
a result of scientific advance, come into conflict with other cultural themes and
values. The idea of manifest destiny and the unlimited exploitation of natural
resources are examples of the first kind of contradiction that can arise, and the issue
of human cloning is an example of the second. Human cloning raises troubling
questions about the sanctity of life, but it also offers the possibility of medical cures
for serious diseases and of organ transplants that will not be rejected.
Cultural values do change, but often on a long time scale, while a scientific discovery
with radical implications for cultural values may spread throughout a culture almost
overnight.
Science does present us with moral issues and serious questions of value, but this
fact in itself does not imply that science is not value free. While knowledge is
considered a good, is not knowledge in itself neutral? Is not the scientist required to
maintain a neutral and detached position of objectivity? The very nature of these
questions points to the importance of values for science.
Since science seeks to gain rational knowledge of the world, it can only flourish in a
culture where this sort of knowledge is valued, and where the methods required for
gaining such knowledge can be freely employed. And the practice of these methods
imposes certain conditions on the nature of scientific activity, and on individual
scientists. The scientist and humanist Jacob Bronowski put it very well when he
wrote, “[Y]ou cannot know what is true unless you behave in certain ways.”[123]
This injunction holds true not only in science, but also in all human activity. In order
to discover truth, we must behave in certain ways, and not in others. In science, we
seek to gain knowledge of the world. In this enterprise, certain methods have been
found useful, and other forms of belief or behaviour have been shown to be
destructive.
A very simple illustration of this idea comes from experiments carried out by the
psychologists H. H. Kelley and A. J. Stahelski.[124] They were studying cooperative
behaviour, making use of a game called “the prisoners dilemma.” In this game, two
subjects play without being able to communicate with each other. Each subject has
the choice of one of two strategies, cooperation or competition. The game is very
simple: each subject chooses a strategy and announces it to the experimenter. Based
on the choices, various payoffs are then given. If both subjects choose to cooperate,
each receives a small reward. If both choose to compete, each loses a moderate
amount. If one subject chooses to cooperate and the other to compete, then the
subject who chose to compete receives a large reward, while the subject who chose to
cooperate loses a large amount.
Many students were used as subjects in the experiment. The rules of the game were
explained, and subjects were quizzed to ensure that they understood the rules. They
were also asked their opinion of the aim of the game, and on that basis were
separated into two groups: cooperators and competitors. The cooperators believed
that the aim of the game was to cooperate so that each round of play would generate
a small profit. The competitors believed that the object of the game was to out-guess
the other player so as to win the largest possible amount in each round.
Once the players had been separated into groups, they were mixed up, and each
subject played a number of games with a number of other subjects. Three patterns of
play were observed: if two competitors played together, they simply competed; if two
cooperators played together, they cooperated. What was interesting was that if a
cooperator played with a competitor, the cooperator would start out cooperating, but
would quickly switch to a competitive strategy in order to minimize losses.
The important conclusion drawn was that competitors learned something false about
the world, namely that everybody was a competitor. The cooperators, on the other
hand, learned something true—the world contains both cooperators and
competitors. Generalizing this conclusion, the way that we interact with the world
determines how we see the world and what we are able to learn about it. “In order to
discover what is true, one must behave in certain ways.”
The general principle is as follows: truth is more likely to be found by open-minded
exploration and vigorous debate than by adherence to dogma or authority. This
principle imposes certain conditions on how these explorations and debates are
carried out. You cannot, for example, attempt to kill an opponent, or even prevent
her or him from being heard; you can only do the best you can to provide answers to
their argument—and be willing to change your mind if the weight of evidence is
against you.
Since science is a search for truths about the world, carried out by a social
community, this search can succeed only if each individual scientist is truthful about
his or her work. Thus, there is an ethical injunction that underlies success in science.
Bronowski states it this way, “We OUGHT to act in such a way that what IS true can
be verified to be so.”[125]
Bronowski concludes that if science is to flourish in a culture, this implies the need
for certain specific social values. Discussion 2.2 presented two examples of highly
advanced cultures where science could not develop fully, in part because of
constraining cultural values. Since science has become such an important tool of
human survival, there is an evolutionary advantage to cultures that support sciencefriendly values. Western science is becoming world science, not because of any
imperialistic design on the part of the West, but because it is effective in enhancing
peoples’ quality of life. It is opposed in cultures whose values are in conflict, not with
the results of science itself, but with the values required for successful science, or
with Western values that are not required for science, but are mistakenly associated
with it.
Scientists are bound to be truthful about their work, and thus science flourishes best
in communities where such truthfulness is supported. This assertion means that
communication within scientific communities must be free and uninhibited, and this
condition, in turn, implies the need for intellectual freedom and independence, and a
tolerance for dissent. Freedom of inquiry allows scientists to follow up whatever
clues are found in the search for truth, and the freedom to dissent opens the door for
originality and creativity.
Bronowski identifies the values of free inquiry, free thought, free speech and
tolerance as essential prerequisites for science and remarks that, “These values . . .
seem self-evident, that is, they are logical needs, only when men are committed to
explore the truth. . . . These freedoms of tolerance have never been notable in a
dogmatic society.”[126]
Tolerance also implies the need to respect human dignity, and a need for mutual
respect among scientists as workers seeking to discover what is true. While
individual scientists may feel jealousy, envy, even hatred of scientific rivals, this is a
matter of individual egos. To the extent that such feelings obscure the deeper respect
for the underlying search for truth, they obscure the ideals of scientific interaction.
Scientists may engage in spirited disputes, but as J. M. Ziman observes,
[O]ne finds oneself rather emotionally engaged in a controversy upon a scientific matter;
but this after all, is one’s job, and one must learn to advocate powerfully, but to concede
defeat gracefully, without making a personal issue of it. In a sense, a well-fought controversy
between two spirited champions is a form of cooperation.[127]
A popular Middle Eastern teaching story relates how a famous scholar challenged a
Sufi teacher to a debate.[128] The Sufi arrived at the appointed venue with several of
his students, and it was decided that the scholar would speak first. He stood at the
podium and looked out at the audience. Then he began his prepared oration. After
listening for several minutes the Sufi stood up and pointed a finger at the scholar.
This so unnerved the man that he stumbled in his speech, panicked and ran from the
stage.
Later, one of the Sufi’s students asked him why he had acted in this way, rather than
simply engaging the scholar in debate and refuting his arguments. “That would not
have been productive” the Sufi replied, “He was only interested in winning the
argument, not in discovering the truth.”
Since science is a community activity, the work of each scientist relies on results
obtained by other scientists. If a scientific community finds that the results claimed
by one of its members cannot be trusted, that member is effectively excluded from
the community. Perhaps not officially—scientific communities are usually not that
tightly organized—but that person’s theories or empirical data will not be used by
other scientists, and while papers submitted to scientific journals may be published,
they will not be read.
Some of the common norms for acceptable scientific behaviour are listed below.
o
o
o
o
o
Knowledge is public.
There are no privileged sources.
Ideas are to be judged on their intrinsic merit, not their source.
Claims are not taken on faith, they must be substantiated.
All contributing sources must be acknowledged.
This being said, it remains true that the potential for success in science is influenced
by certain elements of character. Scientists are subject to the same human foibles as
everybody else. The values of science act to minimize the effect of personal error or
subjectivity, and to correct mistakes. Scientific studies are carried out within the
social context of a community of scientists. The values within this community are
directed toward the discovery of accurate knowledge of the world. These values rely
on judgements about what is and is not important for science, and about the
acceptable methods for doing science. These judgements can be influenced by social
factors, such as the availability of funding and current social needs. What is
important is the maintenance of scientific responsibility and integrity. In this regard
Gerald Holton describes four principles of integrity in science.[129]
1. Try to get it right at all costs, sparing no effort.
2. Try to be a scientist first, a specialist second.
3. Try to explore and contribute to the role of science as part of our overall
worldview.
4. Try to do your duty as a scientist and as a human being.
Holton’s final point directly connects us to the social and historical nature of human
existence. The Russian mystic George Gurdjieff (1877-1949) once remarked that we
do not begin to understand the true meaning of duty until we realize that everything
we possess, even the language we speak, is the result of the efforts and sacrifices of
others who often did not personally benefit from their work and their suffering.[130] As
Robin Dunbar notes, science is a long-term project.
[S]cientists operate on a different time-scale. Their concern is not to try to solve all the
problems today (although occasionally being able to solve one problem in a lifetime comes
as a bonus); rather, their concern is to understand how and why the world is as it is, and if
this takes five hundred years of collective effort by a thousand individuals scattered in
laboratories all around the world, only a handful of whom actually get to meet, then so be it.
Success in science comes only from a long slow methodical working through of all the ins
and outs of a very complex phenomenon, checking and double checking everyone else’s
calculations, because only by patience and careful testing will we avoid mistakes.[131]
Individual scientists also have sets of personal values, based on personal judgements
about what is important in their lives, and how they can best go about achieving
personal goals. To the extent that these personal goals involve scientific success, they
will influence a person’s behaviour as a scientist. Fear of failure in research,
combined with an excessive desire for the appearance of success can lead to the
commission of scientific fraud—falsifying data or stealing another scientist’s ideas.
People who commit such actions have completely lost contact with the actual values
of science, and have fallen into the illusion of identifying appearance and reality.
Even if a scientist is careful to respect the values and methods of science, their
personal goals, values and character traits may hinder them in their work. Secrecy is
a good example of how this can happen. Because credit for scientific discoveries is
based on priority of publication, scientists sometimes find it necessary to keep
aspects of their research secret until they are ready to publish. With some people,
however, secrecy becomes an obsession. Fearing theft of their ideas, they hoard and
conceal them like misers. This behaviour means that they end up missing the public
discussion and criticism that is important for the development of ideas. In addition,
with attention fixated on protecting the ideas they have, the likelihood that they will
get new ideas is drastically reduced.[132]
The same effect occurs with people who refuse to discuss their ideas because they are
afraid they may be criticized or shown to be wrong. Productive scientists have
discovered that the more generous they are in sharing ideas, the more good ideas
they will have. If an idea is wrong, a good scientist wants to know it as quickly as
possible to avoid wasting time.
Another personal characteristic that can stand in a scientist’s way is lack of humility.
Without humility, learning is difficult if not impossible.
The writer Idries Shah alludes to the effect of negative character traits in his
book Knowing How to Know. He identifies greed as one of the prime obstacles to
knowledge, and discusses the kinds of behaviour that are required for real learning,
“Any form of greed—even for knowledge—effectively prevents real learning. . . .
Honor, behavior, discipline, truthfulness, sincerity, these are preparations, not
ultimate objectives.”[133] And Jacob Bronowski remarks, “A scientist who is
emotionally immature is like a poet who is intellectually backward: both produce
work which appeals to others like them, but which is second-rate.”[134]
Summing up the importance of values in science, Bronowski says,
There are, oddly, no technical rules for success in science. There are no rules even for using
test tubes which the brilliant experimenter does not flout; and, alas, there are no rules at all
for making successful general inductions. . . . Instead, the conditions for the practice of
science are found to be of another and an unexpected kind. Independence and originality,
dissent and freedom and tolerance: such are the first needs of science; and these are the
values which, of itself, it demands and forms.[135]
Discussion 7.2 The Rise of
Western Science
TOP
For as water, whether it be the dew of heaven, or the springs of the earth, doth scatter and
leese itself in the ground, except it be collected in some receptacle, where it may by union
comfort and sustain itself . . . so this excellent liquor of knowledge, whether it descend from
divine inspiration, or spring from human sense, would soon perish and vanish to oblivion, if
it were not preserved in books, traditions, conferences, and places appointed, as
universities, colleges, and schools, for the receipt and comforting of the same.
—Francis Bacon (1561-1626)[136]
Discussion 5.2 introduced the concept of cultural relativism, the idea that we cannot
judge a culture from the outside, but only in terms of its own internal criteria. This
view arose in anthropology in the early 20th century as a reaction against the
unquestioned belief in European superiority displayed by anthropologists in the 19th
century.
This shift in point of view proved very fruitful in allowing different cultures to be
seen in terms of their own unique individuality, rather than just as steps in a
developmental sequence whose highest exemplar was late 19th century Europe.
However, a form of hidden Eurocentric bias remained, manifested in attempts that
are still made to show that non-literate cultures also have “science,” and that people
living in these cultures are just as “logical” and “scientific” as Westerners.
The motive behind such attempts is the ideal of tolerance and human equality. They
are aimed at countering the attitude that because some peoples do not have science
and do not think according to Western logic, they are somehow inferior as human
beings. This attitude was behind the 19th century concept of the “white man’s
burden” and the abuses of European colonization and conquest.
However, while it is true that Western science is the best way currently known to
gain factual understanding of the external world, this in itself says nothing about the
relative merits of scientific and non-scientific cultures. This point has been strongly
emphasized by the anthropologist C. R. Hallpike, who points out that simply
possessing science and analytic rationality is no guarantee of wisdom, or of the
ability to function effectively in the world.[137] From an evolutionary point of view,
every successful culture has developed its own means of survival in its environment.
Discussion 5.2 showed that while Azande witchcraft might superficially appear to
share certain features with science (e.g., a coherent conceptual framework, appeals
to experience in support of beliefs, an experimental method), this appearance is
misleading. We might interpret use of the poison oracle as an experimental
procedure, but for the Azande it is a process of divination, undertaken with a very
different attitude and point of view.
In Western science, reasoning is used to test theories, to explore the limitations of
current theory, and to develop alternatives that provide a better fit to data. The
Azande use reason to fit experiences into the existing belief system, which itself is
never questioned. As we noted in Discussion 5.2, this is the difference between
evidence and instances.
If the belief systems and conceptual frameworks used by non-literate cultures to
explain the world are not science, we must return to the question raised in
Discussion 1.2—how did science as we know it evolve out of more basic human
characteristics?
Given our discussions of the Azande, it is apparent that this question must be
addressed both in terms of conceptual framework and of attitude or point of view. It
is also clear that in understanding the transition from traditional belief to science,
attitude will be a key factor. The important step in this transition is the realization
that at least some forms of experience can be taken as evidence; that is, rather than
simply being an instance illustrative of an unquestioned belief, it can be used as a
test of that belief. Once this move has been made, it becomes possible to explore the
conditions required for belief testing, the nature of beliefs, and the forms of
argument required to draw valid conclusions from beliefs. Beliefs can be viewed as
working hypotheses rather than immediate, God-given truths. As Charles Darwin
wrote in an 1861 letter to his friend Henry Fawcett, “all observation must be for or
against some view if it is to be of any service.”[138]
While the actual history of how this transition occurred is obscure, we can offer a
plausible “just so story.”
Discussion 5.2 pointed out that the distinction between evidence and instance
cannot be made unless there is an awareness of possible alternatives. In traditional
cultures, there is no alternative to the conceptual framework used by the group,
although in social terms, everybody is aware that people may have different
opinions, and that people are sometimes deceptive. In addition, disputes will arise
that require some form of resolution. Such issues may be settled by a convincing
argument, by tests for veracity, by the decision of a village chief or council of elders,
and so on.
In early state societies, another level of judgement emerges with legal disputes. In
many cases, legal questions are settled by a king or by one of his appointed
representatives, who would listen to conflicting claims, perhaps apply some tests to
determine the truthfulness of the claimants, and then rely on knowledge of law, and
on personal experience and understanding, in order to render a judgement. The
legendary wisdom of Solomon is a good example of this process.
Nevertheless, when it comes to explaining the world, there is still no acceptance of
alternative possibilities. Alternative belief systems are rejected, and strong sanctions
are imposed on those who would “stray from the fold.”
A prime example of this process is the story of the Egyptian pharaoh Akhenaten (r.
1352-1336 BCE). The son of Amenhotep III, who ruled Egypt at the pinnacle of its
power, he came to the throne under the name Amenhotep IV. His chief wife was the
strikingly beautiful Nefertiti, renowned today for the magnificent bust in the Berlin
museum. Through another wife, he was the father of Tutankhamun.
Very early in his reign, Amenhotep IV changed his name to Akhenaten, and initiated
a radical religious revolution, a move that would be made again by Henry VIII of
England almost 3000 years later. Whether it was a matter of fanatic personal belief
or part of a power struggle with the Egyptian priesthood is unknown. Whatever the
motive, Akhenaten abolished worship of the traditional gods, in particular Amun,
and declared that there was only one universal god, the Aten, or Sun-Disk.
Furthermore, only he as pharaoh was empowered to act as intermediary between the
Aten and the Egyptian people. Some historians have dubbed him the first
monotheist.
Akhenaten ruled Egypt for 16 years. With his death, the religion that he had founded
came to an end. More than that, it was intentionally eradicated. The city he had built
as the centre of the new religion was abandoned and his statues smashed. His name
was chiseled off of all public monuments and stricken from the list of Egyptian kings.
The only references to him were not by name, but as the Great Heretic. The Egyptian
culture, as liberal as it was in many ways, simply could not accept the possibility of
an alternative religious point of view.
The Amarna period, which is what archeologists today call Akhenaten’s reign; saw a
sharp decline in Egyptian power. With the exception of a brief revival during the
long reign of Ramses II (r. 1299-1232 BCE), this decline continued, with periods of
domination by Libyan, or Nubian, kings, until the final end of Egypt as an
independent state following the conquest by the Persian king Cambyses in 525 BCE.
From the time of Cyrus the Great (580-529 BCE), religious tolerance was practised
within the Persian Empire, not by default, but as an official policy—the first such
policy in the Middle East. The various conquered peoples worshipped a wide variety
of gods, and each group would have strongly resisted attempts to impose external
beliefs. Thus the Persian Empire was a mosaic of different religions, each with its
particular myths and stories about the nature of the world.
In the Persian Empire, legal disputes were settled in the traditional way, by the king
or his appointed representatives. On the fringes of this empire, however, another
form of legal process could be found. In the Greek cities, both the independent citystates of Greece and the Persian controlled cities of Asia Minor, legal disputes were
often adjudicated by jury. Claimants would argue their case, and the jury would then
debate the merits of each side and vote on the resolution. Juries in such cases could
be very large (e.g., in the trial in which Socrates was condemned to death, the jury
numbered upwards of 500 members).
The wealthiest of the Greek cities of Asia Minor was Miletus, and it was here that a
new way of viewing the world first blossomed: the idea that human reason could be
used to understand nature. Prior to this time, the implicit assumption had always
been that the world was controlled by the gods; whose behaviour was humanly
inexplicable. This view can be summarized in three basic mythico-magical
assumptions, given below.
1. Nature is capricious, the unknown is unknowable.
2. Everything is determined by external, willful powers.
3. The future can be controlled by magic.
Operating under such beliefs, what was important was to learn the will of the gods
and other supernatural powers, and how to placate them through offerings and
submission, or control them through magical practices.[139] It was the role of the
magician, priest, soothsayer, oracle or prophet to provide insight into the future.
We can imagine citizens of Miletus, however, spending afternoons discussing
commercial and civic affairs, and also the various myths of the different cultural
groups they had encountered on trading journeys throughout the Persian Empire.
Under such circumstances, differences between myths would certainly be noted, and
in a culture where legal disputes were settled by jury, it would be natural for the
parties to a discussion to believe that it ought to be possible to explain these
differences through debate and the application of reason. From there, the next step—
to making up rational explanations of the world—is easy.
The credit for being the first scientist usually goes to Thales (625-546 BCE), a citizen
of Miletus who traveled widely in Babylon and Egypt studying mathematics and
astronomy with priests and magicians. He measured the height of the Great Pyramid
by measuring the length of its shadow and comparing it to the length of the shadow
of a stick of known height; he also correctly predicted a solar eclipse that occurred in
585 BCE. He was apparently a shrewd businessman as well, for he is reputed to have
cornered the olive press market in Miletus.
In mathematics, Thales introduced abstraction into geometry by thinking in terms of
idealized lines with no thickness, rather than lines marked out on papyrus or sand,
and he also introduced the idea of a deductive proof.[140] His most important scientific
contribution, however, was to ask a question. He asked what the world was made of.
As an answer, he suggested that there was a single basic substance, which he thought
was water. In suggesting this theory, he made no reference to the gods or other
supernatural entities, and he thus proposed the world’s first purely physical theory.
While his suggestion may seem silly to us today, it contains the profound assumption
that there actually is some fundamental substance of which all else is composed. We
still believe this today, but we call this substance energy.
Because Thales’ theory was of human origin, it was open to discussion, criticism, and
comparison with alternate theories that were quickly proposed. Two of Thales’
students put forward their own ideas. Anaximenes (d. 528 BCE) suggested that the
fundamental substance was air, while Anaximander (610-546 BCE) referred to it as
the arche, the unlimited, and in a famous phrase said that everything comes from
the unlimited, and returns to it in order to “repay the injustice of its
arising”[141] [Indeed, the legal source of the idea of a rational argument is hinted at in
many references to justice as a warrant for acceptance of a claim.] Another early
thinker, Heraclitus of Ephesus (c. 535-475 BCE), called “The Obscure,” held that fire
was the basic substance, and other theories of the fundamental constituents of
matter were quickly developed. Best known today is the theory of the four elements
of earth, air, fire and water attributed to the Sicilian Greek philosopher-magician
Empedocles (fl. 450 BCE).
What is most important in terms of the origin of science is that all of this theorizing
was based on appeals to human reason rather than to myth or to the gods. Without
actually saying so, the mythico-magical assumptions about nature had been replaced
by the basic assumptions of science given in Discussion 6.1.
1. Nature is ordered.
2. This order is comprehensible to human reason.
3. Our understanding of this order can be communicated without subjective bias.
Another thread in the emerging tapestry of science was spun by Pythagoras of Samos
(582-497 BCE). We remember Pythagoras today primarily for the Pythagorean
theorem: the square of the hypotenuse of a right triangle equals the sum of the
squares of the other two sides (a result that was actually known both in
Mesopotamia and Egypt long before Pythagoras). In fact, Pythagoras’ contributions
are woven into almost every aspect of modern science and culture. –
After years traveling in Egypt and the Middle East, Pythagoras settled in the
southern Italian city of Croton in about 529 BCE. There he established a mystical
school whose influence spread throughout the Greek world. While the Pythagoreans
kept their teaching secret, they were responsible for much of the early development
of Greek science and mathematics. As Peter Kingsley describes it,
When we pick our way through the over-simplifications we repeatedly encounter the
scenario of Pythagoreans teaching highly creative individuals who can in a sense be
described as perpetuating aspects of Pythagorean tradition through their own writings, but
who themselves can only questionably be referred to as Pythagoreans.[142]
The major areas of study in the Pythagorean School were music, mathematics and
astronomy. Perhaps the greatest scientific achievement of Pythagoras himself was
his study of the sound produced by vibrating strings. We still use the seven-tone
scale that he devised. The result of these studies, perhaps the first consciously
conceived and executed scientific experiments, showed that, when subjected to the
same tension, vibrating strings of different lengths produced harmonious sound
together when their lengths were related by simple numerical ratios. For example, a
2 to 1 ratio characterized the octave, a 3 to 2 ratio the musical fifth, and a 4 to 3 ratio
the musical fourth.
By analogy to music, the Pythagoreans believed that the entire universe was a
harmonious unity, a cosmos, in which every aspect was based on numerical
relationships. This belief motivated an intense program of mathematical research
into the abstract properties of numbers, and it is from the Pythagoreans that we get
the idea of expressing scientific laws in mathematical form.
These early natural philosophers had taken on a monumental task, and it is not
surprising that many of their efforts now strike us as simple-minded. But many of
the basic ideas guiding their speculations remain with us today—the quest for
fundamental elements; the importance of mathematics for descriptions of nature;
the search for natural cycles—all of these and many more are as important today as
they were 2500 years ago.
One aspect of special concern had to do with the use of reason itself. There were no
established criteria for what constituted a reliable argument. This gap led to the first
crisis of science—the need to discover such criteria, how to carry out a reasoned
argument without simply falling into nonsense. The crisis was brought to a head by
the emergence of the Sophists—teachers who made their living by teaching rhetoric
and “virtue” to the children of the Greek aristocracy—in the latter part of the 5th
century BCE.
Since political power and influence in the Greek city-states often depended on being
able to persuade assemblies of citizens, what the Sophists taught was the art of
persuasion—in a sense, simply a return to the magical idea that the truth is whatever
the most persuasive speaker (or most powerful magician) says it is. The Sophists
were a threat to science because they were only interested in winning arguments, not
in discovering if the winning argument was actually true.[143] Discussion 6.1 showed
how the Sophist Gorgias argued against the basic metaphysical assumptions of
science. Without a way to refute these arguments conclusively, the very possibility of
science was in question.
While first Socrates and then Plato strongly opposed the Sophists, it remained for
Plato’s student, Aristotle, to resolve the crisis by determining the laws of formal
logic. These laws form the topic of Discussion 9.2. Aristotle also instituted the
principle that philosophical and scientific conclusions could only be accepted if they
were supported by logically sound arguments. Appeals to personality, authority, the
gods and so on could not be accepted, while analogy and metaphor could be used to
illustrate a point, but not to demonstrate its correctness.
The scientific method that developed in the Hellenistic[144] world following Aristotle
can be called “speculative-deductive.” It involves top-down explanation deduced
from abstract metaphysical principles or from commonly agreed upon premises or
beliefs. This does not mean that experimental and observational science did not
exist, but the concepts of using observational or experimental results as a means of
refuting a theory, and of designing experiments to test a theory, were either
unknown, or not used in any systematic way.
Conditions in 17th century Europe, catalyzed by the wealth and outgoing spirit of
exploration arising from the discovery of the Americas, were extremely favourable
for science. Here, the idea of experimental testing of theories could flourish, as could
the development of mathematical tools, such as probability theory and statistics,
which allowed accurate control over interpretation of experimental results. The
distinction between experiences that were simply instances of a theory, and those
that could provide evidence for or against a theory was clarified, and the
hypothetico-deductive method was formalized. In addition, the importance of open
communication, impartial peer review, and accurate assignment of credit were
recognized as essential aspects of the social side of science.
As the values that arose entered European culture, they resonated with and
reinforced similar cultural values, thus contributing to the 18th century
Enlightenment. It is worth noting in this regard that the areas of Europe that
remained most resistant to the Enlightenment were also the areas that remained
most scientifically backward and most firmly under the control of the churches.
In the late 19th and early 20th centuries, this situation led to scientism—the belief
that science, as it existed in the late 19th century, was the final step in human
intellectual development, and that it would soon provide the solution to all human
problems and the answer to every question. As the saying goes, to a man with a
hammer, everything looks like a nail.
Today we are more modest. It has become clear that while modern Western science
is at present unsurpassed as a means of understanding the external world, it has
difficulty dealing with the inner worlds of conscious human experience, where the
tools of third person objective observation are ineffective. This does not mean that
science will never be able to provide understanding of this inner territory, but if it is
to do so, new methods of investigation will be required. Not that current scientific
methods will have to be abandoned—the development of the inductive method did
not mean the abandonment of deductive reason. Rather, methods developed for the
study of the inner realms of human consciousness will need to be integrated with
existing methods to produce a more comprehensive approach to all of human
experience, rather than just the external world.
Unit 7 Study Questions
TOP
Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. Write a short essay (300-500 words) discussing the relation between the
Einstein quote given at the beginning of this unit and the Bronowski quote that
heads Discussion 7.1.
2. Based on the readings from Chapter 11 and Chapter 15 of What Science Is,
and Discussion 7.1, draw up a list of rules for professional conduct for scientists.
3. Write a short essay (300-500 words) discussing Jacob Bronowski’s statement,
“[Y]ou cannot know what is true unless you behave in certain ways.”[145]
4. Write a short essay (300-500 words) discussing Darwin’s statement, “[A]ll
observation must be for or against some view if it is to be of any service.”[146]
5. What are the three basic mythico-magical assumptions about the world?
6. Find three examples of mythico-magical thinking in the world today (e.g., from
news stories, television or popular belief systems).
7. The physicists Alan Sokal and Jean Bricmont list four common abuses of science
that are found in some areas of new age and postmodern discourse:[147]
a. longwinded discussions of scientific theories with little or no
understanding of the actual meaning of what is being discussed.
b. taking natural science concepts and using them in the humanities or
social sciences with no rational or empirical justification to support the
implied analogy.
c. using technical scientific language in a context where it is irrelevant.
d. using scientific sounding language to talk nonsense.
Find examples of each of these abuses in the popular press, on television or in
other domains of popular culture.
8. Try to give a simple, one-sentence summary of this paragraph by the
postmodern French sociologist and philosopher Jean Baudrillard. [It is possible
to do so!]
Our complex, metastatic, viral systems, condemned to the exponential
dimension alone (be it that of exponential stability or instability), to eccentricity
and indefinite fractal scissiparity, can no longer come to an end. Condemned to
an intense metabolism, to an intense internal metastasis, they become
exhausted within themselves and no longer have any destination, any end, any
otherness, any fatality. They are condemned, precisely, to the epidemic, to the
endless excrescences of the fractal and not to the reversibility and perfect
resolution of the fateful.[148]
9. Write a short essay (300-500 words) discussing the apparent contradiction
between the idea that all science is carried out within, and hence is influenced
by, a particular cultural context, and the idea that science produces objective
knowledge of the world.
FOOTNOTES
[115]
Einstein, Albert. “Science and Religion.” in Ideas and Opinions, pp. 41-49. New York:
Three Rivers Press, 1995. Retrieved July 31, 2002.
http://www.stcloudstate.edu/~lesikar/einstein/Einstein2.html/
[116]
Interestingly enough, people who hold this view usually don’t give scientists credit for the
positive results of scientific research.
[117]
Consider, for example, all of the ways that ideas from quantum theory have infused into
everyday cultural discourse.
[118]
This is not to say that computers have not created many issues of social concern, but the
basic ideas involved do not conflict with central cultural values.
[119]
Appleyard, Brian. Understanding the Present, p. 9. New York: Doubleday, 1992.
[120]
This argument has been made very eloquently by Jacob Bronowski in his book Science
and Human Values. New York: Harper and Row, 1965.
[121]
Quoted in Jones, Shirley, ed. The Mind of God, p. 92.
[122]
Bronowski, Jacob. Science and Human Values, p. 73. New York: Harper and Row, 1956.
[123]
Bronowski, Jacob. The Origins of Knowledge and Imagination, p. 129. New Haven, CT:
Yale University Press, 1978.
[124]
Kelley, H. H., and A. J. Stahelski. “Social Interaction Basis of Cooperators and
Competitors.” Journal of Personality and Social Psychology, 16 (1970): 66-91.
The game takes its name from the police practice of separating prisoners suspected of
committing a crime and trying to get one or the other of them to confess and incriminate
their partner.
[125]
Bronowski, Jacob. Science and Human Values, p. 58.
[126]
Ibid., p. 62.
[127]
Quoted in Hull, David L. Science as a Process, p. 15. Chicago: University of Chicago
Press, 1988.
[128]
The Sufis have been highly regarded in the Middle East as exponents of practical wisdom
and spiritual illumination.
[129]
Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein, rev. ed., pp.
460-469. Cambridge, MA: Harvard University Press, 1988.
[130]
For a discussion of Gurdjieff’s views, see Bennett, J. G. Talks on Beelzebub’s Tales, pp.
128-130. York Beach, ME: Samuel Weiser, 1988.
[131]
Dunbar, R. The Trouble with Science, p. 102. Cambridge, MA: Harvard University Press,
1995.
[132]
One area of serious concern about secrecy is that many government or corporate
organizations attempt to keep the results of scientific work conducted under their auspices a
secret. This strategy may be necessary for military or commercial reasons, but it is almost
always detrimental to the advancement of science.
[133]
Shah, Idries. Knowing How to Know, pp. 59, 108. London: Octagon , 1998.
[134]
Bronowski, Jacob. A Sense of the Future, p. 7. Cambridge, MA: MIT Press, 1977.
[135]
Bronowski, Jacob. Science and Human Values, p. 62.
[136]
Bacon, Francis. The Second Book of Francis Bacon of the Proficience and Advancement
of Learning Divine and Human, “To the King,” para. 3. Retrieved August 4, 2002, from
http://www.uoregon.edu/~rbear/adv2.htm/
[137]
Hallpike, C. R. The Foundations of Primitive Thought, p. 491. Oxford: Clarendon Press,
1979.
[138]
Darwin, Charles. “Letter 133: To Henry Fawcett,” In More Letters of Charles Darwin: A
Record of His Work in a Series of Hitherto Unpublished Letters, Volume I, p. 195. F.
Darwin, ed. New York: Appleton; 1903.
[139]
The parallel with social behaviour on a human level is obvious. In a social group
controlled by a dominant elite, other group members quickly learn to read signs indicating
the desires of members of the elite, to offer their allegiance to one or another faction in this
elite, and to seek favour through various means of persuasion.
[140]
He is reported to have sacrificed a bull to the gods to celebrate his proof that any
diameter of a circle separates it into two equal parts.
[141]
Quoted in Smith, T. V., ed. From Thales to Plato, p. 6. Chicago: University of Chicago
Press, 1960.
[142]
Kingsley, Peter. Ancient Philosophy, Mystery and Magic, p. 329. Oxford: Clarendon,
1995.
[143]
Lawyers today are often tarred with this same brush.
[144]
This term refers to those areas of Europe, Africa and the Middle East that came under
Greek influence following the conquests of Alexander. The intellectual and scientific centre
of this world was the city of Alexandria on the coast of Egypt.
[145]
Bronowski, Jacob. The Origins of Knowledge and Imagination, p. 129.
[146]
Darwin, Charles. “Letter 133: To Henry Fawcett.”
[147]
Sokal, Alan, and Jean Bricmont. Fashionable Nonsense: Postmodern Intellectuals’ Abuse
of Science, pp. 4-6. New York: St. Martin’s, 1999.
[148]
Quoted in Sokal, A., and Bricmont, J. Intellectual Impostures, p. 142. London: Profile
Books, 1998.
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Unit 8 What Is Reason?
Discussion 8.1
Discussion 8.2
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Discussion 8.3
Unit 8 Study Questions
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STUDY GU IDE
Unit 8 What Is Reason?
In this unit, we begin to examine the way reason is used in science. Before we can
start, however, we must become more specific in our definition of reason. There is a
general prejudice equating reason and rationality with formal deductive logic.
However, while formal logic is a tool of reason, it is not the same thing as reason, any
more than a hammer is the same thing as carpentry.
Discussion 8.1 gives a very broad definition of reason, briefly mentions several
different forms of reasoning, and provides some general non-analytic criteria for
evaluating reasons. Discussion 8.2 describes three basic heuristics, or rules of
thumb, that the mind automatically uses in thought, and some of the ways they can
lead to error. It also suggests that the evolution of science has in large part been a
process of discovering tools and techniques to assist us in avoiding the errors of
using these heuristics.
Readings in What Science Is introduce critical thinking. Critical thinking is simply
the use of specific tools and strategies of thought to come to clear conclusions on
whether to accept or reject claims that are made, and to determine the degree of
reliability of our conclusions. It is an essential skill not only in science, but also in
everyday life. Discussion 8.3 describes the basic elements of a particular technique
for diagramming arguments that is a useful tool for critical thinking.
Objectives
When you have completed Unit 8, you should be able to
1. explain reasoning as a process of mentally fitting things together, and describe,
briefly, three sorts of reason—empathetical, analogical and analytical.
2. describe seven criteria for a “good fit,” and apply them to reasoning.
3. define the three decision heuristics, and give examples of how they operate in
thought.
4. give examples of some of the cognitive illusions that result from use of the three
decision heuristics.
5. state the two foundations of a well-constructed scientific argument.
6. define “deductive reasoning” and “inductive reasoning,” and describe each
briefly.
7. describe some ways in which the use of critical analysis in everyday life can
benefit from scientific input.
8. discuss the importance of identifying hidden assumptions in both science and
everyday life.
9. discuss some of the issues involved in the evaluation of causality.
10. apply the affective and cognitive strategies of critical thinking.
11. define the three general categories of fallacies, identify the fallacies in each
category and give examples of each fallacy.
12. diagram simple arguments.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 8.1, “Reason—Finding a Good Fit.”
2. Answer Unit 8 Study Questions 1 and 2.
3. Read Discussion 8.2, “How Rational Are We, Really?”
4. Answer Unit 8 Study Questions 3-8.
5. Read Chapter 7, “Thinking Straight: Evidence, Reason, and Critical Evaluation,”
pages 89-106 of What Science Is.
6. Answer Unit 8 Study Questions 9-14.
7. Contact the Office of the Registrar at Athabasca University and request the
midterm examination. Plan on giving yourself about two weeks to review
materials from Units 1-8 in preparation for this exam.
Discussion 8.1 Reason—Finding a
Good Fit
TOP
[T]he critique of reason leads . . . naturally and necessarily to science; and, on the other
hand, the dogmatical use of reason without criticism leads to groundless assertions, against
which others equally specious can always be set, thus ending unavoidably in skepticism.
—Immanuel Kant[149]
Tell me how you are searching, and I will tell you what you are searching for.
—Ludwig Wittgenstein (1889-1951)[150]
The purpose of this discussion is to introduce a definition of the term “reason.” The
question of what reason is has been a central issue in philosophy for over 2000
years, and may well continue to be a bone of contention for 2000 more. It is not the
intention of this course to go into the many opinions expressed on this issue, or the
arguments used to support them. The goal is only to provide an introduction to some
of the more common practices and methods involved in scientific reasoning.
We begin our consideration of reason with a process of directed free association. The
aim is to generate ideas about the nature of reason, and its concomitant associations,
as a means of developing an empathetic connection with the subject matter. To
begin, we want only a rough definition. Attempting to be too precise at this point
would impose overly rigid limitations.
A little work with a good dictionary (e.g., the American Heritage Dictionary of the
English Language, unabridged edition) tells us that the word “reason” (and the word
“ratio”) derives from the proto-Indo-European root ar, “to fit together.” Since we
think of reason as a mental process, this leads to a working understanding of
reasoning as a mental process of fitting together.
Let us look at some of the modern definitions of the word reason. Each of us would
recognize “reason,” or “a reason” or “to reason” as
o
o
o
The basis or motive for an action, decision, or conviction . . . .
A declaration made to explain or justify an action, decision, or conviction . . . .
An underlying fact or cause that provides logical sense for a premise or
occurrence . . . .
o
o
o
o
The capacity for [rational thought, inference, or discrimination] . . . .
Good judgement, sound sense, [intelligence] . . . .
To talk or argue logically or persuasively . . . .
To determine or conclude by logical thinking . . . .[151]
In each of these cases there is a process of fitting together. We even acknowledge this
in everyday language—when somebody suggests a “reasonable” idea we may well
respond, “yes, that fits.”
Note that several of the meanings of reason involve the word “logic.” Returning to
the dictionary, we find that logic derives from the proto-Indo-European root leg, “to
collect,” or “to speak.” Modern definitions of this word include
o
o
o
o
The study of the principles of reasoning . . . .
A system of reasoning . . . .
The formal guiding principles of a discipline, school, or science . . . .
Valid reasoning . . . .[152]
So, speaking roughly, logic is the study of the ways in which a collection of things
may be fitted together and spoken about. The appropriateness of this definition will
become more apparent in Discussion 10.2, which deals with classification and the
construction of taxonomies.
Finally, recall the definition of reason as the capacity for thought, inference or
discrimination. Proto-Indo-European roots for the words “think,” “infer” and
“discriminate” are tong “to think or feel,” bher “to carry, to bear children,”
and skeri “to cut or separate, to sift.” Interestingly, the last of these words is closely
related to the word skei, also meaning to cut or split, which is a root word for the
modern English words “consciousness” and “science.”
We can summarize this etymological research by saying that reasoning is the way
that our mind fits things together. In this process a mind uses, as tools, logics that
tell it how to discriminate which things need to fit together to be collected into a
common category. Once this category has been identified as a coherent collection by
a process of inference, it is “reborn” as an object of consciousness. We may give it a
name and speak about it. The philosopher Immanuel Kant has given what many
consider to be the deepest analysis of how this can occur, but we can also recall
Plato’s injunction about carving nature at the joints.
An alert student might wonder at this point how it is that although reasoning is
being taken as a process of fitting together, some of the main ideas introduced so far
involve discrimination and distinction. Are these activities not the opposite of fitting
together?
Think for a moment about a watch: if we want to know how a watch works, we need
to know, among other things, what the pieces are (i.e., we need to distinguish them
as basic components), and we need to know how they fit together. It is not possible
to know how the pieces fit together if we have not already determined what they
are.[153]
More generally, we must distinguish something from its environment before we can
understand how it fits into its environment.[154] The distinction between identifying
components and fitting them together also relates to the difference between analysis
and synthesis, deduction and induction, and the top-down and bottom-up
approaches you will encounter in Discussion 9.1.
You will recognize that, throughout the preceding paragraphs, we have been engaged
in both making distinctions (i.e., our rough definitions), and fitting them together.
But the form of reasoning we used here is not “scientific reasoning.” It is called
“scholastic reasoning,” because it is based on inferences from dictionary definitions
made according to the rules of analogy and Aristotelian syllogistic logic. You will
study these rules in detail later in the course. Scholastic reasoning may sometimes be
used in scientific work, but it is an error to assume that scientific reasoning is only
an elaboration of scholastic reasoning. There is much more to it than that.
Often when we speak of reasoning, or being rational, we have an image of cold
calculation based on an analysis of a situation and its possibilities. There are other
forms of reasoning, in the general sense of fitting together, however, that are also
important in science. We call these forms “empathetical” and “analogical” reasoning.
It may seem strange to think of empathy as a form of reasoning, but it is what gives
us the real feeling of a fit. In one form, it involves placing ourselves into another
point of view, not simply from an analytical or analogical standpoint, but
experientially. The saying that to understand a person we must walk a mile in their
shoes is an expression of this form of reasoning—it allows us, at least to an extent, to
experience the world in the way that another experiences it, to see how their
particular world fits together.
The formal binomial designation for human beings is Homo sapiens. The
word Homo comes from the Latin word for man. It traces back to the proto-IndoEuropean root dhghen, “earth.” Sapiens derives from the Latin for wisdom. The
English form is “sapient.” Both words come from the proto-Indo-European sap,
“taste, perception.” Analogically, then, we can say that a human is a being of the
Earth who has “tasted” the world and thus become perceptive. This idea is even
expressed in a traditional saying: “He who tastes, knows.”
A number of associations can be made to help with this analogy. It is only necessary
to think of various metaphors that employ the sense of taste. Metaphorically
speaking, taste itself is a form of empathy. We might say that a particular building is
designed in good taste, meaning that there is something about the design that is
pleasing to us, something that “feels” right and fitting. Or, we might say that
something is in poor taste, meaning that it leaves an unpleasant feeling that
something doesn’t fit or is out of place—it evokes sensations with which we do not
want to empathize[155].
So we might imagine that one who has “tasted” the world has learned “good taste,”
which, in combination with the acquired capacity to perceive the nature of
situations, results in a certain ability to make judgements. This ability has been
deemed of sufficient importance that it is used to define human beings.
It is important because it is our main tool for survival. We do not survive on the basis
of brute strength, speed, camouflage, or any of the other strategies found in the plant
and animal kingdoms. We survive because we are able to learn from experience and
make judgements on the basis of what we have learned. That is, we have the capacity
to reason. Having good taste means having a sense of what is fitting, and reasoning
is fitting things together. It also means having discriminating taste, that is, the
ability to make fitting distinctions.
There are significant philosophical points that have been glossed over in the
preceding paragraphs. Indeed, we have been playing rather fast and loose with some
very deep concepts that philosophers have argued over for centuries. These concepts
involve such things as the relationship between sensation, perception and thought;
the various mental structures necessary for sensations to become perceptions which,
in turn, become involved as components of thinking; and the role of reason in all of
this. Since our current interest is simply in developing an introductory feeling for the
subject, we will not go any further here. It is important, however, to recognize that
we are, philosophically speaking, skipping lightly over some very slippery rocks.
For our purposes, it is sufficient to recall that at the most basic level reason is
understood as a process of fitting together. Thus sensations are “fit together” to
become perceptions; perceptions are “fit together” in thinking; and thoughts are “fit
together” in the process of making judgements and deciding on actions. And, at the
most basic level of deciding on actions in a primitive and dangerous environment, if
our action is in bad taste (i.e., is not “fitting”), then we may end up as a tasty morsel
in some predator’s diet.
If reasoning is seen as a process of mental fitting together, then we can ask how to
determine when something actually “fits.” How do we tell a good fit? With respect to
perceptions, gestalt psychologists took up this question in the early 20th century.
They determined a number of criteria for a “good form,” and although these criteria
were concerned with our ability to distinguish forms in the visual field, they also
apply to our ability to recognize “good forms” in theoretical systems of ideas. The
gestalt criteria, adapted to apply to scientific hypotheses, are listed below. They are
not to be considered fixed rules, however, merely “rules of thumb.”
Regularity: All other things being equal, a hypothesis that can be seen as an
extension of already existing and accepted ideas is to be preferred to one that
requires radical changes in these ideas.
Symmetry: All other things being equal, a hypothesis that is symmetrically related to
existing accepted ideas is to be preferred. Examples include cases in which an idea
that has been successful in one field of science is taken over into another field by
analogy.
Harmony: All other things being equal, a hypothesis that is in harmony with, or
resonates with, existing accepted ideas is to be preferred to one that does not, and an
argument whose structure is harmonious with existing patterns of reasoning is to be
preferred.
Simplicity: All other things being equal, the simpler hypothesis or argument is to be
preferred.
Conciseness: All other things being equal, a concise hypothesis or argument is to be
preferred.
Inclusiveness: All other things being equal, the hypothesis that explains the most is
to be preferred. We can observe that this criterion might appear to oppose the
criteria of simplicity and conciseness. One of the remarkable things about science is
that this is not the case. Often it turns out that theories that are simple and concise
are also the most inclusive. Newton’s law of universal gravitation is an example, as is
Darwin’s theory of evolution. Newton’s theory, based on his three laws of motion and
his gravitational force equation, explained not only the motion of the planets, but
also the behaviour of falling bodies and the tides. Darwin’s theory explains the vast
diversity of life on earth on the basis of the simple and concise ideas of variation and
natural selection.
Unity: Theories that provide a sense of closure, a unified understanding of a field of
phenomena, are to be sought as the goal of science.
The conditions of simplicity and conciseness are recognizable as implications of
Occam’s razor, the philosophical injunction that “entities are not to be multiplied
unnecessarily.”
Note that the first six of these criteria are qualified by the phrase, “all other things
being equal,” and that in actual scientific practice, it is often necessary to make
trade-offs. For example, it may be that a hypothesis that is not a regular extension of
an existing theory is more inclusive, or that the simplest hypothesis is not
harmoniously related to existing theory. In other words, finding a good fit is more a
matter of aesthetic sensitivity than of applying fixed rules. But then, the words
“harmony” and “art” share the same proto-Indo-European root as the word “reason.”
Discussion 8.2 How Rational Are
We, Really?
TOP
What I tell you three times is true.
—Lewis Carroll (Charles Dodgson, 1832-1898)[156]
In optical illusions, we see something that is not there, or fail to see something that
is there, or see an ambiguous figure that can appear to us in different ways.
Psychologists and cognitive scientists use these illusions to study the nature of visual
perception.[157] Philosophers have known for close to 2500 years that the senses do
not necessarily provide accurate information about the world, and shamans and
magicians have known this for far longer. In the past 100 years or so, we have begun
to understand how sensory illusions occur, based on the way the brain processes
sensory input. We have come to understand that we do not see the world directly, as
it is. Instead, our perceptual world is constructed within the nervous system and the
brain, and in some cases this process of construction can be fooled.
When we think about the world, however, we tend to believe that our conclusions are
correct. After all, we are in charge of our own thought, and if a conclusion seems
sensible to us, then there is no apparent reason to doubt it. In psychology, it was
believed until the early 1960s that people, by and large, made rational decisions; that
is, decisions based on logical analysis and objective estimates of probabilities.
Mistakes, in this view, were attributed to ignorance, emotional interference, or faulty
data.
Research over the past 50 years has undermined this view. Human beings in general
appear to be rather poor at making judgments involving objective critical standards.
This is especially so in reasoning about novel situations and in estimating
frequencies and probabilities. We form confident opinions based on insufficient,
often biased, data, and hold onto these opinions even in the face of strong contrary
evidence. We imagine non-existent connections between events and overlook
important details. Such effects are called cognitive illusions.
Three general heuristics, or rules of thumb, have been identified as the basis on
which decisions and judgments are usually made.
Representativeness: The quick selection of a few obvious, easily recognizable
features of a thing, and the use of these features as characteristic of the thing itself.
We can never keep all of the features of a person, animal or object in mind, and so,
when thinking of them, we must choose those few features that are relevant to our
purpose. There are a number of ways that the automatic use of the
representativeness heuristic leads to error.
There is a natural tendency to choose vivid, striking features even if they are highly
unrepresentative.[158] For example, experiments have shown that the percentage of
women in a group will be estimated higher for groups in which the women are
famous than for identically composed groups containing no famous women. More
generally, in considering statistical samples there is a tendency to assume that the
most vivid and obvious features of a small number of group members are
representative of the group as a whole.[159]
In explaining an event or situation, we tend to look for reasons that are
representative of the outcome. When looking for causes, we tend to consider those
whose features match features in the effect. The English philosopher John Stuart
Mill described the “most deeply-rooted, perhaps” of all fallacies of causal reasoning
as the prejudice that “the conditions of a phenomenon must, or at least probably
will, resemble the phenomenon itself.”[160] For example, great events must have great
causes, complex events must have complex causes, emotional events must have
emotional causes, and so on.
Another aspect of errors involving the representativeness heuristic is an effect called
the “fundamental attribution error.” When observing a person’s behaviour, we tend
to assume that it is characteristic of their personality, rather than a product of the
context in which we observe them, even though we explain our own behaviour in
terms of the situation.
Availability: The mind is lazy; it does not like to engage in complicated analysis or
long memory searches. Thus it has come up with shortcuts to avoid as much work as
possible. We have already discussed the representative heuristic, which allows the
mind to save time and effort in recognizing and categorizing things, people and
situations. The second heuristic is availability, referring to the fact that we tend to
make comparison judgments on the basis of what is most easily available to the
mind, either in sensory input or in memory. This heuristic finds particular use in
judgments of frequency, probability and causality. That is, objects and events are
judged as frequent, as probable or as causes based on their ease of recall.
If every experience was registered in memory with equal weight in an unbiased way
and if ease of recall was proportional to the accumulated weight in memory, then, in
the absence of other relevant information, the availability heuristic would be a valid
decision strategy.[161] But many factors act to bias the way that events are perceived
and remembered.
The availability heuristic is especially vulnerable to vivid and highly emotional
experiences that tend to stick in the mind with far greater intensity than more
neutral experiences.[162] Vivid information and experience stays in the mind longer,
evokes more widespread associations, and is more easily available in recall.
In fact, there does not have to be all that great a difference between two events for
one to become favoured in memory. This fact can be illustrated by what can be called
the “traffic light phenomenon.” When coming to a red light, many people find
themselves thinking, “Why do I always hit the red lights?”
Careful analysis, of course, would show this to be false. Over the long run, red lights
and green lights tend to balance out. But when the light is green, we drive on
through. When it is red, we must make an extra effort to stop, and may also
experience a slight feeling of irritation. These factors are enough to bias memory, so
that it provides a background recollection of encountering more red lights.
A similar bias shows up in thinking about the causes of events. If we base our belief
about the cause of an event on a search through the possible causes available in
memory, we need to be careful that these possibilities have not been pre-selected for
us by hidden biases built into the way these memories were stored.
It is important to note that both negative and positive associations can distort
memory: pain, fear, anger, depression, pleasure, desire and happiness will all
influence the way memories are recalled, or if they are recalled at all. We may not
want to recall painful memories, and may be too eager to recall pleasant ones.
Anchoring: This heuristic involves the quick selection of an interpretation, belief,
theory or scenario to interpret an event, which is then maintained even in the face of
strong contrary evidence. Without some interpretive context, we do not have that
comfortable feeling of “understanding” and this lack produces insecurity. Thus,
rather than changing the “anchor,” people make minor modifications or offer ad
hoc explanations to deal with cases that do not fit. Such behaviour can lead to
reinforcement of prejudices. For example, there is an illustrative joke about a man
who, on seeing a car being driven poorly would say, “Must be a woman driver.”
When he discovered the driver was male, however, rather than changing his belief
about women drivers, he would say, “He drives just like a woman!” And, seeing a
woman driving well, he would say, with a hint of disapproval, “Drives like a man!”
Aristotle noted one aspect of anchoring long ago when he observed that the ability to
doubt is rare. Modern research in psychology confirms this. People tend to initially
believe what they hear or read, and only question it later, after thinking about
it.[163] Advertisers are well aware of this fact, and try to deliver their message with
accompanying cognitive noise designed to distract us from critical analysis.
In the language of Discussion 5.2, the anchoring effect introduces a bias in favour of
treating new experiences as instances of our beliefs rather than as evidence that
might test those beliefs.[164] Without our preconceptions, theories and beliefs, we
would feel lost. There would be no explanatory framework we could use to interpret
the world, which is why we hold on so tightly. Furthermore, the penalty for holding a
false belief is often not that serious, because in most cases, as mentioned in
Discussion 5.1, a belief does not need to be true to fact to be useful—it must only be
true to function.
Some of our anchors are built in. Very young children, for example, will interpret
covariance between events as implying causality. But even trained adults have
difficulty with evidence that indicates a lack of covariance, and will interpret it
causally, or ignore it, rather than taking it as showing that the factors involved are
independent. We seem to be genetically programmed to identify causal connections
on the basis of very weak evidence, but find it extremely difficult to infer a lack of
causal connections. This inherent tendency at least partially explains the fact that
many people still believe in astrology. A few coincidental agreements between events
in a person’s life and his or her horoscope, and the horoscope quickly takes on causal
power in terms of the anchoring belief, while the many cases of non-agreement are
simply not seen as contrary evidence.
The discovery of the decision heuristics and their associated cognitive illusions
shows that, in general, humans are not logical in the traditional sense of the word. In
fact, we find it difficult to learn to think logically. This situation has led some people,
who identify rationality with formal logic, to claim that people are irrational.
In this course, however, reason has been defined as a process of fitting together, and
this is just what the decision heuristics are used for. In fact, they are essential for
science. Things must be represented, placed in a context of available information for
comparisons, and viewed within some conceptual framework. The evolution of
science can be seen as a process of developing tools for the accurate application of
these heuristics, rather than their replacement by something new.
Thomas H. Huxley was correct in his claim that scientific thought uses our ordinary
everyday modes of thinking, made more precise. But, as we will see later in this
course, the tools required to achieve this precision are often difficult to learn. Often
they may even seem counter-intuitive because they go against the apparently
“obvious” cognitive illusions.
As an analogy, consider a champion gymnast. Her performance uses only the
everyday capacities we all use—the ability to move, our kinesthetic awareness,
muscular strength, and sense of balance and coordination. But the training required
to perform at a championship level is long and arduous. It would be foolish to think
that the tools of clear and precise thinking, in science or in any other intellectual
activity, could be acquired without effort. In the words of Norman MacLean, “All
good things come by grace,. . . and grace comes by art, and art does not come
easy.”[165]
In science, formal logic tells us how to think accurately in terms of representative
categories and concepts. Studies in the methods of classification emphasize the
importance of finding significant categorization criteria, and the significance can be
tested experimentally and observationally. The method of comparison described in
Discussion 10.2 proves exceptionally valuable in this regard. Likewise, the insistence
on repeatability in observations and experiments, and the statistical methods that
have been developed to give unbiased estimates, act to provide a background of
unbiased empirical data. Research into the nature of theories and paradigms, and
recognition of the importance of conceptual frameworks, have given us a deeper
insight into the value of such anchors, while alerting us to their limitations. In
particular, this work allows us to acknowledge that all human beings use similar
forms of reasoning, while the tools employed in using these forms have undergone a
long evolutionary development. This realization, in turn, allows the distinction
between open and closed cultures, and hence the distinction between cultures that
have science, in some form or another, and those that do not.
Discussion 8.3 Argument Analysis
TOP
Lay on, Macduff, and damn’d be him that first cries “Hold, enough!”
—Macbeth, Act V, Scene VII
The reading from What Science Is introduced some of the important aspects of
critical thinking. This discussion provides a brief introduction to argument analysis.
Argument analysis is not something distinct from critical thinking, but it makes use
of a technique of diagramming arguments that can help us to apply the critical tools.
As an introduction, we can consider the argument equating Azande witchcraft to a
form of science, and the counter-argument provided in Discussion 5.2.
The initial argument can be diagrammed as shown in Figure 8.3.1, below.
Figure 8.3.1: Argument that Azande witchcraft is equivalent to science
Diagramming it in this way lays bare the structure of the argument. We have, as it
were, carved it at the joints. The goal of the argument is to conclude that science and
Azande witchcraft are equivalent systems.[166]
To reach this goal, the argument begins with a definition of science. It is then
claimed that Azande witchcraft satisfies this definition: it explains the world, is
internally consistent, has great generality, and uses an experimental method. On this
basis, it is claimed that witchcraft is a form of science.
A further argument is required, however, to counter the condescending reply that
although the Azande do have a form of science, it is inferior to Western science. This
possible response is excluded by the claim that both systems are culturally closed,
and neither is capable of accepting arguments considered valid in the other.[167] This
statement supports the claim that there is no way to make objective judgments about
the relative merits of the two systems. Thus, it is concluded that all we can do is
consider them as equivalent.
By looking at the diagram of Figure 8.3.1, we can see the points where the argument
can be criticized. The definition of science can be criticized because it leaves out the
important idea of the inductive attitude, and the arguments used to show that
Azande witchcraft satisfies this definition can be challenged, as in Discussion 5.2. In
addition, the claim that Western science and Azande witchcraft are both closed
systems can be disputed by pointing to the in principle open nature of science. In
addition, the connecting links between the different parts of the argument can be
challenged.
The full argument and the counter-argument given in Discussion 5.2 can be
diagrammed as shown in Figure 8.3.2, below. This diagram makes clear the major
points of both argument and counter-argument. It indicates that the major issues
revolve around the importance of the inductive attitude for the definition of science,
and the question of what actually constitutes the experimental method. Thus, it
points to further directions in which both argument and counter-argument might be
developed.
Figure 8.3.2: Argument that Azande witchcraft is not equivalent to science
There are no firmly established rules for diagramming arguments, but a few general
suggestions are listed below.
1. Identify the conclusion and the basic assumptions of the argument. In the
example shown in Figure 8.3.1, the conclusion is that Azande witchcraft and
modern science are equivalent explanatory systems; the major assumptions are
the definition of science and the claim that both systems are culturally closed.
2. Write brief summary statements of the major assumptions and conclusion and
place them inside box (or other) frames. [Where you will actually locate them in
the eventual diagram will depend on various factors relating to the need to have
as simple a diagram as possible.]
3. Identify supporting assumptions and items of evidence that are offered in the
argument. Write short summaries for each, and place them inside box (or other)
frames. You may decide to identify different elements of an argument (e.g.,
major assumptions, supporting assumptions, evidence, etc.) with different types
of frames.
4. Identify any hidden assumptions that may be involved in the argument, write
short summaries, and place them in box (or other) frames.
5. If necessary, identify possible objections and their counters, write short
summaries, and place them in box (or other) frames.
6. Arrange the various framed summaries on a page with arrows connecting
related elements of the argument. Label these arrows according to the role
played by the related elements. Some possible labels
are supports, implies, proves (useful mainly in mathematical arguments), is
evidence for and suggests. For arguments in which some elements act to
counter possible objections, connecting lines can carry labels such
as contradicts, opposes, counters, refutes, disproves and is evidence against.
As another example of diagramming an argument, consider the following debate:
Creationism should be taught in the public schools as a legitimate scientific
alternative to evolution by natural selection.
Pro: Creationism is a scientific theory. It is based on a search for empirical evidence,
and offers a theoretical explanation for evidence that exists. The only reason it is not
included in the biology course in public schools is that it is opposed by established
biologists who are following a secular agenda. Furthermore, there are many
problems with the theory of evolution by natural selection and even biologists who
accept evolution argue over the theory. Not only that, but evolution cannot make any
experimentally testable predictions, such as when a new species will appear or what
form new species might take. Since evolution is so weak a theory, and a legitimate
scientific alternative exists in creationism, this alternative ought to be given equal
time in the public school curriculum.
Con: Almost all biologists accept evolution by natural selection as the explanation for
the diversity of species in the world. There is a great deal of observational evidence
in support of evolution, both in the fossil record, and in such things as the
appearance of antibiotic-resistant strains of bacteria. Arguments between biologists
are not over the basic idea, but only over matters of detail, as is the case with every
accepted scientific theory. Furthermore, creationism is not science. It is based on an
untestable hypothesis, and ignores or misinterprets the empirical evidence. It is
nothing more than an attempt to sneak a religious belief into science classes.
For the pro side in this debate, the basic assumptions are as follows:
o
o
o
an activity is scientific if it searches for evidence in support of a theory, criticizes
other competing theories, and attempts to offer an explanation of some aspect
of the natural world.
there is a conspiracy of secular humanists to impose the theory of evolution as
the only scientific explanation of human origins.
scientific theories ought to be able to make exact experimental predictions.
On this basis, creationist supporters claim that creationism is as scientific as
evolution by natural selection, and attack evolution as being an unscientific theory
promoted on the basis of a social agenda. As evidence, they point to disputes
between biologists over aspects of evolutionary theory, and to such things as gaps in
the fossil record. Finally, they appeal to the idea that all scientific theories of a
phenomenon deserve an equal hearing and equal time in textbooks and in the
classroom. This idea can be seen as a hidden assumption.
This argument can thus be diagrammed as shown in Figure 8.3.3, below. We leave
the construction of the corresponding diagram for the con side of the debate for a
study question.
We note that in most ongoing scientific disputes, it is not possible to construct
complete diagrams of the arguments and counter-arguments because all of the
evidence may not be in. The failure to observe stellar parallax, for example, was an
argument against the Copernican theory until instruments sensitive enough to
observe this effect were developed in the 19th century.
It is also important to emphasize that this sort of diagrammatic argument analysis is
an aid in critical reasoning, not a replacement for it. The diagrams themselves must
be considered critically, as tools providing insight into the structure of arguments. Of
themselves, they cannot provide support for or against any argument or conclusion;
their function is illustrative rather than normative.
Figure 8.3.3: Argument that creationism should be given equal classroom time
with evolution
Unit 8 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. List and describe the seven gestalt criteria for good form.
2. Choose one of the two examples of the development of a scientific theory given
in Chapter 2 of What Science Is, and discuss ways in which it exemplifies the
idea of a good fit.
3. What are the three heuristics that are used in making decisions? Give examples
of each.
4. Give an example from your own experience of how each of these heuristics has
led to a cognitive illusion.
5. Which fallacy of causal reasoning did John Stuart Mill describe as “the most
deeply-rooted, perhaps”?
6. What is the “fundamental attribution fallacy”?
7. It seems that the perception of events in terms of cause and effect is a built in
aspect of the way the human mind interprets the world, to the extent that we
automatically assume causal relations even in completely random sets of
experiences. Write a short essay (300-500 words) discussing why such an inbuilt bias might be of value for a hunter-gatherer.
8. Write a short essay (300-500 words) defending the contention that the three
decision heuristics are necessary for any use of language to think about or
communicate experience.
9. Identify the two foundations for a well-constructed scientific argument
discussed in Chapter 7 of What Science Is.
10. Define “deductive reason” and “inductive reason,” and describe the limitations
of each.
11. Identify the three general categories of fallacies, and describe one fallacy from
each category. Give examples of each category of fallacy from the popular press,
television or other public sources.
12. In Discussion 8.3, we analysed and diagrammed an argument in favour of giving
creationism equal time with evolution in the public schools. Carry out a similar
analysis of the argument against this idea, and construct a diagram based on
your analysis. Then, combine the pro and con diagrams into a single diagram,
and from this diagram list two additional directions in which each side in the
debate might continue to develop their arguments.
13. In Unit 3, you read about the way that John Snow came to his conclusion about
how cholera was spread. Review this story, and then construct a diagram of
Snow’s argument. Clearly identify his basic assumptions, evidence, responses to
alternate theories and final conclusions.
FOOTNOTES
[149]
Kant, Immanuel. (1781). Preface to the First Edition. The Critique of Pure Reason, trans.
J. M. D. Meiklejohn. Retrieved July 3, 2002, from
http://eserver.org/philosophy/kant/critique-of-pure-reason.txt/
[150]
Wittgenstein, Ludwig. Philosophical Remarks, edited from his posthumous writings by
R. Rheis, trans. Raymond Hargreaves and Roger White, p. 67. Oxford: Blackwell, 1975.
[151]
Pickett, Joseph, ed. American Heritage Dictionary of the English Language, 4th ed.
Boston: Houghton Mifflin, 2000.
[152]
Ibid.
[153]
Note that we also need to know that an analogy is being made between the movement of
the hands on the face of the watch and the apparent movement of the sun about the Earth.
[154]
To distinguish a thing may be as simple as isolating it as an object of attention and giving
it a name. Making distinctions is such a basic operation of thought that we generally do it
unconsciously, according to patterns we have learned from our parents and society, without
realizing how the kinds of distinctions we make control our thinking. For example, at the
most basic level, opponents of abortion do not distinguish between an embryo and a human
being. Those who favour a woman’s right to choose whether or not she will bear a child
claim that this distinction is essential.
[155]
Research in neural imaging has shown, for example, that when a person is exposed to an
idea they disagree with, neural activity also increases in regions of the brain associated with
experiencing unpleasant smells.
[156]
Carroll, Lewis. “The Hunting of the Snark: An Agony in Eight Fits.” In The Humorous
Verse of Lewis Carroll, pp. 271-307. New York: Dover, 1960.
[157]
There are other sensory illusions as well, involving hearing, touch, taste and smell.
[158]
This is why first impressions are so important.
[159]
If, on our first encounter with a member of a social group, the person is rude and
obnoxious, the impulse is to assume that all members of that group are rude and obnoxious,
even though it is likely that we simply encountered a single rude and obnoxious individual.
[160]
Mill, J. S. “Book V: On Fallacies,” p. 765. In The Collected Works of John Stuart Mill,
Volume VIII: A System of Logic, pp. 735-830. Toronto: University of Toronto Press, 1974.
[161]
It would come under the principle of sufficient reason given in Discussion 9.1.
[162]
This is why prosecutors in murder trials use graphic pictures of the victim as a means of
influencing the jury, and why scenes of gratuitous sex and violence are included in some
movies.
[163]
Unless, of course, it contradicts an already held belief, in which case disbelief is
automatic.
[164]
In psychological terms this is a bias in favor of assimilation (of events to an existing
belief) rather than accommodation (of beliefs to new evidence).
[165]
MacLean, Norman. A River Runs Through It and Other Stories, p. 5. Chicago: University
of Chicago Press, 1976.
[166]
We can also observe that the reason for wanting to reach this conclusion is to reinforce
the moral point of tolerance of different cultures. But the use of science to support moral
ideals is as dicey a proposition as its use to support religious ideals.
[167]
In technical language, the claim is that the two systems are “incommensurable.”
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Unit 9 What Are the Basic Tools of Formal Reason?
Discussion 9.1
Discussion 9.2
Unit 9 Study Questions
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STUDY GU IDE
Unit 9 What Are the Basic Tools
of Formal Reason?
We really don’t know how to formulate all the rules for correct language use. The formal
logic that humans have developed deals only with the correct use of certain very special
kinds of statements.
—Rudy Rucker[168]
In Unit 8, we discussed the nature of reason and the basic heuristics used in all
thought. We noted that development of science could be seen in large part as
involving the development of methods to make use of these heuristics without falling
victim to the associated cognitive illusions.
We also looked at three forms of reason: empathy, analogy and analysis. While
empathy and analogy are important for our personal understanding of science and
the communication of scientific ideas, only analytic reason can be used in support of
a scientific truth claim. In other words, we may favour a theory or hypothesis
because it “feels right,” or because it is similar to some other theory or hypothesis
that we accept as true, but its acceptance as scientifically valid depends on analytic
and empirical demonstration. This is most apparent in mathematics, where a
conjecture may appear to be intuitively obvious, and may be analogous to an
established mathematical result, but does not become a theorem until it is proved.
In this unit, we consider some of the principal logical tools used in science. We do
not consider empirical tools of analysis, since these relate to experimental and
laboratory techniques. But, it is worth mentioning that laboratory methods of
analysis and experimental design are often based on the need to formulate the
empirical results of observations or experiments in such a way that they can be
treated theoretically using the tools of rational analysis.
Discussion 9.1 presents a brief overview of three general principles of reason and
four polarities that relate to the use of top-down vs. bottom-up reasoning. This
overview is followed by an introduction to formal logic in Discussion 9.2. In the
Aristotelian view, logic applied to the world, to the way things are. The modern view
is that logic applies to language usage, to the way that we can legitimately use words
and evaluate true or false statements. It applies to the world only to the extent that
our linguistic descriptions fit the world. This limitation shows up in the fact that
logical validity (which is based on coherence) and empirical truth may disagree. For
example,
Pigs are mammals
All mammals can fly
Therefore pigs can fly
is a logically valid argument, but is empirically false. It is a serious mistake to
confuse empirical truth and logical validity. When a logically valid argument yields
an empirically false conclusion, it means that at least one of the premises of the
argument must be empirically false. In the example above, for instance, it is false
that all mammals can fly. This is what is behind the hypothetico-deductive method.
If I make a hypothesis, and from that hypothesis deduce a logically valid conclusion
that can be empirically tested, then, if the conclusion turns out to be empirically
false, it must be the case that either the hypothesis, or some other assumption
involved in the deduction, is false.
Objectives
When you have completed Unit 9, you should be able to
1. state the principles of non-contradiction, sufficient reason and the identity of
indiscernibles, and describe some of the ways they are applied in scientific
thinking.
2. define the polarities top-down/bottom-up, deduction/induction,
analysis/synthesis and functional/structural, and describe how they
interconnect in scientific reasoning.
3. describe the difference between deductive, inductive and abductive inference.
4. describe the difference between logical validity and empirical truth.
5. state the three laws of Aristotelian logic, and discuss their importance for the
establishment of intelligible meanings for words.
6. define the contrary, subcontrary, contradictory and sublternation forms of a
proposition.
7. state the form of the four figures of syllogisms; define the major and minor
premises of a syllogism; identify the subject, predicate and middle terms of a
syllogism; and determine the quality and quantity of statements in syllogisms.
8. assess the validity of simple logical arguments given in syllogistic form.
9. construct Venn diagrams for syllogisms, and use them as a means of
determining validity.
10. write out the truth tables for the logical connectives “and,” “or” and “implies,”
and use them to construct truth tables for complex logical statements.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 9.1, “Principles and Polarities of Reason.”
2. Answer Unit 9 Study Questions 1-3.
3. Read Discussion 9.2, “Formal Logic.”
4. Answer Unit 9 Study Questions 4-11.
Discussion 9.1 Principles and
Polarities of Reason
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[W]hen two opposite points of view are expressed with equal intensity, the truth does not
necessarily lie exactly halfway between them. It is possible for one side to be simply wrong.
—Richard Dawkins[169]
In Unit 8, we discovered that reason comes in three varieties. We also discovered
that the mind employs three general heuristics in making judgments about the
world, and that it is very easy for users of these heuristics to be led astray. In many
ways, science is based on methods to avoid such straying; that is, to use the cognitive
heuristics, which are essential for any thought, in a way that allows the user to think
accurately about the nature of the world. In this discussion, we examine some of the
basic principles of reason, and the way that reason can be used in constructing
rational models of reality.
Principles of Reason
All the world over and at all times there have been practical men, absorbed in irreducible
and stubborn facts; all the world over and at all times there have been men of philosophic
temperament, who have been absorbed in the weaving of general principles.
—Alfred North Whitehead[170]
Three principles have been claimed to be fundamental for accurate reasoning. The
actual significance of these principles, and their role in providing a foundation for
reason as a means of understanding the world, are matters of philosophical debate.
From the perspective of this course, however, we are more interested in the way
these principles are related to the kinds of reasoning found in actual scientific
practice.
In Discussion 6.1, we noted that, while deductive inferences are certain, the
knowledge gained through reason alone is not, because it is relative to the
acceptance of some set of initial assumptions. Nevertheless, as we found in
Discussion 8.1, reason is the way that we fit together a rational picture of the world
in which we live, the rest of humanity, and ourselves. Thus, it is necessary to give
some thought to the question of whether or not our internal constructs bear any
relation to the external world, and what assumptions are necessary to give us the
best chance of a fit.
Because of the contingency of knowledge obtained by reason alone, we can never
guarantee a fit on strictly rational grounds. What we can do, however, is to base our
reasoning on principles that are the analogues of things that appear to be universally
true in experience; that is, principles of reasoning that appear to satisfy the
correspondence theory of truth, recognizing that this can never provide certain
grounds since empirical experience is always limited.
Philosophers have expended tremendous effort seeking such principles. Three that
have received general support are the “principle of non-contradiction,”[171] the
“principle of sufficient reason” and the “principle of the identity of indiscernibles.”
These three principles were strongly emphasized by Gottfried Wilhelm von Leibniz
(1646-1716), who considered the first two, in particular, to be the ultimate
foundations of both reason and science.[172] Other philosophers have not taken such
an extreme view, although the central importance of these principles is generally
acknowledged. Leibniz’s primary goal was to discover laws of reason that would
allow all humans to communicate without subjective biases that would distort their
understanding. He hoped that it would be possible to resolve all disputes by simple
logical and mathematical calculation. He envisioned the creation of a universal
language, the characteristica universalis, and a science of reason, the calculus
ratiocinator, that would allow disagreeing parties to sit together and say, “Let us
calculate.”[173] It is worth noting that Leibniz occasionally acted as a diplomat.
Principle of Non-contradiction
The principle of non-contradiction asserts that every statement involving a
contradiction is to be judged false, and every statement that is the negation of a
falsehood is to be judged true. For example, the statement “All triangles have four
sides” is false, because it asserts the contradiction 3=43=4. Likewise, the negation of
this statement, “No triangle has four sides” is necessarily true. In Discussion 9.2, we
discuss the fact that in syllogistic reasoning the grounds of all valid inference is that
the particular is contained in the universal. What is true of all men (e.g., mortality) is
necessarily true of each individual man (e.g., Socrates). This is simply a matter of the
definition of the word “all” and its use in statements. Thus, knowledge of the
universal category to which a particular subject belongs allows inference that that
subject possesses all the essential properties of that category. Triangles belong to the
category of three-sided objects; hence, we can infer that every triangle has three
sides. Therefore, no four-sided object can be a triangle, and any attempt to claim
that triangles have four sides must be false.[174]
In the practice of science, the principle of non-contradiction shows up in two ways.
In analytic cases, the appearance of a contradiction in an argument indicates an
error in the argument. Somewhere, a definition or rule of deduction has been
violated. In empirical work, the occurrence of a contradiction indicates that an
empirically derived assumption needs to be checked. It may be wrong, or we may
have gone beyond its limit of applicability. The assumption that all swans are white,
for example, is contradicted by the observation of a non-white swan, but this
contradiction does not mean that we must abandon the idea of white swans. Instead,
we must change our assumption and seek to discover the conditions under which a
non-white swan will appear.
The equivalent of the principle of non-contradiction in experience is that, although
we may subjectively believe that a contradiction has occurred, in experience itself
there is no contradiction. Nobody will ever experience a square triangle, a black
white colour, or freezing heat. This idea can be illustrated with the example of the
Necker Cube, an optical illusion shown in Figure 9.1.1 below.
The Necker cube, shown in part (a) of the figure, is a well-known example of an
optical illusion. What we perceive is a cube viewed either from slightly above (b), or
from slightly below (c).
Figure 9.1.1: Necker cube
Now, before continuing to the next paragraph, practise viewing this figure from each
perspective until you are able to switch from one to the other with ease. Once you
can do this (it should not take more than a minute), try to see both views
simultaneously and make note of the result.
The fact that you cannot see both views simultaneously illustrates the impossibility
of real contradiction in experience. Rather than a contradiction (seeing both at once)
what appears is either a blurred figure or a two-dimensional set of lines on the
page—what is actually there. This is an important point: by looking from only one
perspective (without taking other possible perspectives into account) we may see
apparent contradictions rather than “what is actually there.”
Thus, in eliminating contradiction from legitimate reasoning, we are mirroring the
fact that we do not find it in experience. This fact suggests that all apparent
contradictions are based in personal or social subjectivity. They are projected onto
sensations rather than found in them. In the Necker cube example, the apparent
contradiction arises when we project a third dimension (which our brain
automatically does in constructing our three dimensional visual world) into a twodimensional figure that does not, of itself, contain sufficient information to eliminate
all ambiguity of representation.
In attempts to reason in ways that fit reality, the principle of non-contradiction tells
us that any statement that contains a contradiction must be dismissed. Noncontradictory statements may or may not fit, but those that are contradictory
definitely will not. When dealing with quantum mechanics, we might think that real
contradictions appear. We might say, for example, that light is both a wave and a
particle. But that would be false. The accurate statement is that we need both the
concept of wave and the concept of particle to think about light, and in any given
situation light will act as either a wave or as a particle, but it is situation-dependent—
in circumstances in which light manifests as a particle, it will never manifest as a
wave, and vice versa. This is captured in the comment: “Ask a wave question and get
a wave answer; ask a particle question and get a particle answer.” The conclusion is
that it is incorrect to say light is both wave and particle. Rather, it is neither, but can
appear as one or the other when experimentally observed. Waves and particles are
not light itself, rather they are manifestations of light when it interacts with a
measuring apparatus.
Principle of Sufficient Reason
A general statement of this principle, given by Leibniz, is that “there can be no fact
real or existing, no statement true, unless there be a sufficient reason, why it should
be so and not otherwise.”[175] We can recognize in this principle the fundamental
assumption of science—that the world is ordered, and that this order is
comprehensible to human reason. In Pythagorean terms, this principle states that
everything has a logos, or rational principle of being. There is always a reason “why
it is so and not otherwise.”
Various interpretations of the principle of sufficient reason exist. In one
interpretation, the principle is a consequence of the nature of rational truth in
general, namely, that in every true affirmative proposition, the predicate is contained
in the subject. In Aristotelian terms, the particular is contained in the universal:
what is true of all human beings in general must of necessity be true of each
individual human being. This might be called the logical or analytical view of
sufficient reason, the reason a thing has certain characteristics is that it is contained
in a category defined by those characteristics.
In another interpretation, which can be called the empirical view, the principle of
sufficient reason applies to our experience with the statement that “nothing ever
comes to pass without a cause, or at least a reason.” A standard example of this is a
fair balance with equal weights on each scale. The balance does not move because
there is no reason for it to prefer one direction of motion to the other. Sometimes
this version of sufficient reason has been interpreted as a principle of causality,
stating that every change has a cause, but this is not necessarily the case. We need
not look for a single cause, only for a sufficient reason that the change occurred. For
example, in carrying out an analysis of the collapse of a complex ecosystem, it may
not be useful to think in terms of single causes. Is “the cause” of the ozone layer’s
depletion the chemical fact that certain chlorine bearing molecules interact
destructively with ozone?—or is it that these same compounds were used in
refrigeration and air conditioning devices?—or is it that people were willing to buy
such devices, so that their manufacture was profitable?—or . . . ? We do know,
however, that there is a reason—a working out of natural law—rather than, for
example, the capricious whim of some supernatural being.
This version of sufficient reason also shows up in the statistical principle of equal a
priori probabilities: given a set of possibilities with no prior information to
distinguish between them, the best guess is that each is equally probable. (This is
connected to the principles of simplicity and conciseness as well, which you will
encounter in Discussion 13.1.)
A third interpretation of sufficient reason, the efficient interpretation, suggested by
Leibniz for theological reasons, is a principle of optimality: everything in nature is
determined according to a principle of maximization and minimization—the greatest
result for the smallest effort. The French scientist Pierre-Louis Moreau de
Maupertuis (1698-1759) espoused this view. Stripped of its theological associations,
and no longer directly related to the idea of sufficient reason, it is found throughout
modern physics as the “principle of least action,” i.e., that of all possible evolutions
for a physical system, that taken will be the one for which the “action” is minimal.
One of the generally accepted ideas in physics is that a sufficient reason for accepting
a theory as at least a possible explanation of some set of facts is that the theory
accounts for them on the basis of an optimality principle.
Sufficient reason and symmetry are closely related when the occurrence or nonoccurrence of an event is considered. For example, the balance does not move
because there is symmetry between the two scales. We discuss symmetry in Unit 10,
and consider symmetry and optimality principles in Unit 13.
A fundamental use of the principle of sufficient reason is found in the idea that the
task of science is to explain deviations from an ideal of natural order, and if there is
no deviation, no explanation is required. For example, Newton introduced the idea
that the natural motion of bodies in empty space is straight-line motion at constant
speed. Deviations from this order are to be explained by the action of a force. Thus, a
force on a body, acting according to Newton’s equation of motion
(F=maF=ma; force == mass ×× acceleration), is the sufficient reason for deviation
from straight-line motion.
There is a deep relation between sufficient reason and non-contradiction. Recall the
Necker cube example. The drawing as given does not provide a sufficient reason for
preferring either of the two possible views one can have of the cube. It is symmetric
between the possible views. [In the (b) and (c) images, this symmetry has been
broken, and a particular view is highlighted.] Therefore, all of these possibilities can
be seen, although not at the same time, yielding the possibility of a contradiction in
opinion. Thus, the desire to eliminate contradiction from reasoning is, in itself,
sufficient reason for seeking sufficient reasons for the events of experience.
Supporters of two conflicting scientific theories, for example, will seek to strengthen
their side of the debate by providing further reasons why their theory should be
chosen as correct. At some point, the scientific community makes a judgement to the
effect that theory AA is supported by sufficient evidence to be accepted. The term
“sufficient,” however, is to be taken in the sense of sufficient reason, not as simply
the quantity of evidence. Einstein’s theory of general relativity, for example, was
accepted on almost no evidence. His predictions of the shift in position of stars near
the sun as a result of the bending of light rays in the solar gravitational field was
confirmed, but only within 10 to 20 per cent, by measurements made during a solar
eclipse in 1919. This body of evidence is not much when compared to the great mass
of evidence that seemed to support the Newtonian theory of gravitation, but in the
context of the times, it was sufficient reason to accept general relativity.
Principle of the Identity of Indiscernibles
A general statement of this principle is that two things that cannot be in any way
distinguished from each other are to be considered identical. One way of seeing that
the principle holds is in terms of the principle of sufficient reason: if there is no
possibility of distinguishing two things, then there is no reason to consider them as
different things. As applied to reasoning, this principle is found, in a weaker form, in
category formation and naming (two things are entitled to the same name if there is
no essential difference between them). Here the term “essential difference” relates to
the Aristotelian concept of the essential qualities of a thing: “those qualities which it
must possess in order to be entitled to its name.”[176]
In Unit 10, we consider the means of constructing classification systems in detail.
Here, we will only mention that in constructing classification schemes, it is
important to be aware of the level of discourse. For example, lions and tigers belong
to different species, but both are feline. Thus, we imagine that there are certain
general characteristics that are essential to both lions and tigers, while at a more
refined level of distinction there are essential differences between lions and tigers. If
we are speaking at the higher level (that of genus), we can consider lions and tigers
to be the same; at the lower level (of species), we cannot.
What this example suggests is that in thinking in terms of the identity of
indiscernibles, it is important to be aware of the criteria of distinction being
employed. These criteria will define, for the purposes of the given discussion, what is
to be considered identical and what is not. This point is extremely important,
because distinctions that are essential in one context may be irrelevant in another. If
a sociologist is conducting a study of the income distribution of university graduates
5 years after graduation as a function of race, then race is the important distinction.
If the study is considering income distribution as a function of major field of study,
race may be irrelevant.
This point about what is relevant and what is not is brought out in a telling way in a
traditional Middle Eastern tale. A man of very modest means and low social standing
needed a favour from the city magistrate who was an arrogant social climber. The
magistrate could not find time or spend the effort to even see this man, whom he
dismissed as being of an inferior class. After several months of attempting to gain an
audience without success, the man wrote him a letter. In this letter, he outlined his
needs, and concluded with the remark that the provincial governor would be willing
to vouch for his character because “he is under an obligation to me.”
Sensing an opportunity to ingratiate himself with the governor, the magistrate, who
made distinctions only on the basis of social status, quickly saw to the satisfaction of
the man’s needs.
Later, attending a state banquet, the magistrate encountered the governor and
described the actions he had taken on the man’s behalf. He said that he was curious,
however, how it was that a person of so high a rank as the governor could be obliged
to such a non-entity. The governor replied, “I have never heard of that man until this
moment, but I am under an obligation to him because he is a human being.”[177]
In terms of thinking about experience, the principle of the identity of indiscernibles
relates to the question of what we consider to be identical in experience. In one
sense, each experience is unique—in the words of the philosopher Heraclitus, “you
cannot step in the same river twice.”[178] Thus, given two experiences that appear to be
indistinguishable, we must ask when they are different experiences of the same
thing, and when they are the experience of two different things. In some cases, it is
easy to tell. If we see two apparently identical white billiard balls rolling on a pool
table, we know that we are experiencing two different things. If we see one white
billiard ball rolling on this table, turn away briefly, and see one white billiard ball
when we turn back, we infer that we are seeing the same ball on two separate
occasions. In the first case, in saying that the billiard balls are identical, we are not
making the claim that they are exactly the same ball. Rather, we are assigning both
to the same general category of “cue ball,” and only claiming identity insofar as to
assert that there is no essential distinction between them with respect to the game of
billiards. In the second case we are asserting that it is the same ball, displaced in
time and space.[179]
This strategy works for a billiard ball, which does not change over the time during
which it is being considered. Suppose, however, that we plant a tree and return 10
years later. Is this the same tree? It has certainly changed; it can easily be
distinguished from the seedling that was planted. What kind of identity criteria
would then allow us to say that it really is the same tree? Or, for that matter, what
identity criteria allow you to say that you are the same person today that you were
yesterday, or last year?
What the principle of the identity of indiscernibles tells us is that if we want to assign
an object to a specific category based on the assumption of essential properties, and
speak of two different objects as belonging to the same category, we must specify the
level of discourse and the criteria of identity that we are using to define the category.
That is, we need to think in terms of a hierarchy of categories, and must always be
careful to be aware from which level of this hierarchy the concepts we use in thought
are drawn, and how they relate to concepts at the same and at other levels.
Polarities for Reasoning
[A]ll true and fruitful natural philosophy hath a double scale or ladder, ascendant and
descendent, ascending from experiments to the invention of causes, and descending from
causes to the invention of new experiments . . . .
—Francis Bacon[180]
One view is that the world consists of things, and that any changes we notice are really
secondary, arising from the way things interact with one another. The alternative is that the
world consists of processes, and that the things we discern are only stills out of what is
essentially a movie.
—C. H. Waddington[181]
We have often emphasized the importance of the complementary use of bottom-up
and top-down reasoning. In this discussion, we extend this idea by viewing the
process of scientific reasoning in terms of the interaction of four bipolar pairs of
concepts: top-down/bottom-up, deduction/induction, analysis/synthesis, and
structural/functional. Superficially, these pairs can be viewed as different aspects of
a single dichotomy, with correspondences as indicated in Table 9.1.1.
Table 9.1.1: Bipolar pairs of concepts in reasoning
Approach to
systems
Inference
method
Reasoning
process
Form of
explanation
Top-down
Deduction
Analysis
Functional
Bottom-up
Induction/
Abduction
Synthesis
Structural
The actual process of scientific reasoning involves a continual interplay among these
complementary pairs. It is not a matter of choosing one or the other, but of having a
functional balance employing both.
Top-down/Bottom-up Reasoning
In a top-down approach, we begin with a complete system and assume that the unity
of the system, and the associated necessities of its functioning, are sufficient reason
for the behaviour and characteristics of its components. The nature of the system
components and their interactions are assumed to be constrained by the conditions
required for overall system unity. For example, in evolutionary psychology a basic
assumption is that psychological functions and social behaviour are genetically
determined for the purpose of enhancing the chances of passing genetic material to
the next generation. From a top-down perspective this means that observed social
behaviours must be interpreted in terms of their contribution to individual
reproduction.
Bottom-up reasoning begins with basic components and builds up a system out of
the components and their interactions. For example, in genetics, the search for
specific genes that determine characteristics, such as behaviour or susceptibility to
certain diseases, is a bottom-up approach.
Inference Method
Induction is the inference of a general, universal conclusion from a finite set of
examples. For example, we conclude that the sun will rise in the east every morning
on the basis of our past experiences of this happening in a finite number of cases.
Similarly, Snow’s conclusion that cholera is transmitted by ingestion of
contaminated water was an inductive inference. Induction is subject to the “problem
of induction,” the fact that there is no certain justification for inductive inference
except experience. There is no formal “logic of induction” (or, in the usual
terminology, “logic of discovery”) within which inferences can be proven true.
Deduction is inference according to a formalized logic, generally, the Aristotelian
syllogisms or a propositional calculus. Valid syllogisms, for example, are valid forms
of deductive inference. In deduction, we begin with the universal and derive from it
the properties of the particular. Assume, for example, that all furry carnivores are
either canine or feline, and that we are confronted with a particular species of furry
carnivore. If we argue as follows,
It is either canine or feline
Observation shows that it is not canine
Therefore, it must be feline
then we are making a valid deductive inference.
There is a third form of inference, first distinguished by the American logician
Charles Sanders Peirce, called abduction, or, inference to the best explanation.
Abduction is often used, both in science and in everyday life. When something
unusual occurs the mind automatically tries to find an explanation (i.e., an answer to
the question: why has the expected order of things been altered?) either by carrying
out a memory search or generating possible explanations. The brain is tuned to look
for the causes of surprising and unusual events and will seek through available
explanations until one that seems satisfactory is found, or in desperation will make
up an explanation. In carrying out this process deductive and inductive reasoning
may be used, but the basic form is abductive: simply trying to determine which of the
possible explanations for an event gives the best fit. When a choice is made between
several possibilities based on a single clue, or a variety of apparently unrelated clues,
this is an example of abductive inference.
The form of inferential argument for deduction, induction, and abduction are:
Deduction (Categorical Inference)
* All of the balls in that bag are black.
* These balls are from that bag.
* Therefore, these balls are black.
Induction (Statistical Inference)
* These balls are from that bag.
* These balls are black.
* Therefore, all of the balls in that bag are black.
Abduction (Inference to the best Explanation)
* All of the balls in that bag are black.
* These balls are black.
* Therefore, these balls are from that bag.
A more detailed example of induction would be:
* In 1000 samples of balls drawn from that bag, the ratio of black balls to white balls
is rr. Therefore the ratio of black to white balls in that bag is rr (with an appropriate
estimate of statistical error).
A similar example for abduction would be:
* There are more black balls than white balls in bag AA; and more white balls than
black balls in bag BB.
* This ball is black.
* Therefore this ball is probably from bag AA.
We can compare these three forms of inference and note interesting connections.
1. The pattern for inductive inference is obtained from the deductive pattern by
making the following exchanges in the deductive form: major
premise →→ conclusion; minor premise →→ major premise;
conclusion →→ minor premise.
2. The pattern for abductive inference is obtained from the deductive pattern by
making the following exchanges in the deductive form: major premise remains
the same; minor premise ↔↔ conclusion.
3. The pattern for inductive inference is obtained from the abductive pattern by
making the following exchanges in the abductive form: minor premise remains
the same; major premise ↔↔ conclusion.
Analysis/Synthesis
Analysis is the division of a whole into its component parts so that they may be
studied both individually and in their interactions. In the chemical analysis of a
substance, for example, the substance is broken down into its component molecules
and atoms to determine the elements of which it is made. Another example is the
medical breakdown of the body into component systems (e.g., gastrointestinal,
cardiovascular, lymphatic, etc.), of these systems into organs (stomach, heart, lungs,
etc.), of organs into tissues, and of tissues into cells. Analysis asks, “What are the
essential components of a system?”
Synthesis is the combination of separate elements so as to form a coherent whole. A
chemist may synthesize a compound from its component elements, combining, for
example, two moles[182] of hydrogen with one mole of oxygen to produce one mole of
water. Likewise, in reasoning, synthesis is the combination of a number of distinct
arguments to reach a single conclusion. Snow’s conclusion on the spread of cholera,
for example, was a result of his synthesis of a number of different strands of
evidence. Synthesis asks, “How do all of these different factors, aspects, or elements
fit as an overall system?” “What is the unifying principle or idea?”
Structure/Function
Structural explanations give the reasons for a system’s possession of a particular
property or set of behaviours in terms of the system’s structure; that is, the nature of
the system components, their interactions, and the ways in which they fit into the
overall system. For example, the explanation of the X-ray diffraction patterns
produced by crystals on the basis of the geometry of the crystal lattice is a
structuralist explanation.
Functional explanations give reasons for system properties in terms of the functions
that the system must fulfill within its environment. Saying that the brilliant plumage
of the male peacock serves as a display to attract a mate, for example, is a functional
explanation.
One point to notice is that when causal explanations are given, the direction of
causality is determined by the direction taken in the approach. In the top-down
approach, causality operates from top to bottom; that is, lower-level behaviour is
explained in terms of higher-level causes. In the bottom-up approach, causality
operates from the bottom to the top; that is higher-level behaviour is explained in
terms of lower-level causes.
Correspondences among Complementary Pairs
In a top-down approach, we seek global laws governing system behaviour, and on
the basis of these laws, we carry out an analysis in which we deduce what properties
may be expected of the system components. This process yields a functional
explanation of lower-level behaviour in terms of higher-level requirements.
In a bottom-up approach, we begin with system components and their interactions,
and attempt to use induction to generate from them a synthetic view of the whole
system. This process yields a structural explanation of higher-level system structure
and behaviour in terms of lower-level components and interactions.
In actual practice, there is always an interaction between these sets of pairs. In the
hypothetico-deductive method, for example, the prescribed rules of procedure are to
begin with data, and by induction or abduction, to generate an explanatory
hypothesis. (In general, it is a good idea to consider several distinct hypotheses, and
choose that which seems most likely for further testing.) Then the reasoner assumes
the truth of the hypothesis, and deduces further consequences that can be tested
observationally or experimentally. That is, we begin with a bottom-up approach,
studying the basic components of a system and attempting to generate a synthesis
that describes and explains the system’s structure and behaviour. Once this
synthesis is formalized as a hypothesis, we are in a position to take a top-down
approach in which we can deduce further system properties, and make predictions
about components and their interactions. These, in turn, are compared to empirical
data. If there is a fit, we feel more confident in our hypothesis. If there is a misfit
(contradiction), we need to change or adjust the hypothesis until we are able to find
a fit.
The situation is actually more complicated—the initial data we take as a starting
point has been collected and is interpreted within a particular conceptual framework
that specifies what sort of data is relevant and what questions are significant (i.e., the
data is “theory-laden”), so there are also top-down aspects at the beginning, often
found in the posing of the question or questions we need to answer.
The point to remember is that there is an ongoing interplay between apparently
opposite approaches, methods of induction, analytical processes and forms of
explanation, and that the process of science depends on this interplay for its
continued vitality and fruitfulness. The job for an individual scientist is to be aware
of the techniques and approaches that are required, and more basically, the
questions that should be asked in each moment of research effort.
A good example is found in the field of evolutionary psychology. A basic assumption
of this field is that the human mind is structured by genetically determined patterns
that represent adaptations to survival in the Paleolithic, hunting and gathering era of
human evolution. This assumption is justified with the argument that almost all of
our history as a species took place in the Paleolithic, and that insufficient time has
passed since that time for genetic adaptation to our new technologically based
environment.
Taking a top-down approach, evolutionary psychologists attempt to deduce what
sorts of cognitive modules would be required for Paleolithic survival—very much in
the way that a computer programmer might attempt to determine the appropriate
subroutines for a programming task—and the possible ways that these modules
might be related. They try to construct flow charts for the mind.
This analytic effort is complemented by a bottom-up synthesis in which specialized
modules are postulated on the basis of empirical studies of cognitive abilities,
interpreted within the context of the basic assumption of evolutionary psychology.
In this work, the top-down and bottom-up approaches form a self-reinforcing cycle.
Whether explanations produced by this cycle are valid, however, depends on the
extent to which the basic assumptions made correspond to reality. In other words, in
attempting to evaluate this research program, we must ask to what extent cognitive
abilities and social behaviours are genetically determined. Some researchers insist
that this determination is almost total, while others are equally insistent on the
dominant role of environmental and social factors. At present, there is no clear
resolution of this debate, although it is likely that the more extreme positions of
either side will prove false.
Discussion 9.2 Syllogisms and
Propositional Calculus
TOP
[I]f you would answer me, said Dionysodorus, you will admit these things yourself,
Ctesippus. Just tell me, have you a dog?
Yes, and a very bad one, said Ctesippus.
Has he got puppies?
Very much so, he said, as bad as he is.
Then the dog is their father?
I have seen him myself, he said, on the job with the bitch.
Very well, isn’t the dog yours?
Certainly, he said.
Then being a father he is yours, so the dog becomes your father and you the puppies’
brother.
—Plato[183]
“Contrariwise,” continued Tweedledee, “If it was so, it might be; and if it were so, it would
be; but as it isn’t, it ain’t. That’s logic.”
—Lewis Carroll[184]
In his famous book Critique of Pure Reason, the philosopher Immanuel Kant
lamented that although logic was the first science to be developed, preceding even
mathematics, it had changed not at all in the more than 2000 years since its
birth.[185] Kant would have been amazed at the changes that have taken place in logic
since his death in 1804.
The roots of logic can be found in the Presocratic philosophers, in particular
Parmenides, who attributed the validity of their arguments to “justice,” meaning by
this term what we would call logical necessity. The basic principles of rational
argument had not at that time been worked out, however, and empty rhetoric often
took the place of valid reasoning. Plato parodies this sort of rhetoric in the quote that
heads this discussion.
In the fifth century BCE, a group arose—the Sophists—consisting of teachers who
made their living by training the sons of wealthy Greek families in how to speak well
in public. Their basic motto was that truth is whatever the most persuasive speaker
says it is, and virtue is a matter of presenting the right appearances. Ever since then,
“sophistry” has meant the use of invalid arguments to present the semblance of
truth, rather than truth itself.
The major Greek philosophers spoke against the Sophists, but it was Aristotle who
finally showed how they could be refuted, by formalizing the rules for valid argument
in what is now called “Aristotelian logic.” Since that time, Aristotelian logic has
changed little. It was studied in detail and elaborated slightly in the Middle Ages by
the scholastic philosophers known as the “Schoolmen,” who referred to the logical
rules as the “Laws of Thought” since they believed that any valid thinking must be
carried out according to them. Indeed, for deductive argument the Aristotelian laws
are still basic.
In the 19th and early 20th centuries, however, logic expanded tremendously. The
Aristotelian laws remain valid, but they now compose only a limited portion of the
subject matter of logic. Nevertheless, they are an important part, since they form a
basis for the remainder of the subject. They are, for example, essential for the
construction of truth tables, which play a major role in the modern forms of
propositional and predicate calculus that in turn make up what is known as firstorder logic. In mathematics, the Aristotelian laws are also essential as the basic rules
for the construction of proofs.
Laws of Thought
Aristotle called logic the organon, or tool of thought. His three laws are the rules
that must be followed if the conclusion of an argument is to follow as a necessary
consequence of the initial premises.
In Aristotle’s philosophy, real knowledge of a thing is obtained by gaining an
intuition of its essence: “that which it must have in order to be entitled to its name.”
Essences were to be intuitively abstracted from an accumulation of particular
experiences. Once the essence of a thing had been grasped, all other properties of
that thing could be derived by logical deduction. Aristotle’s rules of logic were
formulated as the proper way to carry out this deductive process. Thus, logic was
seen as a tool for the validation of intuitions, and for working out their
consequences, not as a replacement for intuition itself.
Aristotle was interested in the rules for distinguishing objects of thought, and for
talking accurately about them. Thus, at the basic level, Aristotelian logic deals with
questions of identity, and in particular, universal and unchanging identity: that
which a thing must have in order to be entitled to its name. What is it, for example,
that makes a rose a rose, and not a carnation; or allows us to talk about roses and
know that we are both talking about the same thing?
The three basic laws of Aristotelian logic are outlined below.
1. Law of Identity: A thing is equal to itself. For Aristotle, this law was a
metaphysical proposition relating to the fundamental nature of identity. Taken
in this sense, it asserts that things in the world have an invariant identity that
can be cognitively distinguished. By contrast, in the modern approach to logic,
this law is taken as a condition for the proper use of language—once a term has
been defined, its meaning cannot legitimately be changed in further discussions.
2. Law of Contradiction: No thing is equal to anything other than itself.
3. Law of the Excluded Middle: No thing can have both a property and its
opposite, or negation.
The law of identity asserts that things have an invariant identity. It is supported by
the law of contradiction, which asserts that no thing can ever be other than it is; and
by the law of the excluded middle, which asserts that for any given thing and any
property, either the thing has that property, or it does not. In the modern view, the
law of contradiction means that no statement can be both true and false, or more
generally, that any statement that predicates of a thing both a property and its
negation, is necessarily false. In this view, the law of the excluded middle requires
that every statement have a specific truth-value: either it is true, or it is false. There
can be no in-between, grey area. It should be obvious that these laws can apply to
only a restricted class of statements. Questions, commands, and so on do not fall
under the Aristotelian laws. Statements that can have a definite truth value are
called propositions.[186]
As indicated, for Aristotle, logic applied to the world, and this view was held for
almost 2000 years. It was not until the 19th century that this belief was replaced by
one in which logic is purely formal, dealing only with the proper use of the logical
connectives and quantifiers. Today, logic is no longer assumed to deal directly with
things in the world, but only with the forms of propositions. In the current view,
logic determines if a properly phrased set of propositions yields a valid conclusion,
without regard for the content of the statements involved. Modern logic deals only
with the form of statements—with syntax, not semantics.
Although not sufficient as a basis for modern logic, the Aristotelian laws are
necessary, since they are required for the construction of truth tables, and the
assignment of truth-values to propositions. The rules for assigning truth-values to
propositions are given below.
1. Every proposition has a constant truth-value (law of identity).
2. No proposition can be both true and false (law of contradiction).
3. Every proposition is either true or false (law of the excluded middle).
Note how these rules select out only particular forms of statements as legitimate
propositions. Questions and injunctions, for example, are not propositions. Even the
statement “It is three o’clock” is not a proposition because its truth value depends on
what time it actually is. In the remainder of this discussion, we will be concerned
with classical or Aristotelian logic.
Immediate Inference
In this section, we consider statements called “categorical propositions”: statements
about classes (categories) which either affirm or deny that a class SS is included in a
class PP, either in whole or in part. Our concern will be to classify the types of
categorical propositions, and determine what sort of inferences can be drawn if we
know that a proposition is true or false. Since only one proposition is involved in
these inferences, they are called “immediate inferences.” If we again recall that the
root of the word “logic” means “to collect” or “to speak about,” we see how closely the
discussion of these sorts of propositions still fits with these meanings. They are how
we speak about membership in collections of things.
By definition, there are only four kinds of categorical propositions, determined by
their quality, which may be either affirmative or negative, and by their quantity,
which may be either universal or particular. Thus the four types of categorical
propositions have the forms, called moods, indicated in Table 9.2.1.
Table 9.2.1: The four moods of categorical propositions
Proposition Designation
Traditional Notation
Universal Affirmative (UA)
A
Universal Negative (UN)
E
Particular Affirmative (PA)
I
Particular Negative (PN)
O
Form
All SS are PP
No SS is PP
(SS is not-PP)
Some SS are PP
Not all SS are PP
(Some SS are not-PP)
The term SS is called the “subject” of the proposition, and the term PP is called its
“predicate.” A proposition is said to distribute a term if it refers to all members of the
class designated by that term, and in this case, the term is said to be “distributed” in
the proposition. For example, in the proposition
All men are mortal
the subject term (men) is distributed, since it refers to all members of the class
“men,” but the predicate term (mortal) is not distributed, since the proposition does
not necessarily refer to all mortals.
The forms for universal propositions are “All SS are PP” and “No SS is PP.” In these
statements, the subject term is distributed, since the propositions refer to the entire
class designated by SS. On the other hand, neither of the forms for particular
propositions, “Some SS are PP” and “Not all SS are PP,” refers to the entire class
designated by SS, so in particular propositions the subject term is not distributed.
Analysis of the four different types of propositions shows that the quantity of a
proposition determines if the subject term is distributed, while the quality of the
proposition determines if the predicate term is distributed. In affirmative
propositions the form is “All SS are PP” or “Some SS are PP” and neither of these
forms refers to the entire class designated by PP. Thus the predicate term is
undistributed in affirmative propositions. For negative propositions, the forms are
“No SS is PP” or “Not all SS are PP.” In both of these cases, the predicate term is
distributed, since the assertions are that no SS, or at least not all SS are included in
the entire class designated by PP. For example, the proposition
No gods are mortal
refers to the entire class of mortals, since it asserts that the class of gods is excluded
from membership in this class, and for such exclusion to hold, it must be an
exclusion from the entire class.
These relationships are shown in Table 9.2.2, below.
Table 9.2.2: Distribution of terms in categorical propositions
Affirmative
Negative
Universal
SS is distributed PP is undistributed
SS is distributed PP is distributed
Particular
SS is undistributed PP is undistributed
SS is undistributed PP is distributed
This table can be used to draw certain immediate inferences if the truth-value of a
proposition (i.e., true or false) is given. The obvious example is that if a
proposition TT is true, then the proposition not-TT must be false (from the laws of
contradiction and excluded middle). More generally, the immediate inferences that
can be drawn can be shown in a diagram called the “square of opposition.” To
construct this diagram we need some terms.
Definitions
1. Two propositions are “contradictory” if they necessarily have opposite truthvalues. Thus, assuming the same subjects and predicates, the following pairs of
propositions are contradictory:
AA: (All SS are PP) and OO: (Not all SS are PP)
EE: (No SS is PP) and II: (Some SS are PP)
2. Two propositions are “contraries” if both cannot be true (but both could be
false), and are “subcontraries” if both cannot be false (but both could be true).
Contraries: AA: (All SS are PP) and EE: (No SS is PP)
Subcontraries: II: (Some SS are PP) and OO: (Not all SS are PP)
3. Clearly, if the universal form of a proposition is true or false, this fact implies the
truth or falsity of its particular form. (All SS are PP) implies that
(Some SS are PP), while (No SS is PP) implies that (Not all SS are PP). This
relationship is called “subalternation.”
These relationships are diagrammed in the square of opposition, shown in Figure
9.2.1, below.
Figure 9.2.1: Square of opposition
There are other forms of immediate inference in addition to those drawn directly
from the square of opposition. Given a categorical proposition, we can also form its
converse, obverse and contrapositive.
Suppose that r(S,P)r(S,P) is a categorical proposition with
subject SS and predicate PP.
Definitions
4. The “converse” of r(S,P)r(S,P) is the categorical
proposition r′(P,S)r′(P,S) where r′r′ is obtained from rr by changing its quality.
For example, the converse of “All men are mortal” is “Some mortals are men.”
5. The “complement” of a class SS consists of everything not in that class; It is
called not-SS, or non-SS. It is usually denoted S¯¯S¯ or S′S′
6. The “obverse” of r(S,P)r(S,P) is the categorical
proposition r′(S,P¯¯¯)r′(S,P¯) where r′r′ is obtained from rr by changing its
quality. Thus, “No man is not mortal” is the obverse of “All men are mortal.”
7. The “contrapositive” of r(S,P)r(S,P) is the categorical
proposition r(P¯¯¯,S¯¯)r(P¯,S¯). Thus, the contrapositive of “All men are mortal”
is “All non-mortals are not men.” Or, in better English, “No immortals are
human.”
In some cases, these forms will be invalid inferences.
Table 9.2.3 shows the converse, obverse and contrapositive forms of the four
different types of categorical propositions, and indicates those that are invalid
inferences.
Table 9.2.3: Forms of converse, obverse and contrapositive
Categorical
Proposition
Converse
Obverse
Contrapositive
All SS are PP
(limitation)
Some PP are SS
No SS is non-PP
All non-PP are non-SS
No SS is PP
No PP is SS
All SS are non-PP
Some non-PP are not nonSS (limitation)
Some SS are PP
Some PP are SS
Some SS are not
non-PP
Invalid
Some SS are not PP
Invalid
Some SS are non-PP
Some non-PP are not nonSS
All of the forms given in each row of Table 9.2.3 are logically equivalent to all other
forms in the same row, and can be used as replacements for each other in logical
proofs. The most often used case is the contrapositive for a universal affirmative
proposition. In many cases it is easier to prove “All SS are PP” by proving instead
that “All non-PP are non-SS,” but the other forms of immediate inference are
sometimes used as well.
Diagramming Categorical Propositions
Diagrams are often useful as aids in thinking. Propositions can be represented as
diagrams by diagramming the classes to which they refer. This is done by drawing a
closed curve, with the convention that the region inside the curve represents the
members of the indicated class. This is just a two-dimensional representation of the
mental process by which a class is defined. The logician G. Spencer-Brown puts it
eloquently in his book Laws of Form:
Distinction is perfect continence.
That is to say, a distinction is drawn by arranging a boundary with separate sides so that a
point on one side cannot reach the other side without crossing the boundary. For example,
in a plane space a circle draws a distinction.
Once a distinction is drawn, the spaces, states, or contents on each side of the boundary,
being distinct, can be indicated.
We take as given the concept of distinction and the concept of indication, and that we
cannot make an indication without drawing a distinction.[187]
Mathematics itself can be defined as the study of how to make formal distinctions
and prove theorems about those distinctions.
In Figure 9.2.2, below, the class labeled SS is diagrammed as a circle. To indicate
that SS has no members, the circle is shaded, as in Figure 9.2.3a, while to indicate
that this class has some members, an xx is placed in the interior, as in Figure 9.2.3b.
To diagram two different classes, two overlapping circles are used, as in Figure 9.2.4.
The region where the circles overlap indicates the region where there is membership
in both classes, so if this region is shaded, the classes are mutually exclusive. These
types of diagrams are called “Venn diagrams” after the logician John Venn (18341924).
Figure 9.2.2: Representation of the class SS
Figure 9.2.3: Representation of exclusion (a) and of at least partial membership
(b)
Figure 9.2.4: Venn diagram of two propositions
Recall that the complement of a class is defined as everything not contained in that
class. In Figure 9.2.2, the class SS is represented by the interior of the circle, and its
complement S¯¯S¯ is represented by everything outside of the circle. With this
convention, each region of Figure 9.2.4 can be labeled with the class membership it
represents, as shown in Figure 9.2.5, below.
Figure 9.2.5: Venn diagram of two propositions labeled by membership
Figure 9.2.6, below, shows the Venn diagrams for each of the four possible moods of
a categorical proposition.
Figure 9.2.6: Venn diagrams for categorical propositions
Mediated Inferences—Syllogisms
Aristotle was interested in propositions that dealt with the properties of things. The
simplest non-trivial cases of such propositions involve two distinct properties,
described by two categorical propositions. Aristotle classified the ways in which two
such propositions could be combined in order to yield a conclusion. Such
combinations are called “syllogisms.” The inference in this case is mediated rather
than immediate, since it results from a combination of propositions. A valid
syllogism is an inference in which two propositions are combined to yield a valid
conclusion that is more than just their sum.
This last requirement means that the two propositions linked in a syllogism must
share a common term. For example, the propositions “All cats are furry” and “On a
clear day the sky is blue” yield nothing other than their sum: “All cats are furry, and
on a clear day the sky is blue.” On the other hand, the propositions “All men are
mortal” and “Socrates is a man” yield the further deductive inference “Socrates is
mortal.”
The term common to both propositions in a syllogism is called the “middle term,”
and it does not appear in the conclusion; it is, in a sense, divided out. The first term
appearing in the conclusion (its subject) is called the “minor term,” and the second
term appearing in the conclusion (its predicate) is called the “major term.” The
proposition that contains the middle term and the major term is called the “major
premise,” and the proposition that contains the middle term and the minor term is
called the “minor premise.” For example, in the syllogism concerning the mortality
of Socrates, man (referring to all men) is the middle term, mortality is the major
term and Socrates is the minor term.
A simple count shows that there are only four possible forms, called “figures,” for the
way in which predicate, middle term and subject can be ordered in a syllogism. They
are shown in Table 9.2.4, below.
Table 9.2.4: The four figures of syllogisms
Major Premise
First Figure
Second Figure
Third Figure
Fourth Figure
M-P
P-M
M-P
P-M
Minor Premise
S-M
S-M
M-S
M-S
Conclusion
S-P
S-P
S-P
S-P
In addition, as we discussed earlier in this section, every proposition can be either
affirmative or negative, and universal or particular.
A count shows 16 possible combinations, called “moods,” for the major and minor
premise in each of the four syllogistic figures. Hence, there are 64 possible syllogistic
forms, although most of them are not valid.[188] Table 9.2.5 lists the possible
combinations of major and minor premise. In this table, UU stands for
Universal, AA for Affirmative, PP for Particular, and NN for Negative. The first two
letters in each case refer to the major premise, and the second two to the minor
premise. For example, (UAUA, PNPN) indicates a form in which the major premise
is universal affirmative and the minor premise is particular negative. This pattern
occurs in the second figure syllogism
All cats are mammals
Some animals are not mammals
∴ Some animals are not cats
Table 9.2.5: Possible moods for premises in syllogisms
(UA, UA)
(UN, UA)
(PA, UA)
(PN, UA)
(UA, UN)
(UN, UN)
(PA, UN)
(PN, UN)
(UA, PA)
(UN, PA)
(PA, PA)
(PN, PN)
(UA, PN)
(UN, PN)
(PA, PN)
(PN, PN)
The condition for validity of a syllogism is that the conclusion follows necessarily
from the premises on the basis of self-evident principles of reason. A valid
conclusion must always be in agreement with, and cannot be more general than, its
premises. The basic principle of syllogistic reasoning is that what is true universally
must also be true in particular. What is true of all cats, for example, must be true of
each individual cat.
There are six general rules for ensuring that a syllogism will be valid. Violation of
any one of these rules results in an invalid syllogism, or “fallacy.”
1. Terms that appear in a syllogism must retain the same meaning in both
premises and conclusion.
This rule is an immediate consequence of the laws of identity and contradiction.
A term refers to a thing that retains its identity. To change the reference of the
term is to say that the thing referred to has changed; it has become what it was
not.8
Violation of this rule results in the “fallacy of equivocation,” which has the three
subtypes identified below.
Fallacy of the Ambiguous Major
Example: Light eliminates darkness
Feathers do not eliminate darkness
∴ Feathers are not light (i.e., feathers are heavy)
Fallacy of the Ambiguous Minor
Example: Limitation is not freedom from restriction
All license is limitation
∴ License is not freedom from restriction
Fallacy of the Ambiguous Middle
Example: All rats are eaten by cats
Rats squeal on their friends
∴ Some of those who squeal on their friends are eaten by cats
The fallacy of equivocation is also called the “fallacy of four terms,” since the
ambiguous use of one of the terms means that the syllogism really contains four
terms rather than three.
2. The middle term must be distributed in at least one of the premises.
If this rule is violated, the middle term cannot form a link between the subject
and predicate in the conclusion, since each may refer to a different part of the
class indicated by the middle term. The result is the “fallacy of the undistributed
middle.”
Example: All cats are mammals
All dogs are mammals
∴ All dogs are cats
3. If the subject or predicate is distributed in the conclusion, it must be distributed
in its premise.
This rule expresses the idea that the conclusion of a syllogism cannot be more
general than its premises. Violation of this rule leads to the “fallacy of overgeneralization,” which has the two subtypes described below.
Fallacy of the Illicit Major
In this fallacy, the predicate is distributed in the conclusion, but not in the
major premise.
Example: All reptiles are animals
No dog is a reptile
∴ No dog is an animal
Fallacy of the Illicit Minor
In this fallacy, the subject is distributed in the conclusion, but not in the minor
premise.
Example: All primes greater than 2 have no proper divisor
All primes greater than 2 are odd numbers
∴ All odd numbers have no proper divisor
4. A valid syllogism cannot have two negative premises.
This rule follows since two negative premises would assert that both subject and
predicate were excluded, wholly or in part, from the class indicated by the
middle term, and so the middle term provides no necessary link between subject
and predicate (compare this to Rule 2, above). Violation of this rule leads to the
“fallacy of exclusive premises.”
Example: No dog is an alligator
No cat is an alligator
∴ Every dog is a cat
5. If either premise in a valid syllogism is negative, the conclusion must be
negative.
This rule follows since a negative premise will mean that either the subject or
the predicate will be excluded, wholly or in part, from the class indicated by the
middle term; while the other premise, which must be positive from Rule 4, will
assert that either the predicate or the subject is included, wholly or in part in
this class. Thus the conclusion can only be a statement to the effect that the
subject is excluded, wholly or in part, from the class indicated by the predicate.
Violation of this rule leads to the “fallacy of trying to draw an affirmative
conclusion from a negative premise.” Fallacies of this type are generally obvious,
since their conclusion will often assert exactly the opposite of a valid conclusion.
Example: All dogs are animals
No animal is a tree
∴ All dogs are trees
6. If the conclusion is negative, one of the premises must be negative.
Rule 5 states that it is not possible to draw an affirmative conclusion from a
negative premise. Rule 6 states that it is not possible to draw a negative
conclusion from strictly affirmative premises. This result follows since two
affirmative premises will assert that subject and predicate are both included,
either wholly or in part, in the class indicated by the middle term, so the
conclusion can only assert the complete or partial inclusion of subject class in
predicate class, and this is an affirmative proposition. Violation of this rule leads
to the “fallacy of exclusive inclusion.” Again, examples of this fallacy are easily
recognizable.
Example: All cats are animals
Some cats are tabby
∴ Some tabby cats are not animals
These six rules are both necessary and sufficient for insuring the validity of classical
syllogisms.
On the basis of these general rules, we can prove a number of results.
Lemma 1: The conclusion of a valid syllogism is affirmative if and only if both
premises are affirmative.
Lemma 2: No valid syllogism has two particular premises.
Lemma 3: If one premise of a valid syllogism is particular, the conclusion must be
particular.
Lemma 4: No valid syllogism can have a particular major premise and a negative
minor premise.
Lemma 5: The number of terms distributed in the conclusion of a valid syllogism
must be at least one less than the number of distributed terms in the premises.
Lemma 6: In a valid syllogism in the first figure, the major premise must be
universal and the minor premise must be affirmative.
Lemma 7: In a valid syllogism in the second figure, the major premise must be
universal, and one of the premises must be negative.
Lemma 8: In a valid syllogism of the third figure, the minor premise must be
affirmative and the conclusion must be particular.
Lemma 9: In a valid syllogism of the fourth figure
o
o
o
if the major premise is affirmative, then the minor premise must be universal.
if either premise is negative, then the major premise must be universal.
if the minor premise is affirmative, then the conclusion is particular.
Aristotle developed syllogistic reasoning as a way of determining how we might
make proper deductive inferences, and syllogisms provide the reasoning formula for
Aristotelian logic.
Venn diagrams can be very useful in illustrating syllogisms. We begin by drawing
three circles, as shown in Figure 9.2.7, below, one circle labeled SS, one PP and
one MM. Each circle represents the class indicated by its label. Hence, the
circle SS indicates all possibilities for the subject, the circle PP all possibilities for the
predicate, and the circle MM all possibilities for the middle term. Everything outside
circle SS is not-SS, everything outside circle PP is not-PP, and everything outside
circle MM is not-MM. So, for example, if the subject is Socrates, the predicate is
mortal, and the middle term is men; then the SS circle represents Socrates,
the PP circle all mortals, and the MM circle all men.
Figure 9.2.7: Venn diagram form for syllogisms in the first figure
Each of the seven regions indicated in Figure 9.2.7 corresponds to a different
combination of the variables SS, not-SS, PP, not-PP, and MM, not-MM.
Furthermore, for every valid syllogism, the constraints of the major and minor
premise restrict us to a specific one of these regions (this restriction is indicated in
the diagram by shading the excluded regions). Dropping the middle term from the
labeling of the region then yields the conclusion. This method gives a topological way
of evaluating syllogisms without having to worry about the conclusiveness
requirement on the middle term—it is taken care of automatically by the way in
which the diagrams have been constructed.
The brief discussion below shows how one would construct the Venn diagram for
two cases—one of a valid syllogism and one of an invalid syllogism (see Figure 9.2.8,
below). The syllogisms to be considered are
Every cat likes to eat mice
Felix is a cat
∴ Felix likes to eat mice
and
Every cat likes to eat mice
Every dog likes to eat mice
∴ Every dog is a cat
In the first syllogism, we draw three overlapping circles, as in Figure 9.2.7, above.
We label these “cats,” “Felix,” and “likes to eat mice.” The major premise says that
every cat likes to eat mice. Therefore we shade in all of the circle labeled “cats” that is
outside the circle labeled “likes to eat mice.” This is because points in that part of the
“cat” circle stand for cats that do not like to eat mice. The result is shown in Figure
9.2.8a, below. The minor premise says that Felix is a cat. Therefore we shade in
every part of the “Felix” circle that is outside of the “cat” circle, because these points
represent things named Felix that are not cats. We now have Figure 9.2.8b. The only
unshaded area left that contains both “Felix” and “likes to eat mice” is the region
marked with the xx in Figure 9.2.8b, and this places Felix directly within the “likes to
eat mice” circle, so the syllogism is valid.
In the second case, we label the three circles “cats,” “dogs” and “likes to eat mice.”
Again, the major premise tells us to shade in every part of the “cat” circle that is
outside of the “likes to eat mice” circle, giving the diagram of Figure 9.2.8a′. Now,
however, the minor premise tells us that “every dog likes to eat mice,” so we shade in
every part of the “dog” circle that is outside of the “likes to eat mice” circle. This gives
us the diagram of Figure 9.2.8c. Now we see that there are parts of the “dog” circle
that are both outside of the “cat” circle (and vice versa), and still within the “likes to
eat mice” circle. Hence, there is no unique region specified that restricts dogs to
being cats.
Figure 9.2.8: Example of Venn diagram construction
Figure 9.2.9, below, shows examples of Venn diagrams for several syllogisms and
false syllogisms. Note that a lighter shading is used for particular propositions, as
compared to universal propositions. This strategy helps us keep track of the quantity
of the proposition.
Figure 9.2.9: Sample Venn diagrams
Propositional Calculus
Between the 17th and 19th centuries, many attempts were made to revise the theory
of logical inference. These attempts culminated in the development of modern logic
as a propositional calculus. The modern view of the Aristotelian syllogisms is that
they are a restricted form of a class inclusion inference; that is, they refer to things,
and infer properties on the basis of the classes to which those things belong. In
contrast, a propositional calculus does not deal with things. Rather, it deals with the
truth or falsity of propositions based strictly on their form.[189]
A proposition is any statement to which can be assigned a value of “true” or “false.”
The transition from class inclusion inference to propositional calculus is associated
with George Boole (1815-1864), who developed an algebraic formalism for Aristotle’s
logic.
The basic element for a propositional calculus is the truth table, first developed by
Ludwig Wittgenstein. The truth table consists of a listing of the terms in a logical
proposition, together with their logical connectives, in such a way that the truth or
falsity of the proposition is established for all possible assignments of true or false to
the terms.
The logical connectives together with their standard symbols are as follows:
“and,” ∧∧; “or,” ∨∨; “implies,” →→; and “if and only if,” ↔↔.
Truth-values for each of these connectives are given in Table 9.2.5.
Table 9.2.5: Truth values for logical connectives
PP
QQ
P∧QP∧Q
P∨QP∨Q
P→QP→Q
P↔QP↔Q
T
T
T
T
T
T
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
F
T
T
Thus, we see that both PP and QQ must be true if P∧QP∧Q is to be true,
while P∨QP∨Q is true if PP is true, or if QQ is true, or if both are true. The truthvalues for P→QP→Q may seem strange, but what is being said is that a true
statement cannot imply a false statement, but a false statement can imply a true
statement or a false statement. The values for P↔QP↔Q can be derived from the
“implication” table and the “and” table, if we note the logical
equivalence P↔Q=(P→Q)∧(Q→P)P↔Q=(P→Q)∧(Q→P). Note in Table 9.2.6, below,
that the columns for (P→Q)∧(Q→P)(P→Q)∧(Q→P) and for P↔QP↔Q are identical,
and statements that have the same truth-value are equivalent.
Table 9.2.6: Truth values for the logical
equivalence P↔Q=(P→Q)∧(Q→P)P↔Q=(P→Q)∧(Q→P)
PP QQ
P→QP→Q
Q→PQ→P
(P→Q)∧(Q→P)(P→Q)∧(Q→P)
P↔QP↔Q
T
T
T
T
T
T
T
F
F
T
F
F
F
T
T
F
F
F
F
F
T
T
T
T
The same method can be used to work out the truth table for more complicated
combinations as well. For example, the truth tables
for (P∧Q)→R(P∧Q)→R (PP and QQ together imply RR; or “Given PP and QQ,
then RR”) and (P∧Q)∨R(P∧Q)∨R are given in Table 9.2.7, below.
Table 9.2.7: Truth values for the
propositions (P∧Q)→R(P∧Q)→R and (P∧Q)∨R(P∧Q)∨R
PP
QQ
RR
P∧QP∧Q
(P∧Q)→R(P∧Q)→R
(P∧Q)∨R(P∧Q)∨R
T
T
T
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
T
F
In general, a truth table is constructed in a bottom-up fashion. First, there is a
column for each of the variables involved. The total number of rows will just be equal
to two, raised to the power of the number of variables (so for two variables there will
be four rows, for three variables there will be eight rows, and so on). Then all
possible combinations of true and false for these variables are written into these
columns. You will need to work out several examples of this by hand to become
familiar with the process. Table 9.2.7 illustrates an easy model for you to follow:
starting with the variable column furthest on the right (the RR column in our
example), write in “T” and “F” (or true and false) alternately. In the column to the
left of it, write in T in the first pair of rows, F in the second pair, T in the third, and
so on. In the next column to the left, write in T in the first group of four rows, F in
the second, and if necessary T in the third, and so on. Note that the number of rows
in which you write T or F doubles in each column. Then, based on this initial setup,
compute the truth-values for the combinations of pairs of variables and from these
truth-values, compute the truth-values for larger groups of variables in which these
pairs appear. The process may be tedious, but eventually gives the complete truth
table.
Although truth tables are useful for logical computations, we do not find them used
much in the actual reasoning used in scientific research. Rather, we find a much
looser kind of argument based on plausibility. Nevertheless, knowing how to
construct truth tables is an important aspect of scientific thought, particularly useful
in experimental design. For example, in setting up an experiment it may be
important to conduct the experiment in such a way that all of the logically possible
conclusions are tested in a way that allows exclusion of those possibilities that do not
match the collected data. If the goal is to test the hypothesis “PP causes QQ” then,
consulting the truth table for PP implies QQ (Table 9.2.6) we need only test to see
if PP is present then QQ always follow (since PP followed by not-QQ would falsify the
hypothesis PP implies QQ), and the contrapositive: are there any cases
where QQ occurs but there was no PP. No other tests (PP and QQ, not-PP and not-
QQ) are necessary since none of these are able to falsify “PP implies QQ.” (Note,
however, that in an experimental situation, the observation that QQ always follows
from PP may suggest that PP causes QQ but it does not warrant a certain conclusion
that PP causes QQ because causation is not the same as logical implication.)
Unit 9 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. The Aristotelian law of identity is often stated in the form “AA equals AA,” or “A
thing is identical with itself.” Aristotle also defined the essence of a thing as
“that which it must have in order to be entitled to its name.” Write a short essay
(about 300-500 words) discussing the possible relations between the law of
identity and the definition of an essence, in light of the three principles of
reason given in Discussion 9.1.
2. Chapter 1 of What Science Is describes Edward Jenner’s discovery of a
vaccination against smallpox. Analyze Jenner’s work in terms of the four
polarities of reason presented in Discussion 9.1.
3. Describe the way that the four polarities of reason given in Discussion 9.1 are
employed together in the hypothetico-deductive method.
4. Cats, dogs, tigers and wolves are all mammals; cats and tigers are feline, while
dogs and wolves are canine; cats differ from tigers and dogs differ from wolves.
Write a short essay (300–500 words) describing the formal structure that is
imposed on the set of cats, dogs, tigers and wolves by the naming process taken
together with the three laws of Aristotelian logic. You may wish to draw a
diagram.
5. State the three laws of Aristotelian logic, and describe how they provide a
natural way to think about unchanging independent objects.
6. Determine which of the following syllogisms are valid:
a. Mice have sharp teeth
Some animals with sharp teeth eat cheese
Therefore mice eat cheese
b. Dogs are not giraffes
Cats are not giraffes
Therefore dogs are cats
c. Species in danger of extinction ought to be protected
Smallpox virus is in danger of extinction
Therefore the smallpox virus ought to be protected
d. All fish have legs
All animals with legs can jump over the moon
Therefore fish can jump over the moon
7. Construct truth tables for each of the following propositions:
a. (p∧q)∨−p(p∧q)∨−p
b. (p∨q)∧−(p∧q)(p∨q)∧−(p∧q)
[In computer applications this is written as pXORqpXORq where the
connective XORXOR is the “exclusive or,” defined as pp or qq, but not
both.]
c. [(p→q)∨(q→r)]∧(p→r)[(p→q)∨(q→r)]∧(p→r)
d. [(p→q)∧(q→r)]∧−r[(p→q)∧(q→r)]∧−r
8. Draw a Venn diagram for each of the syllogisms in Question 6, above.
9. You are shown four cards, two labeled with the letters A and K and two with the
numbers 2 and 7:
You are told that these cards follow the rule that if there is a vowel on one side,
then there is an even number on the other side. Which cards must you turn over
in order to test this rule?
10. Prove that p→qp→q and −q→−p−q→−p are logically equivalent.
Hint: Show that they have the same truth tables; −q→−p−q→−p is the contrapositive form
of p→qp→q.
What is the importance of the contrapositive in the hypothetico-deductive
method?
11. A tautology is a proposition that is always true regardless of the truth-values of
its component parts. For example, p∨−pp∨−p is a true whatever the truth-value
of pp. Use truth tables to verify each of the following tautologies:
a. −(p∨q)↔(−p∧−q)−(p∨q)↔(−p∧−q)
b. −(p∨q)↔(−p∧−q)−(p∨q)↔(−p∧−q)
c. p∧q→p∨qp∧q→p∨q
d. [−p→(r∧−r)]→p[−p→(r∧−r)]→p
e. [p∨(q∧r)]↔[(p∨q)∧(p∨r)][p∨(q∧r)]↔[(p∨q)∧(p∨r)]
FOOTNOTES
[168]
Rucker, Rudy. Mind Tools: The Five Levels of Mathematical Reality, p. 201. Boston:
Houghton Mifflin, 1988.
[169]
Quoted in Parker, Ian. “Richard Dawkins’s Evolution: An Irascible Don Becomes a
Surprising Celebrity.” New Yorker, September 6, 1996. The text of this article is available on
the site below. Retrieved August 4, 2002.
http://archives.newyorker.com/?iid=15625&startpage=page0000047#folio=040/
[170]
Whitehead, A. N. Science and the Modern World: Lowell Lectures, 1925, p. 3. New York:
Macmillan, 1928.
[171]
Ironically, this principle is sometimes called the principle of contradiction!
[172]
Leibniz, G. W. von. “Monadology,” p. 235. In Leibniz: Monadology and Other
Philosophical Writings, trans. Robert Latta, pp. 215-277. London: Oxford University Press,
1898.
[173]
Quoted in Latta, Robert. Introduction, p. 85. In Leibniz: Monadology and Other
Philosophical Writings, pp. 1-211. London: Oxford University Press, 1898.
[174]
In 1949, the British science fiction writer William F. Temple (1914-1989) published a
novel with the title Four Sided Triangle. It involved two men in love with the same woman,
who was “duplicated,” but this is a metaphorical use of the term triangle.
[175]
Leibniz, G. W. von. “Monadology,” p. 235.
[176]
This definition is a paraphrase of the argument given in Chapter 4 of Book VII of
Aristotle’s Metaphysics. See for example, A New Aristotle Reader, J. L. Ackrill, ed., pp. 287289. Princeton, NJ: Princeton University Press, 1987.
[177]
This story was related in a recorded reading of comments of Idries Shah.
[178]
This statement is a paraphrase of the text of Fragment 91a. See Heraclitus. Fragments: A
Text and Translation with a Commentary by T. M. Robinson, pp. 55, 139-141. Toronto:
University of Toronto Press, 1987.
[179]
In philosophical language we would say that these two experiences of the same cue ball
differ only in numerical identity (i.e., in their space-time location).
[180]
Bacon, Francis. The Second Book of Francis Bacon of the Proficience and Advancement
of Learning Divine and Human, Section 7, para. 1. Retrieved August 4, 2002, from
http://www.uoregon.edu/~rbear/adv2.htm/
[181]
Waddington, C. H. Tools for Thought, p. 18. New York: Basic Books, 1977.
[182]
Gram molecular weights: 1 mole = 6.022 × 1023 atoms or molecules.
[183]
Plato. “Euthydemus,” 298d, e. trans. W. H. D. Rouse. In Plato: Collected Dialogues,
edited by Edith Hamilton and Huntington Cairns, pp. 385-420. Princeton, NJ: Princeton
University Press, 1961.
[184]
Caroll, Lewis. “Chapter 4: Tweedledum and Tweedledee.” In Through the Looking Glass.
The text of this work is available at the site below. Retrieved August 9, 2002.
http://www.literature.org/authors/carroll-lewis/through-the-looking-glass/chapter04.html/
[185]
Kant, Immanuel. (1787). “Preface to the Second Edition.” The Critique of Pure Reason,
trans. J. M. D. Meiklejohn. Retrieved August 9, 2002, from
http://eserver.org/philosophy/kant/critique-of-pure-reason.txt/
[186]
Note that the Aristotelian laws eliminate the possibility of change, because if something
changes, it has become other than itself. In order to deal with change, Aristotle introduced
the concept of the life cycle as an ideal of natural order.
[187]
Spencer-Brown, G. Laws of Form, p. 1. New York: Dutton, 1979.
[188]
There are 19 valid syllogistic forms.
[189]
Stoic logic was a forerunner of the propositional approach.
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Unit 10 Mathematics and Science
Discussion 10.1
Discussion 10.2
Unit 10 Study Questions
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STUDY GU IDE
Unit 10 Mathematics and Science
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins
to appear as a pure mathematician.
—Sir James Jeans (1877-1946)[190]
To those who do not know mathematics it is difficult to get across a real feeling as to the
beauty, the deepest beauty, of nature.
—Richard Feynman[191]
Mathematics has a premier role in modern science. In the physical sciences all
theories are expressed in mathematical language. Mathematics is an essential tool in
the biological sciences. In population biology, genetics, and ecology mathematical
models and simulation models provide deep insights into the nature and behaviour
of biological systems. Mathematical models are also essential in economics,
especially in economic forecasting. In the human and social sciences, on the other
hand, mathematics plays a somewhat smaller role, often involving only the
application of statistics to determine correlations between variables. Biology and the
physical sciences are very well developed as theoretical sciences, while the social and
human sciences lack general theories that are sufficiently developed to be cast into
mathematical form.
Recognition of the value of mathematics for understanding the world goes back to
Pythagoras; and Plato, who published a geometrical theory of the four elements
(earth, air, fire and water) in his dialogue Timaeus, can be considered the
grandfather of mathematical physics.[192] But ancient science did not develop in any
systematic way, and with the decline of Greece and the rise of Rome, mathematics
fell into disrepute.[193]
Science and mathematics flourished again in the early Islamic world, but as
indicated in Unit 2, went into decline by the 13th century. When works of Greek
philosophy were reintroduced into Europe, primarily from Moorish Spain, natural
philosophy followed Aristotle, and Aristotle had little use for mathematics. The
mathematical sciences we know today came into being, as sciences, only during the
scientific revolution of the 17th century.
In this unit we will begin by describing the nature of a mathematical system, and the
ways that such systems relate to the material world. Newton’s theory of mechanics
and gravitation provides an excellent example. This is followed by a historical
discussion of the rise of the “mixed mathematical sciences” in the 17th century.
Finally we will study some basic aspects of probability and statistics.
Objectives
When you have completed Unit 10, you should be able to
1. discuss some of the ways that mathematics is used in science, and describe some
forms of mathematical thinking.
2. describe the elements composing a mathematical system, in particular the
system of Newtonian mechanics and gravitational theory.
3. describe some of the advantages of being able to formulate a mathematical
theory for some field of science.
4. discuss the history of how mathematics became important in science during the
17th century, and in particular give the justification provided by Newton for
mixing mathematics and science.
5. outline some of the basic principles of probability, and use them in the
computation of simple probabilities.
6. discuss some basic ideas of statistics, including the importance of having
unbiased samples, the use of the standard deviation to measure the spread of a
sample about its average, and the meaning of “statistically significant result.”
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 10.1, “Mathematics in Science.”
2. Answer Unit 10 Study Questions 1-4.
3. Read Statistics, pp. 102-103 in What Science Is.
4. Read Discussion 10.2, “Probability and Statistics.”
5. Answer Unit 10 Study Questions 5-8.
Discussion 10.1 Mathematics in
Science
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I have hardly ever known a mathematician who was capable of reasoning.
—Plato[194]
A mathematical system consists of a set of definitions; a set of initial assumptions,
called axioms; and all statements that can be logically deduced from these axioms
and definitions. The axioms must be mutually independent, in the sense that no
axiom can be deduced from the remaining axioms of the axiom set.[195]
The set of axioms must also be finite, and it must be consistent. Consistency is
needed to avoid contradiction; the theorems of the system must satisfy the
coherence criterion of truth, and if the axioms do not satisfy this condition, the
theorems cannot do so either. The axiom set must be finite, because the axioms must
be available for use in deductions. If the axiom set were infinite, there would always
be an infinite set of unknown assumptions.[196]
Discussion 6.1 stated that the theorems of a mathematical system are true, relative to
the set of axioms that define the system. If we want to use a mathematical system as
a framework for describing or modeling some aspect of the real world, however, then
the axioms of that system must satisfy the correspondence criterion of truth, at least
within an acceptable margin of error. In this case, the truth of the proposed
description or model will be relative to the “facts” the axioms purport to describe. A
“continuity assumption” is also required, to the effect that if the axioms fit the world
to within a set error margin, then results derived from those axioms will also fit
within a similar (but perhaps not identical) error margin.[197]
For example, as a mathematical system, Newtonian mechanics is based on the
axioms listed below.
Axioms of Mathematics: Specifically, the axioms of differential and integral calculus.
Axioms of Space and Time: Space is a three-dimensional Euclidean manifold; that
is, space is three-dimensional, homogeneous, isotropic, and described by the axioms
of Euclidean geometry.
All motion in space can be referred to a universal time parameter.
Axioms of Motion: Newton’s “Laws of Motion,” listed below.
1. An object at rest remains at rest, and an object in motion remains in motion
with uniform velocity, unless acted on by some external force.
2. An object acted on by an external force FF reacts with acceleration equal to the
applied force divided by the mass of the object.
3. For every action there is an equal and opposite reaction.
Simplifying Assumption: All solid bodies may be treated as point masses.
The axioms of space and time assert that space and time are independent, that
objects in space do not influence space itself, and that space does not exert any
influence on the objects it contains. Newton’s laws state what is to be taken as the
ideal of natural motion (straight lines), and define forces as any influence that
produces a deviation from this natural motion.
The axioms of differential and integral calculus allow us to define velocity as the time
derivative of position, and acceleration as the time derivative of velocity, and hence,
as the second time derivative of position. On this basis, we can write second-order
differential equations whose solutions represent the spatial positions of point masses
subjected to specific forces. Solutions of these equations give either the orbits of the
particles (kinematics), or their position and velocity as functions of time (dynamics).
Thus, by inspection of the given axioms, we can say that the “truth” of Newtonian
mechanics is relative to the conditions listed below.
1. Space and time are independent, and there is a universal absolute time
parameter.
2. Space is not influenced by the bodies it contains; nor does it influence those
bodies.
3. Extended solid bodies may be considered as point masses.
4. The natural (i.e., force free) motion of a body in space is in a straight line at a
constant speed.
5. With isolated exceptions (e.g., when two point masses collide) the motion of
bodies in space is sufficiently smooth that at least first and second time
derivatives of position exist.
These conditions corresponded to the “facts” as they were known until about the
middle of the 19th century. With increased accuracy of measurement, however,
experimental results began to indicate that these assumptions were accurate only to
the extent that terms which were of the order of (v/c)2(v/c)2 or higher could be
neglected in the equations of motion and their solutions. Here vv is a measure of the
maximum speeds that might occur in an experimental situation, and cc is the speed
of light, equal to approximately 300,000 km per second.
In the language introduced in Discussion 6.1, we would say that the theorems of
Newtonian mechanics are truths of reason, relative to the axioms listed above, and
that Newtonian mechanics is a truth of experience, relative to an experiencer whose
perceptual capacities are limited to the extent that terms of second order and higher
in the ratio (v/c)(v/c) are not distinguishable.
In terms of our constructive viewpoint, a scientific theory is a rational mental
construct that is posited to stand in analogy to some aspect of the world. In the
example above, the limits of accuracy of Newtonian mechanics tell us just how far
the analogy can be pushed.
A number of advantages are gained when a theory can be stated in mathematical
form. Among these are simplicity, economy of effort, generality and precision. The
Newtonian force law (F=maF=ma), in which the acceleration (aa) is expressed as a
second derivative of position with respect to time, greatly simplifies descriptions of
bodies moving under the influence of a force, and allows prediction of future
positions in time. It also shows great generality—it is applicable to a tremendous
variety of cases from falling apples to projectile motions to planetary orbits.
Mathematical formulation is economical because once a hypothesis is expressed
mathematically, the tools of mathematics become available for its analysis. Likewise,
mathematical formulation allows for greater precision than could be obtained
otherwise, including the use of statistical and probabilistic methods estimate exactly
how accurate a result can be expected to be.
Not only is mathematics the primary tool of much scientific work, but mathematics
“commands” science, in the sense that a theorem derived in a mathematical system
that is accepted as a frame for a scientific theory must be interpreted in terms of the
theory, and the conclusion drawn must be accepted either as an indication that the
mathematical system is not an appropriate framework, or as a consequence of the
theory. The mathematician E. T. Bell captured this dual role with the saying that
“Mathematics is the queen and servant of science.”[198]
To acquire facility in mathematical reasoning, it is necessary to work through many
examples and problems. Mathematics has the reputation of being a hard subject, but
it is an essential one for science, and every science student needs to develop a strong
mathematical background.
A mathematical text is . . . not an end in itself, but a key to a world beyond the compass of
ordinary description.
—G. Spencer-Brown[199]
The Physio-Mathematical Sciences
When we think of science today, we think of experiments. Science, for us, is based on
experiment and all theoretical conclusions or hypotheses must be experimentally or
observationally tested. The philosopher of science Karl Popper went so far as to
insist that a theory that could not be falsified could not be scientific. Popper realized
that no amount of empirical evidence could definitively establish the truth of a
theory, but a single counter-instance could show it was false. Hence he suggested
that for a theory to be scientifically acceptable, it must have the possibility to be
shown false. If no counter-instances were possible, even in principle, then the theory
could not be accepted. Before the 17th century this was not the way that science was
done.
From the Medieval period to the Renaissance the term “science” referred to
Aristotelian “natural philosophy,” which was based on syllogistic deduction from
speculative initial premises called “hypotheses.” These initial starting points were
taken as matters of common knowledge, or supported by biblical or ancient
authority. For this reason, most of the activity in natural philosophy was purely
speculative. Based on the idea that God could create anything, so long as it did not
involve a logical contradiction, medieval philosophers had felt free to speculate on
possible worlds, making assumptions and deducing conclusions from these with
little regard as to whether or not they related to the actual world. What was required
was only that the logical deductions be accurate so that the conclusions reached were
absolutely certain, given the initial hypotheses.[200]
In addition, the idea of setting up an experiment to test a theory was foreign to
Aristotelian science. Its goal was a more-or-less good description of what would
usually happen in the normal course of nature, acknowledging that accidents or
miracles could always disrupt nature’s normal behaviour. From this point of view, a
contrived experiment could not provide information about what was normal in
nature because the experimental situation automatically distorted nature into an
abnormal state.[201]
Thus, there was a sharp distinction between “scientific knowledge,” which was
certain (even if not applicable to the natural world), and “opinion,” which could
apply to the world, but could never be certain. Physics consisted of syllogistic
deduction from speculative initial hypotheses and counted as “science.” Fields such
as mechanics, optics, astronomy and medicine were not certain because they
required measurements and other input that could never be exact. They did not
derive their conclusions from initial causes by a series of logical deductions.
In modern science, especially physics and chemistry, mathematical description and
causal explanation go hand in hand because causal entities such as forces are
represented by terms in mathematical equations that are supposed to describe laws
of nature. But this way of understanding the link between mathematics and our
descriptions of the world only arose in the 17th century.
When Galileo was teaching mathematics at Padua (1592-1610) he participated in
vigorous debates on whether a quantitative mathematical description could be
acceptable in a real science in place of a causal explanation. At that time, “real
science” referred to Aristotelian science, which did not employ mathematics. The
ideal for Aristotelian science was syllogistic deduction from initial premises that
were taken either as obvious to all, or as supported by common agreement or
“weighty authority.” Only in this way could deductively certain conclusions be
drawn, and Aristotelian science required certainty.
Thus, one of the major issues that had to be resolved during the scientific revolution
of the 17th century was how to justify applying mathematics to studies of nature. For
Aristotelians, mathematics was a useful tool for calculation, but had little or no place
is “real” science. The French Jesuit theologian and natural philosopher Honoré Fabri
(1607-1688), for example, argued that physical hypotheses and mathematics could
not be combined in science because sense data, being imprecise, could never provide
support for exact mathematical conclusions. Criticizing Galileo’s work on inclined
planes and falling bodies, he wrote in 1646 that “sensory experience cannot vindicate
mathematically precise ratios. . . . only doubtful hypotheses can be drawn from
doubtful experiences . . . and doubtfulness is not science.”
Other natural philosophers in the 17th century, however, began to accept
mathematics as an appropriate language for the description of nature. Niccolò Cabeo
(1586-1650), a Jesuit teacher of natural philosophy and mathematics,
published Philosophia magnetica in 1629 in which he insisted he was doing physics
(that is, natural philosophy), not mathematics, even while using mathematical
methods. He acknowledged that his results were not certain, and claimed that if
somebody could produce a better physical theory he would accept it over his own,
with the exception of those aspects of his theory that were based on mathematical
proof.
With this, Cabeo moved in the direction of recognizing the stability of mathematical
theories—that any new theory must incorporate, or in some way explain, the
mathematically certain aspects of the theory it replaces. In addition, he made a
direct appeal to the predictive power of a theory as evidence for acceptance of its
underlying assumptions. In speaking of astronomy, he pointed out that since some
astronomical events occur on a long time scale their observation requires synthesis
of the observations of many individuals working over thousands of years. He argued
that the laws of celestial motion that have emerged from such observations give an
excellent fit to the data and went on to say that acceptance of these laws arises from
the continual fulfillment of predictions.
Another important figure in promoting the use of mathematics in science was Isaac
Barrow (1630-1677), Newton’s immediate predecessor as Lucasian professor of
mathematics at Cambridge. In Barrow’s view, what was coming to be called the
physico-mathematical approach was superior to the Aristotelian method. Barrow
argued that all science requires mathematics in one degree or another since they all
involve quantitative considerations. This is directly opposed to the Aristotelian idea
of what a science is—Barrow is admitting as sciences many areas of study that, in
Aristotelian terms, could only be thought of as “opinion.”
By the mid-17th century the term “physico-mathematics” had come into use,
describing mathematics as a tool for gaining genuine physical knowledge. While
Aristotelians fought this incursion of mathematics into natural philosophy, it
continued gaining credence as a new way to obtain knowledge of nature and the
belief that the “mixed mathematical disciplines” could produce genuine causal
understanding of nature soon became commonplace.
The French monk Marin Mersenne (1588-1648), for example, used the mixed
mathematical sciences as paradigmatic models for understanding nature, playing a
major role in the development of 17th century-science along the mixed mathematical
lines through extensive correspondences with the major figures of the age. (Much of
Mersenne’s attraction to these physico-mathematical models was that they were not
Aristotelian and did not claim absolute certainty, which, in an echo of medieval
concern over the possibility of absolutely certain demonstration, he feared could
pose a threat to religion.)
Medieval natural philosophers had sought to fit their observations into pre-assumed
forms based on metaphysical speculation about the nature of the divine order. They
sought to deduce phenomena from models based on first principles such as the
primacy of circular motion.[202] With the new experimental method, scientists sought
to go in the opposite direction, to inductively infer first principles from empirical
data. They could not rely on assumptions of an a priori metaphysical order.[203]
Clear illustration of the difference in point of view is found in the difference between
the Danish astronomer Tycho Brahe and Johannes Kepler. Tycho rejected the
Copernican system and argued that the planetary orbits must be circular because the
perfection of the heavens could only arise if the planetary orbits could “return upon
themselves,” which required circular orbits. Kepler, on the other hand, accepted
Copernican theory and, making use of the masses of data that Tycho had collected on
the orbit of Mars, produced the inductive hypothesis that the orbit of Mars (and of
the other planets) was better modeled as ellipses.
While Galileo and Kepler were transitional figures, it remained for Isaac Newton to
definitively formulate the idea of scientific explanation in terms of mathematical law.
Galileo had sought to maintain a version of the Aristotelian idea of common
experience as the basis for conclusions. In his writings on falling bodies he did not
describe specific experiments; rather, he described the experimental apparatus and
asserts that he has done the experiments “many” times, always obtaining the same
results, thus appealing to the Aristotelian condition of memory of many instances of
the same thing. And while Kepler had used empirical data to arrive at the hypothesis
of elliptical planetary orbits, and believed that the planets were held in their orbits
by some sort of “magnetic” force, he could not produce a connection between this
force and the shape of the orbits and believed that the orbits were as they were
because they fit within nested Platonic solids.
Newton finalized the marriage between mathematical description and causation by
providing a way to ground universal propositions, in the form of mathematically
described laws, in finite physical observations. Newton’s argument was that if there
were laws of nature that could be given mathematical form, then an equation that
was obeyed in every known case of empirical observation could be taken as an
expression of the underlying universal law, even though it arose from empirical
induction.
The philosophical point of view necessary for this transition was radical: it required
that inductive generalizations from finite cases to mathematical equations be
accepted as legitimate, and that these equations be accepted as expressing a truth
about nature. Newton’s famous and, at the time, highly controversial 1672 letter to
the Royal Society on light and colour[204] provided the final synthesis of the modern
method. He used contrived experiments with prisms to validate his physicomathematical theorizing and claimed mathematical certainty for his results. This
bold move did not go unchallenged; in particular by Robert Hooke (1635-1703), who
argued that Newton’s assumption that a prism split white light into a spectrum of
colours was not the only hypothesis explaining his experimental results, and doubted
that Newton’s conclusions could be mathematically certain.
Replying to Hooke’s criticism, Newton wrote to Henry Oldenburg (1619-1677),
secretary of the Royal Society, “I said, indeed, that the science of colour was
mathematical and as certain as any other part of optics; but who knows not that
optics, and many other mathematical sciences, depend . . . on mathematical
demonstration? And the absolute certainty of a science cannot exceed the certainty
of its principles. Now the evidence by which I assert the propositions of colour is . . .
from experiments, and so but physical, whence the propositions themselves can be
esteemed no more than physical principles of a science. And if these principles be
such that on them a mathematician may determine all the phenomena of colours
that can be caused by refractions and that, by disputing or demonstrating after what
manner and how much, those refractions do separate or mingle the rays in which
several colours are originally inherent, I suppose the science of colour will be granted
mathematical and as certain as any part of optics.”[205]
In other words, Newton realized that the mixed mathematical sciences, relying on
empirically derived physical principles as well as on mathematical demonstrations,
could never provide absolute certainty, but he believed that he had found the way of
producing knowledge that was as secure as was humanly possible.[206] The basic
definitions and propositions of a mathematical science are mathematical, hence
anything derived from them is mathematically certain, but these definitions and
propositions themselves are derived by induction from experiment and so are not
certain in the Aristotelian sense. Nevertheless, if the mathematical theories produce
results that are continually in agreement with experiment, those theories can be
taken as having a high degree of certainty.
The final seal of approval for this new method came with publication of Newton’s
monumental Principia. “After [this] the [Royal] Society began to associate its
experimental practice with Newton’s physico-mathematical justifications. Only then
. . . did it begin to appear as an original mover of a new kind of natural
philosophy. . . . When the Fellows adopted Newton as their champion, the
mathematical sciences achieved their final triumph over scholastic natural
philosophy. But in the process, those sciences had themselves mutated into
something new, becoming explicitly experimental.”[207]
Discussion 10.2 Probability and
Statistics
TOP
There are three kinds of lies: lies, damn lies, and statistics.
—attributed to British P. M. Benjamin Disraeli (1804-1881)
Much reasoning in science involves probability arguments. In this discussion only
the bare bones of this subject are considered. To begin, we must distinguish between
probability and frequency. When we ask about the probability of an event occurring
(e.g., drawing an ace from a deck of 52 cards) we are asking about the most
reasonable expectation and would argue that with 4 aces in the deck, the probability
is 4/52=1/13∼.0076924/52=1/13∼.007692. . . . So we would say that
the probability is about 7.7%. That means if we were to draw a single card from the
deck, check to see if it was an ace, then repeatedly replace it and draw again, we
would expect to see about 77 aces per thousand trials, on the basis of this computed
probability.
The frequency of an event, on the other hand, is defined only with respect to an
“ensemble” (i.e., a very large number of identical cases). This is what occurs in actual
experimental trials. The experiment is conducted many times over and
the frequency of results is observed and used to estimate the probability that might
appear in a mathematical equation. This is what was done in the previous paragraph
when a single card was drawn from a 52-card deck for one thousand trials. From that
we compute a frequency but almost immediately turn it around and take it as a good
estimate of the probability.
On the other hand, if you were to draw a single card from a 52-card deck one
thousand times, and repeat this experiment one hundred times, you would not
expect that in every one of these hundred repetitions you would find that you drew
an ace seventy-seven times out of a thousand. Rather, you would observe a variety of
numbers whose average was approximately 77. Further, with some knowledge of
statistics, you could estimate not only how close this average was to .007692, but
also the expected distribution of averages around that number. What is important to
remember is that experimental results determine frequencies, but theoretical
analysis often uses probabilities. Part of the analysis of empirical results will often
involve determination of how close the measured frequencies come to predicted
probabilities.
The formal connection between frequency and probability comes from a
mathematical theorem called the Law of Large Numbers. This theorem proves that
in a process of random trials the frequency of a particular result will converge to the
theoretically predicted probability as the number of trials becomes large. In terms of
the example of drawing an ace, this means that as the size of the ensemble of trials
grows the average of the frequencies will approach 1/131/13.
If AA and BB are the two cases to consider, the basic axioms for working with
probability are:
1. If AA and BB are independent events then P(A∧B)=P(A)P(B)P(A∧B)=P(A)P(B).
2. The probability of AA, BB or both
is P(A∨B)=P(A)+P(B)−P(A∧B)P(A∨B)=P(A)+P(B)−P(A∧B).
The probability of AA if BB occurs is called the conditional probability of AA, or the
probability of AA given BB, denoted P(A|B)P(A|B). It
is P(A|B)=P(A∧B)P(B)P(A|B)=P(A∧B)P(B).
If AA and BB are mutually exclusive this means that the probability
of AA and BB together is zero. Thus in setting up experiments it is often desirable to
ensure that the set of possible outcomes consists of events that
are independent and mutually exclusive.
If the probability of an event AA is P(A)P(A) the probability of not-
AA is 1−P(A)1−P(A). Many times it may be easier to work with the probability of
something not occurring rather than of its occurrence. This shows up, for example,
in the Birthday Problem. Given NN people chosen at random, what is the probability
that two of them have the same birthday? If we denote this probability
as P(A)P(A) then we can only do a direct calculation of this by considering all of the
possible combinations of NN pairs of people. Suppose, however, that we calculate
the probability P*(N)=1−P(N)P*(N)=1−P(N) that none of the NN people have the
same birthday. This is easier. For the first person, P(1)=0P(1)=0 since nobody can
not have the same birthday as they have, hence P*(1)=1P*(1)=1. Ignoring twins, leap
year, and so on, we can continue. Given two people (and assuming that birthdays are
independent) P*(2)=P*(1)(364/365)P*(2)=P*(1)(364/365). This is because the
probability that the second person has a different birthday is 364/365364/365.
Adding a third individual, this person must have a different birthday from both of
the first two, hence P*(3)=1(364/365)(363/365)P*(3)=1(364/365)(363/365).
For NN individuals, none of whom have the same birthday, the probability will be
P*(N)=1⋅(364365)⋅(363365)⋯(366−N365)P*(N)=1⋅ (364365)⋅ (363365)⋯(366
−N365)
The next table shows 1−P*(N)1−P*(N) for differing values of NN.
NN
P(N)P(N)
10
20
23
30
50
95
100
11.7
41.1
50.7
70.6
97.0
99.9
99.99997
Amazingly, this probability is over 1/21/2 for N=23.N=23.
As indicated, when considering empirical measurements it is frequencies that are
directly given and there is a need to quantify not only the average values arising but
also the spread of measurement results about these averages. It is also important to
have some idea of how much trust can be placed in the results. This takes us into the
realm of error analysis and statistics.
Error Analysis and Statistics
Empirical science depends entirely on measurement, and measurements, at least of
continuous variables, are never exact; there will always be limiting factors that
influence their accuracy and precision. The terms “accuracy” and “precision” refer to
different aspects of measurement processes but they are often confused. The
accuracy of a measurement describes how close it is to the actual value being
measured. The precision of a measuring instrument, on the other hand, refers to
how well it is able to distinguish values. A simple example of this difference is a
bathroom scale marked off into units of one pound. This determines its precision—
the expectation is that in a series of measurements of a weight of, say, 100 pounds,
the values obtained will not differ by more than about one pound. On the other
hand, if the scale is biased, these measurements will not be accurate.
In scientific measurements, extreme care is taken to design precise instruments and
eliminate bias. Nevertheless, all measurements contain some degree of error. When
measurements are reported in a scientific paper, they are accompanied by error
estimates or by statistical measures of their reliability. One of the major issues in
late-18th and early-19th century science was development of statistical methods to
deal with error estimates.
Given a quantity WW to be measured, its value may have been predicted by a
theoretical formula involving a number of parameters: W=f(x1,…,xn)W=f(x1,…,xn).
Here only a single parameter is considered. Consider a series {x1,…,xN}{x1,…,xN} of
measurements of this parameter with true (but unknown) value xx. These results
differ by some amount from xx, and the task is to determine the best estimate for the
actual value xx.
The initial assumption is that xi=x+eixi=x+ei where eiei is the error associated with
the ii-th measurement. This error has various components: the known precision of
the measuring instrument, possible errors made by the scientists making the
measurements, random environmental factors, and so on. Assuming that there is no
bias in the measurements and that the various sources of error are independent, the
most natural assumption is that the errors of each source are randomly distributed—
any other assumption would imply that there was additional knowledge available
(remember the principle of sufficient reason and its form as the assumption of
equal a priori probabilities).
In the absence of further knowledge it is reasonable to suppose that the best guess as
to the actual value xx is given by the average over the xixi.
μ=1N∑i=1Nxiμ=1N∑i=1Nxi(1)
Note that this average is denoted μμ rather than xx because it is not the actual value
but our estimate of this value. If the errors eiei are randomly distributed about the
value 00, some will be positive and some negative and as nn increases, according to
the Law of Large Numbers, the expected value for the average error
e=1N∑i = 1Neie=1N∑i = 1Nei(2)
will tend to 00. This will generally not occur for a finite set of measurements and the
term in equation (2) is called the first moment of the error distribution. Since the
estimated value of the parameter being measured is given by μμ, the assumption is
that xi=μ+eixi=μ+ei. But μμ will have a standard error as well. The measurement
error associated to μμ is not the average error ee, it is given by the standard
deviation, defined as
σ=[1N∑i=1Ne2i]1/2=[1N∑i=1N(μ−xi)2]1/2σ=[1N∑i=1Nei2]1/2=[1N∑i=1N(μ−xi)2]1/2(
3)
where σ2σ2 is called the variance, or the second moment of the distribution. Note,
however, that since μμ itself is not the true value but only a best guess, it is necessary
to compute the error ΔμΔμ attached to this mean and write the estimated value
as μ±Δμμ±Δμ. Since the standard deviation depends on μμ, it will have an associated
error as well. For statistical reasons, it turns out that the best estimate for the
standard deviation is not that given in equation (3), but
σest=[1(N−1)∑i=1N(μ−xi)2]1/2σest=[1(N−1)∑i=1N(μ−xi)2]1/2(4)
and the best error estimate for the mean error is
Δμ=σestN−−√=[1N(N−1)∑i=1N(μ−xi)2]1/2Δμ=σestN=[1N(N−1)∑i=1N(μ−xi)2]1/2(5)
The important point is that the description of the errors in the original data set is
determined by the first two moments of the error distribution function. The question
then is: what is the best mathematical form for the error distribution function?
Karl Friedrich Gauss (1777-1855) answered this question in 1809. Gauss proved that
if the maximum likelihood estimate from a series of measured values of a parameter
is required to equal the arithmetic mean of those measurements, as given in
equation (1), this determines a unique error distribution now called the Gaussian, or
Normal distribution. For a single variable one form of this distribution is given by
f(x)=12π−−√e−x2/2f(x)=12πe−x2/2(6)
Aside from the fact that it is determined by a very natural criterion, the Normal
distribution has a number of other properties that make it a natural choice for an
error-distribution function. Prime among these is stability; that is, given any family
of normally distributed random variables, any linear combination of these variables
will also be normally distributed. Mathematically, this is formalized in the Central
Limit Theorem: Given a set of independent, identically distributed variables with
finite variance, the limit as the number of variables increases approaches a normal
distribution.
It is important, however, to realize the limits of using a Gaussian distribution as the
error distribution. It depends on the assumption that the variables involved are
independent. Correlations between variables can introduce substantial deviations.
We won’t go into this, other than to mention that in studies of complex systems, it
turns out that power law error distributions are often better error-distribution
functions than a Gaussian, and power law error functions greatly increase the
probability of extreme deviations.
Often experiments are conducted to test a hypothesis or to differentiate between two
hypotheses. Recall that the standard hypothetico-deductive procedure is to generate
a hypothesis (either as a prediction of a theory or from empirical data), then make a
prediction that can be empirically tested based on that hypothesis. Suppose that our
hypothesis is that the result of measuring a certain parameter is xx. Carrying out the
experiment we obtain a result μ±Δμμ±Δμ. If xx is in the
interval [μ−Δμ,μ+Δμ][μ−Δμ,μ+Δμ] we have some confidence in our hypothesis.
If xx is outside of the interval we are not in a position to automatically reject it; there
may have been errors in the experimental procedure, or some sort of external
influence which needs to be checked. But it could also be that the hypothesis needs
revision.
What we need to know is the probability of obtaining that result assuming that the
predicted value xx is correct and the empirical result is an outlier rather than a
reason to reject the hypothesis, in conditional probability terms it
is P(μ|H0)P(μ|H0) where H0H0 is the hypothesis being tested (called the null
hypothesis). This is where what is usually called a pp-value enters. This is the
probability of obtaining the outcome μμ given that H0H0 and the predicted
value xx are correct. This gives a way to determine whether or not to reject the
hypothesis. Two different forms of hypothesis testing arise, which can be illustrated
by the following examples.
In the first, a physicist uses a new theory of condensed matter to compute a
prediction for the melting point of a new material as a function of pressure. Then, in
experimental tests this melting point is measured several thousand times and the
results averaged. If the results are found to lie within the expected error range, this is
published and this increases confidence in the theory. The goal is not to reject the
theory but to test whether it can give good predictions. If the predicted value is not
within the expected error of the measured results the researcher would first recheck
the experimental apparatus and the calculations behind the prediction. If no
problem could be found with these the conclusion would be that the theory needed
modification.
In the second example, a sociologist wants to study the influence of income level on
political belief. In this case, a well-developed theory is not available. A survey of a
large number of people is carried out. The null hypothesis is that there is no
influence (i.e., no correlation exists between income level and political belief) and
the hope of the researcher is that this hypothesis can be rejected. If it cannot be
rejected, it is useless to continue looking for an effect and the results are probably
not going to be published. The researcher has learned “something that doesn’t work”
and will move on to more fruitful areas of research. If some degree of correlation is
found, results of the survey and the computed correlation coefficient between
income and political belief will be published together with the ppcoefficient indicating the degree of confidence in the results. If, for
example, p=.04p=.04 this says that the results obtained showing a correlation
between income and political opinions have only a 4% chance of arising at random.
In carrying out this sort of analysis two types of error can arise. The first, called a
type I error (or error of the first kind) is to reject the null hypothesis when it is
actually true. The second (a type II error, or error of the second kind) is to accept the
null hypothesis when it is actually false. A type I error, in legal analogy, is equivalent
to finding an innocent person guilty. The conservative nature of science shows up in
that we try to minimize type I errors. Hence in practice, a very stringent criteria is set
for rejection: the null hypothesis is generally rejected if p≤.05p≤.05. But if our
measurements differ substantially from predictions, but not so much as to result in
automatic rejection of the null hypothesis, more experiments are certainly called for.
Unit 10 Study Questions
TOP
1. What are the basic components of a mathematical system?
2. Why did Aristotelian natural philosophers object to applying mathematics in
natural philosophy?
3. Write a short essay (300-500 words) discussing the justification for using
mathematics to study physical systems.
4. Tycho Brahe objected to the Copernican heliocentric theory on both
observational and theoretical grounds. What were these objections?
5. A lottery ticket contains six distinct numbers between 1 and 49. What is the
probability that they will be the winning numbers?
6. The back cover of a popular diet book claims that 72% of the people who
followed its dietary recommendations, according to reader feedback, had lost at
least 5 pounds in the first week. Out of interest you contact the publisher, who is
happy to show you the letters that have been received from the readers. There
are 246 letters praising the diet, and saying the writers have lost weights of 5
pounds or more within a week. There are 96 letters from people asking for their
money back, because they actually gained weight in the first week.
a. What do you think of the publisher’s claim?
b. Based on reader feedback, it appears to be true that 72% of these readers
lost at least 5 pounds in the first week. What do you think is the catch in
this number?
7. Two new drugs, Generabex and Zycolon, are being marketed as cures for the
common cold. Both claim that a cold will be gone within two days of starting to
use the drug. A friend of yours is a doctor, who has carried out tests of the
effectiveness of each drug. He has compiled the table presented below.
Cured
Not Cured
Serious Side Effects
Generabex
176
24
17
Zycolon
353
51
31
8. Based on this data, which drug would you take? (Neither is a legitimate answer
as well, but in a forced choice, which would it be?)
FOOTNOTES
[190]
Jeans, James. The Mysterious Universe, p. 122. Cambridge: Cambridge University Press,
1933.
[191]
Feynman, Richard. The Character of Physical Law, p. 58. Cambridge, MA: MIT Press,
1965.
[192]
The physicist Werner Heisenberg said that the Timaeus and Newton’s Principia
Mathematica are the two most influential works of science ever published. To this, we
might add Darwin’s Origin of Species.
[193]
In large part this was a cultural effect. Romans were not particularly interested in
abstract theory; their interests were directed more toward engineering and politics.
Mathematicians often cast horoscopes as well— a dangerous, sometimes illegal, occupation
in Rome. Emperors were afraid of anybody learning their horoscope, and in some cases
would have possible rivals with favourable horoscopes murdered.
[194]
Quoted in Rose, N. Mathematical Maxims and Minims, p. 69. Raleigh, NC: Rome Press,
1988.
[195]
This requirement is an efficiency condition. We want the axiom set to consist of a
minimal set of statements from which the remainder of the theorems of the system can be
deduced. It is not necessary to “reinvent the wheel.”
[196]
It might be suggested that an infinite set of axioms could be generated by a recursive
procedure, but in such a case, the entire set would be replaced by the first axiom of the
sequence, and a second axiom to the effect that every statement generated by the given
recursive procedure was true.
[197]
For example, Euclidian geometry is normally used in surveying plots of land, even
though we know that it cannot apply exactly because the earth is not flat.
[198]
Bell, E. T. Mathematics: Queen and Servant of Science. Redmond, WA: Tempus, 1988.
[199]
Spencer-Brown, G. Laws of Form, p. xxix.
[200]
In part, this was a way of getting around charges of heresy from the Church. A Scholastic
natural philosopher could propose some hypothesis that went against Church teaching and
claim that he was merely indulging in hypothetical speculation. This cover was even used by
Descartes in Le Monde, a book published after his death. He asserted that he was providing
a model of a world that was not ours, but was one God could have created. He then provided
a completely mechanistic world model in which divine creation was unnecessary.
[201]
You may want to compare this belief to the uncertainty principle, discussed in Unit 13.
[202]
In other words, they were rationalizing experience within a given worldview rather than
testing that worldview, they were treating phenomena as instances rather than as evidence.
[203]
This is the background for Newton’s famous statement that “I do not feign hypothesis.”
Some modern commentators think Newton was confused, pointing out that there are a
number of hypotheses in his work. But the meaning of the word hypothesis has changed
since Newtonian times, when it meant a speculative assumption used to start a syllogistic
argument, while today it means an inductively generated conjecture.
[204]
Newton, Isaac. The New Theory About Light and Colors. Philosophical Transactions of
the Royal Society 80 (Feb. 19, 1672): 3075-3087.
[205]
Cambridge, July 11, 1672, in H. S. Thayer, ed. Newton’s Philosophy of Nature: Selections
from His Writings, p. 81. New York: Harper, 1974.
[206]
Newton and Huygens were apparent the first natural philosophers to completely
abandon the idea that science ought to seek absolute certainty.
[207]
Dear, Peter. Discipline and Experience, p. 247. Chicago: University of Chicago Press,
1995.
TOP
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Unit 11 What Are Some General Approaches in Scientific Reasoning?
Discussion 11.1
Discussion 11.2
Discussion 11.3
Unit 11 Study Questions
TOP
STUDY GU IDE
Unit 11 What Are Some General
Approaches in Scientific
Reasoning?
[I]n order to discover where the professors of any branch of knowledge have erred, one
must make a profound study [and] must equal and even surpass those who know most of it.
—Abu Hamid al Ghazali (1058-1111)[208]
Given an unknown system to investigate, how does a scientist begin? Of course,
there is always a background of theory and/or practice that limits the possible
approaches that can be taken, but within these constraints, how does the scientific
study of a phenomenon get going?
The answer, prosaically enough, involves the observation of similarities among
elements of the system, or aspects of the phenomena in question, and the making of
distinctions between them. We begin with Plato’s injunction to “carve at the joints.”
We gather together collections of those things that are similar and give each distinct
collection a name so that we can talk about it. The name of a collection becomes an
abstract label for the particular entities it contains—it refers specifically to the
characteristics on which those entities have been judged to be similar. This reference
may be very precise and easy to define, as in the use of the word “triangle” to label
the collection of all triangles, but in fact, it is often rather difficult to give a precise
definition. Consider, for example, using the word “cat” to indicate the collection of
all cats. We all know a cat when we see one, but actually giving a precise definition is
rather difficult and time consuming.
Recall from Discussion 8.1 that the root word for logic meant “to collect” or “to speak
about.” Thus, in constructing collections of similar things and giving names to these
collections, we are carrying out a logical exercise. It may or may not follow the rules
of ordinary logic, but we must have some criteria for judging similarities and making
distinctions, and these criteria define the rules of the logic that is being used. In
science, this logic will generally adhere to the Aristotelian laws presented in Unit 9.
They are the laws for talking about fixed, categorical identities, and it would be
difficult to do science if the identity of the object of study was undefined or kept
changing.
The activity of collecting entities into categories, labeling the categories and
searching for structural relations between them is called “systematics,” or more
commonly, “classification.” Entities are placed into categories on the basis of
similarities to other entities in the category, and distinctions that differentiate them
from entities in other categories. The interplay between similarities and distinctions
in this process, and the important role of symmetry are treated in Discussions 11.1
and 11.2; Discussion 11.2 also includes a brief introduction to the “method of
comparison.”
Once the basic elements to be studied are at least partially classified, it becomes
possible to look for patterns in the classification scheme, and to study the various
forms of interaction between the different sorts of entities. One very useful method
in this process is model building. The term “model” is rather vague, and the nature
of models in science is considered in the reading from Chapter 6 of What Science
Is and in Discussion 11.3.
Objectives
When you have completed Unit 11, you should be able to
1. define “symmetry,” and explain the difference between internal, external and
dynamic symmetries.
2. explain the difference between natural and artificial classifications.
3. describe the role of top-down and bottom-up approaches in the construction of
classification schemes, and construct classifications of simple collections of
objects.
4. describe the three aspects of the type concept, and explain the difference
between a typology and a taxonomy.
5. identify the conditions that must be satisfied by typological variables, and
describe the four different scales on which these variables may be evaluated.
6. describe the method of comparison, and discuss its use in science.
7. explain how phylogenetic taxonomies are constructed, and how they can be used
to estimate evolution times.
8. discuss the value of models in science, and describe some of the dangers to be
avoided when using models.
9. describe the different types of models that can be used in science.
10. define “strategic” and “tactical” models, and describe the role of each.
11. describe the relation between theories, models, hypotheses, experiments and
data.
12. state and describe six criteria for the evaluation of models.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 11.1, “Symmetry and Distinction, Similarity and Difference.”
2. Read Chapter 18, “Riding Blake’s Tiger: Symmetry in Science, Art, and
Mathematics,” pages 252-273 of What Science Is.
3. Answer Unit 11 Study Questions 2-3.
4. Read Discussion 11.2, “Classification: Categories, Typologies and Taxonomies.”
5. Answer Unit 11 Study Questions 4-8.
6. Read Chapter 6, “A Universe in a Bottle: Models, Modeling, and Successive
Approximation,” pages 69-88 of What Science Is.
7. Read Discussion 11.3, “Models: Mathematical and Otherwise.”
8. Answer Unit 11 Study Questions 9-12.
Discussion 11.1 Symmetry and
Distinction, Similarity and
Difference
TOP
Tyger! Tyger! burning bright
In the forests of the night
What immortal hand or eye
Dare frame thy fearful symmetry.
In what distant deeps or skies
Burnt the fire of thine eyes?
On what wings dare he aspire?
What the hand dare seize the fire?
—William Blake (1757-1827)[209]
Symmetries and Distinctions
Information consists of differences that make a difference.
—Gregory Bateson[210]
Before we can begin work in any field of science, we need to know what it is that we
want to talk about. That is, we need to have a vocabulary of terms that are
understood to have relatively specific meanings. In addition, however, we need to be
able to state general laws; that is, we do not want to talk in such a way that our
statements are automatically limited to particular cases. For example, it may be
great art to give a precise depiction of a particular cat sleeping on that windowsill, as
in a painting or photograph, but it is not science. The picture of the cat on the sill
might bear the title “Fritz Takes a Nap.” The subject is not really a cat, in the generic
sense, it is Fritz. In science, on the other hand, we are interested in what can be said
about all cats. The “Fritzness” of one particular cat is not a scientific concept.[211]
Use of the term cat, as in Fritz the Cat, implies that there is a general category, “cat,”
of which Fritz is a particular example. And this fact raises a question: Why is it that
we are justified in grouping all of these uniquely individual beasts under the generic
term of cat? Or, in general, what justification is there for generic names? Why can we
talk about particular individuals using names that only refer to classes of
individuals?[212]
The answer is that by ignoring some differences as irrelevant, we notice other
general characteristics that are shared by all members of a distinguished class. We
then justify assigning an identity to this class on the basis of the principle of the
identity of indiscernibles. There are similarities between Fritz and Felix that do not
exist between either Fritz or Felix and Rover, the dog.
It is the existence of similarities between individuals that allows us to group them
into categories, and the existence of differences that yields distinct categories. Let’s
consider again the elegant formulation by G. Spencer-Brown, quoted in Unit 9.
Distinction is perfect continence.
That is to say, a distinction is drawn by arranging a boundary with separate sides so that a
point on one side cannot reach the other side without crossing the boundary. For example,
in a plane space a circle draws a distinction.
Once a distinction is drawn, the spaces, states, or contents on each side of the boundary,
being distinct, can be indicated.
There can be no distinction without motive, and there can be no motive unless contents are
seen to differ in value.
If a content is of value, a name can be taken to indicate this value.
Thus the calling of the name can be identified with the value of the content.[213]
We note from this quotation that the ideas of similarity and distinction are
inextricably connected. In assigning different values to the different sides of a
distinction, we are assuming at the same time that the value on each side of the
distinction is uniform; that is, that all points on a given side of a distinction are
assigned the same value. In this regard, they are all similar. Without this assumption
more distinctions than the given one would be present.[214] In the extreme case, each
point would have a different value, and we would be back at the case of disassociated
individuals.
To make a distinction, we must ignore many other potential distinctions. What we
are looking for, in making definitions of general categories, are the most informative
distinctions, the “differences that make a difference.” What these differences are will
depend on the conceptual framework within which we frame our questions. That
Fritz is tabby while Felix is black with a white nose and paws makes no difference
with respect to the scientific concept of cathood, but may matter vitally if we are
planning to enter them in a cat show. On the other hand, the fact that both Fritz and
Felix are lone hunters who chase birds and mice, while Rover, at least in his wild
state, would hunt in packs and chase deer, does make a scientific difference.
Similarities are established through the concept of symmetry. The proto-IndoEuropean roots for similar and symmetry are respectively sem (one, same),
and me (to measure). Thus, intuitively, two things are similar if they are in some way
the same, if they come under a single name, and this sameness is recognized as a
symmetry, a “likeness of measure.”[215]
We tend to think of symmetry in terms of visual objects that directly exhibit some
form of symmetry. However, there are many other forms of symmetry as well. The
general idea of symmetry relates to invariance under a transformation. This
characteristic is essential for classification in science, which is always based on the
grouping of objects or phenomena together on the basis of common features that are
preserved (i.e., remain invariant) under some set of transformations. For example, in
biology, species are defined in terms of invariance of form under reproduction: the
form of a species is retained in the transformation from one generation to the next.
In the periodic table of elements, each column lists elements contained in a
particular family, having similar chemical properties that are preserved as we move
down the column (i.e., as we increase the atomic weight).
As mentioned, the word symmetry derives from the root me, which has the meaning
“to measure.” The verb “to distinguish” derives from the root steig, meaning “to
stick, pointed, to pierce.” There is a symmetry between two things if they are of like
measure, and distinct things are indicated by pointing. What is being suggested is
that there is a profound complementary relationship between the concepts of
symmetry and distinction. In appearance, they are opposites: if two things are alike
(symmetrical) then they cannot be distinguished, and any distinction made between
two things must, by the identity of indiscernibles, rely on some aspect of those things
where they differ, and hence, lack symmetry. Every distinction involves the breaking
of some symmetry and the highlighting of those symmetries that remain. For this
reason, every scientific classification is based on both the symmetries of the objects
being classified as well as the distinctions between them. Here is a simple
illustration: a baseball is always a baseball; regardless of its location in space—it is
symmetric under spatial translations. But when a pitcher throws it toward a batter, it
is a “ball” or a “strike” depending on its spatial relationship to home plate. That is, a
distinction is defined by the breaking of translation symmetry.
According to the mathematician Hermann Weyl (1885-1955), the way to begin
studying a structure is to determine the transformations that leave it invariant; that
is, to determine its symmetries.[216] For example, if I distinguish a disk against a
uniform background, then I have broken the homogeneity of the background, and
simultaneously, have highlighted the circular symmetries of the disk. I can then use
those symmetries to group together all other disk-like things. Making a distinction, I
have highlighted certain symmetries that can then be used in order to generate
further distinctions by construction of the equivalence class of all things sharing the
highlighted symmetries. This idea applies to qualities as well as to spatial geometric
symmetries. If we distinguish between red and not-red, for instance, we are breaking
a colour symmetry; in distinguishing among bugs with six legs, bugs with eight legs,
and bugs with many legs, we are breaking the “symmetry” of the classifying term
“has legs.”
If we take a bottom-up approach, the fact that a particular object has certain
symmetries allows us to consider the class of all objects having these symmetries,
and this ability may be helpful in finding an inductive or abductive synthesis. Taking
a top-down approach, the requirement that a component of a system fulfil a
particular function may impose limits on the symmetries allowed, and indicate the
kind of distinctions needed to define this component.
For example, suppose that we want to develop a game of chance in which a solid
object having equal faces labeled with distinct symbols is tossed onto a table with
bets placed as to which face will be “up” when the object comes to rest. We want the
game to be fair, we want each face to have an equal chance of coming up, and we
want the object tossed to be as simple as possible. The well-known science writer
Martin Gardner has pointed out that these functional requirements for games of dice
impose a symmetry restriction on the die that is cast: it must be a regular
polyhedron—a volume with congruent faces which are regular polygons and with
congruent internal angles.[217] But it has been known since the time of the
Pythagoreans that only five distinct regular polyhedra exist. Called the “platonic
solids,” they are shown in Figure 11.1.1, below.
Figure 11.1.1: Platonic solids
Any fair die, anywhere in the universe, must have the shape of one of the platonic
solids. A more scientific example is found in elementary particle physics, where the
existence of new particles is predicted on the basis of symmetry conditions. Work
has also been done on the symmetries of living organisms and the relation of
symmetry to the kinds of perceptions an organism would have of the world.
Different Kinds of Symmetries
[T]he qualities of measure and proportion invariably, I imagine, constitute beauty and
excellence.
—Plato[218]
Given two theories, physicists feel that the more symmetrical one, generally, is the more
beautiful.
—Anthony Zee[219]
Symmetry is defined in terms of invariance under transformations. The kind of
transformation determines the kind of symmetry. There are transformations of
geometric figures; more abstract algebraic, permutation and topological
transformations; dynamical transformations; color transformations; and other,
perhaps even metaphorical, transformations, such as those involved in the
establishment of an analogy. The mathematical language for talking about many of
the possible kinds of symmetry is group theory; that is why group theory is so
important in the mathematically sophisticated sciences. The underlying idea,
however, is just that of invariance under some set of transformations.
The easiest symmetries to understand are the geometric symmetries, and we will
focus on them. Consider, for example, the sequence of descriptions provided below.
S1S1 a square of a definite size, with a definite orientation, and with a centre having a
definite location
S2S2 a square of definite size and a definite orientation
S3S3 a square of definite size
S4S4 a square
In each case, we can list the symmetries of the thing described in terms of the
operations (transformations) that can be performed without violating the terms of
the description. These operations are indicated in Table 11.1.1, below. Note that as
the level of description becomes more abstract, the symmetry increases. With this
increase in symmetry, the set of things that we are able to indicate becomes broader,
but the specificity and precision, the information conveyed by our indication,
decreases. That is, the greater the degree of symmetry, the fewer the number of
distinctions that have been made in the description.
Table 11.1.1: Symmetries under transformation
Description Preserves
S1S1
S2S2
S3S3
S4S4
size, orientation,
location
size and orientation
size
“squareness”
Symmetries
a. rotations about centre by 90°, 180°, 270°, 360°
b. reflections about diagonals and bisecting lines
parallel to the sides
a. rotations about centre by 90°, 180°, 270°, 360°
b. reflections about diagonals and bisecting lines
parallel to the sides
c. any translation of the centre
a. all rotations about the centre
b. reflections about diagonals and bisecting lines
parallel to the sides
c. any translation of the centre
a. all rotations about the centre
b. reflections about diagonals and bisecting lines
parallel to the sides
c. any translation of the centre
d. any magnification, expansion or contraction.
Two kinds of geometric symmetries are illustrated in Table 11.1.1. The first are
“external” symmetries, those that are properties of space, or at least of Euclidean
space. In Euclidean space the geometric properties of objects are unchanged if we
move them from one location to another, expand or contract their size, or rotate
them in any way whatever. Or, in an equivalent way of speaking, when we establish a
coordinate system in Euclidean space, we are free to locate the origin at any point
(translation symmetry), to locate the axes in any orientation (rotation symmetry), to
choose the positive and negative directions on the axes at will (reflection symmetry),
and to choose any scale of measurement (dilation; i.e., magnification or contraction
symmetry).
The second set of symmetries is found in the rotations by multiples of 90° and in the
specific reflection symmetries of the square. These are “internal” symmetries,
characteristics of squares in general. A closed figure constructed from four straightline segments is a square if and only if it has these particular symmetries.[220]
In general, then, objects in space will be characterized by both internal and external
symmetries, the former relating to intrinsic properties of the object and the latter to
the freedom of movement and scaling of the object in space. In terms of
classification, it is the intrinsic symmetries that are of interest, since we do not want
the category in which an object is placed to depend on spatial location.[221]
The top-down way of making use of symmetries is to specify some set of symmetry
transformations on the basis of theoretical considerations, and then look
experimentally for objects in the world that are equivalent under these
transformations. That is, a selected set of symmetries is used to define a distinction.
Recall, for example, the symmetry restrictions placed on the possible shape of a die.
If we have reason to believe, or to require, that a system will have certain
symmetries, we can restrict the possible forms that system may assume. This shows
up in elementary particle physics where experimental workers are instructed to
search for new particles having properties predicted from abstract symmetry
considerations. Such symmetry requirements may be imposed for a variety of
reasons, including intuition, structural and functional requirements, or both. What
remains is to work out the consequences of the assumptions made, and to compare
these consequences to experiment and observation. This procedure is described by
the Nobel prize-winning physicist Richard Feynman.
You have an approximate symmetry, so you calculate a set of consequences supposing it to
be perfect. When compared with experiment it does not agree. Of course—the symmetry . . .
is approximate, so if the agreement is pretty good you say, “Nice!”, while if the agreement is
very poor you say, “Well, this particular thing must be especially sensitive to the failure of
the symmetry.” Now you may laugh, but we have to make progress in this way. When a
subject is first new . . . this jockeying around, this “feeling” way of guessing at the results, is
the beginning of any science.[222]
In a bottom-up approach, we begin with a collection of distinct objects and attempt
to group them into equivalence classes on the basis of similar characteristics. To do
so, we must first ask what the significant differences are, and which differences can
be ignored. By this kind of construction, we end up with a set of classes, each of
which is defined in terms of a certain set of similarities—symmetries—and we
associate these symmetries with the identity assigned to each class. In this sense, the
class name carries with it a connotation of all the various symmetries used in
defining the class.
Another kind of symmetry is temporal symmetry. Under this heading we have
displacements in time, corresponding to moving the origin of time; temporal scaling,
corresponding to changes in the unit of measure; time reversal, corresponding to
changing the direction of time; and periodicity, corresponding the periodic
recurrence of an event or state. For example, the periodicity of the solar year
introduces the temporal symmetries of the four seasons, which classify the different
portions of a year.
There are also permutation symmetries that arise in relation to questions such as
“Given NN components of a system, how many different interchanges of these
components leave the system invariant?” The set of interchanges satisfying this
condition form a subgroup of the group of all possible permutations of NN objects.
For example, consider a square digital array that has nn pixels to a side so that there
are N=n2N=n2 pixels in all. Furthermore, suppose that each pixel can be one of three
colours, say red, blue and green. This situation might arise in processing images
broadcast back from a video camera on a space probe. Clearly the image will be
invariant under any permutation of the pixels that exchanges only pixels having the
same colour.[223]
Finally, there are “dynamic symmetries”: symmetries that do not manifest in a
system’s structure, but through its motion. Generally such symmetries are only
useful in sciences that are sufficiently developed that it is possible to write
mathematical equations of motion. In such cases, it is found that these equations
remain invariant (i.e., their form does not change) under certain transformations.
For example, consider Newton’s equation of motion is F=maF=ma, where FF is
applied force, mm is mass, and aa is acceleration. Taking a one-dimensional
example, we will measure position with a variable xx. The acceleration is the second
derivative of position with respect to time, which we will denote x′′x″.
Suppose that we change coordinates to a new coordinate system, defined
by y=x−vty=x−vt, where vv is a constant. The yy-coordinate system is moving with
respect to the xx-coordinate system with a constant velocity vv. The equation of
motion in the xx system is F=mx′′F=mx″. But now x=y+vtx=y+vt,
so x′′=(y+vt)′′=y′′+(vt)′′x″=(y+vt)″=y″+(vt)″, but since vv is independent of the
time tt, the second time derivative of this last term is 00. Hence F=my′′F=my″, and
the equation of motion has the same form in both coordinate systems. That is, there
is a dynamic symmetry under transformations between coordinate systems that
move with constant velocity with respect to each other.
One of the major values of dynamic symmetries is that they are directly related to
conservation laws. The German mathematician Emmy Noether proved that every
dynamic symmetry of a system corresponds to a conserved quantity of that system.
We will not go into this topic, however, except to note that translation symmetry
corresponds to conservation of momentum, rotational symmetry corresponds to
conservation of angular momentum, and time displacement symmetry corresponds
to conservation of energy.
A concept closely related to symmetry is analogy. In an analogy, similarities between
two distinct things or concepts are presented as support for drawing certain
conclusions about one of them from what is known about the other. The analogy,
“Life is like a box of chocolates,” was used effectively in the movie Forrest
Gump.[224] In science, analogies need to be much more precise, although they may be
equally “down home.” In Discussion 14.1, you will find several analogies used in the
early 20th century to think about the structure of atoms. A well-known analogy
found in biology is the “tree of life,” which gives us the image of all living entities as
branches of an evolutionary process that began with a single primitive organism.
Discussion 11.2 Classification:
Categories, Typologies and
Taxonomies
TOP
We take as given the idea of distinction and the idea of indication, and that we cannot make
an indication without drawing a distinction.
—G. Spencer-Brown[225]
The fundamental problem of each science is the establishment of the identity of its
phenomena.
—Bronislaw Malinowski (1884-1942)[226]
These ambiguities, redundancies and deficiencies remind us of those which doctor Franz
Kuhn attributes to a certain Chinese encyclopedia entitled ‘Celestial Empire of Benevolent
Knowledge.’ In its remote pages it is written that the animals are divided into: (a) belonging
to the emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray
dogs, (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with
a very fine camel hair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that
from a long way off look like flies.
—Jose Luis Borges (1899-1986)[227]
One very fruitful method in scientific research is the method of comparison, which
involves making comparisons of similarities and differences between entities
thought to be related in some way. This is a big name for something that we do every
day. When purchasing a car, for example, we may compare five or six different
makes and models in terms of things like price, performance, style, record of
reliability and available options. Our goal in this case is to optimize our personal
satisfaction with the eventual choice. In science, comparisons are made for a variety
of purposes. We may compare different equipment or experimental methods to find
an experimental design best suited to our goals. We may compare two or more
hypotheses to determine which to accept. We may compare different sets of data in
an attempt to discern a pattern of relationships, or compare two theories to
determine which one gives the best account of observations.
In every case, however, the conditions to be met are the same. There must be a goal,
some reason for making comparisons; we will need tools appropriate to this goal;
and we will need to have dimensions for comparisons that are both relevant to our
goals and accessible with the available tools. If the necessary tools are unavailable,
then we will need to change our goals, find new dimensions of comparison, or invent
new tools.[228]
Some of the primary reasons for use of the comparative method are classification,
historical reconstruction, a search for generalizations, and hypothesis testing.
In searching for possible generalizations or hypotheses, the use of comparison often
involves comparisons between deviant and normal states of the system being
studied. Recall that scientific investigations are usually grounded in ideals of natural
order, with theories developed to explain deviations from these ideals. Comparisons
made between different sorts of deviations, and between deviations and the expected
natural state of the system, can provide a basis for the generation of general theories
by induction or abduction of hypotheses about the causes of the deviations. Newton,
for example, was able to show that his general force law—the hypothesis that force
equals the product of mass and acceleration—could explain deviations from uniform
motion in terms of the concept of force, and that this applied to such diverse things
as apples falling from trees, projectiles shot from cannons, and the orbits of the
planets.
Another current example of this method is found in studies of the brain based on
selective lesioning. In studies with rats, for instance, they are partitioned into groups
and a different brain segment removed is different for each group. Furthermore, the
experiment is set up so that overall, the equivalent of an entire brain has been
removed. Prior to all of this, the rats will have been tested to establish a measure of
their normal behaviour and capacities. A comparison between pre-operative and
post-operative performance then yields information on the function of the various
areas of the rat brain.[229] Similar types of studies with stroke patients and braininjured individuals were responsible for much of the early mapping of the human
brain. Perhaps the major use of the method of comparison, however, is in
classification.
The goal in classification is the construction of a taxonomy or typology that allows
the items or entities under consideration to be grouped into classes sharing similar
properties. It should be obvious that science is not possible if we cannot agree on
what we are talking about. Therefore, identification of the objects of study is the first
step in any scientific endeavour. We must “name all of the animals.” But before we
do that, we must collect them into similar groups so that we can name each group.
We do not want to name each individual separately, but according to the group to
which it belongs. That is, we want to classify. Classification and measurement are the
two basic requirements for doing science.
Systematics is the science of classification. Its intellectual pedigree begins with
Aristotle, but the concept of classification is as old as language.[230] All discourse is
conditional on the existence of an adequate set of semantic categories. We cannot
talk very well about something for which we have no name.
There are three fundamental aspects to the identification of objects of sensation as
members of an abstract category: the material or sensory basis, the actually
perceived characteristics of an object; the abstract concept of the category to which
an object is to be assigned; and the linguistic label, the name of the category.
There is also the question of why a particular identification is being made. That is,
there is a goal-oriented (teleological) element in every classification scheme. The
categories defined have been defined for some reason. The biological classification of
animal species, for example, serves a different purpose than a library classification,
but each is intended for a purpose.
Taking a high-level overview, we could say that sometimes we want to talk about
things that are actual objects of our perceptions or thoughts, while at other times we
simply want a convenient way of making a reference. In the first case, we want a
natural classification, that is, one in which the names of the distinguished categories
can be taken as indicating something that exists in the world. In the second case, we
want an artificial classification, one in which categories are established according to
some rule that only becomes useful when accompanied by an index. In reality,
however, most classification schemes involve both natural and artificial aspects.
The process of constructing a classification can be viewed as involving the cyclic use
of the different approaches to reasoning described in Discussion 9.1. Taking a
bottom-up approach, we begin with a collection of samples. By grouping members of
this collection into categories based on their more or less obvious similarities and
differences, we may be able to generate some inductive principles of organization.
Turning to a top-down approach, we superimpose these principles on the original
collection of objects, and apply them to any new objects that are believed to fall
within the bounds of the set of things to be classified.
Although we may try in this process to make distinctions and indications in a way
that maps real objects in the world, success in this attempt will not, in general, be
perfect. Indeed, the extent to which we even try for a natural classification depends
on our purpose. In archaeology, for example, a typology of “North Western Arizona
American Indian Pottery” was developed in 1937[231]. One of the types in this system
is called Tsegi polychrome. Archaeologists find this type useful for tracing trade
contacts with the Tsegi Canyon culture, but for the maker of a particular object
classified as Tsegi polychrome, the meaning of the object would have been “kitchen
utensil,” something used, for example, to mix cornmeal mush. Whatever a member
of the Tsegi Canyon culture called one of these bowls would be more natural than the
basically artificial term “Tsegi polychrome.”
The threefold nature of types (material basis, abstract concept, name) should warn
us that they are theoretically complex entities. In order to speak with more precision
about the science of systematics, we need to make some distinctions so that when a
name is called, we will have some accurate idea of what is being indicated.
The noun “classification” refers to a system of categories and labels constructed for
some purpose. A classification may be natural or artificial. A “typology” is a
classification that has the purpose of segregating all entities classified into nonoverlapping groups called “types.” That is, a typology is a classification that is used
for sorting. The verb “to classify” refers to the activity of distinguishing categories;
“sorting” is the act of putting things into categories that have been distinguished.
Hence, classification involves definition and sorting involves attribution. Finally, a
“taxonomy” is a hierarchically ordered classification in which each level is a
typology.[232]
As a classification designed for the purpose of sorting, a typology must have
properties that are not necessary in other kinds of classification schemes.
Clear Boundaries: Since it must be clear what is to be sorted and what is not, the
boundaries of the typology must be clearly specified. Usually these boundaries are
described, at least implicitly, in the name of the typology itself. For example, the
types of animal species, or the North Western Arizona American Indian Pottery
typology.
Completeness: Every object or thing that is within the boundaries of a typology must
be sorted into a specific category; that is, the set of types composing the typology
must exhaust the possibilities to be categorized. We say that it must be “exhaustive”
or “complete.” In practice, this often means that there will be a catch-all category
labeled “none of the above.” It also means that there must be a way to include new
types in order to account for the possibility of new discoveries.
Exclusive Types: Each entity being sorted must be assignable to only one category.
That is, the types making up the typology must be mutually exclusive.
The structure of a typology follows the axioms of Aristotelian logic. That is, suppose
that an object XX is identified as Type AA. Then, from the law of identity we are
justified in saying that XX is an AA. For example, Felix is a cat. Likewise, the
condition that types are mutually exclusive corresponds to the law of contradiction.
If we know that XX is Type AA, then we also know that it cannot be not-Type AA;
that is, it cannot be Type BB. And the completeness of the typology corresponds to
the law of the excluded middle. XX is Type AA, or it is Type BB, or it is Type CC . . .
but it must fall into one of the categories.
In addition to the three structural criteria of specified boundaries, completeness and
exclusive types, there are four criteria of definition and interpretation that a typology
needs to satisfy.
Consistency: All types must be defined on the basis of the same criteria of identity. It
would not do, for example, to try to define types on the basis of whether the entity
being classified was large or small or green.
Equivalence: All types in a typology are of equal importance.
Equidistance: All types are considered as being equally distinct from each other.
Independence: The presence or absence of any type in a sample is not dependent on
the presence or absence of any other type.
We might want to question the final two criteria. Are we to consider cats, for
example, as being equidistant from lynxes and from iguanas? Certainly not! What
about parasites? If their host species is not found in a sample, then the parasite
species is not likely to be found either. What is needed is some way of taking such
relationships into account.
To indicate relationships between types it is necessary to go to a higher level of
discourse—to develop a taxonomy. Beginning with a given typology, we consider
each type as an individual, and we group the types into categories that become new
types at the next higher level. We proceed with this process until we have reached
either a single highest-level type, or we have run out of significant data. The process
is illustrated in Figure 11.2.1.
In developing a typology, the first step is to establish the boundaries by deciding just
what is to be classified. Arrowheads, pottery, chemical elements, elementary
particles, animals, plants, ships, diseases, minerals, climate, stars, cabbages and
kings are just a few of the possibilities.
Once this decision is made, the next step is to decide how the classification is to be
done. That is, we need to determine what will be considered significant variables.
The variables of a typology are the dimensions of variability on which entities within
the boundaries of the typology may be distinguished and sorted. The variables
chosen are ones that are determined to be relevant for the purposes for which the
typology is being constructed. They are the dimensions on which a difference makes
a difference. Colour, for example, is not relevant for distinguishing cats from dogs,
but it is highly relevant in distinguishing different types of stars (e.g., red giant,
white dwarf, blue giant).
Figure 11.2.1: Developing a taxonomy
In addition to relevance, the variables of a typology must satisfy the three criteria
identified below.
Universality: All relevant variables are relevant to each type. Even if a particular
type does not display a variable this absence is, itself, of significance. For example,
even though fish do not manifest the variable “legs,” it is significant for biological
classification that they do not do so.
Qualitative Difference: Different variables must be qualitatively distinct dimensions
of variation.
Logical Independence: Variables may be chosen independently, even if they happen
to be correlated. The xx and yy coordinates in a two-dimensional Cartesian
coordinate system, for example, are logically independent, even if there is a
function y=f(x)y=f(x) that relates them.
The “attributes” of a variable are the possible values it may manifest. Each entity
being classified will exhibit a single attribute (including the possibility of “not
present”) for each variable. In classifying animal species, for example, the variable
“legs” can have values none, two, four, six, eight or many (as in a centipede).
Attributes are usually quantitative rather than qualitative, and are defined with
respect to a scale that will generally be one of four kinds, described below.
Nominal: Attributes display qualitative differences (e.g., colours).
Ordinal: Attributes are rank ordered along a linear scale (e.g., lining up by height).
Interval: Attributes fall along a uniform number scale with a measure defined by a
specific interval (e.g., there are 100 Celsius degrees between the freezing and boiling
points of water).
Ratio: Attributes fall along a uniform number scale with a measure defined by
reference to a standard unit (e.g., the standard kilogram [kg]).
Once variables are selected and attribute scales established, we have a “parameter
space.” Entities to be classified are represented as points in this space, determined
by their particular attributes on each variable. If the variables have been chosen
carefully, these individual points should fall into easily identifiable clusters, each
cluster corresponding to a type. If this does not happen, it may indicate that new
variables need to be considered. A simple illustration of this phenomenon is shown
in Figure 11.2.2, where the two clusters shown in (a) appear as the single cluster of
(b) when the vertical dimension is omitted. The actual determination of type clusters
is a matter of interplay between variable selection and data analysis, involving the
kind of good intuitive feel for the data that can only come through experience.
Figure 11.2.2: Influence of attributes on clustering
Once type clusters are obtained, they must be named. If we are only interested in an
artificial classification, it is sufficient to assign arbitrary labels or numbers. But if the
types are intended to be natural, then the name is important. The type name will
exert an influence on the choice of what entities are subsequently classified as being
of that type, and so should be chosen with care. Selection of type names involves the
attributes defining the type cluster, the purpose of the typology, and whatever
theoretical background may have been involved in selection of the variables. Ideally
a type name will be an indicator of the relevance or function of the type in the
context of the purpose and theoretical background of the typology. If so, then the full
set of type names for a typology will provide a linguistic map of the parameter space.
Once the types are named, we can incorporate the names into thinking. We order
this linguistic structure, and look for regularities and relationships between the
types. This activity may lead to a redefinition of the typology in terms of inductively
derived theoretical principles. For example, in the early days of particle physics it
was noted that only certain types of particles seemed to exist, and these seemed to
fall into specific families. A phenomenological (i.e., these all seem to fit together)
classification of elementary particles was constructed based on this information.
Eventually the physicists Yuval Ne’eman and Murray Gell-Mann independently
discovered that this phenomenological typology could be theoretically generated on
the basis of certain symmetry principles, and furthermore, that the resulting
typology could be used to predict the existence of new types of particles.
Types have two basic properties—identity and meaning. The identity of a type relates
to its location as a region of a parameter space; that is, to its attributes. The meaning
of a type relates to the variables that define the parameter space, and the
information provided by knowing that a given entity is a particular type of thing.
Meaning also relates to the way the type concept is woven into our tapestry of
empirical thought—how it relates to other concepts and ideas.
We must distinguish between empirical types, determined from observation and
comparison, and ideal types, posited for theoretical purposes. The classification of
dogs into different breeds is empirical, based on examination of the dogs’ pedigrees.
But the judges in a dog show will judge each dog by how close it comes to
exemplifying the ideal type of its breed. Ideal types are sometimes stated in terms of
polar opposites. These opposites can be absolute, as with positive and negative
electric charge, or they may denote the opposite ends of a spectrum of possibilities,
as with red and violet in the visual spectrum. In the latter case, any actual instance
will fall somewhere between the poles.
One method of classification based on ideal polar types is known as the Greek cross.
The ancient Greeks identified two qualitative polarities, wet/dry and hot/cold. They
then constructed a 2×22×2 array as shown in Table 11.2.1, below, labeled by their
four elements.
Table 11.2.1: 2×22×2 array of Greek elements
Wet
Dry
Hot
Air
Fire
Cold
Water
Earth
On this basis, things could be classified as being relatively fiery, airy, watery or
earthy. In a more modern application of this same idea, the psychologist Carl Jung
defined the ideal polar types of thinking/sensing and reasoning/intuiting. These
polarities are used in the widely used Myers-Briggs personality inventory to classify
people into one of 16 categories based on their scores on each of the four
dimensions.
Table 11.2.2, below, shows an array of polarities relevant to construction of
typologies; compare it to Table 9.1.2 in Discussion 9.1.
Table 11.2.2: Polarities important in the construction of typologies
bottomup
induction/
abduction
discovery
natural
intuitive
essential
top-down
deduction
invention
artificial
rational
instrumental
Reading across the two rows of this table, we could say that we inductively or
abductively discover natural types by our intuitive recognition of their essential
features, and we deductively invent artificial types by rationally constructing them
for instrumental purposes. As in the dichotomies of reason, however, the way
typologies are actually constructed involves the interplay of both of these
approaches. The classification of elementary particles mentioned earlier offers a
good example of this dual process. Indeed, one of the signs of a good theory is that it
predicts an “artificial” typology that matches and extends an inductively generated
“natural” typology.
In modern systematics, there are three general ways of defining types: the
Aristotelian, the statistical and the phylogenetic.
An Aristotelian typology contains only a few variables, chosen because they are
believed to relate directly to the “essential” characteristics of the entities being
classified. Each entity of a given type must possess all of the essential characteristics
of that type. In practice, however, it is often difficult or even impossible to determine
essential characteristics. What, for example, are the “essential” characters that
distinguish a cat from a lynx, or either from an iguana?
As an alternative to Aristotelian classification, types may be distinguished
statistically, on the basis of a large number of variables, each of which is considered
relevant, but none of which is necessarily considered essential, for type identity.
Instead it is assumed that each entity to be classified possesses attributes on a large
but unspecified number of the chosen variables, and that each variable is relevant to
almost all entities, but no variable is necessarily possessed by every entity in the
domain of the typology. This kind of classification is based on statistical measures of
similarity and difference. It has become popular since the development of large
computers that are able to handle the tremendous amounts of data necessary for
good statistical results.
Phylogenetic classification is basically a family tree: all members of a particular type
are assumed to have descended directly from an earlier type. We have already
questioned the equidistance requirement for typologies. Cats and lynxes are closer to
each other than either is to an iguana. As mentioned earlier, this kind of relationship
is treated by converting our typology into a taxonomy by introducing a hierarchical
ordering. Taking a bottom-up approach, we would group certain types together on
the basis of apparent similarities, and on this basis define higher-level types. In a
top-down approach, we would make distinctions within a type to define lower-level
types. Both of these approaches are illustrated in Figure 11.2.3, below.[233]
Figure 11.2.3: Top-down and bottom-up approaches to phylogenetic classification
Research in cognitive science indicates that in cognition, the mind most easily uses a
“base-level” typology in which the types are defined at the most inclusive level at
which they remain able easily to fit the structure of attributes perceived in the
world.[234] Thought then abstracts higher-level types, or differentiates more refined
levels, from this base level, as required by circumstance. Two examples of this
process are shown in Figure 11.2.4, below.
Figure 11.2.4: Perception and thought in the development of typologies
In going from a type at some hierarchical level to subtypes at the next lower level, we
treat the entities classified under that type as the domain for a new typology,
introducing variables and attributes as before. In going to a supra-type at the next
higher level, we treat types at the given level as individual entities and cluster them
into groups. Hierarchy is particularly important in phylogenetic classification, where
similarities between types are treated as representing descent from a common
ancestor. Some of the fields in which this approach has proved particularly useful are
evolutionary biology, demography, epidemiology and linguistics.
One way in which a phylogenetic taxonomy is particularly useful is when it is
possible to estimate distances between types assumed to have a common ancestor,
and also to estimate the rate of divergence between the types after their separation.
On this basis, it is possible to establish a timeline and estimate how long ago the
separation took place. In this way, studies of differences in mitochondrial DNA
between existing human groups has been used to estimate that all living humans
share as one of their ancestors a woman who lived in northern Africa about 200,000
years ago. Similarly, this method has been used in linguistics to estimate that protoIndo-European, the ancestor of all the Indo-European languages, was spoken in
Central Asia up until about 6000 to 8000 years ago. The example that follows
illustrates this kind of analysis.
Anthropologists note that there are two tribes, living on either side of a wide desert,
with no intercommunication. Nevertheless, it appears that there are many
similarities between their languages. It is hypothesized that both tribes are the
descendants of an earlier nomadic clan. The question that the anthropologists want
to answer is, “How long ago did the branching take place?” As stated, this question
would be difficult to answer. We will consider a very much simpler problem that
illustrates the method.
Assume that each tribe has a vocabulary of eight words, all of which are four-digit
strings of 00s and 11s. Also, assume that when the original group divided into two
parts, each part started out speaking the same language, and that the change of the
language in each of the two tribes occurred as described below.
o
o
o
A word could mutate only by the flip of a single digit (e.g., a 00 becomes
a 11 or vice versa).
Once a mutation has occurred, the word becomes fixed in the new tribal
language and does not mutate again.
The average mutation rate is one mutation every three generations (say, about
every 75 years).
We now have a basis for analysis. We will proceed by making comparison of “words”
with similar meanings in the two languages. Suppose that we obtain the following
list of words with equivalent meanings:
Language 1
Language 2
0001
0101
0101
1100
1001
0011
0110
0110
0111
1110
1011
0111
1101
1111
0111
0111
We know that both members of each pair of words can only be derived from an
earlier word in the “proto-language” by the flip of a single digit. Therefore, for each
pair, we can list the possible cases.
(0001, 0101) The original word is either 0001 or 0101.
(0101, 1100) The only possible original word is 0100.
(1001, 0011) The only possible original word is 0001.
(0110, 0110) No mutation has occurred.
(0111, 1110) The only possible original word is 0110.
(1011, 0111) The only possible original word is 1111.
(1101, 1111) The original word is either 1111 or 1101.
(0111, 0111) No mutation has occurred.
Going back over this list, we see that if there is a single possible original word then
two mutations must have taken place (one for the new words in each of the two
languages); while if there are two possible initial words then only a single mutation
can have taken place (i.e., one of the two words is the original word). Thus we can
count the number of mutations which, given our initial assumptions, have occurred
since the original clan divided into two tribes. There are 10 of them, and by
assumption, they have occurred on the average at intervals of about 75 years. Thus,
we estimate that the original clan divided into the two tribes about 750 years ago.
It should be clear that this example is highly simplified. In a more realistic case, it
would be necessary to consider many words; to estimate the probabilities of changes
in many different phonemes; to take into account the fact that several mutations
may have occurred, some of which might return words to earlier states; to assume
that many words may have been incorporated from other languages, or just made
up; and to use an analysis based on both the mean and variances of the resulting
probability distributions.
In other words, it would be a very messy job. But this kind of work has been carried
out to obtain a reconstruction of many language families. For example, the family of
proto-Indo-European languages is shown in the phylogenetic tree of Figure 11.2.5,
below.
In these examples, the reconstruction is diachronic, referring to a historically
generated family tree. In other cases, the reconstruction might be synchronic,
referring to spatial distribution over a short time. The reconstruction of the spread of
an epidemic, for example, does not aim at finding a historical family tree, but rather
at establishing a chain of transmission that will reveal the original source, potential
carriers and mode of transmission of the disease. Snow’s study on the spread of
cholera was work of this kind.
Recall that the three elements of a type are the material base (i.e., the set of
exemplars), the concept of the type, and the name. While it is easy to see how a
collection of similar entities may be grouped together and given a name, there are
serious philosophical questions about how an abstract concept is derived. Indeed, it
does not seem that this derivation is something that can be accomplished by a
machine. Computer-generated statistical typologies are often rather bizarre in the
way that they define types, unless watched over by a person who has experience of
the entities being classified.
Figure 11.2.5: Proto-Indo-European language tree
In direct terms, the question is how we abstract from a finite, limited set to an
abstract universal concept. How is it, for example, that we can recognize that “tree”
right there—which we have never seen before and which is different from any other
tree we have ever seen—as a tree? But we can all recognize a tree when we see one.
Somehow we have gained an understanding of the concept of “tree” that goes beyond
the finite number of trees we have seen in our lives. In Aristotelian language, we
have grasped the essence of “treeness.” Furthermore, we are able to use the concept
metaphorically and analogically, as in statements such as “The Tree of Liberty must
be refreshed from time to time with the blood of patriots and tyrants.”[235]
Finally, we note that the use of a generic name to refer to distinct individuals can be
justified on the basis of the principle of the identity of indiscernibles. That is, in
considering things of the world, we are not interested in all the differences that
distinguish them in all of their unique and particular individuality. Rather, we are
interested in “the differences that make a difference.” There is an inherent
purposefulness in naming, and only variables that are relevant to this purpose need
to be considered. Things that are indistinguishable on the basis of these variables
can be grouped under the same name. They have the same typological identity
although not, of course, the same individual identity.
One of the classic errors in thinking is to assume that, because two things have the
same name in a typology constructed with regard to one purpose, they will still be
the same in a typology constructed for some entirely different purpose. And one of
the hallmarks of creative thought is the ability to see the thing rather than the thing
named, and thus see alternative possibilities in it that are obscured by the standard
associations connected to its ordinary typological identity. In the words of Lao Tzu,
“The name that can be named is not the enduring and unchanging name.”[236]
“I never knew what a good doorstop I would make,” said the dictionary.
Discussion 11.3 Models
TOP
Philosophy is written in that great book which ever is before our eyes—I mean the
universe—but we cannot understand it if we do not first learn the language and grasp the
symbols in which it is written. This book is written in the mathematical language, and the
symbols are triangles, circles, and other geometric figures, without whose help it is
impossible to comprehend a single word of it, without which one wanders in vain through a
dark labyrinth.
—Galileo Galilei (1564-1642)[237]
The map is not the territory.
—Alfred Korzybski (1879-1950)[238]
The menu is not the meal.
—Alan Watts (1915-1973)
One of the most useful tools of science is the model. There are both mathematical
and non-mathematical models, and models can be both theoretical and mechanical.
A model of a new aircraft design, used to carry out wind tunnel tests, is an example
of a mechanical model, while the Freudian model of mind, based on an analogy to a
steam engine, is an example of a non-mathematical, theoretical model. Some of the
value of using mathematical models in science is indicated in the following quotation
taken from the ecologist Robert May’s book Stability and Complexity in Model
Ecosystems:
This work seeks to gain general ecological insights with the help of general mathematical
models. That is to say, the models aim not at realism in detail, but rather at providing
mathematical metaphors for broad classes of phenomena. Such models can be useful in
suggesting interesting experiments or data collecting enterprises, or just in sharpening
discussion.[239]
In general, we distinguish between models and theories, although this distinction is
not sharp and the same formal system may be referred to by some scientists as a
theory, and by other scientists as a model. In Kuhnian terms, we might think of a
scientific paradigm as a specification of an open set of canonical models that provide
exemplars of how the paradigm is to be applied in explanation and prediction,
together with a general theory that justifies them in terms of the paradigm.
For purposes of this discussion, we define a model as a mechanical or theoretical
construct designed to study a particular real world situation or phenomenon. Often,
models are stated in terms of a given theory or compatible collection of theories.
Even for mathematical models, these theories do not need to be mathematical
theories. Indeed, one of the uses of mathematical models is as a step toward
development of mathematical formulations of theories. Mathematical models are
those based on mathematical formalisms. They are often the most useful sort of
model, because they allow precise numerical computations. In some cases, however,
even a mathematical model will only be used in a heuristic way.
For example, consider the Freudian model of the mind, in which the mind is
compared to a steam engine.[240] In this model, the id (the complex of basic
instinctual drives, such as survival and sex) is compared to the fire; the superego
(learned social constraints) to the boiler, which confines the steam produced by the
fire; and the ego to the control, which directs the output into useful work (or, if the
situation requires, “lets off steam” in a socially approved way).
If we take this model as more than just an analogy, we could turn it into a
mathematical model based on the ideal gas law (the thermodynamic law that
describes the operation of steam engines). According to this
law, PV=kTPV=kT, PP being pressure, VV volume, TT temperature,
and kk Boltzmann’s constant, which we ignore for the purposes of this exercise. To
obtain a model, we must interpret the variables of this equation in terms of Freud’s
theory. For example, we could define the “temperature” TT as a measure of libido
(sexual energy);[241] the volume VV as a measure of superego constraints (relating, for
example, to the “volume” of “psychological space” left available to libido); and the
“pressure” PP as a measure of energy available for “sublimation,” that is, channeling
into socially productive work.
Writing P=T/VP=T/V then allows us to draw conclusions relating these variables.
For example, the energy a person has available for productive work is proportional
to the strength of their sex drive and inversely proportional to the “volume of psychic
space” available for the sexual energy. We might then recognize that we can relate
this rather vague idea of a volume of psychic space to the Freudian notion of
“cathexis,” which refers to the fixation of sexual energy on particular ideas, emotions
or objects. The equation would then say that the narrower the focus of this fixation
(i.e., the smaller the “volume” VV), the more energy will be directed towards the
cathected ideas, emotions or objects. The role of the ego, then, is to direct this
“pressure” in productive ways. On the other hand, if there are few constraints on
possible objects of sexual fixation (i.e., the volume is large), then the energy available
for productive work is low—it is dissipated into the larger volume and lost. Thus, the
need for superego constraints to prevent the “volume” from becoming too great (that
is, they limit the range of possible objects for the sexual energy). This means that if
“temperature” increases beyond a certain level the pressure must increase, and if the
pressure gets too great for the ego to handle, neurosis will result.
This model gives a superficial overview of Freud’s ideas, but it may aid in thinking
out general aspects of Freud’s theory. Otherwise, however, it is trivial. The reason is
that the concepts of “libido as energy,” “volume of psychic space” and “productive
work” are too vague to quantify. The model’s only usefulness is that it shows certain
relationships in an easy to understand form, but it must not be taken seriously.
Another example of a model based on an analogy is the Bohr model of the hydrogen
atom.[242] In this model, the hydrogen atom is viewed as a miniature solar system,
with the proton as “sun,” the electron as a “planet,” and the inverse square
gravitational force replaced by the inverse square electromagnetic force. Bohr found
it necessary to modify this analogy, however, to deal with the fact that in important
respects the electromagnetic force between proton and electron is not analogous to
the gravitational force between sun and planets, even though both obey inverse
square laws.
The difference is that an accelerating electrical charge radiates electromagnetic
energy. Thus, although we expect a planet moving in a circular orbit to continue in
that orbit forever, an electron moving in a circular orbit would, on the basis of
classical electromagnetic theory, be expected to radiate energy. This process would
lead to decay of the orbit, and the electron would spiral into the proton. This
conceptual difficulty led many early atomic physicists to reject the solar system
model for atoms in favour of what was called the “plum pudding model.” In this
model, the electrons and protons were assumed to be lumped together in the atomic
nucleus (the electrons embedded like raisins in a plum pudding). However, the
physicist Ernest Rutherford (1871-1937) carried out a series of experiments at the
Cavendish Laboratory that discredited the plum pudding model, while Bohr showed
that for the hydrogen atom (the simplest case) the solar system model predicted the
emission spectrum of hydrogen.[243]
The Bohr model was based on Newtonian mechanics combined with
an ad hoc modification of Maxwell’s theory of electromagnetism.
The ad hoc assumption—that electrons could only occupy orbits for
which L=nh/2πL=nh/2π, and would only radiate energy in jumping from one to
another of these orbits—resulted in a model that worked. That is, it allowed for
predictions (of the frequencies of radiation emitted by hydrogen atoms) that fit
experimental results. This success motivated efforts to find a justification for Bohr’s
angular momentum assumption.[244] Such a justification was obtained about a decade
later, with the discovery of quantum mechanics.
The quantum mechanical Schrödinger equation, applied to the hydrogen atom,
predicts that electron states in this atom will have angular momentum given by an
integer multiple of h/2πh/2π. Thus, the Schrödinger equation for the hydrogen
atom provides an example of a mathematical model based directly on a
mathematical theory. But quantum mechanics also undercuts the solar system
model, since in quantum theory it is not possible to pin electrons (or anything else
for that matter) down to a specific position and momentum. So, it is not legitimate in
quantum mechanics to think of an electron as a point mass following a definite orbit.
We have now seen three examples of mathematical models. In the Freudian model,
all that is captured is certain rather vague relationships, while the Bohr model has
the possibility of quantification, hence of comparison with experimental results. And
the Schrödinger equation for the hydrogen atom is an integral part of quantum
mechanical theory: both the model and its predictions are mathematical through
and through.
Figure 11.3.1, below, diagrams of some of the relations among theories, models and
experiments, as these terms are used in this course. This figure is, of course, highly
schematized. What it suggests is that scientific activities are always carried out
within a context of certain basic assumptions, often of a metaphysical nature, which
correspond in the figure to the box labeled “Conceptual framework.” These
assumptions include ideals of natural order, and they determine the nature of the
paradigm employed. The paradigm, in turn, determines the theories that are
considered, the criteria for valid experiments and the sorts of data that are relevant.
Results of these experiments, after analysis, become data that are assimilated into
the web of paradigmatic assumptions.[245] It may turn out that some of these
assumptions need to be changed to accommodate new data, and a change of
paradigm may occur.
Figure 11.3.1: Theories, models, experiments and data
Application of theory to specific situations yields a set of models that act as
descriptions of the world and exemplars of the paradigm. The models can be used to
predict experimental results, or can be adjusted to fit these results. If there is a lack
of fit, or if a prediction is wrong, one attempts to reinterpret the data, change the
model or modify the theory. Only in extreme cases will a change of paradigm take
place.
In cases where a developed theory does not exist, models, coupled with some basic
paradigmatic assumptions, can provide a basis for theory construction. In such cases
models are constructed to fit experimental results (e.g., the Bohr atom) in the hope
that it will be possible to discover a theory either by abduction from the models, or
by exploring ideas suggested by contemplation of the models. If a developed theory
exists, it will be accompanied by a number of models as paradigmatic exemplars.
Some of these models will be those from which the theory was developed, while
others will be models deduced from the theory. We call models used in an attempt to
discover a theory “strategic models,” and models deduced from a theory as
exemplars “tactical models.” Alert students will immediately recognize the bottomup/top-down reasoning cycle here.
With strategic models, we are engaged in a bottom-up approach, seeking models that
can be generalized to suggest new theoretical possibilities. With tactical models we
are engaged in a top-down approach, seeking to apply general theory to a specific
situation. So, for example, we would consider the Bohr atom to be a strategic model
and the Schrödinger equation for the hydrogen atom to be a tactical model.
A third type of model, which can be used either strategically or tactically, is the
“simulation model.” Simulations have become increasingly popular with the
development of powerful computers. They are particularly useful in cases in which
experimentation is not possible. In meteorology, for example, it is not possible to
conduct experiments to determine whether a particular change in the jet stream will
result in more or less rainfall on the West Coast of North America. Instead, a
computer model is constructed which attempts to simulate North American weather
patterns.
Simulation models are explicitly based on theoretical considerations (which
determine, for example, the equations fed into the computer), and are tested against
known data to determine the range of their validity. An issue with simulation models
is determining whether or not they are valid. Just because a simulation produces
output that appears similar to known data this, of itself, is insufficient to imply that
the assumptions going into the simulation are descriptive of the reality being
simulated. These assumptions require other sources of validation.
Constructing Models: An Overview
This section presents a brief description of the modeling process. It may seem
complicated, and will require some effort to learn, but it is important that you do so
because, suitably adapted, the process also provides a good description of the
bottom-up/top-down cycle that plays a central role in the process of scientific
reasoning.
The modeling process is divided into nine stages that are given simple mnemonic
names. It is important to realize that the name assigned to each stage is intended
only to characterize the kind of activities in that stage in a very general way. Specific
activities carried out in any given stage will relate to the name of the stage, but the
form of this relation will depend on the particular context, and on the characteristics
of the model being constructed. The nine stages are listed below.
1. Posing the question
2. Selecting variables and parameters
3. Existing empirical and theoretical resources
4. Analytical tools (conceptual, mathematical, etc.)
5. Descriptive outline of basic model
6. Empirical and observational methods, preliminary testing
7. Formalization, simplification, enrichment
8. Conjectures and hypotheses
9. Conceptual framework, paradigm, ideals of natural order
Modeling begins with the recognition of a question and the decision to construct a
model as a means of seeking an answer to this question. One may have in mind other
models, or cases, which may provide clues or analogies, and in asking a specific
question (even if it is only “What would happen if . . .”), one has some idea of the
parameters and variables that will be involved in a satisfactory answer. Actual
selection of the variables and parameters, however, involves knowledge of the
available analytic tools. For example, if a mathematical model is to be constructed,
then the choice of continuous or discrete variables will depend in part on whether
the existing analytical tools in the field involve differential equations or difference
equations. Other questions could include “Will it be useful to use linear regression
analysis?” “Is set theory relevant?” “What about vector algebra?” “Would a computer
simulation be useful?” Thus, the move from Stage 1 to Stage 2 is mediated by Stage
4.
Once the basic variables and parameters to be included are chosen in Stage 2, the
input from Stage 3 is considered. What empirical results are already known? What
are the theoretical interpretations of this data? What analogies might be available?
This information leads to the formulation of some tentative conjectures or
hypotheses in the analytical language available. That is, Stages 4 and 8 both are
brought into operation, resolving in Stage 5 with the development of a descriptive
overview of the model being constructed. With this overview, it becomes possible to
carry out some simple experimental or observational tests (Stage 6), and on the basis
of the results to formalize the model further, simplifying it by eliminating
unnecessary variables or assumptions, and enriching it by extending its scope to
more complex cases (Stage 7). With the enriched model, the original conjectures and
hypotheses may be modified, and new ones will be generated (Stage 8 again). At the
same time, the process cycles back to Stage 1, since these new conjectures and
hypotheses will lead to new questions. And in this ongoing process, the results
obtained are assimilated into the existing paradigm, unless some radical change in
the paradigm becomes necessary to accommodate the new information.[246]
The sequence of stages is valid for both strategic and tactical models, although the
actual processing at each stage depends on the type of model being constructed.
William T. Morris has listed six criteria for evaluation of models.[247]
Accuracy: How closely does the model fit the empirical data?
Relatedness: How closely is the model connected to known theories and results?
Transparency: Is the model easy to understand? Does it appeal to our intuition?
Robustness: How sensitive is the model to changes in its basic assumptions?
Fertility: Can a rich variety of consequences be deduced from the model?
Ease of Enrichment: Has the model reached the limits of its possible development,
or can it be enriched; that is, does it lead to more comprehensive models?
The criterion of robustness is of particular interest. In one sense, it is not a good idea
to have models that require very specific assumptions that cannot be changed—such
models cannot be easily enriched. On the other hand, a model without such
assumptions may be useful as a guide to intuition, but it can provide little
information—it will not be fruitful. Thus, it is necessary to distinguish two sorts of
modeling assumptions, those that are essential and those that can be modified. The
Bohr model of the atom, for example, posited that electrons moved around the
atomic nucleus in circular orbits with an angular momentum that was an integer
multiple of Planck’s constant divided by 2π2π. In further developments of this
model, it turned out that it was robust under a change from the assumption of
circular orbits to an assumption of elliptical orbits. Any change whatever in the
angular momentum assumption, however, would destroy the predictive power of the
model. In other words, the angular momentum assumption contained the essential
new information of the model.
Finally, we can consider some of the values of models, mathematical and otherwise,
and some of their dangers. The list below outlines some of the reasons modeling is
useful in science, for it can
o
o
o
o
o
suggest general principles.
indicate new directions for empirical research.
suggest conclusions that are not possible to reach through observation alone.
highlight relationships in large quantities of data.
provide an overview for organizing and interpreting data.
On the other hand, it is important to realize that there are also dangers in modeling.
It can
o
o
lead to “mistaking the map for the territory”; that is, focusing on the model
rather than the reality it claims to model.
result in abstract arguments and analyses unrelated to the questions of
significance in the field in which the modeling is being carried out. For example,
there is a famous mathematical model in ecology known as the Lotka-Volterra
model. While this model provided much useful insight into ecological systems, it
was also over-extended by some researchers, who deduced from it that certain
ecological systems had to have an even number of species. This conclusion led to
o
much theoretical speculation about why nature would arrange things so that
there was always an even number of species in an environment.
force data into an inappropriate form and obscure data of significance (like the
“bed of Procrustes” in Ancient Greek mythology).
On this basis, we see that modeling is an important tool for science, but it is equally
important for a scientist who is engaging in this activity to maintain a detached and
critical attitude toward the models being used. As the aphorisms at the beginning of
this discussion have it, the map is not the territory and the menu is not the meal.
Unit 11 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. Write a short essay (300-500 words) on the relationship between symmetries,
distinctions and the principle of the identity of indiscernibles.
2. Many organic compounds in nature come in both right-handed (dextro) and
left-handed (levo) forms, yet all life is based only on the left-handed
form.[248] When organic compounds are found in nature, they are almost always
of the left-handed form, but when they are prepared in the laboratory, half are
left-handed and half are right-handed. Furthermore, the difference between the
two forms can be determined by the way that solutions containing them refract
light. Based on this information, suggest a solution for the following problem
facing Sherlock Holmes.
Martha Weatherby died after eating a soup containing some mushrooms
that may have been poisonous. Holmes suspects, however, that the soup
was artificially doped with a synthetic mushroom poison concocted by her
husband, the fiendish chemist Walter Willingdon Weatherby. How is he
to determine whether his suspicion is correct?
3. Mathematically speaking, a “necklace” consists of some number of “beads”
strung in a circle. Thus, it has a rational symmetry about the centre of the
circle—it does not matter which bead is at the top. It may or may not matter if
one moves clockwise or counter-clockwise about the circle.
a. For five beads that can be either black or white, construct an example of a
necklace other than one with all black or all white beads.
b. Construct a classification of all necklaces that can be made with three
beads that can be red, blue or yellow. Put all necklaces that are related by
a symmetry into the same category.
4. Listed below are three different categories of psychological traits on which
individuals can be measured. People who score high on one trait will tend to
score high on other traits in the same category, suggesting that all traits in a
category are related to a single, underlying variable. By considering the traits in
each category, suggest possible type names for the three categories. Make these
names as descriptive as possible of what you think might be the underlying
variable.
Category 1: visualization; speed of recognition; spatial scanning; recognition of
embedded figures; spatial relations; memory for designs; associative memory
Category 2: deduction; induction; number recognition; syllogistic reasoning;
verbal comprehension
Category 3: use of synonyms; associational fluency; expressional fluency; facility
with words; originality
5. On the basis of the type names you made up in Question 3, think of three or
more other traits for each type that might correlate with the traits listed.
6. Suppose that a collection of plants is to be classified in terms of four distinct
characteristics. The classification is based on whether a plant has a particular
characteristic or not. Thus, each plant can be represented by a four digit binary
number. The number 1011, for example, would label plants having the first,
third and fourth characteristics, but lacking the second. A total of seven fourdigit binary numbers end up as labels for the categories of plants: 1111, 1110,
1011, 1010, 0110, 0011, 0010.
Assuming that all of the plants have evolved from the original species labeled
0010; that any given mutation will result in the presence of only one new
characteristic; and that once a given characteristic is present, it remains,
construct a phylogenetic tree for these plants.
Note: It may not be possible to determine all links uniquely; for those that cannot be so
determined, use question marks.
7. An anthropological expedition discovers two civilizations living on different
planets in the same solar system. On the basis of similarities in the languages
spoken by the two peoples, it is hypothesized that both planets were settled by
members of the same colonization expedition, and that they subsequently lost
contact with each other. By making some estimates about rates of phoneme
substitution and comparing the two existing languages, the linguists hope to
work out an estimate of how long ago the separation took place.
Suppose that each planet has a language in which the vocabulary consists of
eight four-digit binary numbers. Matching up words with similar meanings
yields the lists shown below.
Language 1
Language 2
0001
0101
0101
1100
1001
0011
0110
0110
0111
1110
1011
0111
1101
1111
0011
0011
Assume that a mutation in a word of the original language consists of a flip of
any single digit of that word, and that once a mutation has occurred, the new
word does not mutate. Assume also that the average mutation rate is one word
every 100 years. Estimate how long ago the two civilizations were separated.
8.
a. What three properties or structural criteria are necessary for a typology?
b. What a criteria of definition and interpretation must a typology satisfy?
c. What criteria must be satisfied by the variables of a typology?
d. On what four different scales are the variables of a typology usually
evaluated?
9. Write a short essay (300-500 words) comparing the process of model
construction described at the end of Discussion 10.3 and the general application
of the principles and polarities of reason introduced in Discussion 9.1.
10. Consider an example of a specific model used in science (a model with which
you are familiar, or one of the models from your readings in this course).
a. Describe the uses of this model in advancing scientific understanding.
b. Describe potential or actual misapplications of this model.
11. Choose one of the models described in Chapter 6 of What Science Is, and write a
short essay discussing this model in terms of the process of model construction
described at the end of Discussion 10.3.
FOOTNOTES
[208]
Ghazali, Abu Hamid al. Confessions, or Deliverance from Error. c. 1100 CE. Retrieved
July 3, 2002, from http://www.fordham.edu/halsall/basis/1100ghazali-truth.html/
[209]
Blake, William. “The Tyger.” In Songs of Experience. Retrieved August 9, 2002.
http://www.geocities.com/~spanoudi/poems/blake02.html/
[210]
Bateson, Gregory. Mind and Nature, p. 99. New York: E. P. Dutton, 1979.
[211]
This is why science is not able to make judgements on ethical and moral issues where the
circumstances of the individual case are important. It may matter little to a person that a cat
is being used in a medical experiment, unless that cat happens to be his or her own pet.
Then it matters a great deal.
[212]
The philosophical position of nominalism holds that no higher-level categories exist, only
individuals exist, and names like “cat” are merely linguistic conveniences. You might stop
for a moment to consider the relation of this question to the problem of induction, and to
the recognition of objects from memory.
[213]
Spencer-Brown, G. Laws of Form, p. 1. New York: Dutton, 1979.
[214]
This comment has the same flavour as Parmenides’ argument that “what is” must be the
same everywhere. Now we argue that we are only allowing one distinction. Therefore, there
can only be a difference in value across that distinction, not on any one side of it, since that
would define a second distinction.
[215]
In the Pythagorean sense, there will be a simple transformation between the “matrix of
proportions,” which describes the logos of one, and that which describes the logos of the
other.
[216]
See, for example, Weyl, Hermann. Symmetry. Princeton, NJ: Princeton University Press,
1952.
[217]
Gardner, Martin. “Chapter 1: The Five Platonic Solids.” In More Mathematical Puzzles
and Diversions. Harmondsworth, UK: Penguin, 1966.
[218]
Plato. “Philebus,” 64e, trans. R. Hackforth. In Plato: Collected Dialogues, edited by
Hamilton and Cairns, pp. 1086-1150.
[219]
Zee, Anthony. Fearful Symmetry: The Search for Beauty in Modern Physics, p. 13.
Princeton, NJ: Princeton University Press, 1999.
[220]
See if you can find a counter-example that demonstrates the need for the condition that
the four line segments be straight.
[221]
Differences in size may often be significant for classification, but size is a relative
concept. Without a standard of comparison we would not know what size an object was.
[222]
Feynman, Richard. The Character of Physical Law, p. 159. Cambridge, MA: MIT Press,
1965.
[223]
Permutation symmetries have particular significance in quantum mechanics, but we will
not go into this topic here.
[224]
Zemeckis, Robert, director. Forrest Gump. Hollywood, CA: Paramount, 1994.
[225]
Spencer-Brown, G. Laws of Form, p. 1.
[226]
Malinowski, B. A Scientific Theory of Culture, p. 69. Chapel Hill: University of North
Carolina Press, 1944.
[227]
Borges, Jorge Luis. “The Analytical Language of John Wilkins,” in Other Inquisitions
1937-1952. Ruth L.C. Simms, trans. Austin, TX: University of Texas Press, 1993.
[228]
This is one of the areas where creativity enters in experimental work.
[229]
Interpretation of such data can be complicated and difficult, because a particular brain
area may not be directly involved in the behaviour that is influenced by its removal. A rat
trained to press a bar when a bell rings will cease pressing the bar if it can no longer hear the
bell, but we would not say that the brain area involved in hearing controlled the ability to
press the bar.
[230]
Cuneiform tablets used for teaching scribes in ancient Sumer have been found that begin
with a general word such as sheep, then underneath it, list various categories of sheep
qualities.
[231]
For example, see Ceramic Field Identification Manual, Museum of Northern Arizona at
http://www.musnaz.org/cfim/.
[232]
In general practice the terms systematics, classification, typology and taxonomy are often
used interchangeably. We are being more precise here to help give an idea of some of the
basic structural relations that are found in classification schemes.
[233]
Many of the statistical clustering programs used today automatically generate a
hierarchical taxonomy.
[234]
See, for example, Rosch, E., and B. B. Lloyd, eds. Cognition and Categorization.
Hillsdale, NJ: Erlbaum, 1978.
[235]
Jefferson, Thomas. “Letter to William Smith, November 13, 1787.” The text of this letter
is available at the site below. Retrieved August 9, 2002.
http://www.theatlantic.com/issues/96oct/obrien/blood.htm/
[236]
Lao Tzu. Tao Te Ching. Retrieved August 9, 2002.
http://www.wsu.edu:8080/~wldciv/world_civ_reader/world_civ_reader_1/lao_tzu.html/
[237]
Galilei, Galileo. Discoveries and Opinions of Galileo, Stillman Drake, trans., p. 237-238.
New York: Doubleday, 1957.
[238]
Korzybski, Alfred. General Semantics: Toward a New General System of Evaluation and
Predictability in Solving Human Problems. Retrieved July 23, 2002, from
http://www.gestalt.org/semantic.htm/
[239]
May, Robert. Stability and Complexity in Model Ecosystems, p. v. Princeton, NJ:
Princeton University Press, 1974.
[240]
This is a highly simplified version of Freud’s theory and should not be taken as
presenting anything other than a very superficial view. It is discussed here only to provide
an example of the way in which the construction of a mathematical model can be useful.
[241]
Even in common language, we refer to a person with high libido as “hot blooded.”
[242]
It is sometimes called the Bohr theory, but in our terms, it is a model. Its origin is
described in detail in Discussion 14.1.
[243]
Bohr added a constraint that the angular momentum LL of the electron in its orbit had to
be an integer multiple (n) of Plank’s constant h divided by 2π.
[244]
Bohr did not just pull this assumption out of thin air. In the 1890s, Max Planck (1858-
1947) had explained “blackbody” radiation by assuming that radiant energy came in
multiples of the constant h, and in 1905, Einstein used this same assumption to explain the
photoelectric effect, so there was some background Bohr could draw on for his hypothesis.
[245]
Note that the term paradigm is being used here in a more general sense than in Kuhn’s
works (see Unit 2 for Kuhn’s use of paradigm). The general concept, however, is similar.
[246]
In the example of the Bohr atom described in Discussion 15.1, such a change occurred.
According to classical electromagnetic theory, Bohr’s model could not be stable. But the
model was empirically successful, and so the quantum assumption was adopted and it was
supposed that classical electromagnetism did not apply at the atomic level.
[247]
Morris, W. T. “On the Art of Modeling.” Management Science 13, 12 (1967): B707-B717.
[248]
Left-handers take heart!
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Unit 12 What Are Scientific Facts?
Discussion 12.1
Discussion 12.2
Discussion 12.3
Unit 12 Study Questions
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STUDY GU IDE
Unit 12 What Are Scientific Facts?
[T]he brightest of flashes in the world of thought are incomplete until they have been proved
to have their counterparts in the world of fact.”
—John Tyndall (1820-1893)[249]
The facts will eventually test all our theories, and they form, after all, the only impartial jury
to which we can appeal.
— Louis Agassiz (1807-1873)[250]
Science deals with facts, but just what is a fact is not all that clear. Who determines
whether something is a “fact” or not? By what criteria are facts to be judged? Is there
anything to a fact other than the fact that people believe in it? What is the difference
between a fact of experience and a scientific fact? This unit is devoted to an inquiry
into the nature of scientific fact, and its relationship to theory and to experience. We
discuss the criteria for accepting a fact, and the reasons for rejecting a proposed fact
or changing our view on an accepted fact. In addition, we deal with important
aspects of measurement, since most scientific facts in one way or another ultimately
rest on the results of a measurement.
We begin with a general consideration of some of the issues surrounding the
question of what a fact in science might be. These questions are posed and addressed
in Discussion 12.1. We then take a brief historical detour and consider the
development of the concept of heat. This history gives an excellent illustration of the
interplay between experience, experiment and theory in the construction of scientific
understanding. We see how the invention of a measuring instrument, the
thermometer (i.e., “heat-meter”) led to a long process of experimental and
theoretical research in which there was much debate over the theoretical
interpretation of experimental facts, and which eventually culminated in the modern
kinetic theory of heat.
We next consider the nature of measurement in science. We will consider the
difference between precision and accuracy of measurements, and the importance of
not attempting to be overly accurate in carrying out a measurement. We also
consider some of the dangers of misinterpretation and mismeasurement, and end
with discussions of the use of significant figures and estimates of error in numerical
measurements. Discussion 12.2 describes the different types of measurement scales
and how they are established.
Objectives
When you have completed Unit 12, you should be able to
1. explain the difference between facts of experience, facts of reason and scientific
facts.
2. describe what is meant by the claim that all facts are “theory-laden.”
3. discuss the way scientific facts are constructed in the interplay of theory and
experiment.
4. discuss the relation between relational, metric and linear scales.
5. describe the difference between linear scales for mass and distance on the one
hand, and linear scales for temperature and pressure on the other.
6. describe the value of metric measurement scales in science.
7. discuss the experimental setup Galileo used to study falling bodies.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 12.1, “Measurement.”
2. Answer Unit 12 Study Questions 1-5.
3. Read Discussion 12.2, “Facts.”
4. Read Discussion 12.3, “Galileo and Falling Bodies.”
5. Answer Unit 12 Study Questions 6-10.
Discussion 12.1 Measurement
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Man is the measure of all things.
—Protagoras (ca. 490-ca. 420 BCE)[251]
[S]ince the measuring device has been constructed by the observer . . . we have to remember
that what we observe is not nature in itself but nature exposed to our method of
questioning.
—Werner Heisenberg (1901-1976)[252]
At the beginning of Discussion 11.2, we indicated that classification and
measurement are the basic conditions for doing science. Classification is a tool of
qualitative evaluation; measurement is a tool of quantitative evaluation. In the same
way that a typology provides a basis for the qualitative discrimination of things,
measurement provides a basis for their quantitative discrimination. In addition,
measurement introduces greater precision and provides a means for evaluating the
accuracy of predictions.
As might be expected, there is a strong connection between classification and
measurement. In many cases, although not all, classification is based on the results
of measurements. Classification deals with establishing the identity of the basic
elements of a science, and with the logical relations among these elements.
Measurement introduces a numerical scale for significant variables, allowing us to
determine functional relations between variables in terms of general mathematical
laws. Classification and measurement are essential for the communication of
scientific facts.
In this discussion, we consider the conditions necessary for valid scientific
measurements. Attention will be restricted to “scalar” variables; that is, to variables
measured on a one-dimensional number scale.[253] We considered the four kinds of
scales on which a variable can be evaluated in Discussion 11.2:
1. a nominal scale measures only qualitative differences.
2. an ordinal scale rank orders a linear sequence.
3. an interval scale is a uniform numerical scale with a standard interval.
4. a ratio scale is a uniform numerical scale with a standard unit for comparison.
Measurement, as we are using the term, is based on interval scales or ratio scales.
The temperature scale, for example, is an interval scale. It is a uniform numerical
scale parameterized by assignment of specific values to the two end points of an
interval—the freezing and boiling points of pure water.[254] An example of a ratio scale
is the measurement of mass, which is determined in terms of a standard mass (e.g.,
the kilogram). So, to say that an object weighs 10 kilograms means that 10 standard
kilograms would be required to balance it on a fair scale.
Nominal scales are classifications. They are formed by establishing, in one way or
another, a set of categories that are exhaustive and mutually exclusive for the
variable in question. Colour categories are an example of a nominal scale. Although
we know that specific colours correspond to specific wavelengths of light, we do not
say, “That tree is 5200 angstroms.” Even astronomers classify stars by colour rather
than wavelength of peak energy emission.
An ordinal scale introduces a serial order, giving a comparative measure. For
example, the hardness of minerals is evaluated by the “scratch test.” Mineral AA is
harder than mineral BB if AA will scratch BB but BB will not scratch AA. Such a scale
establishes a non-numeric relational order that satisfies a condition called
“transitivity.” That is, if AA precedes BB, and BB precedes CC,
then AA precedes CC.[255]
To establish a numerical measure, we begin with a comparative (ordinal) scale, and
metricize it. Metrication, or introducing a metric, means establishing a numerical
measure of distance, which allows us to say not only that AA precedes BB, but also
that AA precedes BB by such and such a numerical amount, which can be compared,
for example, to the distance by which BB precedes CC.
A comparative ranking of entities in a set SS with respect to a
variable vv (e.g., ranking minerals by “hardness”) introduces what is called a “partial
order” on SS. [The order is partial because two entities may have the same value.]
This order is metricized by defining a function fv: S→Rfv: S→R, where RR is the set
of real numbers. This function assigns each element ss of SS a unique real
number fv(s)fv(s), called the “value of variable vv for element ss.” The
function fvfv must have the properties listed below.
o
o
If ss and s′s′ are the same with respect to the relational scale defined by the
variable vv, then fv(s)=fv(s′)fv(s)=fv(s′).
If ss precedes s′s′ with respect to the relational scale defined
by vv then fv(s)<fv(s′)fv(s)<fv(s′).
In practice, the function fvfv is defined by the empirical procedure used to define the
comparative scale, by a scaling assumption, and by the choice of a standard of
measure. The standard of measure determines whether the final result will be an
interval scale or a ratio scale. The choice of a specific interval as standard (e.g., the
freezing and boiling points of water) yields an interval scale. Choosing a comparative
standard (e.g., the standard kilogram) yields a ratio scale.
The scaling assumption determines the specific form of the function fvfv. The
condition fv(s)<fv(s′)fv(s)<fv(s′) if ss precedes s′s′ means that this function must be
increasing; that is, its slope must always be positive. Beyond this condition, however,
it is up to us to choose the most convenient form for this function. In the cases of
mass and temperature fvfv is chosen to be linear. In other cases, such as the Richter
scale of earthquake magnitudes, fvfv is logarithmic.
Linear scales have a simple additive property that may be expressed with the
equation
fv(s∧s′)=[afv(s)+bfv(s′)]fv(s∧s′)=[afv(s)+bfv(s′)]
0≤a,b≤10≤a,b≤1(1)
Here s∧s′s∧s′ indicates ss and s′s′ taken together. How this interpreted will depend
on the specific quantity being considered. If we are talking about mass,
then s∧s′s∧s′ is just the sum of the masses; if we are talking about temperature
then s∧s′s∧s′ indicates the result of bringing ss and s′s′ into contact, and allowing
them to come to an equilibrium temperature.
For an important subclass of measurements, a=b=1a=b=1; mass, length, volume and
electric charge are a few of the quantities that satisfy this condition. On the other
hand, the case a,b<1a,b<1 is exemplified by such measures as temperature and
pressure, which describe quantities obeying an equilibrium principle. The actual
values of aa and bb in these last cases are determined by the specific natures of the
entities being combined. If ss and s′s′ are identical except for their differing values
on the variable vv, then a=b=1/2a=b=1/2 (from the principle of sufficient reason).
Heat capacity gives a good example of the more general case.
Suppose that ss is one kg of water at 0°C, while s′s′ is 1 kg of mercury at 100°C.
Then s∧s′s∧s′ turns out to be a mixture of 1 kg of water and 1 kg of mercury at about
3°C. The equation defining the relation between the initial and final temperatures is
Cm(Tm−Te)=Cw(Tw−Te)Cm(Tm−Te)=Cw(Tw−Te)
(2)
where TmTm, TwTw and TeTe are, respectively, the initial temperatures of mercury
and water, and the final equilibrium temperature of the mixture;
and CmCm and CwCw are the respective “heat capacities” of mercury and water.
Solving this equation for TeTe yields
Te=CmCw+CmTm+CwCw+CmTwTe=CmCw+CmTm+CwCw+CmTw(3)
So, for this case, the constants aa and bb in Equation (1) become
a=CmCw+Cm, b=CwCw+Cma=CmCw+Cm, b=CwCw+Cm(4)
Note that another variable has been introduced, namely heat capacity and a ratio
scale for this variable has been established by arbitrarily taking Cw=1Cw=1. The
procedure for measuring the heat capacity CmCm of a material is to take equal
masses of that material and of water, heat them to different temperatures, mix them,
and then solve Equation (2) for CmCm.
Cm=Tw−TeTm−Te, Cw=1Cm=Tw−TeTm−Te, Cw=1(5)
The invention of the uncalibrated thermometer that allowed accurate judgements of
the form AA is hotter (or colder) than BB is an example of a progression from an
ordinal scale (cold, cool, warm, hot), to a comparative scale. The calibration obtained
by choosing the interval between the freezing and boiling points of water as a
standard represents progression to an interval scale. Although it seems so normal
today that we don’t even think about it, we should recognize that the construction of
a temperature scale involved theoretical and experimental work carried out as a
cooperative enterprise by many scientists over a number of years, that having this
scale was instrumental in the development of the theory of heat and heat engines, as
well as in the development of the science of thermodynamics, all of which underlie
much of the technology of modern civilization.
Scientists seek metric scales for several reasons, including those listed below.
o
o
o
o
o
Metric scales allow us to make more refined distinctions than are possible with
classification into types or with comparative scales.
Metric scales allow us to give precise numerical meaning to be given to relative
differences.
The use of metric scales helps in the discovery of general laws.
The use of metric scales facilitates the development of mathematical theories.
Metric scales provide a foundation for abstract concepts (e.g., the concept of
temperature in modern thermodynamics is a very abstract generalization of the
concept of temperature as defined by a mercury thermometer.)
Discussion 12.2 Facts
TOP
What are the facts? Again and again and again—what are the facts? Shun wishful thinking,
ignore divine revelation, forget what “the stars foretell,” avoid opinion, care not what the
neighbors think, never mind the unguessable “verdict of history”—what are the facts, and to
how many decimal places? You pilot always into an unknown future; facts are your single
clue. Get the facts!
—Robert A. Heinlein (1907-1988)[256]
It is generally agreed that in science, facts are relative rather than absolute. Facts are
inherently “theory-laden,” both in their nature and their significance. In this
discussion, we consider the way in which scientific facts are constructed. This topic
was introduced in Discussions 11.2 and 12.1, since scientific facts often have to do
with what taxonomic category a certain entity belongs to, or with the results of some
measurement.
The American Heritage Dictionary defines a fact as “something known with
certainty, something that has been objectively verified, something having real,
demonstrable existence.”[257] On the basis of Discussion 6.1, it should be clear that
each of these three definitions has philosophical problems. We can ask: What is
certainty? What does it mean to verify something objectively? How can we
demonstrate that a thing exists?
In Discussion 6.1, we suggested that the only things we can know for certain are the
logical validity of formal reasoning (the truths of reason) and the fact that we are
having an experience at a given moment (the truths of experience). On this basis, we
can accept true logical and mathematical propositions as facts (relative to an
axiomatic system), and we can accept it as fact that we have experience, although the
content of our experience remains open to question. If we are willing to make the
modest assumption that all healthy human beings have nervous systems that—
because of their physiology—respond similarly to sensory input, then we can agree,
by objective interpersonal communication, that, for example, both Tycho and
Kepler, experiencing the same input as they watch a sunrise, are experiencing the
same immediate fact. But while the sunrise is an empirical fact, it is not a scientific
fact.
The philosopher of science Norwood Russell Hanson (1924-1967) asserts that in this
situation, Tycho and Kepler see different things. Tycho sees the sun rise above the
horizon as it moves about the Earth. Kepler sees a stationary sun appear to rise as
the Earth rotates beneath it.[258] Hanson denies that this difference is only a matter of
interpretation. He is using the word “seeing” to include not only the immediate
awareness of sensory input, but also the way in which this awareness is embedded in
a person’s worldview. In his interpretation, what one “sees” is what one has learned
to see.
Hanson illustrates his view with the example of the Necker cube, which we
considered in Discussion 9.1 as an illustration of the principle of contradiction. In
that discussion, we concluded that the nature of the illusion resulted from the
inability of the two-dimensional image to provide enough information to support the
unambiguous representation of a three-dimensional object. We have been trained to
accept certain conventions of graphic presentation as indicating the presence of a
third dimension in two-dimensional images, and we tend to employ this training
automatically. In cases of insufficient information, however, there can be alternative
possible interpretations of the same perception, or alternative possible perceptions
of the same apprehension, or alternative possible apprehensions of the same
sensation.
On a constructive reading, we can say that both Tycho and Kepler see the same thing
in terms of their immediate sensations, but differ over what they have learned to
make of these sensations. Both do the same thing—stand on a hillside and watch the
sunrise—and this action results in a particular fact of experience, but each makes
something different out of the experience by locating it in a particular conceptual
web of paradigmatic assumptions. It is within these constructed conceptual worlds
that the facts of experience are interpreted,[259] and it is at this level that we are able
to speak of “scientific facts.” In the example, Tycho and Kepler share the same
sensations and apprehensions, but differ in their perceptions. The difference
between their beliefs about the sunrise relates to their different views of the relation
between sun and earth.
The word “fact” derives from the Latin root factum, the past participle of facere, “to
produce.” The word “fiction” derives from the Latin root fictum, the past participle
of fingere, “to fashion or form.” So, working along these lines, we could say that
scientific “facts” are convenient fictions that we make up in order to explain the
sensations that result when we do something. Some philosophers claim that this is
all that there is to the matter. This view is the “instrumentalist” position, and it is
useful as far as it goes. Sometimes, however, these “convenient fictions” turn out to
be real once we have discovered a way to extend the limits of our experience. The
posited existence of “germs” is an example. Germs could not be seen before the
invention of the microscope. Of course, sometimes our made-up entities turn out not
to exist as well, as exemplified by the theory of heat based on the idea of a subtle
fluid called “caloric.”
Quite often, however, the hypothesized entities do, on later experiment or
observation, turn out to exist. The existence of radio waves, the planets Uranus and
Neptune and a great variety of elementary particles, to mention only a few cases, has
been demonstrated. These examples raise again the question of what has been called
the “unnatural effectiveness of science in fitting the world.”
In the picture being presented in this discussion, the “facts” of immediate sensation
are molded and shaped to yield the facts of science. This process involves theory as
well as observation and experiment—scientific facts are not just the result of any
arbitrary observation or experiment, they are the results of observations and
experiments undertaken for specific reasons that are theoretically and
philosophically based. It is important to remember this before deciding what “facts”
one will accept, or judging others on the basis of the “facts” they accept or reject.
The complexities that may arise when we attempt to sort out what are the facts, or to
remove apparent contradictions between distinct facts, are illustrated by the wellknown story of the Aristotelian professors who refused to even look through
Galileo’s telescope to check whether or not there were moons around Jupiter.
Galileo claimed that his observation showed that there were four moons circling
Jupiter. This “fact” depended on his use of a newly discovered instrument—the
telescope—the design and operation of which depended on a theory of geometric
optics; that is, a theory about the behaviour of light rays. If one does not accept this
theory, then the validity of Galileo’s “fact” is cast into doubt. In addition, magicians
of the time were noted for the illusions that they produced with lenses, so it was at
least possible, from a skeptical perspective, that what Galileo saw was merely an
illusion.
We might think that this is silly—everybody knows how a telescope works, and
knows how to interpret its images. But in the 16th century this was not so. And
today, we have instruments that are based on massive amounts of theorizing, such as
the scanning electron microscope and the superconducting supercollider at CERN,
which discovered the Higgs boson. Without this theory, there would be no legitimate
basis for accepting and interpreting observations made with such complex
instruments.
If you recall the tables of dichotomies given in Discussions 9.1 and 11.2, we can see
that they also relate to the construction of scientific facts. For example, by induction
and/or abduction from particular examples, we obtain conjectured “facts.” Then, on
the assumption that these are indeed facts, we deduce certain other consequences
for future experience. If these consequences occur when the appropriate
observations or experiments are conducted, then we feel more confident in our facts.
The immediate empirical facts, however, are just that under certain conditions,
certain results follow.
It is a fact, for example, that if certain measurements are made, certain results will
be obtained, within a margin of experimental error. Or that if a set of objects is
observed, it will be possible to group these objects under a particular taxonomy. We
must ask, however, “What is to be made of such facts—what do they mean?” For
example, if objects are dropped near the surface of the earth and the distance they
fall is measured as a function of time, it will turn out that so long as air resistance
can be neglected, the distance will be related to the time of fall by the formula
d=(4.9 m/sec2)t2d=(4.9 m/sec2)t2(6)
with a small, apparently random error. This is a scientific fact, based not only on the
experimental procedure, but also on the theory of distance and time
measurement.[260] The fact is that if the distance fallen is plotted against time in a
Cartesian coordinate system, a set of points like those shown in Figure 12.2.1 is
obtained, and the application of a least squares curve fitting algorithm yields a good
fit for Equation (6). By combining a graphical representation of experimental facts
together with a particular mathematical fact we have obtained a scientific fact.
Figure 12.2.1: Plot of distance fallen against time
A person who wanted to play the role of skeptic could point out that the claimed
experimental facts represented by points on a graph are already strongly theoryladen, because they involve the notions of distance and time, and the
representational device of a Cartesian coordinate system. And our introduction of
numerical measures for distance and time involves an assumption about the fidelity
of our clocks and metre sticks. Thus, our scientific fact is based on very specific
assumptions whose only real justification is the pragmatic one that they seem to
work.
For a scientist it is usually not productive to carry skepticism to this extreme. Most
scientists accept the reality of the things they work with every day. Indeed, when it
comes to such things as distance and time, we all accept not only their existence but
also that they can be measured in a reliable way. We still need to keep a critical eye
on our scientific facts, however, to ensure that they do not mislead us. We need to
keep in mind that they are facts only relative to a particular set of assumptions,
which are valid only within certain limits—that they are theory-laden, and must
always be considered as subject to further revision or reinterpretation.
As a general checklist, we can give six criteria relating to our statement of what we
consider to be scientific (as against rational or mathematical) facts.
o
o
o
o
o
o
Facts are the result of observation.
Facts are expressed through empirical propositions.
Statements of facts must, within our capacity to determine, be empirically true.
The unity of reality requires that there cannot be a contradiction between facts.
Belief and prior knowledge influence what facts will be observed.
If two facts appear to be in contradiction, and if we are certain of the
observations that generated them and of the consistency of the statements of
each of these facts, then we must determine what beliefs must be changed or
what corrections must be made in our prior knowledge in order to remove the
contradiction.
Discussion 12.3 Galileo and
Falling Bodies
TOP
Galileo Galilei conducted a number of experimental studies of motion. While it is
questionable as to whether he really dropped weights from the Tower of Pisa, he did
carry out a number of experiments to show that objects fell with constant
acceleration (hence covering a distance proportional to the square of the time of
fall). These experiments show both the beginning of experimental technique and the
creative way that he was able to overcome obstacles arising from the lack of accurate
instruments.
He reported on these experiments in his book Dialogue Concerning Two New
Sciences, published in 1638:
A piece of wood molding or scantling, about twelve cubits long, three cubits wide, and about
three finger-breadths thick, was taken; on its edge was cut a channel a little more than one
finger in breadth; having made this groove very straight, smooth, and polished as possible,
we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a
sloping position, by lifting one end one or two cubits above the other, we rolled the ball, as I
was just saying, along the channel, noting, in a manner presently to be described, the time
required to make the descent. We repeated this experiment more than once in order to
measure the time with an accuracy such that the deviation between two observations never
exceeded one-tenth of a pulse-beat. Having performed this operation and having assured
ourselves of its reliability we now rolled the ball only one quarter the length of the channel;
and having measured the time of its descent we found it precisely one-half of the former.
Next we tried other distances, comparing the time for the whole length with that for the half,
or with that for two-thirds, or three-fourths; or indeed for any fraction; in such experiments,
repeated a full hundred times, we always found that the spaces traversed were to each other
as the squares of the times, and this was true for all inclinations of the plane, i.e., of the
channel along which we rolled the ball. We also observed that the times of descent, for
various inclinations of the plane, bore to one another precisely that ratio which, as we shall
see later, the Author had predicted and demonstrated for them.
For the measurement of time, we employed a large vessel of water placed in an elevated
position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet
of water, which we collected in a small glass during the time of each descent, whether for the
whole length of the channel or for a part of its length; the water thus collected was weighed,
after each descent, on a very accurate balance; the differences and ratios of these weights
gave us the differences and ratios of the times, and this with such accuracy that although the
operation was repeated many, many times, there was no appreciable discrepancy in the
results.[261]
While this is Galileo’s report of his experiments, there is evidence that at least in his
initial preparations for these experiments he timed the rolling balls using musical
beats, later making use of a water clock for confirmation and a more publically
acceptable presentation of the results. Galileo was an accomplished lute player and
both his father and brother were musicians. Setting gut strings across the inclined
plane in a way such that the note sounding when the bronze ball rolled past came at
equal intervals (something which a musically trained ear can identify with far greater
accuracy than any of the methods of measuring time available in Galileo’s day)
would provide an accurate measure of distances travelled in equal times.[262]
Based on his results, Galileo claimed empirical evidence supporting the idea that
falling objects underwent a constant acceleration, and that the distance covered was
proportional to the square of the time of the fall while the velocity was proportional
to the time.
From our modern point of view a couple of immediate questions arise:
1. The experiments Galileo describes are similar to those using falling steel balls in
elementary physics labs today. In these experiments, the ball is held at a given
height by an electromagnet and drops when the magnet is turned off. Why did
Galileo find it necessary to add the extra complication of an inclined plane?
(Clearly he didn’t have an electromagnet available, but why didn’t he just drop
the balls?)
2. What reason did Galileo have to assume that rolling a ball down an inclined
plane would provide information about a free falling ball and how did he justify
this assumption?
The answer to the first question, of course, is that Galileo did not have accurate
clocks, in both methods described for measuring time—the water clock and musical
timing—he needed time intervals long enough to obtain an accurate measurement.
The inclined plane is described as 12 cubits long, with one end raised “one or two
cubits.” Thus he would have conducted his tests with inclinations ranging between
12:1 and 6:1. In addition, in using the water clock, all of his time measurements are
given in terms of ratios rather than absolute times.
The second question is answered by Galileo’s assumption, justified by considering a
variety of inclinations of the plane (as well as various theoretical arguments and
thought experiments he constructed), that the motion of a body could be partitioned
into horizontal and vertical parts. Further, hidden in this assumption is a secondary
assumption, namely that whatever force is producing the acceleration acts only in
the vertical direction. This is justified by observing the obvious fact that if the plane
is horizontal the ball does not move (or, ignoring friction, moves at a constant
velocity—something which may have influenced Newton when he was formulating
his laws of motion).
Unit 12 Study Questions
TOP
Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. The philosopher W. T. Stace has claimed that potential energy is nothing but a
fiction invented as a means of saving the law of conservation of energy. He
supports this claim by arguing that if an object is thrown into the air and lands
on a roof, then there is no evidence that it has anything like “potential
energy.”[263] Can you find the flaw in this argument?
Hint: In physics, energy is defined as the capacity to perform work.
2. In the study of heat, the calorie theory was eventually discarded in favour of the
kinetic theory. Thus, the theoretical entity known as “caloric” must now be
considered as a mere fiction. Can you think of a reason that the theoretical term
“potential energy” is believed to refer to something real, while the theoretical
term “caloric” is not?
Hint: Energy is defined in terms of a capacity to do work while calorie was defined as a material
substance.
3. Which of the following statements would count as facts, and which would you be
willing to accept as scientific facts? Why?
a. Columbus reached America in 1492.
b. The acceleration due to gravity at the surface of the earth is
approximately 9.8 m/sec29.8 m/sec2.
c. Dante and Shakespeare are the greatest European writers of all time.
d. A spherical ball bearing 0.51±0.0020.51±0.002 centimetres in diameter
was dropped into a fluid, and the time it took to fall metres was
measured. Over five trials the results in seconds were as follows:
1.0±0.11.0±0.1, 1.2±0.21.2±0.2, 0.9±0.10.9±0.1, and 1.0±0.21.0±0.2.
e. The sphere in part (d), above, sinks one metre in one second.
The population of deer on Bowen Island was evaluated over a four-year
period. The results are shown in the figure below.
f. The population crash observed in year four is a result of over-grazing
limited resources.
g. The population increase in year three followed an intensive effort to
eliminate natural predators on the deer population.
h. The product of 3 and 9 is 27.
4. One of the first results Galileo reported after he began to study the planets with
his telescope was that there were mountains on the moon. This “observation,”
however, was not as straightforward as it might seem. The mountains Galileo
claimed to have seen were his interpretation of the patterns of light and dark
that he actually saw when focusing his telescope on the moon. (In his Divine
Comedy, Dante interprets the same patterns of light and dark as resulting from
differences in density of the lunar material.) Write a short essay (300-500
words) discussing the bottom-up and top-down aspects of this discovery. When
do you think that mountains on the moon became a scientific fact?
5. List three cases from your own experience in which beliefs or expectations led
you to accept as a fact something that you later found to be false.
6. Identify each of the following measurement scales as nominal, ordinal, interval
or ratio:
a. the energy of particles in high-energy particle accelerators is measured
in electron volts. One electron volt is defined as the amount of energy
acquired by an electron falling through a potential difference of one volt.
b. the Millen Scale of earthquake intensity defines the strength of
earthquakes on the basis of the amount of damage which they do
(shaking buildings, breaking glass, cracking ground, etc.).
c. interstellar distances are generally measured in light-years, where one
light year is the distance travelled by light in empty space in one year.
d. stars are classified according to their colour.
7. The Fahrenheit and Celsius temperature scales share the same empirical basis
and have the same standard of measure. They differ in the scaling assumptions
which each makes. Explain.
8. In Equation (1) of Discussion 11.1, the constant coefficients aa and bb are both
less than or equal to one. Why would it not make sense to allow either of these
values to be greater than one?
9. “Viscosity is a measure of a fluid’s resistance to flow.”[264] The column below lists
fluids from least to greatest on an ordinal scale based on viscosity. (A ball
bearing dropped into a fluid with higher viscosity will fall more slowly than it
would through a fluid with lower viscosity, and a fluid with higher viscosity will
flow more slowly through a tube than a fluid with lower viscosity.) Devise a
means of metricating this scale.
Note: You will need to come up with an experimental way of getting a numerical measure of
viscosity, and then construct either an interval or a ratio scale.
gasoline
water
salt water
maple syrup
crude oil
tar
10. In Discussion 12.3 the suggestion was made that Galileo used musical timing to
measure his inclined plane experiments. Doing this would provide data in terms
of distance traveled in equal units of time. On the other hand, timing his
experiments with a water clock would provide ratios of time elapsed for varying
units of distance. Write a short essay comparing these two forms of data
presentation.
FOOTNOTES
[249]
Tyndall, John. “Scientific Materialism,” p. 84. Fragments of Science: Part Two, pp. 82-
98. New York: Collier, 1902.
[250]
Agassiz, Louis. “Internal Structure and Progression of Glaciers,” p. 234. In Geological
Sketches, pp. 233-282. Boston: Ticknor and Fields, 1866.
[251]
Quoted in Plato. “Theaetetus,” 152a. F. M. Cornforth, trans. In Plato: Collected
Dialogues, edited by Hamilton and Cairns, pp. 845-919.
[252]
Heisenberg, Werner. Physics and Philosophy, p. 58. New York: Prometheus, 1958.
[253]
No generality is lost in making this restriction. Other kinds of variables, such as spinors,
vectors and tensors are constructed out of scalars.
[254]
The modern temperature scale is a theoretical construct deductively imposed on the old
inductively generated scale. The advantage of the theoretical scale is that it extends from
absolute zero up to temperatures in the hundreds of millions of degrees—far beyond the
limits of the empirical scale.
[255]
A relational order that is not transitive can occur in the game results of a round robin
tournament: team AA may beat team BB, and team B may beat team CC, but team CC might
beat team AA.
[256]
Heinlein, R. A. Time Enough for Love, p. 246. New York: Ace, 1989.
[257]
Pickett, Joseph, ed. American Heritage Dictionary of the English Language, 4th ed.
Boston: Houghton Mifflin, 2000.
[258]
Hanson, N. R. “Chapter 13, Observation.” In Klemke, E. D., Robert Hollinger, and A.
David Kline, eds. Introductory Readings in the Philosophy of Science, rev. ed., pp. 184-195.
Buffalo, NY: Prometheus Books, 1988.
[259]
Hanson uses a restricted meaning for the word “interpret,” limiting it to something one
does to a written text. Here, the term is used more broadly, referring as well to automatic
and unconscious choices made between alternatives.
[260]
These theories are so basic that we accept that metre sticks and accurate clocks exist. But
what assurance do we really have about all of this? Suppose we were to say that the real
standard for time was the pulse beat of the Dalai Lama? Equation (6) would certainly not
hold under those conditions.
[261]
Found at http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_160to243.html/.
[262]
See, for example, Stillman Drake, “The role of music in Galileo’s experiments,” Scientific
American, June 1975.
[263]
Stace, W. T. “Chapter 14: Science and the Physical World.” In Klemke, E. D., Robert
Hollinger, and A. David Kline, eds. Introductory Readings in the Philosophy of Science, rev.
ed., pp. 196-201. Buffalo, NY: Prometheus Books, 1980.
[264]
Definition at https://www.princeton.edu/~gasdyn/Research/T-
C_Research_Folder/Viscosity_def.html/.
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Unit 13 What Makes a Theory Scientific (Instead of Just Opinion or Belief)
Discussion 13.1
Discussion 13.2
Unit 13 Study Questions
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STUDY GU IDE
Unit 13 What Makes a Theory
Scientific (Instead of Just Opinion
or Belief)
Science is the attempt to make the chaotic diversity of our sense-experiences correspond to
a logically uniform system of thought.
—Albert Einstein (1879-1955)[265]
Science is built up with facts as a house is with stone, but a collection of facts is no more a
science than a heap of stones is a house.
—Henri Poincaré (1854-1912)[266]
Three things are to be looked to in a building: that it stand on the right spot; that it be
securely founded; that it be successfully executed.
—Johann Wolfgang von Goethe (1749-1832)[267]
In this unit, we consider the nature of scientific theories. Theories are not just
ordered collections of facts, they are conceptual structures—patterns of thought—
that transcend and illuminate, but also filter and shape, the “content” to which they
are applied.
Historically, the late 16th century saw a radical change in the view of the significance
of theory. In the earlier Aristotelian view, the purpose of theory was as an aid to the
contemplation of nature, through which a person nourished his or her spiritual
growth. Indeed, the Greek word theoria meant “vision.” Thus, a theory provided a
person with a vision of the world.
Francis Bacon criticized the “empty” speculations of Aristotelian academics, and
challenged the Aristotelian idea of the role of theory. In Bacon’s view, the role of
knowledge was to reduce human suffering. In modern society, with its emphasis on
the control of nature and the practical applications of scientific research, the
Baconian view predominates. Advances in cognitive science, however, are beginning
to focus attention on the nature of mind, understanding and consciousness, a change
that may lead to a greater appreciation of the Aristotelian view.
A good expression of the Greek view is found in the works of Plato. In the Republic,
the study of science, and in particular mathematics, is suggested as the appropriate
means of training the mind to rise from its concern with the mundane and illusory
world of experience to the realm of the transcendental ideas, such as Justice, Truth
and Beauty. In the Symposium, this same ascent is described in terms of an everwidening appreciation of beauty. In each case, Plato claimed that the person who
had made this ascent had acquired the wisdom to govern both themselves and the
political state. It is often said that one of the chief problems of the world today is that
scientific knowledge has outstripped moral development. In such a situation, Plato’s
ideas may be highly relevant to a world where a major issue is the proper use of the
power over nature that has been gained through science.
Discussion 4.2 described understanding in science as the ability to interpret a thing
within a conceptual framework, and further discusses both the concept of ideals of
natural order, and the idea that all that ever needs to be explained about a
phenomenon is its deviations from the relevant ideals of natural order. Theories
provide these explanations. They are the conceptual bridges between our ideals of
natural order and our experience. As such, they are responsible to both—a bridge not
firmly anchored at both ends is likely to fall down.
Theories must be both internally coherent and empirically testable, at least in
principle. For example, there are no current ways in which string theory in physics
can be tested, but such tests may become possible with advances in technology. The
belief that the world was created in 4004 BCE (or, for that matter, half a second ago)
can never be tested—part of the belief is that the world was created to look as if it
were billions of years old no matter how accurate our empirical tests, it is a belief
that must be accepted on faith alone.
A scientific theory is an abstract conceptual system that is constructed to explain
some aspect of nature. When we learn a theory, we are engaged in an internal mental
construction, and this process changes us. We learn to use the theory as an
interpretative “window on the world.” In terms of the cognitive heuristics, it
becomes an “anchor,” and we learn to interpret experience in terms of its
representational categories and available concepts. This action of theories is further
considered in Discussion 13.1 and we conclude with a general discussion of the
nature and role of theories in science.
Objectives
When you have completed Unit 13, you should be able to
1. discuss the differences between the Aristotelian and Baconian views on the uses
of scientific theory.
2. discuss the importance of experimental tests of the theories, and the condition
that must be met if a particular experiment is to provide us with a good test of a
theory.
3. describe, and discuss the relationship between, confirmation and refutation, as
they apply to theories in science.
4. describe the nature of a controlled experiment, and relate it to the method of
comparison described in Discussion 11.2.
5. discuss the need for a control group or a baseline expectation in order to
interpret the implications of a controlled experiment.
6. explain how theories allow description, prediction and explanation in science.
7. discuss the relationship between theories and experiments.
8. discuss the relationships among theories, models, paradigms and ideals of
natural order.
9. explain what it means to “understand” a scientific theory, and describe the
differences between scientific theories and beliefs or opinions.
10. describe six types of scientific theories, and state six criteria for a good theory.
11. describe the nature and role of laws in science, and define laws of coexistence,
succession and interaction.
12. state four principles of causality that are important in science.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 13.1, part one, “What the Mind Can Digest.”
2. Answer Unit 13 Study Questions 1-2.
3. Read Discussion 13.1, part two, “Description, Prediction, Explanation,
Understanding.”
4. Answer Unit 13 Study Questions 3-5.
5. Read Discussion 13.1, part three, “Types of Theories in Science.”
6. Answer Unit 13 Study Questions 6-7
7. Read Discussion 13.2, “Big Bang or Steady State?”
8. Answer Unit 13 Study Questions 8-9.
Discussion 13.1 The Role of
Theory in Science
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[I]t is more important to have beauty in one’s equations than to have them fit experiment.
—Paul Dirac[268]
Part One: What the Mind Can Digest
I do not know what “is.” I only know what my mind can digest.
—Robert Anton Wilson (1932-2007)[269]
A learning curve is a graph that displays the degree of mastery of a subject as a
function of the time spent in its study. Naively, we might expect that learning curves
would always look more or less like the curve shown in Figure 13.1.1, below.
Figure 13.1.1: Incremental learning curve
This diagram implies that the amount learned varies incrementally with study time,
and in fact, psychologists do observe such curves. Analogically speaking, it is useful
to think of such curves in terms of the mind as a container that is gradually filled.[270]
It turns out, however, that there is a second kind of learning curve, illustrated in
Figure 13.1.2, below.
Figure 13.1.2: All-or-none-learning curve
In this second kind of learning, almost nothing is learned until there is a sudden
jump, an “Ah Ha!” experience, after which nearly complete mastery is exhibited.
Learning to ride a bicycle often follows this pattern.
With this second kind of learning, the example of a container being filled does not
hold. In popular culture, this kind of learning is often characterized using the
analogy of a light bulb going on, but we can also think of it in terms of the
completion of a mental construct that is unable to function as an information
processing unit, or an organizer of perceptions, until all the pieces are in place.
There is also a significant difference between the types of learning that go on
according to these two curves. Although learners who have mastered a subject by
either curve are able to perform well on tests of subject matter, the person who
learned according to the curve of Figure 13.1.2 has the additional capacity of
generalizing their knowledge. That is, they are able to recognize analogous cases in
areas outside the area of original study. In terms of the two learning metaphors, we
could say that all that can be done with a filled container is to pour out some of what
it already contains, while a functioning information processor can be used to deal
with any information that fits as appropriate input.
Since science seeks generality, this second kind of learning is of particular
significance. It is not enough, for example, for a physicist to learn the statements
that make up quantum mechanics. She or he must also know how to employ the
tools of quantum mechanics analytically and analogically in a broad variety of
circumstances. It is not that it is not important to learn the content of a theory, but
one must exert an equal or even greater effort to grasp the patterns of thought, the
ways of establishing connections and viewing relationships, that are central for
application and understanding of the theory.
In his Seventh Letter, Plato describes five stages in learning: the name, the
definition, the image, true opinion, and “the fifth.” For example, in learning the
theory of evolution a student starts with the name—evolution, and whatever
associations they may have with that name. From that starting point they move to
the definition: change of species and the appearance of new species through
variation and natural selection. The “image” arises as they learn a number of
examples illustrating the theory, and the “true opinion” (i.e., the theory) arises as
they learn the analytic details of the theory and how to apply it. Plato, however, goes
on to say that “the fifth” can only come as a “flash of lightning.” It corresponds to the
internalization of evolutionary thinking to the extent that it becomes automatic,
allowing a person to recognize it whenever it occurs, and to apply it automatically,
without thought. The difference between the “true opinion” and “the fifth” shows up,
for example, in the difference between the proficient individual, who knows all the
rules and how to apply them in different circumstances, and the expert who knows
all of this intuitively, and also knows how and when to break the rules.
Learning a theory in this way involves processes of mental construction that provide
a new point of view on reality. But such a construction can also act as a filter,
excluding those aspects of reality that do not conform to its established patterns of
relationships. It allows us to digest (i.e., understand) certain kinds of sensations and
ideas by providing a means of locating them within a conceptual system, but if we
are not careful, it can lead us into confusion when we are faced with data that does
not fit within the limits of the system. This danger is captured in the folk saying: to a
man with a hammer, everything looks like a nail.[271]
According to the perspective advanced by Thomas Kuhn, science deals with this
issue by a refusal to accept data that is not consistent with an existing paradigm until
the degree of contradiction becomes so great that discovery of a new paradigm
becomes the major goal of the field. Making an anthropological analogy, we could
say that certain kinds of (mental) food (i.e., experimental data, interpretations,
theorizing) are declared taboo until researchers reach a state where hunger (the need
for secure understanding) overwhelms the taboos.
Part Two: Description, Prediction, Explanation, Understanding
There is . . . a rhythm and a pattern between the phenomena of nature which is not apparent
to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call
Physical Laws.
—Richard Feynman[272]
Modern atomic theory shares with the old one the fundamental idea that we explain the
external world’s multiplicity by setting it into correspondence with a multiplicity of forms
that can be distinguished and analyzed. To serve as forms of this kind the Greek
philosophers had only geometric configurations at their disposal, and therefore the old
theory explains the qualities of matter by grouping atoms in different ways.
—Werner Heisenberg[273]
In The Critique of Pure Reason, the philosopher Immanuel Kant develops a useful
model of the mind, distinguishing sensations, apprehensions, perceptions and
empirical thought as its four aspects. Direct sensory input constitutes sensation.
Apprehensions are sensations located in space and time. Perceptions are
apprehensions brought under a conceptual scheme, and empirical thought is
conceptual thought based on perceptions. Certain categorical frameworks that take
input from the lower level and construct the elements of the higher mediate the steps
between each level. So, for example, Kant saw the transition from mere sensation to
apprehension as being mediated by the categories of space and time.
In terms of this model, we would view perceptions as objectified apprehensions, and
as the basic elements of empirical thought. A theory operates at the level of empirical
thought. It is presented as a collection of relationships among abstract categories,
together with certain rules of construction and inference for adding to and
manipulating statements about these categories and their relationships. On this
basis, a theory allows us to describe, explain and predict perceptions (taken here in
the general sense, which includes the results of observation and experiment). It tells
us that if we objectify our apprehensions according to the categories defined by the
theory, then certain other similarly objectified apprehensions are to be anticipated.
By fitting perceptions into a conceptual framework in this way, we understand them.
For example, looking out my window the immediate sensations are simply a display
of colours (what William James referred to as a “booming buzzing confusion”).
These are immediately transformed into apprehensions of spatial patterns of colours
temporarily located “now.” As my mind then “carves at the joints” in this field of
apprehensions to assigns names: “tree,” “car,” “window,” “building” and so on, these
become the now-abstract elements of empirical thought. I can notice, for example,
that there are “buds” on the “limbs” of a particular “tree” and infer that in a few days
these will “blossom.”
As described in earlier discussions in this Study Guide, there is actually a circular
process going on, since the categories and relationships that make up a theory are
derived by induction from earlier apprehensions on the basis of observed
symmetries and distinctions. The idea is to seek a set of categories that fits the
empirical distinctions observed in nature, and that also satisfies simple and
aesthetically pleasing relationships. That is, a scientist engaged in theory
construction is seeking a conceptual pattern within which data becomes intelligible.
The basic processes involved in theory construction have been described in
Discussions 9.1, 11.1, 12.2, 11.3, 12.1 and 12.2. Note in particular that the model of
mathematical modeling given in Discussion 11.3 can be directly adapted to give a
model of theory construction as well.
The remainder of this discussion presents some general ideas that relate theories to
laws, hypotheses, models and paradigms, and show their role in scientific
explanation and understanding.
The general view of natural laws is that they are a description of empirical
regularities observed in nature. As such they are neither true nor false, only more or
less accurate statements of what will occur under given sets of circumstances.
Another view is that natural laws actually exist as relationships between abstract
universals, and our current statements of them are partial expressions in terms of
our best available conceptual frameworks. In this Platonic view, the regularities
observed are just particular instantiations of the abstract laws. In any case, however,
it is clear that our formulation of what we call “natural laws” depends on the
conceptual framework used. A good example is the law of gravity. A common-sense
statement of this law is, “What goes up must come down.” The fact that this
statement is not always true (e.g., chlorofluorocarbons float about in the upper
atmosphere for years, destroying ozone; the Voyager spacecraft will travel into
interstellar space, never to return to earth) does not invalidate the law of gravity. All
that it does is to show that our common-sense statement gives a limited description
of gravity.
Aristotle made another statement of the law of gravity when he asserted that
material objects automatically seek to return to their natural place. Thus fiery
objects seek to rise to the “sphere of fire” in the heavens; airy things move naturally
to the atmosphere; watery things (e.g., rivers) move to the oceans; and earthy things
fall to earth. In contrast to the common-sense statement, Aristotle provided a theory
that gave an explanation of observed phenomena. It “saves the appearances” by
giving a theoretical system within which appearances (i.e., what happens) are
rationalized. That which appears to the mind is rendered comprehensible in terms of
theoretical constructs within the mind. And again, the fact that Aristotle’s theory of
gravity is no longer accepted as an accurate description of nature does not alter the
law of gravity. It is the theory that is faulty, not the phenomena.
Likewise, Newton’s theory of gravity gives a statement of this law in the form that
any two masses will exert a force on each other that is proportional to the product of
their masses and inversely proportional to the square of the distance between them.
But, although this formulation provides an extremely accurate means for computing
the motion and orbits of gravitating bodies, it has been replaced by Einstein’s theory
of general relativity—in which there is no gravitational “force” at all; except that in
attempts at quantum gravity, the concept of force, as mediated by particles called
gravitons, may make a return.
The fundamental assumption that nature is ordered implies that there is a “law of
gravity.” But for us to understand the regularities that follow from this law, and to
communicate this understanding, requires some form of language. Thus all
statements of this law are relative to particular paradigmatic and theoretical
assumptions. There are a variety of theory-based statements of the law of gravity,
but at bottom the law itself is an abstract universal relation that is manifest in the
empirical regularities falling under it. Our statement of this law will always be made
within a theoretical and paradigmatic framework, one in which we have preassumed certain ideals of natural order. The law statement provides an analytical
presentation of the observed regularity, rationalized within the context of the preassumed framework. The map is not the territory, but in some cases it can give a
pretty good image of the territory.
What are given are appearances, which we assume are manifestations of the natural
order covered by the law. Theories give us the conceptual apparatus for making “law
statements” that are designed to fit (i.e., to reproduce within the universe of our
linguistic discourse) the ordering of appearances. In a phrase going back to the
ancient Greeks, they aim “to save the appearances.” Theories provide ordered mental
structures that we take as conceptual analogies for that which is beyond
appearances, which gives the observed manifestations of appearances. In other
words, theories give us explanations by asserting that the existence of certain
abstract laws and relationships (those described in the particular theory) are
sufficient reason for our observations and experiences.[274] And the more that a theory
explains, the more highly it is regarded.
In seeking generality in our theories and laws, we are seeking to explain more with
less, and success in science requires the development of an instinct for this kind of
conceptual parsimony.
You can recognize truth by its beauty and simplicity. It is always easy when you have made a
guess, and done two or three little calculations to make sure that it is not obviously wrong,
to know that it is right. When you get it right it is obvious that it is right—at least if you have
any experience—because usually what happens is that more comes out than goes in. Your
guess is, in fact, that something is very simple.[275]
Table 13.1.1 gives a rough classification of different types of law statements that are
encountered in science. In this table, we distinguish between deterministic and
stochastic laws, and describe three different forms of law statements: laws of
coexistence, laws of succession and laws of interaction. It should be clear that the
language in which various laws are expressed depends on the current state of a given
science. The ideal in most sciences is to obtain mathematical formulations for law
statements, and mathematical modeling is the major tool in this enterprise.
Table 13.1.1: Forms of law statements
Deterministic
Stochastic
Laws of
Coexistence
These laws select the possible
subsets of representations that
are to be allowed.
These laws specify a probability
distribution on the set of possible
representations.
Laws of
Succession
These laws specify the allowed
transitions between states,
representations or both.
These laws give the probability of a
transition between states,
representations or both.
Laws of
Interaction
These laws describe the result of
the allowed interactions between
systems.
These laws give the probability of
specified results of interactions
between systems.
Whether or not mathematical formulation is possible, however, the requirements of
the maximum possible precision, conciseness and simplicity are fundamental, both
for law statements and for theories. Yet, we must be careful not to be overly simple.
The philosopher Alfred North Whitehead said, “Seek simplicity and distrust
it,”[276] and Albert Einstein put it this way, “Everything should be made as simple as
possible, but not simpler.”[277]
In science, the term “elegant” is reserved for theories that successfully combine
simplicity and comprehensive accuracy. We want to explain as much as possible as
simply as possible, and a real feeling of exhilaration comes with the realization that
something is, indeed, “very simple.”
It may seem from what has been said that theories are nothing but mental constructs
serving only the instrumental purpose of allowing us to make computations and
predictions. In this view, they differ from guides on reading entrails only in that,
being more restricted in the phenomena they seek to describe, and more governed by
logic and experimental confirmation, they are more accurate.
What argues against the strict instrumental view of theory is the fact that it is often
the case that some entities originally defined only theoretically turn out to have
empirical existence. Indeed, the view taken on theories, laws and hypotheses by
working scientists depends very much on the context within which they are being
considered.
In its development and use of theories, laws, and hypotheses, science can advance and
employ them with a variety of different statuses—as purportedly true descriptions of classes
of phenomena, as idealizations, simplifications, or approximations of phenomena. Which
status a concept, theory, law, or hypothesis—or a portion thereof—is accorded is determined
on the basis of current scientific knowledge and may vary with the context of what is taken
as knowledge; that is, whether the theory, and so on, is taken as providing a realistic
treatment or as being a conceptual device is established by characteristic reasoning patterns
based on factual claims currently accorded the status of knowledge.[278]
In terms of our model, we will consider theories as mediating between basic
metaphysical and paradigmatic assumptions on the one hand; and appearances,
taken as manifestations of natural law, on the other. They are how we “save,” or
rationalize appearances. (Recalling that “appearances” means how things appear to
us, or to us mediated by our means of observation and measurement.)
Recall the fundamental metaphysical assumptions of science.
1. Nature is ordered.
2. The order of nature is comprehensible to human reason.
3. It is possible to communicate our understanding of this order.
When we say that nature is ordered, we mean that we are assuming that nature is
lawful. That is, there are universal relations that manifest as regularities in nature.
Furthermore, we are assuming that we can discover these regularities and
understand them on the basis of reason by crafting law statements that explain them
in the context of a given conceptual framework. Finally, we are assuming that there
are ways in which we can objectively communicate the nature of our understanding.
We can share conceptual frameworks and the law statements that we make and
agree that we are talking about the same things.
On top of these fundamental metaphysical assumptions, there are paradigmatic
assumptions about the kinds of reasoning that are scientifically acceptable, about the
fundamental entities that exist, about the ways in which these entities interact, and
about the kind of questions that can legitimately be asked.
As indicated in earlier units, some of these paradigmatic assumptions are called
“ideals of natural order.” The idea is that scientific theorizing works on the basis of
certain ideals of natural order that are assumed (i.e., accepted without question),
and seeks to describe, predict and explain deviations from the assumed ideal order.
Following this perspective, we can usefully view theories as abstract conceptual
structures that mediate between our ideals of natural order and the observed
regularities of nature. They provide us with reasons that are sufficient for us to
believe that we understand why things are the way they are, and not otherwise.
If we take a top-down approach to theory, we begin with the ideals of natural order
and seek to formulate our theory so that we can generate hypothetical statements
about anticipated regularities. That is, we make predictions. In Discussion 11.3, we
called this use of theory to fit observed regularities in specific cases “tactical
modeling.”
If we take a bottom-up approach, then we begin with observed regularities and seek
to generate, inductively, a theory that simultaneously embodies our ideals of natural
order and predicts the regularities with which we began. In this process, we may
formulate strategic models that describe specific cases of the observed regularities.
The process is indicated schematically in Figure 13.1.3, below.
Figure 13.1.3: Formulating scientific theories
Theory provides a conceptual taxonomy within which it is possible to formulate
descriptions of observed phenomena, together with means of operating on such
descriptions in order to generate predictions of new phenomena. In generating a
prediction, we do not need to start with an actual description. In fact, we usually
begin with a hypothetical description and generate a statement of the form
“If AA then BB,” where AA is the condition referred to by our hypothetical
description, and BB is the derived prediction. What is predicted is that if
conditions AA are produced, then event BB will occur (at least with a predicted
probability that can be tested through statistical analysis over a number of
experimental trials).
The development of a conceptual taxonomy requires that we distinguish sets of
categories within which at least some of the phenomena can be classified.[279] The
transformation of descriptions allowed in this language requires specification of the
logical and mathematical systems that can be employed—the “rules of reasoning.”
We have explained a phenomenon when we are able to formulate
1. a description of a class of phenomena in terms of a theory in such a way that the
phenomena of that class are only what would have been expected on the basis of
our ideals of natural order.
2. the theoretical mechanisms that describe deviations from those ideals and show
how the phenomenon under consideration fits within these descriptions.
When a phenomenon is only what would have been expected, we feel satisfied—we
feel that we understand it. The phenomenon has been fit into a conceptual
framework. There are no more questions to ask.
Understanding does not mean that some complex system of equations is found that saves all
the phenomena; rather it means, as it has always meant, that some contact has been made
with principles so simple and so natural that one is ready to accept them, provisionally of
course, as representing some version of the way things really are.[280]
One of the main questions in the philosophy of science is: What constitutes an
adequate explanation of a phenomenon? The discussion above describes a general
version of what is called the “covering law” model of explanation. This model asserts
that an explanation requires that what is to be explained be deduced from existing
conditions and natural law. The view presented here differs from the standard
covering law model in that it does not require strictly logical deduction, but rather
allows the kinds of theoretical plausibility arguments that are often made in science
(i.e., if certain conditions are present, then that which was to be explained follows
from natural laws, theoretical principles and our ideals of natural order).
We do not consider the various debates an adequate explanation, but briefly discuss
four forms of explanation that are generally accepted: formal, causal, statistical and
functional. It is important to emphasize, however, that whether or not a particular
form of explanation is acceptable depends very much on context and the state of the
science in which it is offered.
Formal Explanation: Formal explanation takes its name from formal logic. We
explain a property or characteristic of a thing by asserting that the property or
characteristic is an essential quality of the category to which that thing belongs. The
Aristotelian explanation that fire moves upward because it is a fiery material seeking
to return to the “Sphere of Fire” is a formal explanation, as is the explanation that
the charge of an electron is what it is because that is the nature of
electrons.[281] Without careful thought, this kind of explanation can lead to
meaningless tautologies, as for example, the purported explanation of opium’s sleep
inducing property by the statement that opium contains a “dormative (i.e., sleep
producing) principle.”
Causal Explanation: A causal explanation asserts that we have explained a
phenomenon or event when we are able to state its cause. Causal explanation is
modeled on deductive logic, in that we explain an event by positing initial conditions
and the natural laws that lead to the occurrence of the event, given the initial
conditions posited. In other words, our explanation is of the form: given the initial
conditions, the observed event follows from natural laws.
Aristotle defined four different causes: (1) material cause; (2) formal cause; (3)
efficient cause; (4) final cause. The material cause is the actual physical matter
involved. The formal cause is the design, or pattern that a system seeks to attain,
while the efficient cause is that which produces the change or event, and the final
cause is the purpose that change was directed towards producing. For example, the
material cause of a house is the material it is composed of, the formal cause is the
blueprint, the efficient cause is the contractor who builds it, and the final cause is the
goal of producing a dwelling place. In modern science formal and final causes are
not considered, and efficient causes are looked at as the natural laws involved. A
minimal form of final cause does appear in evolutionary theory in the idea that
species form and behaviour is evolved to promote species survival (but note that this
is a consequence of evolutionary processes, not a predetermined final cause).
Some philosophers claim that all explanation is, or can be, reduced to causal
explanation. Aristotle, for example, asserted that the only way to understand a thing
was to know its causes. Be this as it may, we can state four “principles of causal
simplicity” that act as ideals for construction of causal explanations
1. Principle of Uniformity: This principle can be viewed as an application of the
principle of the identity of indiscernibles. It states that similar causes tend to
have similar effects, and vice versa. Deviations in effects must be a result of
specific causal factors. This principle is challenged by modern results, especially
in chaos theory and theories of non-linear systems, where very small deviations
in the cause can produce major differences in effects.
2. Principle of Stability: Change does not occur without a cause. This principle is
obviously related to the principle of sufficient reason, but it is less general,
claiming that the only sufficient reason for change in a system is a causal reason.
3. Principle of Composition: Causal explanations of things, phenomena and
processes should be in accord with known behaviours of the components of that
which is being explained, and not contradict them. For example, a causal theory
of species evolution should not violate the known laws of genetics. This principle
is a consequence of the principle of contradiction.
4. Principle of Common Cause: It is important in causal reasoning to avoid the
mistake of assuming that just because two events are constantly correlated in
time (e.g., event AA always precedes event BB), the first must be a cause of the
second. For example, before a storm the barometer reading will decrease, but
the decrease in barometer reading does not cause the storm. The assumption
that is justified is that if two events or phenomena show a constant
correlation, then they may both be dependent on a common cause. For example,
the barometer falls because the atmospheric pressure is decreasing, and
decreasing atmospheric pressure also results in storms.
A danger of thinking in terms of causal explanations is the tendency to believe that
every phenomenon has only one cause. We considered this problem briefly in
Discussion 9.1, when we asked whether “the cause” of the ozone layer’s depletion
was the chemical fact that certain chlorine-bearing molecules interact destructively
with ozone? —or is it that these same compounds are used in refrigeration and air
conditioning devices? —or is it that people are willing to buy such devices, so that
their manufacture is profitable? —or. . . ? In the case of complex systems, it is
important to avoid the trap of thinking in terms of single causes.
Statistical Explanation: A statistical explanation asserts that given a particular
situation, there is—on the basis of natural laws, our current theories and our ideals
of natural order—a high probability of the event that is being explained. Statistical
explanation is the basis of the rates charged by insurance companies. A person
purchasing insurance is classified into a particular risk group and charged a rate that
gives the company a high probability, but not a certainty, of making money on the
transaction.[282]
Functional Explanation: A functional explanation asserts that an event, thing or
phenomenon is as it is because its being so serves some function. Functional
explanations show up often in biology and sociobiology in discussions of species’
characteristics and behaviours: certain characteristics or behaviours are asserted to
have “survival value” for their possessors. This kind of functional appeal is found, for
example, in the sociobiological and evolutionary psychology claim that altruistic
behaviour is to be expected only toward close genetic relatives, and serves the
purpose of insuring that at least a part of an individual’s genes are passed to the next
generation.
Functional explanations are often accused of circularity or tautology. For example,
the idea of “survival of the fittest” is often attacked because it is circular: the fittest
survive, and we can only tell who is fittest by who survives.
We have described theories in terms of how they provide a conceptual system for
description, prediction and explanation. Understanding, however, is not just a
matter of being able to describe, predict and explain. Understanding comes under
what Plato called “the fifth.”
Understanding requires a human (or at least sentient and self-conscious) subject.
Thus there is not only the question of the external requirements, but also that of
what internal conditions must exist within the subject before he or she can
legitimately claim to understand. One such condition, it would seem, is the ability to
generalize beyond the immediately given materials, which involves going beyond
appearances to the recognition of underlying principles and patterns. That is, it
understanding is associated with the learning curves of Figure 13.1.2, and the “Aha!”
experience of the light bulb going on.
On a more prosaic level, we could say that a person understands a theory when she
or he has incorporated internal mental structures of that theory in such a way that
those structures are able to function as an information processor and an organizer of
perception.
A theory is not pieced together from observed phenomena; it is rather what makes it
possible to observe phenomena as being of a certain sort and as relating to other
phenomena. Theories put phenomena into systems. . . . From the observed properties of
phenomena the [scientist] reasons . . . toward a keynote idea from which the properties are
explicable as a matter of course.[283]
As we have noted, however, theories also act as filters on perception. That is, they
will tend to force new apprehensions to conform to established categories of
perception.
Science rests itself not in the world the scientist beholds at any particular point in time, but
in his mode of viewing that world. A [person] is a scientist not because of what [they] see,
but because of how [they] see it.[284]
This filtering is not a negative thing because it provides stability and a secure
foundation for communication between scientists. It is, however, something that
needs to be borne in mind, because scientific creativity involves, in part, the capacity
to view old ideas or concepts in a new way. Thus, there will always be a dichotomy
for the scientist to balance. Too much to one side, and scientific creativity and
innovation are restricted; too much to the other, and a person loses contact with his
or her scientific community.
Part Three: Types of Theories in Science
The discussion of theories given above can be summarized in N. R. Hanson’s
statement, “theories provide patterns within which data become intelligible.”[285] We
can think of this definition in conjunction with Gregory Bateson’s injunction that in
attempting to understand a thing, we should seek for “the pattern that
connects.”[286] In a study of theories, however, we find that there are different ways in
which these “patterns that connect” can be expressed, depending on the state of
development in particular fields of science and on the available descriptive
languages. We will distinguish six different kinds of theories.[287]
1. Intuitional Theories: These are theories whose purpose is to provide an intuitive
understanding of their subject. A set of general principles is given which can be
used in making explanations, but these theories lack predictive power. Freud’s
theory of neurosis is an example of an intuitional theory. On the basis of this
theory, an observed neurosis can be explained in terms of theoretical entities,
such as fixated defense mechanisms originating in childhood trauma. However,
we cannot predict that particular traumatic events in childhood will result in
neurosis, or if they do, what specific form the neurosis will take.
2. Historical Theories: These are theories that describe the occurrence of unique
historical events by reconstructing the event. A good example is the theory that
the extinction of the dinosaurs was caused by the impact of a large meteor or
comet. Such theories have predictive power, but this power is limited to the
possibility of discovering traces left in the historical record by the hypothesized
event. The meteor theory of dinosaur extinction, for example, predicted the
possibility of discovering evidence of a major meteor impact about 65 million
years ago.[288]
3. Qualitative Theories: These theories are characterized by a focus on the
qualitative organization of observations as a means of gaining heuristic and
predictive power. Theories of weather prediction are of this type, based on
attempts to draw inferences from masses of qualitative data on temperature,
atmospheric pressure, humidity, the jet stream and so forth, which is then used
as input to simulation models.
4. Taxonomic Theories: These theories are concerned with classification and
recognition of phenomena. They deal with the construction of category systems
and taxonomies, as discussed in Discussion 11.2, and with predictions based on
class membership. Biological systematics is an example of this kind of theory.
5. Statistical Theories: Statistical theories are non-deterministic mathematical
theories. Predictions are given in terms of probabilities rather than certainties,
and the theories are formulated in terms of ensembles of events rather than
state spaces. Population genetics, in which predictions are made as to the
expected frequencies of genes in a population, is a good example of a stochastic
(statistical) theory.
6. Dynamical Theories: Dynamical theories are deterministic mathematical
theories in which system change over time follows from a set of equations of
motion. The standard form for such theories is based on the idea of a state space
that contains all possible system states, together with a transition rule (the
equations of motion) that determines possible changes of state. Newtonian
mechanics is a paradigmatic example.
Many current scientific theories have features characteristic of more than one of
these types of theories. The theory of plate tectonics in geology, for example, has
intuitional, historical and qualitative aspects; while current theories in elementary
particle physics have taxonomic, dynamical and stochastic aspects.
The physicist Henry Margenau (1901-1997) has suggested six criteria for a good
scientific theory, which apply to all types of theory discussed above.[289]
1. Constructs found in the theory (e.g., electrons, molecules, biological species,
social classes, psychological states, etc.) ought to possess logical fertility. That is,
they must provide a basis for fruitful deductions and the construction of
hypotheses.
2. Constructs in the theory ought to be multiply connected.
3. The theory needs to be stable. That is, it must be possible to adapt the theory to
new data.
4. Constructs in the theory ought to be extensible. That is, they should be found to
have applications beyond the original confines of the theory itself.
5. The theory ought to provide causal connections between phenomena and the
events it is intended to explain.
6. Theories must strive for simplicity and elegance.
Theories that satisfy these criteria are considered good theories, and much of the
work in theoretical science involves attempting to develop or refine theories so that
they can be considered to be “good” theories.
Discussion 13.2 Big Bang or
Steady State?
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Cosmology, as a modern science, began empirically with the discovery that the
distant “spiral nebulae” observed through existing telescopes were actually galaxies
consisting of hundreds of millions of stars. This was discovered by Edwin Hubble
with the 100-inch telescope on Mount Wilson in California. Using this telescope
Hubble first studied the Andromeda “nebula,” determining that it contained stars,
and then, making use of a distance measure based on Cepheid variable
stars,[290] showed that it was very far outside of our galaxy. In other words, it was
another galaxy, not some feature of our own. By studying other galaxies, Hubble
made the surprising discovery that these galaxies seemed to be receding from us. In
other words, he discovered that the universe was expanding. He published his
results in 1929. Further work allowed estimates of the rate of expansion and,
extrapolating this rate into the past suggested the age of the universe.
The foundation for theoretical cosmology was provided in Einstein’s general theory
of relativity, published in 1916. The Russian physicist Alexander Friedmann (18881925) published a mathematical solution of the Einstein equations that
corresponded to an expanding universe in 1922, but his paper was not recognized as
anything other than a mathematical exercise. Similarly, the Belgian physicist
Georges Lemaître (1894-1966) published a 1927 paper giving solutions that also
corresponded to an expanding universe, but again scant attention was paid until
1930 when his theoretical work was compared to Hubble’s discovery that the
universe was, indeed, expanding. In 1931, Lemaître suggested that the universe had
begun with the explosion of a “cosmic egg,” that our current universe was “the ashes
and smoke of bright but very rapid fireworks.” This “fireworks theory” provided the
first version of the Big-Bang theory for the origin of the universe.
The 1930s were a time of getting used to the radical change in our view of the cosmos
brought about by Hubble’s discovery, but there was little advance in cosmology in
the 1940s while the Second World War was in progress. The late 1940s, however,
saw the reemergence of scientific interest in cosmology, especially centered on the
problem of predicting the distribution of elements (i.e., the proportions of the
different elements in the universe as a whole). New understanding in atomic and
nuclear theory provided the possibility of finding an answer for this question, under
the assumption that the elements had been “cooked up” in the initial explosion of the
“cosmic egg.”
George Gamow, Ralph Alpher, and Robert Herman took up this question. The basic
assumption underlying their work was that the universe had cooled from an initial
state of almost infinite density and temperature in which the only particles present
were protons, neutrons, and electrons immersed in a sea of high energy
electromagnetic radiation. (They called this mix “Ylem,” an ancient philosophical
word for “primordial substance.”) Calculations (carried out with some of the first
electronic computers) were able to give reasonably good estimates of the observed
abundances of hydrogen and helium, but the answers obtained for other elements
were radically in error (Gamow joked that the theory must be pretty good, though,
since it predicted the abundances of 99% of the matter in the universe). Thus, the
question remained as to how the other elements had arisen.
The resolution of this question came with the realization that other elements arose
primarily through the process of nucleosynthesis taking place in the core of stars.
Measurements by the British astronomer Fred Hoyle provided empirical support for
theoretical calculations based on this assumption. Hoyle, however, along with
Hermann Bondi and Thomas Gold, were proponents of another cosmological theory,
the “Steady-State Theory,” which challenged the “fireworks” theory (sarcastically
renamed the “Big Bang Theory”[291] by Hoyle).
According to the steady-state theory, the universe has existed eternally and always
looked much as it does now. Since it is expanding, however, this theory requires that
a certain amount of new matter be continuously created. At the time it was proposed
the steady-state theory seemed to resolve a problem in the big bang theory, namely
that when the expansion of the universe was run backward, based on the best
available data on expansion rates, the age it predicted for the universe was less than
the known age of the Earth! This argument against the big bang theory was
dispelled, however, with the discovery that distance measurements based on the
period-brightness of Cepheid variables confused two distinct types of these stars.
Correcting for this error indicated an age for the universe that was twice the original
estimates and satisfactorily much older than the age of the Earth.
A possible test to distinguish between the two theories involved the age of galaxies.
In the big bang theory, the more distant galaxies ought to be older than nearby
galaxies. In a steady-state universe a more uniform distribution of young and old
galaxies would be expected. Older galaxies did appear to be more distant in a 1948
survey, but a review of the data in 1954 showed that this effect was an artifact of
measurement. The available data wasn’t refined enough to provide a means of
deciding between the two theories.
Other evidence against the steady-state theory began to show up, however, based on
radio astronomy observations initiated by Martin Ryle in 1951 and published in
1955. Ryle had conducted a survey of 2000 extra-galactic radio sources and claimed
evidence indicating that the more distant ones were also older. Steady-state
supporters questioned these results and it took a number of years to develop further
substantial supporting data for the big bang theory.
The deathblow for the steady-state theory came in 1965, but five more years of study
were needed to bury it. This evidence came with the discovery of the cosmic
microwave background by Bell Laboratory physicists Arno Penzias and Robert
Wilson. In 1948 George Gamow had published a paper arguing that as the universe
expanded from its initial state of almost infinite density, the radiation present would
cool to a current estimated temperature of about 5°Kelvin (i.e., five degrees above
absolute zero). In 1963 Penzias and Wilson observed a background microwave
radiation corresponding to about 3°K that seemed to be uniform across the sky. In
1965 the Princeton physicist Robert Dicke and his former student James Peebles,
who had been working along the same lines as Gamow, recognized that Penzias and
Wilson’s data provided empirical support as the archaic signature of the big bang.
A few cosmologists persist in attempts to adjust the steady-state model so that it fits
available data, but the theory was effectively abandoned by the mid-1970s.
Unit 13 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. If we focus attention on the forms of scientific theories, in contrast to their
content we see that a form of theory valuable in one area of science may be
worth little in some other area, and vice versa. The Aristotelian idea of a life
cycle is a particular form for theorizing, based on the assumptions listed below.
o
o
Entities develop according to a natural progression in an organic way,
from birth to death, or completion.
By various means, the rate of this natural development can be accelerated
or retarded, within certain bounds.
This particular form of theory was taken over into alchemy, where it was
assumed that metals develop in the earth toward gold; which, being the most
noble metal, is their final state of completion. Many alchemists sought to
discover a method to speed up this process in the laboratory.
c. Today we know that this form of theory does not fit when applied to the
study of metals. Can you think of an area of modern science where this
life cycle does appear as an ideal of natural order?
d. Write a short essay (300-500 words) discussing an example from your
own experience where the life cycle concept applies.
2. Write a short essay (300-500 words) discussing the following quotation, “There
is only one way of seeing one’s own spectacles clearly: that is, to take them off. It
is impossible to focus both on them and through them at the same time.”[292]
3. Each of the following statements gives an instance of a scientific law statement.
Determine whether the law that describes the instance is a law of coexistence,
succession or interaction, or some combination of these.
a. One mole (gram molecular weights) of oxygen combines with two moles
of hydrogen to yield one mole of water.
b. A pea plant that produces red flowers is crossed with a pea plant that
produces white flowers. The offspring are found to be 25% red flowering,
25% white flowering, and 50% pink flowering.
c. Two electrons cannot simultaneously occupy the same quantum state.
d. If two species compete for the same ecological niche, one of the species
will either become extinct or find a new niche.
e. The only constraint on the particles emerging from a high energy
elementary particle collision is that energy, momentum and all quantum
numbers are conserved.
f. A sufficiently massive star becomes a supernova when the star’s core
burns all of its available fuel so that it can no longer support the weight of
the outer layers of the star, which then collapse into the core.
4. Determine whether each of the following explanations is formal, causal,
statistical or functional, or some mixture of these types.
a. The chameleon’s ability to change colour evolved as a means of protective
camouflage.
b. The litmus paper changed colour because the solution it was dipped into
is acidic.
c. The neurotic behaviour of thoroughbred horses and of some breeds of
dogs is a result of excessive inbreeding.
d. Excessive inbreeding is not recommended because it tends to reinforce
negative genetic traits.
e. Play gives young animals the opportunity to practise behaviours, such as
stalking, fighting and fleeing, which will be important for them in adult
life.
f. HCl is corrosive because it is an acid.
g. If an individual shows a recessive trait, it means that both of that
individual’s parents carried that trait.
h. There was a storm yesterday because a low-pressure zone moved into the
area, and bad weather usually follows a drop in atmospheric pressure.
5. State which of the four causal principles described in Discussion 13.1 applies in
each of the cases outlined below.
a. The British bacteriologist Sir Alexander Fleming (1881-1955) carried out
a series of experiments in which dishes of bacteria were cultured. Fleming
noticed that some sort of mold had contaminated one of the dishes. He
also noticed that there were circles around the spots of mold where the
bacteria had died. He recognized that the mold must be producing some
anti-bacterial agent, and thus discovered penicillin.
b. In the 19th century the orbits of Mars, Jupiter and Saturn showed
irregularities that could not be accounted for using Newton’s theory of
gravity. On this basis, it was predicted that there was another planet
beyond Saturn. This prediction led to the discovery of Uranus. Later, yet
another outer planet was predicted on the basis of further orbital
irregularities, leading to the discovery of Neptune.
c. Unexplained irregularities were later observed in the orbit of the planet
Mercury, and on this basis some astronomers began to search for a new
planet, closer to the sun than Mercury. No planet was ever found, and the
irregularities in Mercury’s orbit were only explained by Einstein’s general
theory of relativity.
d. There are statistically significant correlations between being good at
mathematics, being nearsighted, being left-handed, being susceptible to
allergies and autoimmune diseases, and being male. Investigators have
hypothesized that all of these phenomena are a result of the effect of
testosterone on the developing brain of a fetus.
e. A spray of particle tracks observed in a bubble chamber cannot have been
caused by a collision event in which quantum numbers are not conserved.
6. Give at least one example, other than those already given in the discussion of
each of the six types of theories described in Discussion 13.1, part three.
7. In Discussion 3.2, we considered science as a question-and-answer process, and
described the qualities of a good question and the criteria for a valid answer. A
scientific theory can be seen both as providing an answer to a general question
and as a framework within which answers can be given to more specific
questions. With this in mind, write a short essay (300-500 words) comparing
the qualities of a good question, the criteria for a valid answer, and the six
criteria for a good theory given in Discussion 13.1, part three.
8. Why, in the Big Bang Theory, would we expect more distant galaxies to be older
than nearby ones?
9. Write a short essay (300-500 words) analyzing the history of modern cosmology
as presented in Discussion 13.2. Focus attention on the different factors
contributing to the modern theory and the underlying theoretical assumptions
that were involved in the development of this theory.
FOOTNOTES
[265]
Einstein, Albert. Out of My Later Years, p. 88. Westport, CT: Greenwood, 1971.
[266]
Poincaré, H. Science and Hypothesis, p. 141. New York: Dover, 1952.
[267]
Goethe, Johann Wolfgang von. Elective Affinities, Book I, Chapter 9, 1808.
[268]
Quoted in Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein,
rev. ed. p. 274. Cambridge, MA: Harvard University Press, 1988.
[269]
Wilson, Robert Anton. The Widow’s Son: The Historical Illuminatus Chronicles, Volume
2. New York: New American Library, 1981.
[270]
This analogy is behind much of our educational system. The role of the teacher is seen as
being to pour information from the course textbooks into the receptive minds of students.
Carrying this analogy further, we can also say that we cannot pour anything into an already
full container. A mind filled with preconceptions is incapable of learning.
[271]
The imprisoning nature of belief systems is one of Doris Lessing’s main points in The
Prisons We Choose to Live Inside; the same is true even of scientific theories. There is a
saying in science that the proponents of old theories don’t change their minds, they just
eventually die off.
[272]
Feynman, R. The Character of Physical Law, p. 13. Cambridge, MA: MIT Press, 1965.
[273]
Heisenberg, Werner. Philosophical Problems of Nuclear Science, p. 55. New York:
Fawcett World Library, 1966.
[274]
If we have a theory that does fit, within certain limits (e.g., the limits of accuracy of
observation), then we have our “sufficient reason,” and there is no need to seek for further
theories. In such a situation, scientists will seek to develop and extend the existing theory
rather than continually developing new ones. You might compare this to the Popperian idea
that we should constantly be developing new theories and attempting to refute them, and
recall Pólya’s “virtue of wise restraint.”
[275]
Feynman, Richard. The Character of Physical Law, p. 171.
[276]
Whitehead, Alfred North. The Concept of Nature, p. 163. Cambridge: Cambridge
University Press, 1926.
[277]
Quoted in Reader’s Digest, October 1977.
[278]
Suppe, F. The Structure of Scientific Theories, 2nd ed., p. 708. Urbana: University of
Illinois Press, 1977.
[279]
There will generally be categories of theoretical entities as well.
[280]
Park, David. The How and the Why: An Essay on the Origins and Development of
Physical Theory, p. 392. Princeton, NJ: Princeton University Press, 1990.
[281]
When inappropriately extended to the social world, this form of explanation is often a
basis for intolerance.
[282]
Historically, the development of mathematical statistics was strongly motivated by the
need to compute accurate statistical tables for insurance companies!
[283]
Hanson, N. R. Patterns of Discovery: An Inquiry into the Conceptual Foundations of
Science, p. 90. Cambridge: Cambridge University Press, 1965.
[284]
Roszak, Theodore. The Making of a Counter Culture, p. 213. Garden City, NY: 1969.
[285]
Hanson, N. R. Patterns of Discovery, p. 90.
[286]
Bateson, Gregory. Mind and Nature, p. 8. New York: Bantam, 1980.
[287]
Adapted from Rappaport, A. “Various Meanings of ‘Theory.’” American Political Science
Review (1958): 927-988.
[288]
This theory provides an excellent example of the process of theory construction. It
originated with the observation of a world-wide layer of iridium in sediment dated to about
65 million years ago. The concentration of iridium in this layer was much higher than would
be expected in deposits laid down by known earthly mechanisms, but was consistent with
the known concentration of iridium in meteors. The iridium layer was correlated with the
extinction of the dinosaurs, which was known to have occurred about 65 million years ago,
and on the basis of this correlation, various geological consequences of a large meteor
impact were deduced. Researchers sought whether the geological record showed traces that
these consequences had actually occurred. To date the evidence supports the theory, which
gained corroboration from the discovery of the Chicxulub crater off the shore of southern
Mexico. This crater was produced by the impact of a comet or asteroid, in the range of 5 to
15 km in diameter, about 65 million years ago.
[289]
Margenau, Henry. The Nature of Physical Reality. Woodbridge, CT: Ox Bow Press, 1977.
[290]
Cepheid variables come in two types. For both, however, there is a relation between their
brightness and the period of their variation, so by observing this period it is possible to
measure of their intrinsic brightness. Comparing this measurement to their observed
brightness allows estimation of their distance.
[291]
A religious aspect to the question arose when Pope Pius XII proclaimed in 1952 that big
bang cosmology affirmed the existence of a Creator God. A few years later, Hermann Bondi
published a rejoinder, pointing out that the big bang was based on a singularity in certain
solutions to the Einstein equations and that, in mathematics, one usually dealt with
singularities by finding ways to remove them. Thus, to associate the big bang with creation
“raised the possibility of eliminating God with better mathematics.”
[292]
Toulmin, Stephen. Chapter 5: “Forms and Styles of Theory.” In Foresight and
Understanding: An Enquiry into the Aims of Science, pp. 83-98. Westport, CT: Greenwood,
1961.
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Unit 13 What Makes a Theory Scientific (Instead of Just Opinion or Belief)
Discussion 13.1
Discussion 13.2
Unit 13 Study Questions
TOP
STUDY GU IDE
Unit 13 What Makes a Theory
Scientific (Instead of Just Opinion
or Belief)
Science is the attempt to make the chaotic diversity of our sense-experiences correspond to
a logically uniform system of thought.
—Albert Einstein (1879-1955)[265]
Science is built up with facts as a house is with stone, but a collection of facts is no more a
science than a heap of stones is a house.
—Henri Poincaré (1854-1912)[266]
Three things are to be looked to in a building: that it stand on the right spot; that it be
securely founded; that it be successfully executed.
—Johann Wolfgang von Goethe (1749-1832)[267]
In this unit, we consider the nature of scientific theories. Theories are not just
ordered collections of facts, they are conceptual structures—patterns of thought—
that transcend and illuminate, but also filter and shape, the “content” to which they
are applied.
Historically, the late 16th century saw a radical change in the view of the significance
of theory. In the earlier Aristotelian view, the purpose of theory was as an aid to the
contemplation of nature, through which a person nourished his or her spiritual
growth. Indeed, the Greek word theoria meant “vision.” Thus, a theory provided a
person with a vision of the world.
Francis Bacon criticized the “empty” speculations of Aristotelian academics, and
challenged the Aristotelian idea of the role of theory. In Bacon’s view, the role of
knowledge was to reduce human suffering. In modern society, with its emphasis on
the control of nature and the practical applications of scientific research, the
Baconian view predominates. Advances in cognitive science, however, are beginning
to focus attention on the nature of mind, understanding and consciousness, a change
that may lead to a greater appreciation of the Aristotelian view.
A good expression of the Greek view is found in the works of Plato. In the Republic,
the study of science, and in particular mathematics, is suggested as the appropriate
means of training the mind to rise from its concern with the mundane and illusory
world of experience to the realm of the transcendental ideas, such as Justice, Truth
and Beauty. In the Symposium, this same ascent is described in terms of an everwidening appreciation of beauty. In each case, Plato claimed that the person who
had made this ascent had acquired the wisdom to govern both themselves and the
political state. It is often said that one of the chief problems of the world today is that
scientific knowledge has outstripped moral development. In such a situation, Plato’s
ideas may be highly relevant to a world where a major issue is the proper use of the
power over nature that has been gained through science.
Discussion 4.2 described understanding in science as the ability to interpret a thing
within a conceptual framework, and further discusses both the concept of ideals of
natural order, and the idea that all that ever needs to be explained about a
phenomenon is its deviations from the relevant ideals of natural order. Theories
provide these explanations. They are the conceptual bridges between our ideals of
natural order and our experience. As such, they are responsible to both—a bridge not
firmly anchored at both ends is likely to fall down.
Theories must be both internally coherent and empirically testable, at least in
principle. For example, there are no current ways in which string theory in physics
can be tested, but such tests may become possible with advances in technology. The
belief that the world was created in 4004 BCE (or, for that matter, half a second ago)
can never be tested—part of the belief is that the world was created to look as if it
were billions of years old no matter how accurate our empirical tests, it is a belief
that must be accepted on faith alone.
A scientific theory is an abstract conceptual system that is constructed to explain
some aspect of nature. When we learn a theory, we are engaged in an internal mental
construction, and this process changes us. We learn to use the theory as an
interpretative “window on the world.” In terms of the cognitive heuristics, it
becomes an “anchor,” and we learn to interpret experience in terms of its
representational categories and available concepts. This action of theories is further
considered in Discussion 13.1 and we conclude with a general discussion of the
nature and role of theories in science.
Objectives
When you have completed Unit 13, you should be able to
1. discuss the differences between the Aristotelian and Baconian views on the uses
of scientific theory.
2. discuss the importance of experimental tests of the theories, and the condition
that must be met if a particular experiment is to provide us with a good test of a
theory.
3. describe, and discuss the relationship between, confirmation and refutation, as
they apply to theories in science.
4. describe the nature of a controlled experiment, and relate it to the method of
comparison described in Discussion 11.2.
5. discuss the need for a control group or a baseline expectation in order to
interpret the implications of a controlled experiment.
6. explain how theories allow description, prediction and explanation in science.
7. discuss the relationship between theories and experiments.
8. discuss the relationships among theories, models, paradigms and ideals of
natural order.
9. explain what it means to “understand” a scientific theory, and describe the
differences between scientific theories and beliefs or opinions.
10. describe six types of scientific theories, and state six criteria for a good theory.
11. describe the nature and role of laws in science, and define laws of coexistence,
succession and interaction.
12. state four principles of causality that are important in science.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 13.1, part one, “What the Mind Can Digest.”
2. Answer Unit 13 Study Questions 1-2.
3. Read Discussion 13.1, part two, “Description, Prediction, Explanation,
Understanding.”
4. Answer Unit 13 Study Questions 3-5.
5. Read Discussion 13.1, part three, “Types of Theories in Science.”
6. Answer Unit 13 Study Questions 6-7
7. Read Discussion 13.2, “Big Bang or Steady State?”
8. Answer Unit 13 Study Questions 8-9.
Discussion 13.1 The Role of
Theory in Science
TOP
[I]t is more important to have beauty in one’s equations than to have them fit experiment.
—Paul Dirac[268]
Part One: What the Mind Can Digest
I do not know what “is.” I only know what my mind can digest.
—Robert Anton Wilson (1932-2007)[269]
A learning curve is a graph that displays the degree of mastery of a subject as a
function of the time spent in its study. Naively, we might expect that learning curves
would always look more or less like the curve shown in Figure 13.1.1, below.
Figure 13.1.1: Incremental learning curve
This diagram implies that the amount learned varies incrementally with study time,
and in fact, psychologists do observe such curves. Analogically speaking, it is useful
to think of such curves in terms of the mind as a container that is gradually filled.[270]
It turns out, however, that there is a second kind of learning curve, illustrated in
Figure 13.1.2, below.
Figure 13.1.2: All-or-none-learning curve
In this second kind of learning, almost nothing is learned until there is a sudden
jump, an “Ah Ha!” experience, after which nearly complete mastery is exhibited.
Learning to ride a bicycle often follows this pattern.
With this second kind of learning, the example of a container being filled does not
hold. In popular culture, this kind of learning is often characterized using the
analogy of a light bulb going on, but we can also think of it in terms of the
completion of a mental construct that is unable to function as an information
processing unit, or an organizer of perceptions, until all the pieces are in place.
There is also a significant difference between the types of learning that go on
according to these two curves. Although learners who have mastered a subject by
either curve are able to perform well on tests of subject matter, the person who
learned according to the curve of Figure 13.1.2 has the additional capacity of
generalizing their knowledge. That is, they are able to recognize analogous cases in
areas outside the area of original study. In terms of the two learning metaphors, we
could say that all that can be done with a filled container is to pour out some of what
it already contains, while a functioning information processor can be used to deal
with any information that fits as appropriate input.
Since science seeks generality, this second kind of learning is of particular
significance. It is not enough, for example, for a physicist to learn the statements
that make up quantum mechanics. She or he must also know how to employ the
tools of quantum mechanics analytically and analogically in a broad variety of
circumstances. It is not that it is not important to learn the content of a theory, but
one must exert an equal or even greater effort to grasp the patterns of thought, the
ways of establishing connections and viewing relationships, that are central for
application and understanding of the theory.
In his Seventh Letter, Plato describes five stages in learning: the name, the
definition, the image, true opinion, and “the fifth.” For example, in learning the
theory of evolution a student starts with the name—evolution, and whatever
associations they may have with that name. From that starting point they move to
the definition: change of species and the appearance of new species through
variation and natural selection. The “image” arises as they learn a number of
examples illustrating the theory, and the “true opinion” (i.e., the theory) arises as
they learn the analytic details of the theory and how to apply it. Plato, however, goes
on to say that “the fifth” can only come as a “flash of lightning.” It corresponds to the
internalization of evolutionary thinking to the extent that it becomes automatic,
allowing a person to recognize it whenever it occurs, and to apply it automatically,
without thought. The difference between the “true opinion” and “the fifth” shows up,
for example, in the difference between the proficient individual, who knows all the
rules and how to apply them in different circumstances, and the expert who knows
all of this intuitively, and also knows how and when to break the rules.
Learning a theory in this way involves processes of mental construction that provide
a new point of view on reality. But such a construction can also act as a filter,
excluding those aspects of reality that do not conform to its established patterns of
relationships. It allows us to digest (i.e., understand) certain kinds of sensations and
ideas by providing a means of locating them within a conceptual system, but if we
are not careful, it can lead us into confusion when we are faced with data that does
not fit within the limits of the system. This danger is captured in the folk saying: to a
man with a hammer, everything looks like a nail.[271]
According to the perspective advanced by Thomas Kuhn, science deals with this
issue by a refusal to accept data that is not consistent with an existing paradigm until
the degree of contradiction becomes so great that discovery of a new paradigm
becomes the major goal of the field. Making an anthropological analogy, we could
say that certain kinds of (mental) food (i.e., experimental data, interpretations,
theorizing) are declared taboo until researchers reach a state where hunger (the need
for secure understanding) overwhelms the taboos.
Part Two: Description, Prediction, Explanation, Understanding
There is . . . a rhythm and a pattern between the phenomena of nature which is not apparent
to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call
Physical Laws.
—Richard Feynman[272]
Modern atomic theory shares with the old one the fundamental idea that we explain the
external world’s multiplicity by setting it into correspondence with a multiplicity of forms
that can be distinguished and analyzed. To serve as forms of this kind the Greek
philosophers had only geometric configurations at their disposal, and therefore the old
theory explains the qualities of matter by grouping atoms in different ways.
—Werner Heisenberg[273]
In The Critique of Pure Reason, the philosopher Immanuel Kant develops a useful
model of the mind, distinguishing sensations, apprehensions, perceptions and
empirical thought as its four aspects. Direct sensory input constitutes sensation.
Apprehensions are sensations located in space and time. Perceptions are
apprehensions brought under a conceptual scheme, and empirical thought is
conceptual thought based on perceptions. Certain categorical frameworks that take
input from the lower level and construct the elements of the higher mediate the steps
between each level. So, for example, Kant saw the transition from mere sensation to
apprehension as being mediated by the categories of space and time.
In terms of this model, we would view perceptions as objectified apprehensions, and
as the basic elements of empirical thought. A theory operates at the level of empirical
thought. It is presented as a collection of relationships among abstract categories,
together with certain rules of construction and inference for adding to and
manipulating statements about these categories and their relationships. On this
basis, a theory allows us to describe, explain and predict perceptions (taken here in
the general sense, which includes the results of observation and experiment). It tells
us that if we objectify our apprehensions according to the categories defined by the
theory, then certain other similarly objectified apprehensions are to be anticipated.
By fitting perceptions into a conceptual framework in this way, we understand them.
For example, looking out my window the immediate sensations are simply a display
of colours (what William James referred to as a “booming buzzing confusion”).
These are immediately transformed into apprehensions of spatial patterns of colours
temporarily located “now.” As my mind then “carves at the joints” in this field of
apprehensions to assigns names: “tree,” “car,” “window,” “building” and so on, these
become the now-abstract elements of empirical thought. I can notice, for example,
that there are “buds” on the “limbs” of a particular “tree” and infer that in a few days
these will “blossom.”
As described in earlier discussions in this Study Guide, there is actually a circular
process going on, since the categories and relationships that make up a theory are
derived by induction from earlier apprehensions on the basis of observed
symmetries and distinctions. The idea is to seek a set of categories that fits the
empirical distinctions observed in nature, and that also satisfies simple and
aesthetically pleasing relationships. That is, a scientist engaged in theory
construction is seeking a conceptual pattern within which data becomes intelligible.
The basic processes involved in theory construction have been described in
Discussions 9.1, 11.1, 12.2, 11.3, 12.1 and 12.2. Note in particular that the model of
mathematical modeling given in Discussion 11.3 can be directly adapted to give a
model of theory construction as well.
The remainder of this discussion presents some general ideas that relate theories to
laws, hypotheses, models and paradigms, and show their role in scientific
explanation and understanding.
The general view of natural laws is that they are a description of empirical
regularities observed in nature. As such they are neither true nor false, only more or
less accurate statements of what will occur under given sets of circumstances.
Another view is that natural laws actually exist as relationships between abstract
universals, and our current statements of them are partial expressions in terms of
our best available conceptual frameworks. In this Platonic view, the regularities
observed are just particular instantiations of the abstract laws. In any case, however,
it is clear that our formulation of what we call “natural laws” depends on the
conceptual framework used. A good example is the law of gravity. A common-sense
statement of this law is, “What goes up must come down.” The fact that this
statement is not always true (e.g., chlorofluorocarbons float about in the upper
atmosphere for years, destroying ozone; the Voyager spacecraft will travel into
interstellar space, never to return to earth) does not invalidate the law of gravity. All
that it does is to show that our common-sense statement gives a limited description
of gravity.
Aristotle made another statement of the law of gravity when he asserted that
material objects automatically seek to return to their natural place. Thus fiery
objects seek to rise to the “sphere of fire” in the heavens; airy things move naturally
to the atmosphere; watery things (e.g., rivers) move to the oceans; and earthy things
fall to earth. In contrast to the common-sense statement, Aristotle provided a theory
that gave an explanation of observed phenomena. It “saves the appearances” by
giving a theoretical system within which appearances (i.e., what happens) are
rationalized. That which appears to the mind is rendered comprehensible in terms of
theoretical constructs within the mind. And again, the fact that Aristotle’s theory of
gravity is no longer accepted as an accurate description of nature does not alter the
law of gravity. It is the theory that is faulty, not the phenomena.
Likewise, Newton’s theory of gravity gives a statement of this law in the form that
any two masses will exert a force on each other that is proportional to the product of
their masses and inversely proportional to the square of the distance between them.
But, although this formulation provides an extremely accurate means for computing
the motion and orbits of gravitating bodies, it has been replaced by Einstein’s theory
of general relativity—in which there is no gravitational “force” at all; except that in
attempts at quantum gravity, the concept of force, as mediated by particles called
gravitons, may make a return.
The fundamental assumption that nature is ordered implies that there is a “law of
gravity.” But for us to understand the regularities that follow from this law, and to
communicate this understanding, requires some form of language. Thus all
statements of this law are relative to particular paradigmatic and theoretical
assumptions. There are a variety of theory-based statements of the law of gravity,
but at bottom the law itself is an abstract universal relation that is manifest in the
empirical regularities falling under it. Our statement of this law will always be made
within a theoretical and paradigmatic framework, one in which we have preassumed certain ideals of natural order. The law statement provides an analytical
presentation of the observed regularity, rationalized within the context of the preassumed framework. The map is not the territory, but in some cases it can give a
pretty good image of the territory.
What are given are appearances, which we assume are manifestations of the natural
order covered by the law. Theories give us the conceptual apparatus for making “law
statements” that are designed to fit (i.e., to reproduce within the universe of our
linguistic discourse) the ordering of appearances. In a phrase going back to the
ancient Greeks, they aim “to save the appearances.” Theories provide ordered mental
structures that we take as conceptual analogies for that which is beyond
appearances, which gives the observed manifestations of appearances. In other
words, theories give us explanations by asserting that the existence of certain
abstract laws and relationships (those described in the particular theory) are
sufficient reason for our observations and experiences.[274] And the more that a theory
explains, the more highly it is regarded.
In seeking generality in our theories and laws, we are seeking to explain more with
less, and success in science requires the development of an instinct for this kind of
conceptual parsimony.
You can recognize truth by its beauty and simplicity. It is always easy when you have made a
guess, and done two or three little calculations to make sure that it is not obviously wrong,
to know that it is right. When you get it right it is obvious that it is right—at least if you have
any experience—because usually what happens is that more comes out than goes in. Your
guess is, in fact, that something is very simple.[275]
Table 13.1.1 gives a rough classification of different types of law statements that are
encountered in science. In this table, we distinguish between deterministic and
stochastic laws, and describe three different forms of law statements: laws of
coexistence, laws of succession and laws of interaction. It should be clear that the
language in which various laws are expressed depends on the current state of a given
science. The ideal in most sciences is to obtain mathematical formulations for law
statements, and mathematical modeling is the major tool in this enterprise.
Table 13.1.1: Forms of law statements
Deterministic
Stochastic
Laws of
Coexistence
These laws select the possible
subsets of representations that
are to be allowed.
These laws specify a probability
distribution on the set of possible
representations.
Laws of
Succession
These laws specify the allowed
transitions between states,
representations or both.
These laws give the probability of a
transition between states,
representations or both.
Laws of
Interaction
These laws describe the result of
the allowed interactions between
systems.
These laws give the probability of
specified results of interactions
between systems.
Whether or not mathematical formulation is possible, however, the requirements of
the maximum possible precision, conciseness and simplicity are fundamental, both
for law statements and for theories. Yet, we must be careful not to be overly simple.
The philosopher Alfred North Whitehead said, “Seek simplicity and distrust
it,”[276] and Albert Einstein put it this way, “Everything should be made as simple as
possible, but not simpler.”[277]
In science, the term “elegant” is reserved for theories that successfully combine
simplicity and comprehensive accuracy. We want to explain as much as possible as
simply as possible, and a real feeling of exhilaration comes with the realization that
something is, indeed, “very simple.”
It may seem from what has been said that theories are nothing but mental constructs
serving only the instrumental purpose of allowing us to make computations and
predictions. In this view, they differ from guides on reading entrails only in that,
being more restricted in the phenomena they seek to describe, and more governed by
logic and experimental confirmation, they are more accurate.
What argues against the strict instrumental view of theory is the fact that it is often
the case that some entities originally defined only theoretically turn out to have
empirical existence. Indeed, the view taken on theories, laws and hypotheses by
working scientists depends very much on the context within which they are being
considered.
In its development and use of theories, laws, and hypotheses, science can advance and
employ them with a variety of different statuses—as purportedly true descriptions of classes
of phenomena, as idealizations, simplifications, or approximations of phenomena. Which
status a concept, theory, law, or hypothesis—or a portion thereof—is accorded is determined
on the basis of current scientific knowledge and may vary with the context of what is taken
as knowledge; that is, whether the theory, and so on, is taken as providing a realistic
treatment or as being a conceptual device is established by characteristic reasoning patterns
based on factual claims currently accorded the status of knowledge.[278]
In terms of our model, we will consider theories as mediating between basic
metaphysical and paradigmatic assumptions on the one hand; and appearances,
taken as manifestations of natural law, on the other. They are how we “save,” or
rationalize appearances. (Recalling that “appearances” means how things appear to
us, or to us mediated by our means of observation and measurement.)
Recall the fundamental metaphysical assumptions of science.
1. Nature is ordered.
2. The order of nature is comprehensible to human reason.
3. It is possible to communicate our understanding of this order.
When we say that nature is ordered, we mean that we are assuming that nature is
lawful. That is, there are universal relations that manifest as regularities in nature.
Furthermore, we are assuming that we can discover these regularities and
understand them on the basis of reason by crafting law statements that explain them
in the context of a given conceptual framework. Finally, we are assuming that there
are ways in which we can objectively communicate the nature of our understanding.
We can share conceptual frameworks and the law statements that we make and
agree that we are talking about the same things.
On top of these fundamental metaphysical assumptions, there are paradigmatic
assumptions about the kinds of reasoning that are scientifically acceptable, about the
fundamental entities that exist, about the ways in which these entities interact, and
about the kind of questions that can legitimately be asked.
As indicated in earlier units, some of these paradigmatic assumptions are called
“ideals of natural order.” The idea is that scientific theorizing works on the basis of
certain ideals of natural order that are assumed (i.e., accepted without question),
and seeks to describe, predict and explain deviations from the assumed ideal order.
Following this perspective, we can usefully view theories as abstract conceptual
structures that mediate between our ideals of natural order and the observed
regularities of nature. They provide us with reasons that are sufficient for us to
believe that we understand why things are the way they are, and not otherwise.
If we take a top-down approach to theory, we begin with the ideals of natural order
and seek to formulate our theory so that we can generate hypothetical statements
about anticipated regularities. That is, we make predictions. In Discussion 11.3, we
called this use of theory to fit observed regularities in specific cases “tactical
modeling.”
If we take a bottom-up approach, then we begin with observed regularities and seek
to generate, inductively, a theory that simultaneously embodies our ideals of natural
order and predicts the regularities with which we began. In this process, we may
formulate strategic models that describe specific cases of the observed regularities.
The process is indicated schematically in Figure 13.1.3, below.
Figure 13.1.3: Formulating scientific theories
Theory provides a conceptual taxonomy within which it is possible to formulate
descriptions of observed phenomena, together with means of operating on such
descriptions in order to generate predictions of new phenomena. In generating a
prediction, we do not need to start with an actual description. In fact, we usually
begin with a hypothetical description and generate a statement of the form
“If AA then BB,” where AA is the condition referred to by our hypothetical
description, and BB is the derived prediction. What is predicted is that if
conditions AA are produced, then event BB will occur (at least with a predicted
probability that can be tested through statistical analysis over a number of
experimental trials).
The development of a conceptual taxonomy requires that we distinguish sets of
categories within which at least some of the phenomena can be classified.[279] The
transformation of descriptions allowed in this language requires specification of the
logical and mathematical systems that can be employed—the “rules of reasoning.”
We have explained a phenomenon when we are able to formulate
1. a description of a class of phenomena in terms of a theory in such a way that the
phenomena of that class are only what would have been expected on the basis of
our ideals of natural order.
2. the theoretical mechanisms that describe deviations from those ideals and show
how the phenomenon under consideration fits within these descriptions.
When a phenomenon is only what would have been expected, we feel satisfied—we
feel that we understand it. The phenomenon has been fit into a conceptual
framework. There are no more questions to ask.
Understanding does not mean that some complex system of equations is found that saves all
the phenomena; rather it means, as it has always meant, that some contact has been made
with principles so simple and so natural that one is ready to accept them, provisionally of
course, as representing some version of the way things really are.[280]
One of the main questions in the philosophy of science is: What constitutes an
adequate explanation of a phenomenon? The discussion above describes a general
version of what is called the “covering law” model of explanation. This model asserts
that an explanation requires that what is to be explained be deduced from existing
conditions and natural law. The view presented here differs from the standard
covering law model in that it does not require strictly logical deduction, but rather
allows the kinds of theoretical plausibility arguments that are often made in science
(i.e., if certain conditions are present, then that which was to be explained follows
from natural laws, theoretical principles and our ideals of natural order).
We do not consider the various debates an adequate explanation, but briefly discuss
four forms of explanation that are generally accepted: formal, causal, statistical and
functional. It is important to emphasize, however, that whether or not a particular
form of explanation is acceptable depends very much on context and the state of the
science in which it is offered.
Formal Explanation: Formal explanation takes its name from formal logic. We
explain a property or characteristic of a thing by asserting that the property or
characteristic is an essential quality of the category to which that thing belongs. The
Aristotelian explanation that fire moves upward because it is a fiery material seeking
to return to the “Sphere of Fire” is a formal explanation, as is the explanation that
the charge of an electron is what it is because that is the nature of
electrons.[281] Without careful thought, this kind of explanation can lead to
meaningless tautologies, as for example, the purported explanation of opium’s sleep
inducing property by the statement that opium contains a “dormative (i.e., sleep
producing) principle.”
Causal Explanation: A causal explanation asserts that we have explained a
phenomenon or event when we are able to state its cause. Causal explanation is
modeled on deductive logic, in that we explain an event by positing initial conditions
and the natural laws that lead to the occurrence of the event, given the initial
conditions posited. In other words, our explanation is of the form: given the initial
conditions, the observed event follows from natural laws.
Aristotle defined four different causes: (1) material cause; (2) formal cause; (3)
efficient cause; (4) final cause. The material cause is the actual physical matter
involved. The formal cause is the design, or pattern that a system seeks to attain,
while the efficient cause is that which produces the change or event, and the final
cause is the purpose that change was directed towards producing. For example, the
material cause of a house is the material it is composed of, the formal cause is the
blueprint, the efficient cause is the contractor who builds it, and the final cause is the
goal of producing a dwelling place. In modern science formal and final causes are
not considered, and efficient causes are looked at as the natural laws involved. A
minimal form of final cause does appear in evolutionary theory in the idea that
species form and behaviour is evolved to promote species survival (but note that this
is a consequence of evolutionary processes, not a predetermined final cause).
Some philosophers claim that all explanation is, or can be, reduced to causal
explanation. Aristotle, for example, asserted that the only way to understand a thing
was to know its causes. Be this as it may, we can state four “principles of causal
simplicity” that act as ideals for construction of causal explanations
1. Principle of Uniformity: This principle can be viewed as an application of the
principle of the identity of indiscernibles. It states that similar causes tend to
have similar effects, and vice versa. Deviations in effects must be a result of
specific causal factors. This principle is challenged by modern results, especially
in chaos theory and theories of non-linear systems, where very small deviations
in the cause can produce major differences in effects.
2. Principle of Stability: Change does not occur without a cause. This principle is
obviously related to the principle of sufficient reason, but it is less general,
claiming that the only sufficient reason for change in a system is a causal reason.
3. Principle of Composition: Causal explanations of things, phenomena and
processes should be in accord with known behaviours of the components of that
which is being explained, and not contradict them. For example, a causal theory
of species evolution should not violate the known laws of genetics. This principle
is a consequence of the principle of contradiction.
4. Principle of Common Cause: It is important in causal reasoning to avoid the
mistake of assuming that just because two events are constantly correlated in
time (e.g., event AA always precedes event BB), the first must be a cause of the
second. For example, before a storm the barometer reading will decrease, but
the decrease in barometer reading does not cause the storm. The assumption
that is justified is that if two events or phenomena show a constant
correlation, then they may both be dependent on a common cause. For example,
the barometer falls because the atmospheric pressure is decreasing, and
decreasing atmospheric pressure also results in storms.
A danger of thinking in terms of causal explanations is the tendency to believe that
every phenomenon has only one cause. We considered this problem briefly in
Discussion 9.1, when we asked whether “the cause” of the ozone layer’s depletion
was the chemical fact that certain chlorine-bearing molecules interact destructively
with ozone? —or is it that these same compounds are used in refrigeration and air
conditioning devices? —or is it that people are willing to buy such devices, so that
their manufacture is profitable? —or. . . ? In the case of complex systems, it is
important to avoid the trap of thinking in terms of single causes.
Statistical Explanation: A statistical explanation asserts that given a particular
situation, there is—on the basis of natural laws, our current theories and our ideals
of natural order—a high probability of the event that is being explained. Statistical
explanation is the basis of the rates charged by insurance companies. A person
purchasing insurance is classified into a particular risk group and charged a rate that
gives the company a high probability, but not a certainty, of making money on the
transaction.[282]
Functional Explanation: A functional explanation asserts that an event, thing or
phenomenon is as it is because its being so serves some function. Functional
explanations show up often in biology and sociobiology in discussions of species’
characteristics and behaviours: certain characteristics or behaviours are asserted to
have “survival value” for their possessors. This kind of functional appeal is found, for
example, in the sociobiological and evolutionary psychology claim that altruistic
behaviour is to be expected only toward close genetic relatives, and serves the
purpose of insuring that at least a part of an individual’s genes are passed to the next
generation.
Functional explanations are often accused of circularity or tautology. For example,
the idea of “survival of the fittest” is often attacked because it is circular: the fittest
survive, and we can only tell who is fittest by who survives.
We have described theories in terms of how they provide a conceptual system for
description, prediction and explanation. Understanding, however, is not just a
matter of being able to describe, predict and explain. Understanding comes under
what Plato called “the fifth.”
Understanding requires a human (or at least sentient and self-conscious) subject.
Thus there is not only the question of the external requirements, but also that of
what internal conditions must exist within the subject before he or she can
legitimately claim to understand. One such condition, it would seem, is the ability to
generalize beyond the immediately given materials, which involves going beyond
appearances to the recognition of underlying principles and patterns. That is, it
understanding is associated with the learning curves of Figure 13.1.2, and the “Aha!”
experience of the light bulb going on.
On a more prosaic level, we could say that a person understands a theory when she
or he has incorporated internal mental structures of that theory in such a way that
those structures are able to function as an information processor and an organizer of
perception.
A theory is not pieced together from observed phenomena; it is rather what makes it
possible to observe phenomena as being of a certain sort and as relating to other
phenomena. Theories put phenomena into systems. . . . From the observed properties of
phenomena the [scientist] reasons . . . toward a keynote idea from which the properties are
explicable as a matter of course.[283]
As we have noted, however, theories also act as filters on perception. That is, they
will tend to force new apprehensions to conform to established categories of
perception.
Science rests itself not in the world the scientist beholds at any particular point in time, but
in his mode of viewing that world. A [person] is a scientist not because of what [they] see,
but because of how [they] see it.[284]
This filtering is not a negative thing because it provides stability and a secure
foundation for communication between scientists. It is, however, something that
needs to be borne in mind, because scientific creativity involves, in part, the capacity
to view old ideas or concepts in a new way. Thus, there will always be a dichotomy
for the scientist to balance. Too much to one side, and scientific creativity and
innovation are restricted; too much to the other, and a person loses contact with his
or her scientific community.
Part Three: Types of Theories in Science
The discussion of theories given above can be summarized in N. R. Hanson’s
statement, “theories provide patterns within which data become intelligible.”[285] We
can think of this definition in conjunction with Gregory Bateson’s injunction that in
attempting to understand a thing, we should seek for “the pattern that
connects.”[286] In a study of theories, however, we find that there are different ways in
which these “patterns that connect” can be expressed, depending on the state of
development in particular fields of science and on the available descriptive
languages. We will distinguish six different kinds of theories.[287]
1. Intuitional Theories: These are theories whose purpose is to provide an intuitive
understanding of their subject. A set of general principles is given which can be
used in making explanations, but these theories lack predictive power. Freud’s
theory of neurosis is an example of an intuitional theory. On the basis of this
theory, an observed neurosis can be explained in terms of theoretical entities,
such as fixated defense mechanisms originating in childhood trauma. However,
we cannot predict that particular traumatic events in childhood will result in
neurosis, or if they do, what specific form the neurosis will take.
2. Historical Theories: These are theories that describe the occurrence of unique
historical events by reconstructing the event. A good example is the theory that
the extinction of the dinosaurs was caused by the impact of a large meteor or
comet. Such theories have predictive power, but this power is limited to the
possibility of discovering traces left in the historical record by the hypothesized
event. The meteor theory of dinosaur extinction, for example, predicted the
possibility of discovering evidence of a major meteor impact about 65 million
years ago.[288]
3. Qualitative Theories: These theories are characterized by a focus on the
qualitative organization of observations as a means of gaining heuristic and
predictive power. Theories of weather prediction are of this type, based on
attempts to draw inferences from masses of qualitative data on temperature,
atmospheric pressure, humidity, the jet stream and so forth, which is then used
as input to simulation models.
4. Taxonomic Theories: These theories are concerned with classification and
recognition of phenomena. They deal with the construction of category systems
and taxonomies, as discussed in Discussion 11.2, and with predictions based on
class membership. Biological systematics is an example of this kind of theory.
5. Statistical Theories: Statistical theories are non-deterministic mathematical
theories. Predictions are given in terms of probabilities rather than certainties,
and the theories are formulated in terms of ensembles of events rather than
state spaces. Population genetics, in which predictions are made as to the
expected frequencies of genes in a population, is a good example of a stochastic
(statistical) theory.
6. Dynamical Theories: Dynamical theories are deterministic mathematical
theories in which system change over time follows from a set of equations of
motion. The standard form for such theories is based on the idea of a state space
that contains all possible system states, together with a transition rule (the
equations of motion) that determines possible changes of state. Newtonian
mechanics is a paradigmatic example.
Many current scientific theories have features characteristic of more than one of
these types of theories. The theory of plate tectonics in geology, for example, has
intuitional, historical and qualitative aspects; while current theories in elementary
particle physics have taxonomic, dynamical and stochastic aspects.
The physicist Henry Margenau (1901-1997) has suggested six criteria for a good
scientific theory, which apply to all types of theory discussed above.[289]
1. Constructs found in the theory (e.g., electrons, molecules, biological species,
social classes, psychological states, etc.) ought to possess logical fertility. That is,
they must provide a basis for fruitful deductions and the construction of
hypotheses.
2. Constructs in the theory ought to be multiply connected.
3. The theory needs to be stable. That is, it must be possible to adapt the theory to
new data.
4. Constructs in the theory ought to be extensible. That is, they should be found to
have applications beyond the original confines of the theory itself.
5. The theory ought to provide causal connections between phenomena and the
events it is intended to explain.
6. Theories must strive for simplicity and elegance.
Theories that satisfy these criteria are considered good theories, and much of the
work in theoretical science involves attempting to develop or refine theories so that
they can be considered to be “good” theories.
Discussion 13.2 Big Bang or
Steady State?
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Cosmology, as a modern science, began empirically with the discovery that the
distant “spiral nebulae” observed through existing telescopes were actually galaxies
consisting of hundreds of millions of stars. This was discovered by Edwin Hubble
with the 100-inch telescope on Mount Wilson in California. Using this telescope
Hubble first studied the Andromeda “nebula,” determining that it contained stars,
and then, making use of a distance measure based on Cepheid variable
stars,[290] showed that it was very far outside of our galaxy. In other words, it was
another galaxy, not some feature of our own. By studying other galaxies, Hubble
made the surprising discovery that these galaxies seemed to be receding from us. In
other words, he discovered that the universe was expanding. He published his
results in 1929. Further work allowed estimates of the rate of expansion and,
extrapolating this rate into the past suggested the age of the universe.
The foundation for theoretical cosmology was provided in Einstein’s general theory
of relativity, published in 1916. The Russian physicist Alexander Friedmann (18881925) published a mathematical solution of the Einstein equations that
corresponded to an expanding universe in 1922, but his paper was not recognized as
anything other than a mathematical exercise. Similarly, the Belgian physicist
Georges Lemaître (1894-1966) published a 1927 paper giving solutions that also
corresponded to an expanding universe, but again scant attention was paid until
1930 when his theoretical work was compared to Hubble’s discovery that the
universe was, indeed, expanding. In 1931, Lemaître suggested that the universe had
begun with the explosion of a “cosmic egg,” that our current universe was “the ashes
and smoke of bright but very rapid fireworks.” This “fireworks theory” provided the
first version of the Big-Bang theory for the origin of the universe.
The 1930s were a time of getting used to the radical change in our view of the cosmos
brought about by Hubble’s discovery, but there was little advance in cosmology in
the 1940s while the Second World War was in progress. The late 1940s, however,
saw the reemergence of scientific interest in cosmology, especially centered on the
problem of predicting the distribution of elements (i.e., the proportions of the
different elements in the universe as a whole). New understanding in atomic and
nuclear theory provided the possibility of finding an answer for this question, under
the assumption that the elements had been “cooked up” in the initial explosion of the
“cosmic egg.”
George Gamow, Ralph Alpher, and Robert Herman took up this question. The basic
assumption underlying their work was that the universe had cooled from an initial
state of almost infinite density and temperature in which the only particles present
were protons, neutrons, and electrons immersed in a sea of high energy
electromagnetic radiation. (They called this mix “Ylem,” an ancient philosophical
word for “primordial substance.”) Calculations (carried out with some of the first
electronic computers) were able to give reasonably good estimates of the observed
abundances of hydrogen and helium, but the answers obtained for other elements
were radically in error (Gamow joked that the theory must be pretty good, though,
since it predicted the abundances of 99% of the matter in the universe). Thus, the
question remained as to how the other elements had arisen.
The resolution of this question came with the realization that other elements arose
primarily through the process of nucleosynthesis taking place in the core of stars.
Measurements by the British astronomer Fred Hoyle provided empirical support for
theoretical calculations based on this assumption. Hoyle, however, along with
Hermann Bondi and Thomas Gold, were proponents of another cosmological theory,
the “Steady-State Theory,” which challenged the “fireworks” theory (sarcastically
renamed the “Big Bang Theory”[291] by Hoyle).
According to the steady-state theory, the universe has existed eternally and always
looked much as it does now. Since it is expanding, however, this theory requires that
a certain amount of new matter be continuously created. At the time it was proposed
the steady-state theory seemed to resolve a problem in the big bang theory, namely
that when the expansion of the universe was run backward, based on the best
available data on expansion rates, the age it predicted for the universe was less than
the known age of the Earth! This argument against the big bang theory was
dispelled, however, with the discovery that distance measurements based on the
period-brightness of Cepheid variables confused two distinct types of these stars.
Correcting for this error indicated an age for the universe that was twice the original
estimates and satisfactorily much older than the age of the Earth.
A possible test to distinguish between the two theories involved the age of galaxies.
In the big bang theory, the more distant galaxies ought to be older than nearby
galaxies. In a steady-state universe a more uniform distribution of young and old
galaxies would be expected. Older galaxies did appear to be more distant in a 1948
survey, but a review of the data in 1954 showed that this effect was an artifact of
measurement. The available data wasn’t refined enough to provide a means of
deciding between the two theories.
Other evidence against the steady-state theory began to show up, however, based on
radio astronomy observations initiated by Martin Ryle in 1951 and published in
1955. Ryle had conducted a survey of 2000 extra-galactic radio sources and claimed
evidence indicating that the more distant ones were also older. Steady-state
supporters questioned these results and it took a number of years to develop further
substantial supporting data for the big bang theory.
The deathblow for the steady-state theory came in 1965, but five more years of study
were needed to bury it. This evidence came with the discovery of the cosmic
microwave background by Bell Laboratory physicists Arno Penzias and Robert
Wilson. In 1948 George Gamow had published a paper arguing that as the universe
expanded from its initial state of almost infinite density, the radiation present would
cool to a current estimated temperature of about 5°Kelvin (i.e., five degrees above
absolute zero). In 1963 Penzias and Wilson observed a background microwave
radiation corresponding to about 3°K that seemed to be uniform across the sky. In
1965 the Princeton physicist Robert Dicke and his former student James Peebles,
who had been working along the same lines as Gamow, recognized that Penzias and
Wilson’s data provided empirical support as the archaic signature of the big bang.
A few cosmologists persist in attempts to adjust the steady-state model so that it fits
available data, but the theory was effectively abandoned by the mid-1970s.
Unit 13 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. If we focus attention on the forms of scientific theories, in contrast to their
content we see that a form of theory valuable in one area of science may be
worth little in some other area, and vice versa. The Aristotelian idea of a life
cycle is a particular form for theorizing, based on the assumptions listed below.
o
o
Entities develop according to a natural progression in an organic way,
from birth to death, or completion.
By various means, the rate of this natural development can be accelerated
or retarded, within certain bounds.
This particular form of theory was taken over into alchemy, where it was
assumed that metals develop in the earth toward gold; which, being the most
noble metal, is their final state of completion. Many alchemists sought to
discover a method to speed up this process in the laboratory.
c. Today we know that this form of theory does not fit when applied to the
study of metals. Can you think of an area of modern science where this
life cycle does appear as an ideal of natural order?
d. Write a short essay (300-500 words) discussing an example from your
own experience where the life cycle concept applies.
2. Write a short essay (300-500 words) discussing the following quotation, “There
is only one way of seeing one’s own spectacles clearly: that is, to take them off. It
is impossible to focus both on them and through them at the same time.”[292]
3. Each of the following statements gives an instance of a scientific law statement.
Determine whether the law that describes the instance is a law of coexistence,
succession or interaction, or some combination of these.
a. One mole (gram molecular weights) of oxygen combines with two moles
of hydrogen to yield one mole of water.
b. A pea plant that produces red flowers is crossed with a pea plant that
produces white flowers. The offspring are found to be 25% red flowering,
25% white flowering, and 50% pink flowering.
c. Two electrons cannot simultaneously occupy the same quantum state.
d. If two species compete for the same ecological niche, one of the species
will either become extinct or find a new niche.
e. The only constraint on the particles emerging from a high energy
elementary particle collision is that energy, momentum and all quantum
numbers are conserved.
f. A sufficiently massive star becomes a supernova when the star’s core
burns all of its available fuel so that it can no longer support the weight of
the outer layers of the star, which then collapse into the core.
4. Determine whether each of the following explanations is formal, causal,
statistical or functional, or some mixture of these types.
a. The chameleon’s ability to change colour evolved as a means of protective
camouflage.
b. The litmus paper changed colour because the solution it was dipped into
is acidic.
c. The neurotic behaviour of thoroughbred horses and of some breeds of
dogs is a result of excessive inbreeding.
d. Excessive inbreeding is not recommended because it tends to reinforce
negative genetic traits.
e. Play gives young animals the opportunity to practise behaviours, such as
stalking, fighting and fleeing, which will be important for them in adult
life.
f. HCl is corrosive because it is an acid.
g. If an individual shows a recessive trait, it means that both of that
individual’s parents carried that trait.
h. There was a storm yesterday because a low-pressure zone moved into the
area, and bad weather usually follows a drop in atmospheric pressure.
5. State which of the four causal principles described in Discussion 13.1 applies in
each of the cases outlined below.
a. The British bacteriologist Sir Alexander Fleming (1881-1955) carried out
a series of experiments in which dishes of bacteria were cultured. Fleming
noticed that some sort of mold had contaminated one of the dishes. He
also noticed that there were circles around the spots of mold where the
bacteria had died. He recognized that the mold must be producing some
anti-bacterial agent, and thus discovered penicillin.
b. In the 19th century the orbits of Mars, Jupiter and Saturn showed
irregularities that could not be accounted for using Newton’s theory of
gravity. On this basis, it was predicted that there was another planet
beyond Saturn. This prediction led to the discovery of Uranus. Later, yet
another outer planet was predicted on the basis of further orbital
irregularities, leading to the discovery of Neptune.
c. Unexplained irregularities were later observed in the orbit of the planet
Mercury, and on this basis some astronomers began to search for a new
planet, closer to the sun than Mercury. No planet was ever found, and the
irregularities in Mercury’s orbit were only explained by Einstein’s general
theory of relativity.
d. There are statistically significant correlations between being good at
mathematics, being nearsighted, being left-handed, being susceptible to
allergies and autoimmune diseases, and being male. Investigators have
hypothesized that all of these phenomena are a result of the effect of
testosterone on the developing brain of a fetus.
e. A spray of particle tracks observed in a bubble chamber cannot have been
caused by a collision event in which quantum numbers are not conserved.
6. Give at least one example, other than those already given in the discussion of
each of the six types of theories described in Discussion 13.1, part three.
7. In Discussion 3.2, we considered science as a question-and-answer process, and
described the qualities of a good question and the criteria for a valid answer. A
scientific theory can be seen both as providing an answer to a general question
and as a framework within which answers can be given to more specific
questions. With this in mind, write a short essay (300-500 words) comparing
the qualities of a good question, the criteria for a valid answer, and the six
criteria for a good theory given in Discussion 13.1, part three.
8. Why, in the Big Bang Theory, would we expect more distant galaxies to be older
than nearby ones?
9. Write a short essay (300-500 words) analyzing the history of modern cosmology
as presented in Discussion 13.2. Focus attention on the different factors
contributing to the modern theory and the underlying theoretical assumptions
that were involved in the development of this theory.
FOOTNOTES
[265]
Einstein, Albert. Out of My Later Years, p. 88. Westport, CT: Greenwood, 1971.
[266]
Poincaré, H. Science and Hypothesis, p. 141. New York: Dover, 1952.
[267]
Goethe, Johann Wolfgang von. Elective Affinities, Book I, Chapter 9, 1808.
[268]
Quoted in Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein,
rev. ed. p. 274. Cambridge, MA: Harvard University Press, 1988.
[269]
Wilson, Robert Anton. The Widow’s Son: The Historical Illuminatus Chronicles, Volume
2. New York: New American Library, 1981.
[270]
This analogy is behind much of our educational system. The role of the teacher is seen as
being to pour information from the course textbooks into the receptive minds of students.
Carrying this analogy further, we can also say that we cannot pour anything into an already
full container. A mind filled with preconceptions is incapable of learning.
[271]
The imprisoning nature of belief systems is one of Doris Lessing’s main points in The
Prisons We Choose to Live Inside; the same is true even of scientific theories. There is a
saying in science that the proponents of old theories don’t change their minds, they just
eventually die off.
[272]
Feynman, R. The Character of Physical Law, p. 13. Cambridge, MA: MIT Press, 1965.
[273]
Heisenberg, Werner. Philosophical Problems of Nuclear Science, p. 55. New York:
Fawcett World Library, 1966.
[274]
If we have a theory that does fit, within certain limits (e.g., the limits of accuracy of
observation), then we have our “sufficient reason,” and there is no need to seek for further
theories. In such a situation, scientists will seek to develop and extend the existing theory
rather than continually developing new ones. You might compare this to the Popperian idea
that we should constantly be developing new theories and attempting to refute them, and
recall Pólya’s “virtue of wise restraint.”
[275]
Feynman, Richard. The Character of Physical Law, p. 171.
[276]
Whitehead, Alfred North. The Concept of Nature, p. 163. Cambridge: Cambridge
University Press, 1926.
[277]
Quoted in Reader’s Digest, October 1977.
[278]
Suppe, F. The Structure of Scientific Theories, 2nd ed., p. 708. Urbana: University of
Illinois Press, 1977.
[279]
There will generally be categories of theoretical entities as well.
[280]
Park, David. The How and the Why: An Essay on the Origins and Development of
Physical Theory, p. 392. Princeton, NJ: Princeton University Press, 1990.
[281]
When inappropriately extended to the social world, this form of explanation is often a
basis for intolerance.
[282]
Historically, the development of mathematical statistics was strongly motivated by the
need to compute accurate statistical tables for insurance companies!
[283]
Hanson, N. R. Patterns of Discovery: An Inquiry into the Conceptual Foundations of
Science, p. 90. Cambridge: Cambridge University Press, 1965.
[284]
Roszak, Theodore. The Making of a Counter Culture, p. 213. Garden City, NY: 1969.
[285]
Hanson, N. R. Patterns of Discovery, p. 90.
[286]
Bateson, Gregory. Mind and Nature, p. 8. New York: Bantam, 1980.
[287]
Adapted from Rappaport, A. “Various Meanings of ‘Theory.’” American Political Science
Review (1958): 927-988.
[288]
This theory provides an excellent example of the process of theory construction. It
originated with the observation of a world-wide layer of iridium in sediment dated to about
65 million years ago. The concentration of iridium in this layer was much higher than would
be expected in deposits laid down by known earthly mechanisms, but was consistent with
the known concentration of iridium in meteors. The iridium layer was correlated with the
extinction of the dinosaurs, which was known to have occurred about 65 million years ago,
and on the basis of this correlation, various geological consequences of a large meteor
impact were deduced. Researchers sought whether the geological record showed traces that
these consequences had actually occurred. To date the evidence supports the theory, which
gained corroboration from the discovery of the Chicxulub crater off the shore of southern
Mexico. This crater was produced by the impact of a comet or asteroid, in the range of 5 to
15 km in diameter, about 65 million years ago.
[289]
Margenau, Henry. The Nature of Physical Reality. Woodbridge, CT: Ox Bow Press, 1977.
[290]
Cepheid variables come in two types. For both, however, there is a relation between their
brightness and the period of their variation, so by observing this period it is possible to
measure of their intrinsic brightness. Comparing this measurement to their observed
brightness allows estimation of their distance.
[291]
A religious aspect to the question arose when Pope Pius XII proclaimed in 1952 that big
bang cosmology affirmed the existence of a Creator God. A few years later, Hermann Bondi
published a rejoinder, pointing out that the big bang was based on a singularity in certain
solutions to the Einstein equations and that, in mathematics, one usually dealt with
singularities by finding ways to remove them. Thus, to associate the big bang with creation
“raised the possibility of eliminating God with better mathematics.”
[292]
Toulmin, Stephen. Chapter 5: “Forms and Styles of Theory.” In Foresight and
Understanding: An Enquiry into the Aims of Science, pp. 83-98. Westport, CT: Greenwood,
1961.
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Unit 15 What Are Some General Methods in Science?
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Discussion 15.1
Discussion 15.2
Discussion 15.3
Unit 15 Study Questions
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STUDY GU IDE
Unit 15 What Are Some General
Methods in Science?
There has got to be an easier way.
—Blitzer’s Law[305]
Seek simplicity, and distrust it.
—Alfred North Whitehead[306]
When we think of science, we often have an image of scientists in white lab coats
carrying out carefully calibrated measurements, or perhaps of ivory-tower theorists
engaged in arcane dialogues laced with incomprehensible mathematics. Every so
often a cry of “Eureka” rings out, announcing the appearance of another earthshattering result or monumental theory.
By now, you should realize that this is not the way that science takes place. To
paraphrase Jacob Bronowski, science is hard work that involves effort, sweat and a
good deal of bad language.
It should also be clear that it is impossible to write out a definitive set of rules for
doing science. There is no unique scientific method that if followed, will guarantee
good science. Science takes place when a person with a few tools for clear reasoning
and unbiased observation and experimentation decides to study some phenomenon
from a particular point of view and with a particular attitude. At bottom, science is
grounded in the consciousness of scientists and the scientific community.
In addition to the more formal tools of science—formal logic, mathematics and the
tools for accurate observation and experiment—there are a number of other
strategies of thought that can be used in doing science. These strategies cannot be
used to prove a claim, but they offer hints, point directions, suggest new ideas, and
make conjectures plausible.
In this final unit, we consider some of these heuristics. We begin, in Discussion 15.1,
with a consideration of the way various forms of reasoning are used in science. In
particular, we distinguish between empathetical, analogical and analytical reasoning,
provide examples of each, and describe the role each can play in scientific work. A
specific case is also considered: the development of the Bohr model of the atom,
which illustrates the interplay of these three forms of reason.
In Discussion 15.2, we turn our attention to thought experiments, and follow this
with a reading in What Science Is dealing with the nature of scientific research and
its contributions. Discussion 15.3 elaborates on the ideas of rough estimates and
dimensional analysis, two methods of getting useful, if very approximate,
information about systems.
Objectives
When you have completed Unit 15, you should be able to
1. define empathetical, analogical and analytical reason, describe their role in
science, and give examples of each.
2. discuss the concept of a thought experiment, and describe a number of thought
experiments that have been used in both science and philosophy.
3. discuss the contributions of science, as described in Chapter 7 of What Science
Is.
4. carry out rough order of magnitude estimates and dimensional analyses.
5. apply the material you have learned in this course to the analysis of historical
examples, and also to your own scientific studies.
Indications
As you work through this unit, you will complete the readings and assignments listed
below.
1. Read Discussion 15.1, “The Use of Reason in Science.”
2. Answer Unit 15 Study Questions 1-4.
3. Read Discussion 15.2, “Thought Experiments.”
4. Answer Unit 15 Study Questions 5-8.
5. Revisit and review Chapter 7, “Thinking Straight: Evidence, Reason, and Critical
Evaluation” Pages 87–106 of What Science Is.
6. Read Discussion 15.3, “The Back of the Envelope.”
7. Answer Unit 15 Study Questions 9-11.
8. Contact the Office of the Registrar at Athabasca University and request the final
examination. Plan on giving yourself about two weeks to review materials from
Units 9-15 in preparation for writing this exam.
Discussion 15.1 The Use of Reason
in Science
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If we want to solve a problem that we have never solved before, we must leave the door to
the unknown ajar.
—Richard Feynman[307]
I know the weakness of human reason . . . but it is all we have, and the only safety of man is
to cultivate it and extend his knowledge so that he will be sure to understand life and as
many of the mysteries of the universe as he can possible solve.
—Clarence Darrow (1857-1938)[308]
There are many ways to approach the question of the nature and function of reason
and how it is used in science. For some philosophers, reason is strictly related to
analytical thought, and to formal logic and propositional calculus.[309] For other
philosophers, however, reason is something much more general. As should be clear
already, the view of reason taken in this course is of the more general variety.
In Unit 8, we distinguished three broad categories of reasoning: empathetical,
analogical and analytical. Here, we are concerned with how these three forms of
reasoning fit together in the scientific enterprise. In thinking of scientific reasoning,
we tend to assume that it is always analytical. However, while the presentation of
ideas in science relies heavily on analytical reason, empathetical and analogical
reasoning are also important.
Empathy involves “putting oneself in the place of.” If you have ever decided what to
order in a restaurant by imagining how you would feel after having eaten some of the
dishes on the menu, then you have used empathetical reasoning. You have, as it
were, put yourself in the place of yourself—later, after you have eaten. Many
philosophers and even a few scientists will not admit that empathetical reason is
important in science, or will claim that it is beyond philosophical analysis. This
attitude was particularly characteristic of the logical positivist school, which put
empathy under the heading of “psychologism,” the attempt to support rational
conclusions on the basis of psychological factors such as intuition or feelings.
However, while it is certainly true that no scientific conclusion can be accepted
merely on the basis of intuition, empathy or aesthetic appeal, this fact does not mean
that empathy is not significant in science. Conclusions based on such grounds,
however, need to be treated with extreme care and eventually justified analytically
and empirically—reasoning based on intuition, empathy or aesthetics is difficult to
communicate, and these mental capacities are notorious for leading people astray. In
the words of the French mathematician Henri Poincaré, when an appealing idea
does not work out, “[W]e almost always notice that this false idea, had it been true,
would have gratified our natural feeling for mathematical elegance.”[310] Nevertheless,
such forms of insight are essential in the creative phase of scientific work. An
example of this kind of reasoning is found in the autobiography of Jonas Salk, the
discoverer of the polio vaccine. Salk reports that he used to imagine that he was a
virus and ask himself the question, “What would I do under such-and-such
circumstances; for example, if I wanted to infect a particular kind of cell?”[311]
The role of beauty in science is emphasized by Paul Dirac, called the greatest British
physicist since Newton, who went so far as to say, “[I]t is more important to have
beauty in one’s equations than to have them fit experiment.”[312] He supports this
seemingly non-scientific statement with the story of how Erwin Schrödinger
developed the equation in quantum mechanics that bears his name. At first
Schrödinger wanted an equation that gave the correct quantum results and that was
also consistent with special relativity. He came up with a beautiful equation that was
relativistic, but when he tried to apply it to the case of the electron, its predictions
did not fit experimental values. So he gave up the condition of relativistic invariance
and developed the Schrödinger equation, which did fit. But it later turned out that
the experiments had not taken into account a previously unsuspected property of
electrons—their “spin.” When spin was included, the experiments gave results that
agreed with the first equation Schrödinger had tried. This equation, rediscovered in
1926 by Oskar Klein (1894-1977) and Walter Gordon (1893-1940), is now called the
Klein-Gordon equation.
Empathetical reasoning cannot be used as a means of giving formal support to a
conclusion in science, but it plays an important role in determining which
conclusions a scientist will attempt to support with his or her analogical and
analytical reasoning. It involves what scientists feel as a sense of intuitive rightness,
elegance, simplicity and beauty. The sensitivity of such intuitive forms of thought to
personal bias, however, means that other more rigorous methods must be used to
test our insights and intuitions, and this is where analysis and the experimental test
enter the picture.
Analogical reasoning is based on the assumption that if two things are alike in some
respects, they will be alike in other respects as well. The standard form for an
analogy is AA:BB::CC:DD which reads AA is to BB as CC is to DD. Some examples of
simple analogies are
o
o
o
water is to steam as ice is to water (i.e., if we heat water enough it becomes
steam, and if we heat ice enough it becomes water).
mind is to brain as computer software is to computer hardware.
racism and sexism are to society as cancer is to the body.
In communicating scientific ideas, it is often useful to have a familiar picture, image
or model to provide a basis of reference. Reasoning about the properties of the
model or image can then be extended to the actual subject of interest. Indeed, one
form of understanding is being able to have a mental image.[313]
Analogical reasoning is often found in science when ideas that are known to be
useful in one science disciplineare borrowed by another. In fact, a scientific theory
itself is a form of analogy. That is, the theory is a mental construct that, it is claimed,
stands in analogy to the aspects of reality for which it provides a theory. For
example, suppose that a theory of some real-world system claims that the behaviour
of this system is governed by a particular differential equation, and that solutions of
this equation correspond to possible states of the system. Or, making the analogy
more explicit; solutions of the equation, which exist only in a particular
mathematical space, will have properties characterized by certain parameters that
appear in or can be computed from the governing equation. It is supposed that the
assumptions and constraints employed in deriving the equation, and in setting the
initial and boundary conditions for its solution, are analogous to conditions that are
satisfied by the real system, and therefore that solutions of the equation will
correspond to possible real-world behaviours.
A much simpler example of the use of analogy in science is the Freudian model of
mind compared to a steam engine, which we considered in Discussion 11.3.
Analogies are extremely useful in science, because they allow us to think about the
unfamiliar in terms of the familiar. But this advantage is also a danger. We need to
be aware that the images used in our analogies may mislead as well as illuminate,
especially when they are used to describe aspects of the world that are far from our
everyday experience. As Jacob Bronowski notes, referring to difficulties in
understanding quantum mechanics and relativity theory,
It needs a profound and sustained effort of imagination to project ourselves into these
extreme states, and to draw from the language that describes them an understanding of
their subtle yet radical departures from the analogies that we necessarily borrow from our
daily experience.[314]
A good example of the use of analogy in science is found in the early-20th century
attempts to understand the structure of the atom. Two different models of the atom
were proposed, based on simple analogies, and much debate took place in attempts
to decide between them.
The word atom is from the Greek atomos, “indivisible.” In the Greek view, the
smallest possible units of each element (earth, water, air and fire) were
indivisible, atomos. This view resurfaced in the atomic theory of the chemical
elements put forward by John Dalton (1766-1844), and developed throughout the
19th century. The elements were no longer the analogically and empathetically
defined elements of the Greeks,[315] but the analytically determined chemical elements
we recognize today. However, the belief that they were the smallest, indivisible units
of matter persisted.
The 1897 discovery of the electron by J. J. Thomson (1856-1940) overturned this
belief. It was observed that ultraviolet light shining on a metal plate resulted in the
emission of negatively charged particles from the plate. The problem was to explain
where these particles came from. The emission effect itself was called the
“photoelectric effect.” The first suggestion was that these “electrons” existed in the
interstices (spaces) between the indivisible atoms. From this model, Thomson
concluded that the energy of the electrons emitted in the photoelectric effect would
depend on the intensity of the light. But further experiments showed that, while the
number of electrons emitted was proportional to intensity, their energy was not. The
energy depended only on the frequency of the light. On this basis, Thomson
concluded that the negatively charged “corpuscles” he had discovered were
constituents of atoms, thus initiating a major research program among physicists of
the day to discover the structure of the atoms composing the chemical elements.
The question of how many electrons were contained in the atoms of the different
chemical species was resolved in studies carried out by Antonius van den Broek
(1870-1926), Lord Rutherford (1871-1937), and Henry Moseley (1887-1915). The
number of electrons in the atoms of a given chemical element was found to be
proportional to its atomic weight, and as was later determined, exactly equal to its
atomic number.
Two other questions were essential for assessing the actual role of the electrons in
the structure of the atom. J. J. Thomson had determined that the mass of the
electron was extremely small compared to the mass of an atom (the electron mass
was about 1836 times smaller than the mass of the hydrogen ion), yet atoms contain
only a relatively small number of electrons. It was important to determine the source
of the remaining mass. In addition, non-ionized chemical elements are electrically
neutral, so it was necessary to determine the nature of the positive charge required
to neutralize the negatively charged electrons.
Two almost identical models for the atom were suggested by Lord Kelvin (18241907) in 1902, and by J. J. Thomson in 1903-04. Both models envisioned atoms as
clouds of uniformly distributed positive charge in which electrons were located.
Kelvin’s initial model became known as the “plum pudding model” because it
proposed that electrons were embedded in this cloud of positive charge like currants
in a plum pudding.
It was supposed that collisions between atoms could jar electrons loose from an
atom, and that the electrons would then either escape, leaving a positively charged
ion, or fall back into the atom, emitting electromagnetic radiation in the process.
However, atoms of the different chemical elements were known to emit radiation
only at certain characteristic frequencies, which provided an immediate theoretical
challenge for the model.
Based on the kinetic theory of gases, the atoms in a hot gas would have a statistical
velocity distribution about a mean, related to the temperature; hence the energy
involved in collisions between atoms would be similarly distributed. Therefore,
electrons knocked out of an atom would have a continuous distribution of energies,
and would be expected to emit a continuous spectrum of radiation as they fell back
into the atom. This hypothetical expectation was in direct contradiction to the
observed discrete spectrum of the chemical elements.
Unsuccessful attempts to deal with this problem involved assuming that electrons
were distributed on concentric circles within the sphere of positive charge (proposed
by J. J. Thomson), or that atoms consisted of concentric spheres of alternating
positive and negative charge (proposed by Kelvin).
Other versions of the plum pudding model might have been proposed, but the model
came under attack from another set of experiments based on the scattering of
ionized helium (αα-particles) from atoms.[316] Since the αα-particles were positively
charged, they were repelled by the positive charge in the atoms being bombarded.
The approximate size of these atoms could be determined, so the pattern of
scattered αα-particles could be used to make inferences about the distribution of
positive charge within the atom[317].
On the basis of such scattering experiments, carried out by his collaborators Hans
Geiger (1882-1945) and Ernst Marsden (1889-1970), Rutherford concluded that the
positive charge in an atom was concentrated in a very tiny central region that he
called the nucleus. Rutherford initially suggested a model in which a cloud of
electrons surrounded a small, positively charged nucleus. This model had a major
flaw, however: because of the attraction between positive and negative charges, the
electrons would be expected to fall quickly into the nucleus, and there seemed to be
no mechanism to get them back out. This problem led Rutherford to propose a
model based on an analogy to the solar system: the positively charged nucleus
resided at the centre of an atom and contained almost all of the atom’s mass.
Electrons orbited the nucleus in the same way the planets orbited the sun, being held
in their orbits by the inverse balance of centrifugal force and the inverse square
electrical force, instead of gravitation.
In one sense, this “miniature solar system model” was no better than the plum
pudding model. Indeed, according to classical electrodynamics, it would be highly
unstable. Charged particles undergoing acceleration emit radiation and lose energy.
Thus, from the viewpoint of classical theory, an electron orbiting a positively charged
nucleus would continually emit radiation, lose energy, and spiral into the nucleus in
a very short time. In addition, the electron would emit a continuous spectrum of
radiation as it spiraled in, contradicting the observed discrete spectra. The only
advantage of the model was that it was consistent with the data from scattering
experiments.
A young Danish physicist visiting Rutherford’s laboratory remedied these
deficiencies in a spectacular way in 1913. Niels Bohr (1885-1962) accepted a solar
system type of atomic model, but was concerned about its apparent instability. By
making use of two counter-intuitive assumptions, Bohr was able to produce a model
for the hydrogen atom that was not only stable, but also allowed a derivation of the
experimentally based formula for the possible frequencies of radiation emitted by
hydrogen.
It is worth following Bohr’s line of thought in the development of his model of the
atom. He believed that the image of a positively charged nucleus about which
electrons moved in circular orbits provided at least a good approximation for atoms.
He knew that, according to classical electromagnetic theory, such a model could not
be stable. On the other hand, it was obvious that matter is stable.
All other researchers had considered the contradiction between the stability of
matter and the classical instability of the miniature solar system model to be
evidence against the model. Even Rutherford did not believe that this model could
be correct. Bohr, on the other hand, concluded that the classical theory was invalid at
the atomic level.
While this was a radical step, it was not one taken in the dark. The data from
scattering experiments did imply that the positive charge in an atom was confined to
a very tiny central region, far smaller than the atom itself. This was enough to
eliminate all versions of the plum pudding model. It was also certain that electrons
carried the negative charge in an atom. In addition, Bohr was aware of work that
implied that radiant energy came in discrete packets,[318] and of the theoretical
explanation of the photoelectric effect published in 1905 by Albert Einstein, which
assumed that light was composed of discrete packets of energy.[319]
Bohr proposed two hypotheses about the nature of atomic systems:
1. atoms can only exist in particular “stationary states,” characterized by a discrete
series of energy values, and changes in the energy of an atom can only happen
by a transition from one stationary state to another.
2. an atom can only emit or absorb radiation at frequencies such that the
difference in energy between the initial and final atomic states is an integer
multiple of hvhv. The discrete energy values of an atom’s stationary states were
determined by requiring that the angular momentum of electrons in their
circular orbits be an integer multiple of h/2πh/2π. This fixed the possible values
for the radii of electron orbits, and thus their energy.
The Bohr model was developed and extended over the next dozen years, and is still
discussed in elementary physics classes. It was replaced in the mid-1920s, when
quantum theory—to which it was a major contributor—was developed. This model
has also persisted in popular culture, because it provides an easy image for the mind
to grasp. The quantum theory of atoms provides no such convenient picture.
(Indeed, in response to a complaint by Erwin Schrödinger that the idea of quantum
jumps was unnatural and illogical, and could lead only to paradox, Bohr replied,
“What you say is absolutely correct. But it does not prove there are no quantum
jumps. It only proves that we cannot imagine them.”[320])
In this example, a third type of reasoning—analytical reason—was used to help in
deciding between two analogical models. Analytical reason is what we usually think
of as reason itself. It involves breaking something down into components in an
appropriate way, with the idea that we can gain an understanding of the thing itself
by studying the nature of its components and how they fit together. In business, for
example, a person might have a wonderful, intuitive idea for a new product or
service, and might argue for this idea on the basis of illustrative analogies to other
similar ventures that have been successful. Analysis enters the picture when
somebody says, “Yes, this is all very nice, but now let’s look at the numbers.”
In chemistry, the periodic table of the elements is another example of the analytic
approach. The various chemical compounds are broken down into atoms whose
properties are listed in this table. Knowing what atoms a compound is made of gives
an understanding of its properties. For example, hydrochloric acid (HCl) is
composed of one hydrogen atom and one chlorine atom, which disassociate in
solution. Chlorine is a highly reactive element so we know that hydrochloric acid will
be corrosive. We also know that, because fluorine is even more reactive than
chlorine, hydrofluoric acid (HF) will be even more corrosive.
One of the major problems facing psychology and the social sciences is the
determination of an appropriate analytical framework. What is the best way to cut
up a human psyche, or a society, into components whose interactions may be
studied?
The three forms of reasoning can all be viewed in terms of the working definition of
reasoning as a process of fitting together. In empathetical reasoning the question is:
If I put myself in the place of xx how do I fit, and from that view how do things in
general fit?[321] In analogical reason, the question is: How well does the analogy fit?
How far can it be pushed before it breaks down? To what extent is it really possible,
for example, to think of the mind as a steam engine? Does it make sense to compare
society to the human body? In analytical reason, we are concerned with gaining
understanding by identifying the basic components of a system and discovering how
they fit together to yield observed system properties. Or, in mathematics, with
determining fundamental definitions and axioms and discovering what theorems
can be constructed (i.e., fit together).
Different kinds of reasoning use empathy, analogy and analysis in different ways.
Critical thinking, for example, is strongly analytic. It does not use empathetical
reasoning at all, and subjects all analogies to a strict analysis to ensure that they
conform to the rules of Aristotle’s logic. Scholastic reasoning uses critical thinking,
but can also use analogy, as in the case of theological arguments and literary
criticism. In scientific reasoning, empathy, analogy and analysis are all used, but in
very specific ways. Empathy provides a direction for searching, and gives a basis for
intuition; analogy provides models and thought structures for conceptualizing a
problem or question; and analysis takes care of the details, checking intuitively
generated ideas and hypotheses. Or, turning this description around, a hypothesis
cannot be accepted as scientific if it does not satisfy analytical truth criteria; it
becomes more useful when it can be seen in analogy to other, known concepts; and it
is generally accepted when it satisfies scientists’ intuitive (empathetic) feelings of
correctness.
Discussion 15.2 Thought
Experiments
TOP
Without . . . playing with fantasy no creative work has ever yet come to birth. The debt we
owe to the play of imagination is incalculable.
—Carl Gustav Jung[322]
O Man! If you only knew how many of the false fantasies of the imagination were nearer to
the Truth than the careful conclusions of the cautions. And how these truths are of no
service until the imaginer, having done his work with the imagination, has become less
imaginative.
—Shab-Parak[323]
Two of the claims made in Aristotelian physics are that when a faster moving object
collides with a slower moving object, it slows down, and that heavier objects fall
faster than lighter objects. There seems to be some empirical support for both of
these claims. If a chariot collides with an ox cart, it slows down, and a cannon ball
certainly falls faster than a feather.
Galileo is reputed to have refuted the belief that heavier objects fall faster by
dropping two objects of different weights from the Leaning Tower of Pisa, and
observing that both struck the ground at the same time. It is doubtful that Galileo
ever performed this experiment,[324] but he did refute the Aristotelian doctrine that
heavier objects fall faster.
Galileo’s refutation took the form of a thought experiment—the construction of an
imaginary experimental situation that leads to the same kind of conclusion as one
would draw from a real experiment. Galileo argued as follows: suppose that two
objects, one heavy and one light, are dropped from a high tower. Furthermore,
suppose that the lighter is dropped first, and the heavy one is dropped shortly
afterwards, directly above the lighter one. If the heavier object falls faster, it will
overtake and collide with the lighter object. We know (from Aristotelian physics)
that in such collisions, the faster object slows down. But, after the collision, we will
have a new composite object that is even heavier than the original heavy object, and
so if heavy objects fall faster, it should speed up rather than slow down. This is a
contradiction, so we must assume that (neglecting air resistance) heavy and light
objects fall at the same speed.
A thought experiment presents a situation and then appeals to reason, experience,
intuition or some combination of these qualities to support its conclusion. In
Galileo’s thought experiment, the appeal is to our knowledge that something cannot
fall slower and faster at the same time.
Thought experiments have been used throughout the history of science as arguments
for or against various theoretical claims and presuppositions. When we are seeking
plausible arguments to support some conjecture or hypothesis, it is often useful to
attempt to come up with a thought experiment which shows that our conjecture or
hypothesis is what would be expected under reasonable circumstances. When we
have made a conjecture or deduced a hypothesis, and feel secure about what is
claimed, we immediately think of looking for a proof or for empirical confirmation.
Before we undertake what may be a difficult task, however, it is useful to look for a
counter-example; that is, a situation which will show that our conjecture or
hypothesis is false. Thought experiments can be very useful in this process of selfchecking. We attempt to concoct an imaginary situation in which our hypothesis can
easily be seen to fail. If we are able to do so, we have saved ourselves the trouble of
trying to find an actual confirmation or proof.
The basic idea in a thought experiment is to construct an imaginary situation that
appeals to our understanding in such a way as to support or refute an idea,
statement or conclusion in which we are interested. In the section below, we describe
case histories of several famous thought experiments in science.
Classic Thought Experiments
Zeno’s Paradoxes
Perhaps the earliest known thought experiments are Zeno’s paradoxes. The
philosopher Zeno of Elea (fl. 450 BCE) developed a number of paradoxes as a
challenge to the Pythagorean idea that it was possible to think of space and time as
being composed of discrete points and instants. Zeno proposed forty paradoxes in
all, of which eight are still known.[325] Three of these are the Achilles paradox, the
arrow paradox, and the paradox of size.
In the paradox of Achilles and the Tortoise, Zeno argues that if Achilles (an ancient
Greek warrior renowned for his fleetness of foot) had a race with a tortoise, and if
the tortoise were given a head start, then Achilles could never catch up. The
reasoning behind this seemingly absurd claim is as follows: Suppose that Achilles
starts at a point x0x0, and the tortoise starts at a point x1x1 that is further along the
race course. By the time Achilles reaches the point x1x1 the tortoise has advanced to
a further point x2x2. By the time Achilles reaches x2x2, the tortoise has reached a
point x3x3 (see Figure 15.2.1, below)—and so on, forever. Whenever Achilles reaches
where the tortoise has just been, the tortoise has moved a little further ahead.
Figure 15.2.1: Achilles and the tortoise
In the arrow paradox, Zeno argues that an arrow in flight is not moving. Why?
Because by definition, an object is at rest if it occupies its own place; that is, if it
occupies a unique region of space that coincides with its physical extension. But at
every instant of time, the arrow does occupy its own place, because in an instant (i.e.,
a point of zero temporal duration), it has not moved. Therefore, the arrow is at rest
at each instant, and so must always be at rest.
In the paradox of size, Zeno argues that everything must be either infinitely large or
infinitely small. This is because, if an object is composed of infinitely many points,
and a point has no size, then the object can have no size. On the other hand, if a
point has any size then, since there are infinitely many of them, the object itself must
be infinite in size.
While we may think that these paradoxes are obviously wrong, they are worth
thinking about carefully. Philosophers still argue about whether or not they can
actually be resolved.
Is Gravitation Universal?
Imagine that you are standing on the surface of the Earth, tossing a ball higher and
higher—higher than Everest, higher than the moon. Is there ever a height at which
the ball becomes completely free of the earth’s gravity? Since no specific height can
be determined at which this freeing occurs, we conclude that the earth
gravitationally attracts every body, no matter where it is located. Since the earth has
no privileged position (we could carry out the same experiment on any gravitating
body), we conclude that gravitational attraction is universal.
This thought experiment appeals twice to the principle of sufficient reason: once in
asserting that there is no reason to pick out any height at which the ball is no longer
attracted to the earth, and therefore there is no such height; and a second time in
asserting that there is no reason to assume that the earth has any privileged position.
Does Absolute Space Exist?
Isaac Newton proposed a thought experiment designed to support the idea that
absolute space exists and defines a state of absolute rest that provides a fixed
reference frame for his gravitational theory. He imagined a bucket filled with water,
suspended by a twisted rope as indicated in Figure 15.2.2, below.
Figure 15.2.2: Newton’s suspended bucket
At first the bucket and water are at rest with respect to each other and the surface of
the water is flat (a). When the bucket is released, the rope begins to untwist, and the
bucket starts to rotate with respect to the water. At the very first, this motion is not
communicated to the water, and the surface of the water remains flat, although the
bucket is now rotating with respect to the water (b). After a short time, the rotation
of the bucket is communicated to the water. The bucket and water are again at rest
relative to each other, and the surface of the water is now concave (c). Finally, the
bucket is stopped but the water continues to rotate. Again the water and bucket are
moving relative to each other, just as in (b), but now the surface of the water is
concave (d).
Newton argued that the similarity of the relative motion between states (b) and (d),
together with the dissimilarity of the physical situation—in (b) the surface of the
water is flat, but in (d) it is concave—shows that the behaviour of the system could
not be predicted simply from the relative motions of bucket and water. To
distinguish the two cases, we need to have absolute motion as well, and such motion
could only be described with respect to a reference frame which we can be sure is not
moving itself. Thus, Newton concluded that the situation must be described with
respect to the only reference frame that is not moving (rotating)—namely absolute
space.
The physicist and philosopher Ernst Mach, whose ideas were very influential on the
young Einstein, criticized this conclusion. Mach’s basic criticism was that Newton
had been insufficiently diligent in seeking a reference system other than absolute
space. Mach’s suggestion was that a suitable system could be found in the fixed stars.
That is, in case (b), the water is at rest relative to the fixed stars, while in case (d), the
water is rotating with respect to the fixed stars. This idea was expanded into what
was known as Mach’s principle, which stated that local events in space are subject to
the influence of the distribution of all of the matter in the universe.
Mach’s rather vague intuition was formalized in Einstein’s general relativity theory,
in which the local metrical properties of space-time are determined by the
distribution of matter throughout space-time. This formulation provided a way of
understanding Newton’s thought experiment without requiring the existence of an
absolute space. That is, the shape of the surface of the water in Newton’s bucket is a
result of centrifugal forces. But in Einstein’s theory, those forces are consequences of
the metrical field, and at every point of space-time this field is determined by the
distribution of matter in all of space-time. In case (b) the water is not rotating with
respect to this distribution of matter, hence feels no centrifugal force; in case (d) the
water is rotating with respect to this distribution, and does feel centrifugal forces.
From our point of view, the interesting point of this thought experiment is that it was
devised to argue in favour of absolute space, and ended up helping to support an
argument against absolute space. Newton was making an appeal to Archimedes’
principle of symmetry, pointing out that because two cases that appeared to be
symmetrical (i.e., the water and bucket are in states of relative rotation in both
cases) show a lack of symmetry (i.e., a flat water surface in one case and a concave
water surface in the other), there must be a lack of symmetry somewhere in the
description of the situation. He sought this lack in the rotation or non-rotation of the
water with respect to an absolute space. Mach argued that Newton was prejudiced by
the conclusions he wanted to obtain, and Einstein’s general relativity theory showed
how the results of this thought experiment could be explained without a reference to
absolute space by tying the properties of space directly to the matter and energy
content of space.
Moral: Be careful that you do not assume too much in interpreting thought
experiments, especially if you want the answer to come out in a certain way.
Is the Geometry of Space Euclidean?
The following thought experiment, illustrated in Figure 15.2.3, below, was used to
argue that if Einstein’s special theory of relativity is correct then the geometry of
physical space cannot be Euclidean. This argument was first proposed by the
physicist Paul Ehrenfest to suggest that a special relativistic theory of rigid bodies
developed by Max Born contained a paradox. It was later employed by the
mathematician Herman Weyl to demonstrate the way in which geometry is
influenced by motion.
Figure 15.2.3: Ehrenfest’s thought experiment
Consider a disk in a state of uniform rotation with respect to a Euclidean reference
frame. Now suppose that we make geometrical measurements of the disk with
respect to measuring rods that are moving with the disk (i.e., we measure with
respect to a reference frame in which the disk appears stationary). Our
measurements of the radius of the disk, being made perpendicular to the direction of
motion, will agree with the radius as measured in the original Euclidean frame. But
distance around the circumference of the disk is in the direction of motion, and
hence is subject to the Lorentz contraction of special relativity. Thus the Euclidean
relationship that the circumference equals two times ππ times the radius will no
longer hold, and the geometry of the disk cannot be Euclidean.
This thought experiment is based on our acceptance of the consequences of special
relativity theory and on the principle of non-contradiction. That is, the assumption
of Euclidean geometry in the disk leads to a contradiction, since the Euclidean
relation C=2πrC=2πr fails to hold. The assumption, therefore, must be false.
There Is No Difference between Inertial and Gravitational Fields
The statement above is the conclusion of Einstein’s well-known elevator thought
experiment. Imagine we are in a closed elevator, Einstein proposed, accelerating
at 9.8 m/sec29.8 m/sec2 in empty space. We will be unable to distinguish this
situation from being in the same elevator at rest on the surface of the earth (where
the acceleration due to gravity is 9.8 m/sec29.8 m/sec2).
Figure 15.2.4: Einstein’s elevators
In each case, if we drop a ball, it will fall to the elevator floor with the same
acceleration, and in general, no measurement that we make inside this elevator will
be able to distinguish these two cases (Figure 15.2.4a). Therefore, we cannot
distinguish, strictly on the basis of local measurements, an inertial (i.e., acceleration)
field from a gravitational one. This realization was formalized as Einstein’s principle
of equivalence.
Einstein concluded from this thought experiment that light should bend in a
gravitational field. He reasoned that if a light ray were shining through a small hole
in the side of the elevator when it was accelerating in empty space, it would appear to
follow a curved path, and therefore, from the equivalence principle, it should follow
a curved path when the elevator was at rest on the surface of the earth as well
(Figure 15.2.4b).
This case shows how conclusions drawn from a thought experiment (e.g., the
principle of equivalence) can be generalized to further conclusions that can be tested
in real experiments. Verification of the bending of light by gravitational fields was
one of the major reasons that the general theory of relativity was so quickly accepted.
We might also speculate on the origin of this thought experiment. It is known that in
about 1907, Einstein devised a thought experiment in which he imagined himself
falling freely in a gravitational field, and this scenario evolved into his elevator
thought experiment. He knew that in such circumstances he would feel no
gravitational force. Was this thought experiment suggested by some incident in
which Einstein stepped into an elevator and felt an apparent increase in weight when
rising and an apparent decrease in weight when descending? It certainly helps in
understanding the conclusions drawn if we empathetically put ourselves into the
elevator and feel the accelerations and forces involved.
Is Evolution a Scientific Theory?
At one time the philosopher of science Karl Popper claimed that the theory of
evolution is not scientific because its main postulate, evolution via natural selection,
does not satisfy his criterion of falsifiability. He claimed that evolutionary theory
offers no way of falsifying the hypothesis of evolution through natural selection,
because no case where selection did not act could ever be found. How could we ever
falsify the hypothesis that only the fittest survive, if we start out assuming that
natural selection ensures that all of the animals we find surviving are the fittest?
Popper eventually retracted his claim, partially as a result of the following thought
experiment. Imagine an isolated island populated only by various species of birds.
There is plenty of food on the island, and no competition between species for
survival. A virus, which periodically kills the same fraction of each species, controls
the bird population. Is this not a case in which there is no natural selection?
Here is a case of a thought experiment that posits an ideal case as a response to a
philosophical claim that no such case exists. You might consider this in light of the
ideals of natural order, which determine the nature of our scientific theories.
The Identity Paradox
Suppose that some device is invented which is able to duplicate matter in every
detail, and that you walk through this device. Out the other side come you and an
identical duplicate of you. Which is you?
Figure 14.2.5: Matter duplicator
This scenario is a variation of a group of paradoxes and questions relating to
personal identity that have been used in philosophy to stimulate thought about the
nature of identity. In each of these cases, the purpose of the thought experiment is
not so much to lead to a conclusion as to provoke thought.
Can Machines Think? The Chinese Room
The philosopher John Searle developed this highly controversial thought experiment
as a response to the idea, current in both science fiction and the field of artificial
intelligence, that a sufficiently powerful computer with sufficiently clever
programming will have all the capacities of a human being, including thought and
consciousness.
The mathematician Alan Turing proposed the standard test for attribution of
conscious awareness in artificial intelligence in 1951. The Turing test posits that we
have a computer terminal and can use it to communicate with a hidden room. This
hidden room contains either another person or a computer. We carry on a
conversation on any topic we desire using the terminal, and the computer “passes
the Turing test” if we are unable, after a sufficiently long time for communication, to
decide whether we are talking to a person or to a machine. Searle’s “Chinese Room”
thought experiment attacks the idea that the Turing test is sufficient reason to
attribute thinking to a computer.
Suppose, Searle says, that we have a closed room with communication in and out
only through a slot in the door. Inside the room are millions of written Chinese
ideographs, together with a large instruction book that tells which ideographs should
be put out through the slot in response to any conceivable set of ideographs that are
put through the slot into the room. There is also a person in the room who does not
understand anything of the Chinese language.
When we carry on a conversation with this room by putting various ideographs in
through the door, and getting others back out, we may well decide that we are
conversing with a conscious entity who understands everything that we are saying.
In reality, however, there is nobody in the room who understands Chinese. Searle
draws the conclusion that the Turing test is not sufficient to test for understanding,
and that one cannot get semantics (meaning and understanding) out of syntax
(grammar) alone.
This particular thought experiment is highly controversial in the field of artificial
intelligence. On one side are those who accept Searle’s conclusion, and with it the
conclusion that machines will not be able to think so long as we rely only on
computing power and programming. On the other side are those who believe that all
that is involved in thought is computation; these people have come up with a variety
of objections to Searle’s argument. The most popular response among this group is
called the “systems response,” which asserts that although the person in the Chinese
room does not understand Chinese, the entire room, considered as a unified system,
does. Another often-used response is to argue that the Chinese room set up would
operate so slowly that nobody could ever take it for a thinking machine, so the
analogy to a computer is no good. All of these responses have been met by counterresponses, and the argument continues.
Nature Abhors a Vacuum
In the medieval period, there was a controversy over whether or not pure vacuum
was possible in nature. A thought experiment designed to show that a vacuum could
be produced was central in these arguments. Suppose that a bottle is filled with hot
water and capped, then put outside on a cold day. As the water freezes, so the
thought experimenters claimed, it condenses, leaving a vacuum at the top of the
bottle. Opponents of the idea of a vacuum in nature responded that the space at the
top would fill with vapours from the water, or that the water would never freeze, or
that the bottle would break. The argument went back and forth for many years, with
many subtleties and refinements along the way.
Neither side in the debate realized one fundamental fact: when water freezes, it
expands!
Moral: Your thought experiment is only as good as the scientific assumptions that go
into its construction.
Why Do Thought Experiments?
One thing that is immediately noticeable about thought experiments is that, in most
cases, it would be very difficult, or even impossible, to carry them out as real
experiments. They deal with extreme situations that usually cannot be realized in the
world. Even in cases where a thought experiment could actually be carried out, it is
often better to leave it to the imagination. We understand the thrust of Newton’s
bucket experiment, without actually doing it (although Newton is reputed to have
carried it out), and questions could be raised about the validity of an actual
experiment, relating to things such as experimental errors and errors of observation.
In a thought experiment, these messy aspects of the real world have been idealized
away, and we can get right to the important points of a situation.
As an example we can consider a thought experiment devised by Mach to show that
mass is transitive; that is, that if mass AA equals mass BB, and mass BB equals
mass CC, then mass CC equals mass AA. We proceed to make an argument by
contradiction. Suppose that mass is not transitive, that AA equals BB,
and BB equals CC, but CC is greater than AA.
Now consider each of these three masses as perfectly elastic plastic balls that slide
without friction on a circular ring, as illustrated in Figure 15.2.6, below.
Figure 15.2.6: Mach’s thought experiment
Ball AA is propelled with a certain momentum and it strikes ball BB. Since the balls
are perfectly elastic all the momentum of AA is transferred to BB, which slides with
the same velocity as AA (since the masses of AA and BB are equal) until it strikes
ball CC. Again, all of the momentum of BB is transferred to CC which slides with the
same velocity as BB until it strikes ball AA. But now we are assuming that the mass
of CC is greater than that of AA; hence the momentum transferred to AA is greater
than the initial momentum with which AA started. With each circuit, AA gains in
momentum, and hence velocity, without limit. Since this situation is absurd (if for no
other reason than it violates every conservation law that we know of) we must reject
the initial assumption that the mass of CC was greater than that of AA.
Although we can conceive of carrying out this experiment in practice, it would be
open to several objections. There are no perfectly elastic materials, and there is no
frictionless wire. Thus, we would face questions as to the real validity of our
conclusions. We could say that to within a certain experimental limit we had
demonstrated support for the idea that mass was transitive, but we could not claim
this as a general and exact result. The value of the thought experiment here is that it
allows us to eliminate these messy real-world details, which are not relevant to the
argument, and focus attention on the real crux of the matter, which is that if mass is
not transitive, then we could, at least in principle, violate the law of the conservation
of energy.
In general, thought experiments appeal to our reason, our knowledge of accepted
scientific theory, and our general understanding of and intuition about the world.
Therefore, in order to evaluate a thought experiment, we must consider it from each
of these distinct perspectives. Very often, the sticking point of a thought experiment
will involve the use of one or two of these courts of appeal to point out a
contradiction in the third. Galileo’s thought experiment shows that acceptance of
Aristotelian physics (the then-current theory) leads to a contradiction in our reason
(the same thing cannot move faster and slower at the same time). The rotating disk
thought experiment shows that acceptance of the special theory of relativity can lead
to a contradiction in our experience-based intuition (which would anticipate that
space was always Euclidean). The Chinese room thought experiment shows that use
of the Turing test as a measure of conscious thought violates our common-sense idea
of what it means to understand.
Clearly, thought experiments can be mistaken. Newton’s misinterpretation of the
bucket experiment was based on his theoretical belief that there was an absolute
space, and the medieval “vacuum” thought experiment fails because its construction
ignores an easily discoverable fact about water. It is important to remember that a
thought experiment must be checked against reason, theory and experience before
we can accept its conclusions.
Types of Thought Experiments
We can get a rough classification of thought experiments by considering the
interplay of reason, theory and experience.[326]
Illustrative: These thought experiments make use of reason, within an existing
theory, to provide us with a simple and intuitively obvious illustration of some aspect
of the theory. In other words, they act as an aid to our intuition. The Schrödinger’s
cat thought experiment, for example, is often used to illustrate the strange nature of
the quantum world. In this thought experiment, a cat is in a closed box with a vial of
cyanide gas and a mechanism that will break the vial and release the gas if it detects
the decay of an atom of a radioactive element in the box. This will kill the cat. The
radioactive atom has a half-life TT (i.e., in a time TT the probability that it will have
decayed is 1/21/2). The quantum mechanical description of the system requires a
superposition of two states, one in which the atom has decayed and one in which it
has not. After a time TT we open the box and see either a live cat or a dead cat. But
before opening the box all we can say is that the cat is in a superposition of states:
alive and dead.
Critical: In such thought experiments, reason and intuition are used to challenge
some aspect of existing theory. Galileo’s refutation of the Aristotelian assumption
that heavy objects fall faster than lighter ones is a good example.
Generative: These thought experiments use reason and intuition (or experience) to
suggest new theoretical principles or ideals. They appeal to imagined experiences
(and our intuitions about them), and suggest a theoretical basis for conclusions
about these experiences. The thought experiment offers a basis for a rational
inference. (Since a thought experiment involves a specific case, it cannot provide a
basis for intuitive inferences.) Newton’s bucket thought experiment, used to infer the
existence of absolute space, is a good example.
Many thought experiments combine critical and generative aspects. Galileo’s
thought experiment on falling bodies, for example, pointed to a logical contradiction
in the Aristotelian theory, while simultaneously lending support to the deductive
inference that all bodies fall at the same rate. In other cases, a thought experiment
that started out as critical or generative may eventually be used to illustrate a theory.
The Schrödinger’s cat thought experiment, for example, was originally posed to
criticize certain counter-intuitive aspects of the Copenhagen interpretation of
quantum mechanics, but is now used to illustrate them.
Both critical and generative thought experiments can lead to incorrect conclusions.
In a critical thought experiment, it is argued that some aspect of an existing theory
violates either some principle of reason or our intuition. Galileo’s falling bodies
thought experiment showed that Aristotelian assumptions violated the principle of
contradiction by implying that a body could fall faster and slower at the same time. It
led to the correct conclusion that all bodies fall at the same rate. The Schrödinger’s
cat thought experiment, on the other hand, does not show a contradiction in
quantum mechanics (the cat is both alive and dead), only that the wave function
describing the entire system must contain a superposition of two wave functions,
one corresponding to a live cat and one to a dead cat. This conclusion is highly
counter-intuitive for us, but it is not a contradiction, and rather than leading us to
doubt quantum mechanics, this thought experiment is an illustrative example that
can help us to develop our intuition of the quantum world.
Discussion 15.3 The Back of the
Envelope
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It is the mark of an instructed mind to rest satisfied with the degree of precision which the
nature of the subject permits and not to seek an exactness where only an approximation of
the truth is possible.
—Aristotle[327]
Order of Magnitude Estimates
Often, in beginning scientific investigation, we are not interested in detail. We would
be satisfied with a modest approximation that our studies might be refined with
further work. For example, we might want to begin a study in cosmology by
estimating the total mass of the universe. Clearly we would not want to go to all the
trouble of trying to get an exact estimate—our information is so scant that it would
probably be wrong anyway. But we can do a very simple “back of the envelope”
calculation that gives an adequate answer.
We consult a reference book and find that the density of observed matter in the
universe is about 10−27 grams/cm310−27 grams/cm3. We also find that the age of
the universe since the big bang is estimated to be between 10 and 20 billion years.
Therefore, the radius of the universe can be estimated as 10 to 20 billion light years.
Suppose we use 15 billion light years as our estimate. The speed of light is
about 3×1010 cm/sec3×1010 cm/sec, and (a brief calculation) there are
about 3×107 seconds3×107 seconds in one year. So the distance light travels in one
light year is about 9×1017 cm9×1017 cm. The volume of a sphere is given by the
formula V=4πr3/3V=4πr3/3. Thus the volume of the universe will be
about 3×1054 cm33×1054 cm3. Multiplying this value by the estimated density gives
a mass of about 3×1027 grams3×1027 grams. However, the factor of 3 here is not
trustworthy, and we might well say that an order of magnitude estimate of the mass
of the universe is about 1027 grams1027 grams. This is an example of an “order of
magnitude” calculation.
Here is another: estimate the thickness of rubber worn off an automobile tire in one
revolution. We note that the tread on a new tire is about half a centimetre thick. Now
suppose that the tire is guaranteed for 50,000 kilometres50,000 kilometres. So we
will estimate by assuming that the tire wears down half a centimetre
in 50,000 km50,000 km. The diameter of the tire is about 1 metre1 metre, so its
circumference is about 3 metres3 metres, or
about 0.003 kilometres0.003 kilometres. Therefore, in our estimation, one-half
centimetre of tread wears off
in 50,000/0.003=16,666,667 revolutions50,000/0.003=16,666,667 revolutions of
the tire. We round this value off to
about 20,000,000 revolutions20,000,000 revolutions (we are only considering
orders of magnitude, remember). Then the amount of rubber wearing off in one
revolution is 0.5 cm/20,000,000=2.5×10−8 cm0.5 cm/20,000,000=2.5×10−8 cm, or
in the order of magnitude of about one-hundred-millionth (10−8)(10−8) of a
centimetre.
We encounter order of magnitude calculations all of the time. Suppose that a
population ecologist wants to estimate the number of trees of a certain type in a
forest. She or he will not go through all of the forest counting every tree. Rather, the
ecologist will select a small portion of the forest as “typical,” and will count the
number of trees of the type in question in that region. The number obtained is then
multiplied by an estimate of the ratio of the entire forested area to the area selected
as a typical region.
In making order of magnitude estimates, it is important to make sure that we can
justify the assumptions made about the magnitudes going into the calculation. For
example, is the forest region selected for our count really typical? Perhaps its soil is
different from that usually found in the forest in such a way as to distort our sample.
We would not consider a section of riverbank as typical if we were counting trees
that only grew on riverbanks. We must also be careful not to try to be too precise,
and in reporting our computations we must give reasons that we think they are
adequate for whatever our purpose might be.
Dimensional Analyses
Dimensional analysis is another kind of estimation. It is based on the idea that in
many sciences, the quantities we calculate with are dimensional quantities rather
than pure numbers. Thus, we work with grams, centimetres, seconds, coulombs,
joules, ergs, degrees centigrade, and so on. When we write out an equation, we must
ensure that the dimensions on the right and left sides of the equation balance. On
this basis, we can sometimes guess the kind of formula that will give a quantity of
interest. For example, velocity is measured in distance per unit time (e.g., cm/sec),
so if we are looking for an equation that has the units of velocity, we know that it
must somehow reduce to a distance unit divided by a time unit.
For example, consider the derivation of Poiseuille’s law, which relates the flow rate
and viscosity of fluids flowing through cylindrical tubes. It was originally determined
experimentally by the physician Jean Léonard Marie Poiseuille (1799-1869), who
was studying the flow of blood.
Consider a fluid flowing through a cylindrical tube (as a model of blood flowing in a
blood vessel) slowly enough that there is no turbulence. We can imagine the fluid as
being partitioned into concentric cylinders, with the cylinder closest to the tube walls
flowing slowest, and successive layers flowing faster as we move toward the centre,
where the flow will be the fastest. Suppose v*v* is the flow velocity at the centre. It is
found experimentally that the average flow velocity is v=v*/2v=v*/2 so the flow rate
is R=Av*/2R=Av*/2 where AA is the cross-sectional area of the tube.
In a horizontal tube with constant cross section, the average velocity remains
constant, but the pressure drops, because work is being done to overcome the
viscous force (i.e., the fluid must work to overcome friction at the tube walls). The
pressure drop ΔPΔP along a length of horizontal tube of constant cross section will
be proportional to the viscous force, and hence to the average velocity of the fluid. It
will also be proportional to the length of the tube. That is, ΔPΔP is proportional
to v*L/2v*L/2, where LL is the length of the tube. Thus the average velocity vv is
proportional to the pressure gradient ΔP/LΔP/L. The average velocity will also
depend on the radius rr of the tube, and on the fluid viscosity μμ.
So far we have considered only background information. We could use sophisticated
mathematical techniques to derive an equation for the average velocity. But we can
get an equation that gives the main features of the result by dimensional analysis.
We know that vv is proportional to Δ P/LΔ P/L, and also depends on rr and μμ. We
then assume that there is a formula
v=βraμb(ΔPL)v=βraμb(ΔPL)(1)
that expresses the relation we want. Here ββ is a dimensionless numerical factor. We
must then choose the numbers aa and bb in Equation (1) so that the dimensions on
the right and left sides of this equation balance. The average velocity has dimensions
of length divided by time (l/tl/t, or lt−1lt−1). The dimensions of the other quantities
in (1) are listed below.
ΔPΔP: ml−1t−2ml−1t−2 (mass divided by length multiplying time squared)
rr and LL: ll (length)
μμ: ml−1t−1ml−1t−1 (mass divided by length multiplying time)
Looking only at the dimensions that occur in (1) gives us
lt−1=(la)(mbl−bt−b)(ml−1t−2)l−1=mb+1la−bt−b−2lt−1=(la)(mbl−bt−b)(ml−1t−2)l−1=mb
+1la−bt−b−2(2)
So, we see that if the dimensions are going to balance on both sides of the equation
we must have b=−1b=−1 and a=2a=2, and our heuristic equation relating average
velocity to pressure gradient, viscosity and tube radius is
v=βR2ΔPlμv=βR2ΔPlμ(3)
All that is gained from the detailed mathematical analysis of this situation is that we
find the value of the constant ββ to be equal to 1/8. But if we are working
experimentally, the value of ββ can be computed from our experimental results. In
the lab, we would have used dimensional analysis to generate Equation (3) as our
best guess, and then we would have carried out experiments to see if the result
supported this conclusion.
Unit 15 Study Questions
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Note: If you have difficulty with any of these questions, please contact your tutor to discuss the
problem.
1. Give example, other than any of those cited in Discussion 15.1, of the use of
empathetical, analogical and analytical reasoning in science.
2. Make a list of the pros and cons for each of the plum pudding and miniature
solar system models of the atom.
3. Based on your readings in What Science Is and this Study Guide, write a short
essay (300-500 words) explaining why empathy and analogy cannot provide a
conclusive argument in support of a scientific idea or hypothesis.
4. Write a short essay (300-500 words) comparing the quotes from Paul Dirac,
“[I]t is more important to have beauty in one’s equations than to have them fit
experiment” and Henri Poincaré, “[W]e almost always notice that this false idea,
had it been true, would have gratified our natural feeling for mathematical
elegance.”[328]
5. What are the three things that thought experiments are based on, as described
in Discussion 15.2?
6. How would you suggest resolving the Achilles and arrow paradoxes? Note that
simply pointing out that Achilles does catch the tortoise, and that the
arrow does move, are not sufficient.
7. What are three dangers that can invalidate the conclusions of a thought
experiment?
8. The Zeroth Law of Thermodynamics states that if two systems are in thermal
equilibrium with a third system, then they are in thermal equilibrium with each
other. (This assumption is the basis of the thermometer—it asserts that if two
systems are at the same temperature they will be in thermal equilibrium.)
Construct a thought experiment to show that violation of this law would allow
construction of a machine that could extract heat energy from three systems at
the same temperature.
9. Make the appropriate assumptions to compute order of magnitude estimates for
the following quantities:
a. the number of atoms in one mole of any substance (i.e., one “gram
molecular weight”—a weight of a substance in grams which is equal to its
molecular weight), given that the weight of one hydrogen atom is
about 1.6×10−241.6×10−24 grams. This value is known as Avogadro’s
number. Originally, experimental measurements of this number were
used to estimate the mass of a hydrogen atom.
b. the number of dentists in Canada.
Hint: How much time per year do you estimate that you spend at the dentist’s? Assume
that this is typical for everybody in Canada. The population of Canada is about 30,000,000,
so you can then figure the total time spent in visits to dentists. Then assume that all
dentists are busy 6 hours per day, 5 days per week, 48 weeks per year.
c. the amount of leather worn off of the bottom of one of your shoes in a
single step.
d. the number of grains of sand on a beach.
Hint: You will need to estimate the length and width of the beach, and the depth of sand
on it, as well as the size of a grain of sand.
e. the weight of Canada.
Hint: Estimate the surface area and the average height above sea level and multiply the
volume you compute by an estimate for the average density of the material.
10. What information would you need to devise a thought experiment to estimate
the number of people living on an island who have an infectious disease
(Disease X)? Consider infectivity, fatality, transmission, and treatment.
11. An object moving through a fluid experiences a viscous drag force that, at low
velocities, is proportional to the object’s velocity. Consider a sphere of
radius RR moving with a small velocity vv through a fluid of viscosity μμ and
density dd. Assuming that the force FF is proportional to velocity, and depends
on RR, μμ and dd, determine an expression for FF using dimensional analysis.
Note: The dimensions of RR, μμ and dd are, respectively, length, mass divided by length
multiplying time, and mass divided by length cubed.
FOOTNOTES
[305]
This comment was frequently made by Professor Leon Blitzer to his graduate class in
classical mechanics, University of Arizona, 1964-65.
[306]
Whitehead, Alfred North. The Concept of Nature, p. 163. Cambridge: Cambridge
University Press, 1926.
[307]
Feynman, Richard. The Value of Science. In What Do You Care What Other People
Think?: Further Adventures of a Curious Character, pp. 240-248. New York: Norton, 1988.
Retrieved August 19, 2002.
http://www.cc.gatech.edu/people/home/idris/Speeches/Value_of_Science.html
[308]
Darrow, Clarence. Absurdities of the Bible. Little Blue Book No. 1637, ed. E. Haldeman-
Julius. Girard, KA: Haldeman-Julius Publications, n.d. This text is at the site below.
Retrieved August 19, 2002.
http://infidels.org/library/historical/clarence_darrow/bible_absurdities.html
[309]
In popular culture, the Star Trek character Mr. Spock is an example of this sort of
reasoning.
[310]
Quoted in Koestler, Arthur. The Act of Creation, p. 213. London: Pan, 1966.
[311]
Paraphrase of a comment in Salk, Jonas. Anatomy of Reality, p. 7. New York: Columbia
University Press, 1983.
[312]
Quoted in Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to Einstein,
rev. ed., p. 274. Cambridge, MA: Harvard University Press, 1988.
[313]
Recall that in Plato’s five stages of knowledge, the third stage was “the image.”
[314]
Bronowski, Jacob. Nature and Knowledge, p. 42. Eugene, OR: Oregon State System of
Higher Education, 1969.
[315]
It is not clear that the Greeks even considered their “elements” as completely material. A
more accurate term than element might be force, power or potency. This view is similar to
the way that the ancient Chinese regarded their five elements—wood, fire, earth, metal and
water. As Joseph Needham notes, “The five ‘elements’ were five powerful forces in ever
flowing cyclical motion, and not passive motionless fundamental substances.” [Science and
Civilization in China, Vol. 2, p. 244. Cambridge: Cambridge University Press, 1956.]
[316]
In these experiments, a columnated beam of a-particles would be directed at a thin sheet
of some material such as copper or gold, and the pattern of scattered a-particles would be
observed.
[317]
Note the layers of abstraction and modeling involved in setting up and interpreting these
experiments.
[318]
Max Planck’s 1900 derivation of the correct formula for the spectrum of black-body
radiation was based on the assumption that matter emits radiation with an energy that is an
integer multiple of a quantity hv, where v is the frequency of the radiation and h is a very
small constant, now called Planck’s constant.
[319]
In the photoelectric effect, it is observed that shining light on a metal plate results in the
emission of electrons. The energy of the emitted electrons depends only on the frequency of
the light, not its intensity, and no electrons at all are emitted if the frequency is less than a
threshold frequency that depends on the given material. Einstein followed up on Planck’s
assumption that radiant energy comes in discrete packets that are integer multiples of hv to
give an explanation of this effect. It was for this work that he received the Nobel Prize, not
his work on relativity.
[320]
Quoted in Heisenberg, Werner. “Reminiscences from 1926 and 1927,” p. 165. In French,
A. P., and Kennedy, P. J. Niels Bohr: A Centenary Volume, pp. 163-171. Cambridge, MA:
Harvard University Press, 1985.
[321]
A key technique for generating suspense in literature is to get the reader to identify
empathetically with a character, while indicating that the character’s viewpoint—the way in
which he or she fits together a world-view—is mistaken. We know, for example, that
Desdemona is faithful, but still empathize with Othello’s jealousy.
[322]
Jung, C. G. Psychological Types, p. 82. London: Routledge and Kegan Paul, 1923.
[323]
Quoted in Shah, Idries. Wisdom of the Idiots, p. 153. London: Octagon, 1991.
[324]
While it is doubtful that Galileo ever performed this experiment, many other people in
the early 17th century were involved in dropping objects of different weight from high
buildings, with mixed results.
[325]
See, for example, Robinson, John M. An Introduction to Early Greek Philosophy: The
Chief Fragments and Ancient Testimony, with Connecting Commentary. Boston:
Houghton Mifflin, 1968.
[326]
Students may find the classification given the work cited below of interest. Brown, James
R. The Laboratory of the Mind, London: Routledge, 1993.
[327]
Quote in Harte, John. Consider a Spherical Cow: A Course in Environmental Problem
Solving, p. vii. Los Altos, CA: Kaufmann, 1985.
[328]
Dirac, Paul, quoted in Holton, Gerald. Thematic Origins of Scientific Thought: Kepler to
Einstein, rev. ed., p. 274. Cambridge, MA: Harvard University Press, 1988; Poincaré, quoted
in Koestler, Arthur. The Act of Creation, p. 213. London: Pan, 1966.
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