CDT403 Research Methodology in Natural Sciences and Engineering
Theory of Science
SCIENCE VERSUS PSEUDOSCIENCE AS
CRITICAL THINKING VERSUS WISHFUL THINKING
Gordana Dodig-Crnkovic
School of Innovation, Design and Engineering
Mälardalen University
1
THEORY OF SCIENCE
Lecture
L
t
1 INFORMATION,
INFORMATION COMPUTATION,
COMPUTATION KNOWLEDGE AND
SCIENCE
Lecture 2 SCIENCE AND CRITICAL THINKING.
PSEUDOSCIENCE AND WISHFUL THINKING - DEMARCATION
Lecture 3 SCIENCE, RESEARCH, TECHNOLOGY, SOCIETAL
ASPECTS PROGRESS
ASPECTS.
PROGRESS. HISTORY OF SCIENTIFIC THEORY.
THEORY
POSTMODERNISM AND CROSSDISCIPLINES
Lecture 4 PROFESSIONAL & RESEARCH ETHICS
2
RECAPITULATION OF THE FIRST LECTURE:
HISTORICAL DEVELOPMENT OF THINKING
ABOUT THE WORLD/NATURE/UNIVERSE
3
MYTHOPOETIC THINKING
Mythopoetic (myth + poetry) truth is revealed through
myths, stories and rituals.
Myths are stories about persons
persons, where persons may
be gods, heroes, or ordinary people.
Myth allows for a multiplicity of explanations, where
the explanations are not logically exclusive (can
contradict each other) and are often humorous,
exciting and colorful.
Mythic traditions are conservative. Innovation is slow,
and radical departures from tradition rare.
The Egyptian king Akhenaton and Queen Nefertiti
making offerings to the Aton.
4
MYTHOPOETIC
THINKING
Myths are self-justifying. The
inspiration of the gods was
enough to ensure their validity,
and there was no other
explanation for the creativity of
poets, oracles, and prophets
than inspiration by the gods
gods.
Thus, myths are not argumentative.
5
GREEK NATURAL PHILOSOPHY –
PROTOSCIENCE
Anaximander of Miletus ((c. 610 -c. 546 bc)) a pupil of Thales, and as other
members of the Ionian School, was an early scientist. He constructed the
first geometrical model of the universe, and made maps of the earth and the
skies. He said that the arche ('beginning and basis') of existing things is an
apeiron ('limitless') nature of some kind, from which come the heavens and
the kosmos ('world
( world order')
order ).
Anaximander advocated the idea of biological evolution with human beings,
like other animals,
animals evolved from fish
fish.
Anaximander was the author of the first written work of philosophy in ancient
Greece On Nature,
Greece,
Nature which has been lost
lost. His philosophy was a natural
dialectics.
SCIENCE: THE MECHANICAL UNIVERSE
The mechanicistic paradigm which systematically revealed physical
structure in analogy with the artificial. The self-functioning
self functioning automaton
- basis and canon of the form of the Universe – the clockwork.
7
Newton Principia, 1687
SCIENCES: THE UNIVERSE AS A COMPUTER
We are all living inside a gigantic
computer.
t No,
N nott The
Th Matrix:
M t i the
th
Universe.
Every process
process, every change that takes
place in the Universe, may be
considered as a kind of computation.
K Zuse,
Z
E Fredkin,
F dki S Wolfram,
W lf
S Lloyd,
Ll d
G Chaitin and many more
Konrad Zuse,
Zuse Computing Space
http://www.nature.com/nsu/020527/020527-16.html
8
Three Major Paradigm Shifts
Mytho-poetic,
God-Centric
Universe
(Classical)
Mechanic
Universe
Info-Computational
Human-Centric
Universe
Dodig-Crnkovic G and Müller V, A Dialogue Concerning Two World Systems: Info-Computational
vs. Mechanistic. In: INFORMATION AND COMPUTATION , World Scientific Publishing Co. Series
in Information Studies. Editors: G Dodig-Crnkovic and M Burgin, 2011.
http://arxiv.org/abs/0910.5001 2009
MAKING SENSE – CONSTRUCTION OF
MEANING
10
EMBODIED COGNITION –
KNOWLEDGE AS A PHYSICAL
PHENOMENON (PROCESS/STRUCTURE)
From: http://parliamodisalutegianugoberti.blogspot.se/2011/11/brief-guideto-embodied-cognition-why.html
http://psych.wisc.edu/glenberg/glenberglab/GLindex.html
p py
g
gg
g
Ago Ergo Cogito - "I act, therefore I think“ –
Certesian divide bridged. No separation mind-body
11
BOTTOM-UP (BODY->MIND) VS. TOP DOWN
(MIND->BODY)
(MIND
>BODY) VIEW OF SENSE-MAKING
SENSE MAKING
12
BOTTOM-UP VIEW, COGNITIVE AGENCYBASED
Meaning (1)
All meaning is determined by the method of analysis where the
method of analysis sets the context and so the rules that are
used to determine the “meaningful” from “meaningless”.
C JJ. Lofting
C.
13
Meaning (2)
At the fundamental level meaning is the result of process of
•
•
Differentiation and
Integration
or identification of differences and similarities, recognition of patterns.
14
Meaning (3)
Human brain is not tabula rasa (clean slate) on birth but rather
contains (gene-based, evolutionary acquired)
•
•
morphological structures used for meaning production based on the
distinctions of same/different, what/how, where/when etc.
behavioral patterns, etc.
15
TOP-DOWN VIEW, LANGUAGE-CENTERED
SEMIOTICS (1)
Semiotics, the science of signs, looks at how humans search for
and construct meaning.
Semiotics: reality is a system of signs!
(with an underlying system which establishes mutual
relationships
l ti
hi among th
those and
dd
defines
fi
id
identity
tit and
d diff
difference,
i.e. enables the description of the dynamics.)
16
SEMIOTICS (2)
Three Levels of Semiotics (Theory of Signs)
syntactics
semantics
pragmatics
17
SEMIOTICS (2A)
pragmatics
semantics
syntactics
y
18
SEMIOTICS (3)
Reality is a construction.
Information or meaning is not 'contained'
contained in the (physical) world
and 'transmitted' to us - we actively create meanings (“make
sense”!) through a complex interplay of perceptions, and agency
based on hard-wired
hard wired behaviors and coding-decoding
coding decoding
conventions.
The study of signs is the study of the construction and
maintenance of reality.
19
SEMIOTICS (4)
'A sign... is something which stands to somebody for something
in some respect or capacity'.
Sign takes a form of words, symbols, images, sounds, gestures,
objects, etc.
Anything can be a sign as long as someone interprets it as
'signifying' something - referring to or standing for something.
20
SEMIOTICS (5)
(signified)
(signifier)
CAT
The sign consists of
– signifier (a pointer)
– signified (that what pointer points to)
21
SEMIOTICS (6)
Thi is
This
i Not
N t a Pipe
Pi . . . by
b Rene
R
Magritte.
M itt . . . Surrealism
S
li
22
SEMIOTICS (7)
– Reality is divided up into arbitrary categories by every language.
[However this arbitrariness is essentially limited by our physical
predispositions as human beings. Our cognitive capacities are
defined to a high extent by our physical constitution.]
– The conceptual world with which each of us is familiar with,
could have been divided up in a very different way.
– The full meaning of a sign does not appear until it is placed in its
context, and the context may serve an extremely subtle function.
23
COMMUNICATION
– Communication is imparting of information, interaction
through signs/messages.
– Information is the meaning that a human gives to signs by
applying the known conventions used in their representation.
– Sign is any physical event used in communication.
– Language is a vocabulary and the way of using it.
24
HIERARCHICAL STRUCTURE
OF LANGUAGE
Object-language ←→ Meta-language
In dictionaries of SCIENCE there is no definition of science!
The definition of SCIENCE can be found in PHILOSOPHY
dictionaries.
25
AMBIGUITIES OF LANGUAGE (1)
Lexical ambiguity
Lexical ambiguity, where a word have more than one meaning:
meaning
i
(
(sense,
connotation,
t ti
denotation,
d
t ti
import,
i
t gist;
i t
significance, importance, implication, value, consequence, worth)
– sense (intelligence, brains, intellect, wisdom, sagacity, logic, good
j d
judgment;
t ffeeling)
li )
– connotation (nuance, suggestion, implication, undertone,
association, subtext, overtone)
– denotation (sense, connotation, import, gist) …
26
AMBIGUITIES OF LANGUAGE (2)
Syntactic ambiguity like in “small dogs and cats” (are cats small?).
Semantic ambiguity comes often as a consequence of syntactic
ambiguity. “Coast road” can be a road that follows the coast, or a
road that leads to the coast.
27
AMBIGUITIES OF LANGUAGE (3)
Referential ambiguity is a sort of semantic ambiguity (“it” can
refer to anything).
Pragmatic ambiguity (If the speaker says “I’ll meet you next
Friday”, thinking that they are talking about 17th, and the
hearer think that they are talking about 24th.)
Vagueness is an important feature of natural languages. “It is
warm outside
outside” says something about temperature
temperature, but what
does it mean? A warm winter day in Sweden is not the same
thing as warm summer day in Kenya.
28
AMBIGUITIES OF LANGUAGE (4)
Ambiguity of language results in its flexibility, that makes it
possible for us to cover the whole infinite diversity of the world
we live
li in
i with
ith a lilimited
it d means off vocabulary
b l
and
d a sett off rules
l
that language is made of.
On the other hand, flexibility makes the use of language
complex.
Nevertheless, the languages, both formal and natural, are the
main tools we have on our disposal in science and research.
29
USE OF LANGUAGE IN SCIENCE.
SCIENCE LOGIC AND
CRITICAL THINKING. PSEUDOSCIENCE
• LOGICAL ARGUMENT
• DEDUCTION
• INDUCTION
• REPETITIONS, PATTERNS, IDENTITY
• CAUSALITY AND DETERMINISM
• FALLACIES
• PSEUDOSCIENCE
30
REASONING
•
Use of reason, especially to form conclusions,
inferences, or judgments.
•
Evidence or arguments used in thinking or
argumentation.
•
The process of drawing conclusions from facts,
evidence, etc.
31
LOGICAL ARGUMENT
An argument is a statement logically inferred from premises.
Two sorts
T
t off arguments:
t
– Deductive
general → particular
– Inductive
particular → general
32
LOGICAL ARGUMENT
Constituents of a logical argument:
– premises
i
– inference and
– conclusion
33
JUDGMENT
It is important to notice that all reasoning basically depends on
judgment (the ability to perceive and distinguish relationships; the
capacity to form an opinion by distinguishing and evaluating)
“Now, the question, What is a judgment? is no small question,
because the notion of judgment is just about the first of all the
notions of logic, the one that has to be explained before all the
others, before even the notions of proposition and truth, for
instance.”
Per Martin-Löf
On the Meanings of the Logical Constants and the Justifications of the Logical
L
Laws;
N
Nordic
di JJournall off Phil
Philosophical
hi l L
Logic,
i 1(1)
1(1):11
11 60
60, 1996
1996.
34
INDUCTION
• Empirical Induction
• Mathematical Induction
35
EMPIRICAL INDUCTION
The generic form of an inductive argument:
•
Every A we have observed is a B.
•
Therefore, every A is a B.
36
An Example of Inductive Inference
•
•
Every instance of water (at sea level) that we have observed
has boiled at 100° C.
Therefore, all water (at sea level) boils at 100°
100 C.
Inductive argument will never offer 100% certainty!
A typical problem with inductive argument is that it is formulated
generally, while the observations are made under some
particular, specific conditions.
( In our example we could add ”in an open vessel” as well. )
37
Inductive Inference has Limited Validity
An inductive argument have no way to logically (with certainty,
with necessity) prove that:
•
•
the phenomenon studied do exist in general domain
that it continues to behave according to the same pattern
According to Popper, inductive argument only supports working
theories based on the collected evidence.
38
Counter-example
Perhaps the most well known counter-example was the
discovery of black swans in Australia
Australia. Prior to the point
point, it was
assumed that all swans were white. With the discovery of the
counter-example, the induction concerning the color of swans
had to be re-modeled.
re modeled.
39
MATHEMATICAL INDUCTION
The aim of the empirical induction is to establish the law.
In the mathematical induction we have the law already
formulated. We must prove that it holds generally.
The basis for mathematical induction is the property of the wellordering for the natural numbers.
40
THE PRINCIPLE OF MATHEMATICAL
INDUCTION
Suppose P(n) is a statement involving an integer n.
Than to prove that P(n) is true for every n ≥ n0 it is sufficient to
show these two things:
1.
2.
P(n0) is true.
For any k ≥ n0, if P(k) is true, then P(k+1) is true.
(the basis step)
(the induction step)
41
THE TWO PARTS OF INDUCTIVE PROOF
•
•
the basis step
the induction step
step.
•
In the induction step, we assume that statement is true in the
case n = kk, and we call this assumption the induction
hypothesis.
42
THE STRONG PRINCIPLE OF
MATHEMATICAL INDUCTION
Suppose P(n) is a statement involving an integer n. In order to
prove that P(n) is true for every n ≥ n0 it is sufficient to show
th
these
ttwo thi
things:
1.
2.
P(n0) is true.
For any k ≥ n0, if P(n) is true for every n satisfying
n0 ≤ n ≤ k, then P(k+1) is true.
43
INDUCTION VS DEDUCTION
DEDUCTION,
TWO SIDES OF THE SAME COIN
Deduction and induction occur as a part of the common
hypothetico-deductive method, which can be simplified in the
following scheme:
•
Ask a question and formulate a hypothesis (educated guess) induction
•
Derive predictions from the hypothesis - deduction
•
Test the hypothesis and its predictions - induction.
44
INDUCTION VS DEDUCTION
DEDUCTION,
TWO SIDES OF THE SAME COIN (1)
Deduction, if applied correctly, leads to true conclusions. But
deduction itself is based on the fact that we know something for
sure (premises must be true). For example we know the general
law which can be used to deduce some particular case, such as
“All humans are mortal. Socrates is human. Therefore is
Socrates mortal.”
How do we know that all humans are mortal? How have we
arrived to the general rule that our deduction is based on?
Again, there is no other method at hand but (empirical)
induction.
45
INDUCTION VS DEDUCTION
DEDUCTION,
TWO SIDES OF THE SAME COIN (2)
Even the process of induction implies use of deductive rules.
On our
O
ou way
ay from
o spec
specific
c (pa
(particular)
cu a ) up to
ou
universal
e sa (ge
(general)
e a)
we use deductive reasoning.
We collect the observations or experimental results and extract
the common p
patterns or rules and regularities
g
by
y deduction.
For example, in order to infer by induction the fact that all
planets orbit the Sun, we have to analyze astronomical data
using
g deductive reasoning.
g
46
INDUCTION & DEDUCTION:
Traditional View
47
Deduction-Induction Roller Coaster (A Loop)
general
deduction
induction
particular
48
GENERAL
INDUCTION & DEDUCTION
PARTICULAR
Problem domain
49
INDUCTION & DEDUCTION
“There is actually no such thing as a distinct process of
induction” said Stanly Jevons; “all
induction
all inductive reasoning is but the
inverse application of deductive reasoning” – and this was what
Whewell meant when he said that induction and deduction went
upstairs and downstairs on the same staircase.”
…(“Popper, of course, is abandoning induction altogether”).
Peter
e e Medawar,
eda a , Pluto’s
u o s Republic,
epub c, p 177.
50
INDUCTION & DEDUCTION
In short: deduction and induction are - like two sides of a piece
of paper - the inseparable parts of our recursive thinking
process.
51
FALLACIES - ERRORS IN REASONING
What about incorrectly built arguments? Let us make the following
distinction:
•
A formal fallacy is a wrong formal construction of an argument.
•
An informal fallacy is a wrong inference or reasoning.
52
FORMAL FALLACIES (1)
An example: “Affirming the consequent"
"All fish swim. Kevin swims. Therefore Kevin is a fish", which
appears to be a valid argument. It appears to be a modus
ponens, but it is not!
If H is true, then so is I.
(As the evidence shows)
shows), I is true
true.
H is true
This form
f
off reasoning, known as the fallacy
f
off "affirming
ff
the
consequent" is deductively invalid: its conclusion may be false
even if premises are true.
53
FORMAL FALLACIES (2)
Incorrect deduction from auxiliary hypotheses
If H and A1, A2, …., An is true, then so is I.
But (As the evidence shows)
shows), I is not true
true.
H and A1, A2, …., An are all false
(Comment: One can be certain that H is false, only if one is certain
that all of A1, A2, …., An are all true.)
54
INFORMAL FALLACIES (1)
An informal fallacy is a mistake in reasoning related to the
content of an argument.
Appeal to Authority
Ad Hominem (personal attack)
False Cause (synchronicity; unrelated facts that appear at the
same time coupled)
Leading Questions
55
INFORMAL FALLACIES (2)
Appeal to Emotion
Straw Man ((attacking
g the different p
problem))
Equivocation (not the common meaning of the word)
Composition (parts = whole)
Division (whole = parts)
See more on: http://en.wikipedia.org/wiki/List_of_fallacies
56
SOME NOT ENTIRELY UNCOMMON
“PROOF TECHNIQUES”
Proof by vigorous hand waving
Works well in a classroom or seminar setting.
Proof by cumbersome notation
Best done with access to at least four alphabets and special
symbols.
Proof by exhaustion
Proof around until nobody knows if the proof is over or not…
Read the rest on http://www.pleacher.com/mp/mhumor/proof.html
57
UNDERSTANDING PHENOMENA IN
NATURE: CAUSALITY AND
DETERMINISM
58
CAUSALITY AND DETERMINISM
Causality establishes that one thing causes another.
Practical question (object-level):
what was the cause (of an event)?
Philosophical question (meta-level):
what is the meaning of the concept of a cause?
59
CAUSALITY
Early natural philosophers, concentrated on conceptual issues
and questions (why?)
(why?).
Later natural philosophers and scientists concentrated on more
concrete issues and questions (how?)
(how?).
The change in emphasis from conceptual to concrete coincides
with the rise of empiricism
empiricism.
60
ARISTOTLE S CAUSALITY:
ARISTOTLE’S
The Four Causes
The material cause - constituents, substratum or materials. This
reduces the explanation of causes to the parts (factors, elements,
constituents, ingredients).
The formal cause - form, pattern, essence, whole, synthesis or
archetype. The account of causes in terms of fundamental principles or
general laws - the influence of the form (essence)
(essence).
The efficient cause - 'what makes what is made and what causes
change of what is changed - agency,
agency nonliving or living
living, acting as the
sources of change.
The final cause or telos is the purpose or end that something is
supposed to serve. Omitted from present day causal explanations.
61
CAUSALITY
David Hume was probably the first philosopher to postulate a
entirely empirical definition of causality. Of course, both the
definition of "cause" and the "way of knowing" whether X and Y
are causally linked have changed significantly over time
time.
Some natural philosophers deny the existence of "cause" and
some natural philosophers who accept its existence
existence, argue that
it can never be known by empirical methods.
Modern scientists,
scientists on the other hand
hand, define causality in limited
contexts (e.g., in a controlled experiment).
62
DETERMINISM
Determinism is the philosophical doctrine which regards everything
pp
as solely
y and uniquely
q y determined by
y what p
preceded it.
that happens
From the information given by a complete description of the world at
time t, a determinist believes that the state of the world at time t + 1
can be deduced; or, alternatively, a determinist believes that every
event is an instance of the operation of the laws of Nature.
63
Critique of Usual Naïve Image
of Scientific Method
64
Critique of Usual Naïve Image
of Scientific Method (1)
The naive inductivist idea of scientific inquiry sees scientific
process as consisting of the following steps:
1. All facts are observed and recorded.
2. All observed facts are analyzed, compared and classified,
without hypotheses or postulates other than those necessarily
involved in the logic of thought.
3. Generalizations inductively made about the relations, structural
or causal,
causal between the facts
facts.
4. Further research consists of inferences (deductions) from
previously established generalizations.
65
Critique of Usual Naïve Image
of Scientific Method (2)
This narrow idea of scientific investigation is groundless for
several reasons:
1. A scientific investigation could never get off the ground, for a
collection of all facts would take infinite time, as there are infinite
number of facts.
The only possible way to do data collection is to take only relevant
facts. But in order to decide what is relevant and what is not, we
have to have a theory or at least a hypothesis about what is it we
are observing
observing.
66
Critique of Usual Naïve Image
of Scientific Method (3)
A hypothesis (preliminary theory) is needed to give the direction
to a scientific investigation!
2. A set of empirical facts can be analyzed and classified in many
different ways. Without hypothesis, analysis and classification
are blind.
3. Induction is sometimes imagined as a method that leads, by
mechanical application of rules
rules, from observed facts to general
principles. Unfortunately, such rules do not exist!
67
Why is it not (yet)
(yet)* possible to derive theory
directly (automatically) from the data? (1)
– For example, theories about atoms contain terms like “atom”,
“electron”, “proton”, etc; yet what one actually measures are
spectra (wave lengths), traces in bubble chambers, calorimetric
data, etc.
– So the theory is formulated on a completely different (and more
abstract) level than the observable data!
– The transition from data to theory requests creative imagination
imagination.
* However, we cannot exclude the possibility of intelligent automated process of
discovery!
68
Why is it not (yet) possible to derive theory
directly (automatically) from the data? (2)
– Scientific hypothesis is formulated based on “educated guesses”
at the connections between the phenomena under study, at
regularities and patterns that might underlie their occurrence.
Scientific guesses are completely different from any process of
systematic inference.*
– The discovery of important mathematical theorems, like the
discovery of important theories in empirical science, requires
inventive ingenuity.
*Here it is instructive to study Automated discovery methods in order to see how
much theory must be used in order to extract meaning from the “raw data”
69
KNOWLEDGE AND JUSTIFICATION
Knowledge and Objectivity: Observations
Observations are always interpreted in the context of an a priori
knowledge (Kuhn
knowledge.
(Kuhn, Popper)
“What a man sees depends both upon what he looks at and also
upon what his previous visual-conceptual
visual conceptual experience has taught
him to see”.
70
KNOWLEDGE AND OBJECTIVITY
Observations
– All observation is potentially ”contaminated”, whether by our
theories, our worldview or our past experiences.
– It d
does nott mean th
thatt science
i
cannott ””objectively”
bj ti l ” [inter[i t
subjectivity] choose from among rival theories on the basis of
empirical testing.
– Although science cannot provide one with hundred percent
certainty, yet it is the most, if not the only, objective (intersubjective) mode of pursuing knowledge.
71
Perception and “Direct Observation”
72
Perception and “Direct Observation”
73
Perception and “Direct Observation”
74
Perception and “Direct Observation”
"Reality is merely an illusion, albeit a
very persistent one." - Einstein
75
76
Perception and “Direct Observation”
Checker-shadow illusion
http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html
See even:
http://web.mit.edu/persci/gaz/gaz-teaching/index.html
http://persci.mit.edu/people/adelson/publications/gazzan.dir/gazzan.htm
Lightness Perception and Lightness Illusions
77
Direct Observation?!
An atom interferometer, which splits an atom into separate wavelets, can
allow the measurement of forces acting on the atom. Shown here is the
laser system used to coherently divide, redirect, and recombine atomic
wave packets (Yale University)
University).
78
Direct Observation?!
Electronic signatures
produced by collisions of
protons and antiprotons in
th T
the
Tevatron
t
accelerator
l t
at Fermilab provided
evidence that the elusive
subatomic p
particle known
as top quark has been
found.
79
KNOWLEDGE JUSTIFICATION
– Foundationalism (uses architectural metaphor to describe the
structure of our belief systems. The superstructure of a belief
system inherits its justification from a certain subset of beliefs –
all rests on basic beliefs.)
– Coherentism
– Internalism (a person has “cognitive grasp”) and Externalism
(external
(e
te a just
justification)
cat o )
80
TRUTH (1)
– The correspondence theory
– The coherence theory
– The deflationary theory
81
TRUTH (2)
The Correspondence Theory
A common intuition is that when I say something true, my
statement corresponds to the facts.
But: how do we recognize facts and what kind of relation is this
correspondence?
82
TRUTH (3)
The Coherence Theory
Statements in the theory are believed to be true because being
compatible with other statements.
The truth of a sentence just consists in its belonging to a
system of coherent statements.
The most well-known adherents to such a theory was Spinoza
(1632-77), Leibniz (1646-1716) and Hegel (1770-1831).
Characteristically they all believed that truths about the world could be found by
pure thinking, they were rationalists and idealists. Mathematics was the
paradigm for a real science; it was thought that the axiomatic method in
mathematics could be used in all sciences.
83
TRUTH (4)
The Deflationary Theory
The deflationary theory is belief that it is always logically
unnecessary to claim that a proposition is true, since this claim
adds nothing further to a simple affirmation of the proposition
itself.
It is true that birds are warm-blooded"
warm blooded means the same thing as
"It
"birds are warm-blooded".
For the deflationist,
deflationist truth has no nature beyond what is captured
in ordinary claims such as that ‘snow is white’ is true just in case
snow is white.
84
TRUTH (5)
The Deflationary Theory
The Deflationary Theory is also called the
redundancy
d d
th
theory, th
the di
disappearance th
theory, th
the
no-truth theory, the disquotational theory, and the
minimalist theory.
y
See: Stanford Encyclopedia of Philosophy
http://plato.stanford.edu/entries/truth-deflationary/
85
If you want to learn more here are reach sources of further reading…
A Mathematical Analysis of The Scientific
Method, The Axiomatic Method, and Darwin's
Theory Of Evolution, G. J. Chaitin
htt //
http://www.umcs.maine.edu/~chaitin/ufrj.html
i
d / h iti / f j ht l
http://www.cs.auckland.ac.nz/~chaitin/ufrj.html
86
Chaitin’s work on Epistemology, Information Theory, and
Metamathematics important for understanding of Formal Systems
and their Relationship with Biology:
http://www.umcs.maine.edu/~chaitin/ecap.pdf Epistemology as
Information Theory: From Leibniz to Ω
http://www.umcs.maine.edu/~chaitin/mjm.pdf The Halting Probability
Omega: Irreducible Complexity in Pure Mathematics
http://www.umcs.maine.edu/~chaitin/unm.html
http://www
umcs maine edu/ chaitin/unm html Randomness in Arithmetic and
the Decline & Fall of Reductionism in Pure Mathematics
http://www.umcs.maine.edu/~chaitin/hu.html
http://www
umcs maine edu/~chaitin/hu html The Search for the Perfect
Language
87
Despite the fact that there can be no TOE (Theory Of
Everything) for pure mathematics as Hilbert hoped,
mathematicians remain enamored with formal proof.
See the special issue on formal proof of the AMS Notices,
December 2008 (From Chaitin’s lectures)
htt //
http://www.ams.org/notices/200811/index.html
/ ti
/200811/i d ht l
David Malone, Dangerous Knowledge, BBC TV, 90
minutes Google video vividly illustrates the search for
minutes,
TOE in mathematics
http://video.google.com/videoplay?docid=-5122859998068380459#
88
Truth and Reality
Noumenon "Ding an sich" is
distinguished from
Phenomenon "Erscheinung",
g an
observable event or physical
manifestation, and the two
words serve as interrelated
technical terms in Kant
Kant'ss
philosophy.
89
Whole vs. Parts
•
•
•
•
•
tusk → spear
tail → rope
trunk → snake
side → wall
l → tree
leg
t
The flaw in all their reasoning is that speculating on
the WHOLE from too few FACTS can lead to VERY
LARGE errors in judgment.
90
Science and Truth
With respect to the truth content, there are different
views of science:
– Science
S i
as controversy
t
(new
(
science,
i
ffrontiers)
ti )
– Science as consensus (old, historically settled)
– Science as knowledge about complex systems
– Open
p systems
y
with p
paraconsistent logic
g
91
PROOF
The word proof can mean:
•
•
•
a test assessing the validity or quality of something.
a rigorous
rigorous, compelling argument
argument, including:
– a logical argument or a mathematical proof
– a large accumulation of scientific evidence
– and alike
In mathematics, a proof is a demonstration that, given certain
axioms, some statement of interest is necessarily true.
92
PROOF OF PYTHAGORAS THEOREM
http://www.youtube.com/watch?v=EmBjt0b2BKE&feature=related
Einsteins Proof
http://demonstrations.wolfram.com/EinsteinsMostExcellentProof/
http://www.youtube.com/watch?NR=1&v=CAkMUdeB06o Water proof
http://www.youtube.com/watch?v=e4LuX48rD_k&feature=related
Geometrical proof
http://www youtube com/watch?v=xLkfDdsnpuY&feature=related
http://www.youtube.com/watch?v=xLkfDdsnpuY&feature=related
93
Pressupositions and Limitations
of Axiomatic Logical Systems
Axiomatic theory is built on a set of few axioms/postulates (ideas which
are considered so elementary and obvious that they do not need to be
proven as any proof would introduce more complex ideas). All the
theorems (true statements) are derived logically from those axioms
axioms.
Thus axiomatic theories are closed logical systems. For open
axiomatics,
a
o at cs, see U
Unconventional
co e t o a Algorithms:
go t s Co
Complementarity
p e e ta ty o
of
Axiomatics and Construction, Dodig Crnkovic G. and Burgin M.
http://www.mrtc.mdh.se/~gdc/work/Entropy-DodigCrnkovic-Burgin20120913.pdf
When a system requires increasing number of axioms (as e.g. number
theory does), doubts begin to arise. How many axioms are needed?
How do we know that the axioms aren't
aren t mutually contradictory? Each
new axiom can change the meaning of the previous system.
94
GÖDEL: TRUTH AND PROVABILITY (1)
Kurt Gödel proved two extraordinary theorems. They have
revolutionized mathematics, showing that mathematical truth is
more than bare logic and computation.
Gödel has been called the most important logician since
Aristotle. His two theorems changed logic and mathematics as
well as the way we look at truth and proof.
95
GÖDEL: TRUTH AND PROVABILITY (2)
Gödels first theorem proved that any formal system strong
enough to support number theory has at least one undecidable
statement. Even if we know that the statement is true, the
system cannot prove it. This means the system is incomplete.
For this reason, Gödel's first proof is called "the incompleteness
theorem".
96
GÖDEL: TRUTH AND PROVABILITY (3)
Gödel's second theorem is closely related to the first. It says that
no one can prove, from inside any complex formal system, that it
is self-consistent.
"Gödel showed that provability is a weaker notion than truth, no
matter what axiomatic system is involved.
In other words, we simply cannot prove some things in
mathematics (from a given set of premises) which we
nonetheless can know are true
true. “ (Hofstadter)
97
TRUTH VS. PROVABILITY
ACCORDING TO GÖDEL
Ö
After: Gödel, Escher, Bach - an Eternal Golden Braid
by Douglas Hofstadter.
98
TRUTH VS.
VS PROVABILITY
ACCORDING TO GÖDEL
Gödel theorem is built upon Aristotelian logic.
So it is true within the paradigm of Aristotelian logic.
However nowadays it is not the only logic existing.
However,
existing
99
CRITICAL THINKING (1)
Critical thinking is rationally deciding what to believe or do.
To rationally decide something is to evaluate claims to see
whether they make sense, whether they are coherent, and
whether they are well-founded on evidence, through inquiry
and the use of criteria developed for this purpose.
Critical Thinking http://en.wikipedia.org/wiki/Critical_thinking
100
CRITICAL THINKING (2)
How Do We Think Critically?
A. Question
First, we ask a question about the issue that we are wondering
about.
For example “What is going on?"
B. Answer (hypothesis)
Next, we propose an answer or hypothesis for the question raised.
A hypothesis is a "tentative theory provisionally adopted to explain
certain facts." We suggest a possible hypothesis, or answer, to the
question posed.
For example “A
A phenomenon occurs under certain conditions
conditions."
101
CRITICAL THINKING (3)
C. Test
Testing the hypothesis is the next step. With testing, we draw
out the implications of the hypothesis by deducing its
consequences (deduction). We then think of a case which
contradicts the claims and implications of the hypothesis
(inference).
For example,
example “If
If phenomenon really exists it will systematically
occur under certain conditions.“
C te a for
Criteria
o ttruth
ut
Criteria are used for testing the truth of a hypothesis such as
self-consistency, consistency with existing knowledge, empirical
confirmation, etc.
102
PSEUDOSCIENCE (1)
A pseudoscience is set of ideas and activities resembling
science but based on fallacious assumptions and supported by
fallacious arguments.
Martin Gardner: Fads and Fallacies in the Name of Science
103
PSEUDOSCIENCE (2)
Motivations for promotion of pseudoscience range from simple
lack of knowledge or no skills in the scientific method, to
deliberate deception for winning a power, financial gain or other
benefits.
Some people consider some or all forms of pseudoscience to be
harmless entertainment or sort of counseling.
Others, such as Richard Dawkins, and James Randi consider all
forms of pseudoscience to be harmful, whether or not they result
in immediate harm to their followers.
http://richarddawkins.net/ Richard Dawkins, Professor of the Public
Understanding of Science at Oxford University web page.
104
PSEUDOSCIENCE (3)
•
•
•
•
Typically, pseudoscience fails to meet the criteria met by science
Typically
generally (including the scientific method), and can be identified by one
or more of the following rules of thumb:
asserting claims without supporting experimental evidence;
asserting claims which contradict experimentally established results;
failing to provide an experimental possibility of reproducible results; or
violating
i l ti O
Occam's
' R
Razor (th
(the principle
i i l off choosing
h
i th
the simplest
i l t
explanation when multiple viable explanations are possible.
105
PSEUDOSCIENCE (4)
•
•
•
•
•
•
•
•
•
•
Astrology
Dowsing
Creationism
ETs & UFOs
Supernatural
Parapsychology/Paranormal
New Age
Divination (fortune telling)
Graphology
Numerology
•
•
•
•
•
•
•
•
•
Velikovsky's, von Däniken's,
and Sitchen's theories
Pseudohistory
Homeopathy
Healing
Alternative Medicine
Cryptozoology
Lysenkoism
Psychokinesis
Occult & occultism
106
PSEUDOSCIENCE (5)
http://www.youtube.com/watch?v=pYGjtlgGtY4&feature=related James Randi Exposes
Telekinesis (1:48)
http://www.youtube.com/watch?v=OZeQGld5QBU&NR=1James Randi Tests An Aura
Reader (3:24)
http://www.youtube.com/watch?v=3Dp2Zqk8vHw
http://www.youtube.com/watch?v
3Dp2Zqk8vHw James Randi on Astrology (1:36)
http://www.youtube.com/watch?v=6RtJ0yJL4tg&feature=related James Randi Tests a
Dowser (5:46)
http://www.youtube.com/watch?v=LSOD77clNZM&feature=related James Randi and
Richard Dawkins (6:23)
http://www.youtube.com/watch?v=_VAasVXtCOI&feature=related Dawkins debunks
dowsing (4:57)
http://www.youtube.com/watch?v=IZLKKW2SQoc&feature=related Richard Dawkins on
alternative medicine and the nature of science (1:14)
http://www.youtube.com/watch?v=LgyNNtBHOYc&feature=related Astrology Numerology
and You-4 (3:26)
http://www youtube com/watch?NR=1&v=Iunr4B4wfDA Carl Sagan on Astrology (8:35)
http://www.youtube.com/watch?NR=1&v=Iunr4B4wfDA
http://www.youtube.com/watch?v=TZiLsFaEzog Ben Goldacre on Homeopathy (3:04)
107
PSEUDOSCIENCE AS WISHFUL
THINKING
• No science can predict human future with certainty –
pseudosciences fulfill human wish to know their
future.
future
• No science can cure all diseases – but
pseudosciences fulfill human wish to have cure for
p
every disease.
• No science can what pseudosciences claim to be
able to!
• While sciences support critical thinking,
pseudosciences apply
p
pp y wishful thinking.
g
108
PSEUDOSCIENCE (6)
http://skepdic.com/ The Skeptic's Dictionary,
http://www.csicop.org/si/ Skeptical Inquirer
http://www.physto.se/~vetfolk/Folkvett/199534pseudo.html
The Swedish Skeptic movement (in Swedish)
http://www8.nationalacademies.org/onpinews/newsitem.aspx?RecordID=
11876 SCIENTIFIC EVIDENCE SUPPORTING EVOLUTION
http://www.scientificamerican.com/article.cfm?id=15-answers-top
creationist&page=2 Scientific American, July 2002: 15 Answers to
Creationist Nonsense
Human Genome, Nature 409, 860 - 921 (2001)
109
THE PROBLEM OF DEMARCATION (1)
After more than a century of active dialogue, the question of what
marks the boundary of science remains formally unsettled.
As a consequence the issue of what constitutes pseudoscience
continues to be controversial.
Nonetheless, reasonable consensus exists on a number of issues.
110
THE PROBLEM OF DEMARCATION (2)
Criteria for demarcation have traditionally been coupled to
philosophy of science.
Logical positivism, for example, held that only statements about
empirical observations are meaningful, effectively asserting that
statements which are not derived in this manner (including all
metaphysical statements) are meaningless
meaningless.
111
THE PROBLEM OF DEMARCATION (3)
Karl Popper attacked logical positivism and introduced his own
criterion for demarcation, falsifiability.
Thomas Kuhn and Imre Lakatos proposed criteria that
distinguished between progressive and degenerative research
programs.
112
THE PROBLEM OF DEMARCATION (4)
Read a book by astrophysicist Carl
Sagan against pseudoscience:
http://www.youtube.com/watch?v Cs3S
http://www.youtube.com/watch?v=Cs3S
iKm2iMQ The Demon-Haunted World
(13:56)
http://www.youtube.com/watch?v hGkf
http://www.youtube.com/watch?v=hGkf
s9WU98s (2:49:35)
The book explains the scientific method
and encourage people to learn critical
thinking. It explains methods to help
g
between science,, and
distinguish
pseudoscience by means of critical
113
thinking.
THEORY OF SCIENCE ASSIGNMENTS
– Assignment 2: Demarcation of Science vs.
Pseudoscience (in groups of two)
– Discussion of Assignment 2 - compulsory
– Assignment 2-extra (For those who miss the discussion of the
Assignment 2)
– Assignment 3: GOLEM: Three Cases of Theory
C fi
Confirmation
ti (i
(in groups off ttwo))
– Discussion of Assignment 3 - compulsory
– Assignment 3-extra
3 extra (For those who miss the discussion of the
Assignment 3)
htt //
http://www.idt.mdh.se/kurser/ct3340/ht12/deadlines.html
idt dh /k
/ t3340/ht12/d dli
ht l Deadlines
D dli
114
Assignment 2: Demarcation: Pseudoscience
vs. Science (done in groups of two)
[1]
Hansson, Sven Ove, "Science and Pseudo-Science",
The Stanford Encyclopedia of Philosophy (Fall 2008 Edition),
Edward N. Zalta (ed.),
( )
http://plato.stanford.edu/archives/fall2008/entries/pseudo-science
[2] http://en.wikipedia.org/wiki/Pseudoscience
[3] The Astrotest A tough match for astrologers (Rob Nanninga),
http://www.skepsis.nl/astrot.html
Further reading
Popper on Demarcation (Stanford Encyclopedia): http://plato.stanford.edu/entries/popper/
Astrology Fact Sheet (North Texas Skeptics), http://www.ntskeptics.org/factsheets/astrolog.htm
115
Assignment 2: Demarcation: Pseudoscience
vs. Science (done in groups of two)
•
•
•
•
•
•
•
Use the template (Answer form)
Leave the template unchanged, write down your answer after each
question.
Think critically!
You are expected to work in groups of two.
Yo r text
Your
te t should
sho ld not be shorter than ttwo
o A4 pages written
ritten in usual
s al te
textt
format.
Prepare for the discussion in the class!
Please write the file name in the following format:
name1_name2_a2.doc
116
APPENDIX
For additional reading
117
ISLANDS OF KNOWLEDGE
“You see, you have all of mathematical truth, this ocean of
mathematical truth. And this ocean has islands. An island here,
algebraic truths. An island there, arithmetic truths. An island
here the calculus.
here,
calculus And these are different fields of mathematics
where all the ideas are interconnected in ways that
mathematicians love; they fall into nice, interconnected patterns.
But what II've
ve discovered is all this sea around the islands.
islands ”
Gregory Chaitin, an interview, September 2003
118
Two Examples of Axiomatic Systems Limitations and Developments
119
Pressupositions and Limitations
of Formal Logical Systems
Axiomatic System
y
of Euclid: Shaking
g up
p Geometry
y
Euclid built geometry on a set of few axioms/postulates (ideas
which are considered so elementary and manifestly obvious that
th d
they
do nott need
d tto b
be proven as any prooff would
ld iintroduce
t d
more
complex ideas).
When a system requires increasing number of axioms (as e.g.
number theory does), doubts begin to arise. How many axioms are
needed? How do we know that the axioms aren't mutually
contradictory?
t di t ?
120
Pressupositions and Limitations
of Formal Logical Systems
Axiomatic System
y
of Euclid: Shaking
g up
p Geometry
y
Until the 19th century no one was too concerned about
axiomatization.
Geometry has stood as conceived by Euclid for 2100 years.
If Euclid's work had a weak point, it was his fifth axiom, the axiom
about parallel lines. Euclid said that for a given straight line, one
could draw only one other straight line parallel to it through a point
somewhere
h
outside
t id itit.
121
EUCLID'S AXIOMS (1)
1. Every two points lie on exactly one line.
2. Any line segment with given endpoints may be continued in
either direction.
direction
3. It is possible to construct a circle with any point as its center and
with a radius of any length. (This implies that there is neither an
upper nor lower limit to distance. In-other-words, any distance,
no mater how large can always be increased, and any distance,
no mater how small can always be divided.)
122
EUCLID'S AXIOMS (2)
4. If two lines cross such that a pair of adjacent angles are
congruent, then each of these angles are also congruent to any
other angle formed in the same way. (Says that all right angles
are equal to one another.)
5. (Parallel Axiom): Given a line l and a point not on l, there is one
and only one line which contains the point
point, and is parallel to l.
l
123
NON-EUCLIDEAN GEOMETRIES (1)
Mid-1800s: mathematicians began to experiment with different
definitions for parallel lines.
Lobachevsky, Bolyai, Riemann: new non-Euclidean geometries by
assuming that there could be several parallel lines through the
outside point or there could be no parallel lines.
124
NON-EUCLIDEAN GEOMETRIES (2)
Two ways to negate the Euclidean Parallel Axiom:
– 5-S (Spherical Geometry Parallel Axiom): Given a line l and a
point not on l, no lines exist that contain the point, and are parallel
to l.
– 5-H (Hyperbolic Geometry Parallel Axiom): Given a line l and a
point not on l, there are at least two distinct lines which contains
the point
point, and are parallel to l.
l
125
Reproducing the Euclidean World
in a model of the Elliptical Non-Euclidean World.
126
Spherical/Elliptical Geometry
In spherical geometry lines of latitude are not great circles (except for
the equator),
q
), and lines of longitude
g
are. Elliptical
p
Geometry
y takes the
spherical plan and removes one of two points directly opposite each
other. The end result is that in spherical geometry, lines always intersect
in exactly two points, whereas in elliptical geometry, lines always
intersect in one point.
127
Properties of Elliptical/Spherical Geometry
In Spherical Geometry, all lines intersect in 2 points. In elliptical
geometry lines intersect in 1 point
geometry,
point.
In addition, the angles of a triangle always add up to be greater
than 180 degrees. In elliptical/spherical geometry, all of Euclid's
postulates still do hold
hold, with the exception of the fifth postulate
postulate.
This type of geometry is especially useful in describing the
Earth's surface.
128
Hyperbolic Cubes
129
DEFINITION:
Parallel lines are infinite lines in the same plane that do not intersect
intersect.
Hyperbolic Universe
Flat Universe
Spherical Universe
Einstein incorporated Riemann's ideas into relativity theory to
d
describe
ib th
the curvature
t
off space.
130
MORE PROBLEMS WITH
AXIOMATIZATION…
Not only had Riemann created a system of geometry which put
commonsense notions on its head, but the philosopherphilosopher
mathematician Bertrand Russell had found a serious paradox
for set theory!
He has shown that Frege’s attempt to reduce mathematics to
logical reasoning starting with sets as basics leads to
contradictions.
131
HILBERT’S PROGRAM
Hilbert’s hope was that mathematics would be reducible to
finding proofs (manipulating the strings of symbols) from a fixed
system of axioms that everyone could agree were true.
Can all of mathematics be made algorithmic, or will there always
be new problems that outstrip any given algorithm, and so
require creative mind to solve?
132
AXIOMATIC SYSTEM OF PRINCIPIA:
PARADOX IN SET THEORY
Mathematicians hoped that Hilbert's plan would work because
axioms and definitions are based on logical commonsense
intuitions such as e
intuitions,
e.g.
g the idea of set.
set
A set is any collection of items chosen for some characteristic
common for all its elements
elements.
133
RUSSELL'S PARADOX (1)
There are two kinds of sets:
– Normal sets, which do not contain themselves, and
– Non-normal sets, which are sets that do contain themselves.
The set of all apples is not an apple. Therefore it is a normal set. The
set of all thinkable things is itself thinkable, so it is a non-normal set.
134
RUSSELL'S PARADOX (2)
Let 'N' stand for the set of all normal sets.
Is N a normal set?
If it is a normal set, then by the definition of a normal set it
cannot be a member of itself. That means that N is a non-normal
set, one of those few sets which actually are members of
themselves.
135
RUSSELL'S PARADOX (3)
But on the other hand
hand…N
N is the set of all normal sets; if we
describe it as a non-normal set, it cannot be a member of itself,
because its members are, by definition, normal.
136
RUSSELL'S PARADOX (4)
Russell resolved the paradox by redefining the meaning of 'set'
to exclude peculiar (self-referencing) sets, such as "the set of all
normal sets“.
Together with Whitehead in Principia Mathematica he founded
mathematics on that new set definition.
They hoped to get self-consistent and logically coherent system
…
137
RUSSELL'S PARADOX (5)
… However, even before the project was complete, Russell's
expectations were dashed!
The man who showed that Russell's aim was impossible was
Kurt Gödel, in a paper titled "On Formally Undecidable
Propositions of Principia Mathematica and Related Systems."
138
LOGIC
139
LOGIC (1)
The precision, clarity and beauty of mathematics are the
consequence of the fact that the logical basis of classical
mathematics possesses the features of parsimony and
transparency.
Classical
Cl
i l logic
l i owes its
it success in
i large
l
partt tto the
th efforts
ff t off
Aristotle and the philosophers who preceded him. In their
endeavour to devise a concise theory of logic, and later
mathematics they formulated so-called
mathematics,
so called "Laws
Laws of Thought
Thought".
140
LOGIC (2)
One of these, the "Law of the Excluded Middle," states that
every proposition must either be True or False.
When Parminedes proposed the first version of this law (around
400 B.C.)
B C ) there were strong and immediate objections
objections.
For example, Heraclitus proposed that things could be
simultaneously True and not True
True.
141
NON-STANDARD
NON
STANDARD LOGIC
FUZZY LOGIC (1)
Plato laid the foundation for fuzzy logic, indicating that there was
a third region (beyond True and False).
Some among more modern philosophers follow the same path,
particularly Hegel.
But it was Lukasiewicz who first proposed a systematic
alternative to the bi-valued logic of Aristotle.
142
NON-STANDARD
NON
STANDARD LOGIC
FUZZY LOGIC (2)
In the early 1900's, Lukasiewicz described a three-valued logic,
along with the corresponding mathematics.
The third value "possible," assigned a numeric value between
True and False.
Eventually, he proposed an entire notation and axiomatic
system from which he hoped to derive modern mathematics.
143
NON-STANDARD LOGICS
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Categorical logic
Combinatory logic
Conditional logic
Constructive logic
g
Cumulative logic
Deontic logic
Dynamic logic
Epistemic
p
logic
g
Erotetic logic
Free logic
Fuzzy logic
Higher-order
g
logic
g
Infinitary logic
Intensional logic
Intuitionistic logic
Linear logic
g
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Many-sorted logic
Many-valued logic
Modal logic
Non-monotonic logic
g
Paraconsistent logic
Partial logic
Prohairetic logic
Quantum logic
g
Relevant logic
Stoic logic
Substance logic
Substructural logic
g
Temporal (tense) logic
Other logics
144
MATHEMATICAL INDUCTION
The aim of the empirical induction is to establish the law.
In the mathematical induction we have the law already
formulated. We must prove that it holds generally.
The basis for mathematical induction is the property of the wellordering for the natural numbers.
145
THE PRINCIPLE OF MATHEMATICAL
INDUCTION
Suppose P(n) is a statement involving an integer n.
Than to prove that P(n) is true for every n ≥ n0 it is sufficient to
show these two things:
1.
2.
P(n0) is true.
For any k ≥ n0, if P(k) is true, then P(k+1) is true.
146
THE TWO PARTS OF INDUCTIVE PROOF
•
•
the basis step
the induction step
step.
•
In the induction step, we assume that statement is true in the
case n = kk, and we call this assumption the induction
hypothesis.
147
THE STRONG PRINCIPLE OF
MATHEMATICAL INDUCTION (1)
Suppose P(n) is a statement involving an integer n. In order to
prove that P(n) is true for every n ≥ n0 it is sufficient to show
th
these
ttwo thi
things:
1.
2.
P(n0) is true.
For any k ≥ n0, if P(n) is true for every n satisfying
n0 ≤ n ≤ k, then P(k+1) is true.
148
THE STRONG PRINCIPLE OF
MATHEMATICAL INDUCTION (2)
A proof by induction using this strong principle follows the same
steps as the one using the common induction principle.
The only difference is in the form of induction hypothesis.
Here the induction hypothesis is that k is some integer k ≥ n0 and
that all the statements P(n0), P(n0 +1), …, P(k) are true.
149
Example. Proof by Strong Induction
•
P(n): n is either prime or product of two or more primes, for n ≥ 2.
•
Basic step. P(2) is true because 2 is prime.
•
Induction hypothesis. k ≥ 2, and for every n satisfying
2 ≤ n ≤ k, n is either prime or a product of two or more primes.
150
•
Statement to be shown in induction step:
If k+1 is prime
prime, the statement P(k+1) is true
true.
•
Otherwise, by definition of prime, k+1 = r·s, for some positive
integers r and ss, neither of which
hich is 1 or k+1
k+1. It follo
follows
s that 2 ≤ r
≤ k and 2 ≤ s ≤ k.
•
By the induction hypothesis, both r and s are either prime or
product of two or more primes.
•
Therefore, k+1 is the product of two or more primes, and P(k+1)
is true.
151
The strong principle of induction is also referred to as the
principle of complete induction, or course
course-of-values
of values induction. It
is as intuitively plausible as the ordinary induction principle; in
fact, the two are equivalent.
As to whether they are true, the answer may seem a little
surprising. Neither can be proved using standard properties of
natural numbers. Neither can be disproved either!
152
This means essentially that to be able to use the induction
principle, we must adopt it as an axiom.
A well-known set of axioms for the natural numbers, the Peano
axioms, includes one similar to the induction principle.
153
PEANO'S AXIOMS
1. N is a set and 1 is an element of N.
2 Each element x of N has a unique successor in N denoted x'
2.
x.
3. 1 is not the successor of any element of N.
4. If x' = y' then x = y.
5. (Axiom of Induction) If M is a subset of N satisfying both:
1 is in M
x in M implies x' in M
then M = N.
154
CAUSALITY
What does the scientist mean when (s)he says that event b was
caused by event a?
Other expressions are:
– bring about, bring forth
– produce
– create…
…and similar metaphors of human activity.
Strictly
y speaking
p
g it is not a thing
g but a p
process that causes an
event.
155
CAUSALITY
Analysis of causality, an example (Carnap): Search for the cause of
a collision between two cars on a highway.
•
According to the traffic police
police, the cause of the accident was too
high speed.
•
According to a road-building engineer, the accident was caused
by the slippery highway (poor, low-quality surface)
•
According to the psychologist, the man was in a disturbed state
of mind which caused the crash.
156
CAUSALITY
•
•
•
An automobile construction engineer may find a defect in a
structure of a car.
A repair-garage man may point out that brake-lining of a car was
worn out
worn-out.
A doctor may say that the driver had bad sight. Etc…
Each person, looking
g at the total picture from certain point of
view, will find a specific condition such that it is possible to say:
if that condition had not existed, the accident might not have
happened.
But what was The cause of the accident?
157
CAUSALITY
•
•
It is quite obvious that there is no such thing as The cause!
No one could know all the facts and relevant laws.
(Relevant laws include not only laws of physics and technology
technology,
but also psychological, physiological laws, etc.)
•
But if someone had known,
known (s)he could have predicted the
collision!
158
CAUSALITY
The event called the cause, is a necessary part of a more
complex web of circumstances. John Mackie, gives the following
definition:
A cause is an INSUFFICIENT BUT NECESSARY part of a
complex of conditions which together are UNNECESSARY BUT
SUFFICIENT for the effect.
This definition has become famous and is usually referred to as
the INUS-definition: a cause is an INUS-condition.
159
CAUSALITY
The reason why we are so interested in causes is primarily that
we want either to prevent the effect or else to promote it. In both
cases we ask for the cause in order to obtain knowledge about
what to do.
Hence, in some cases we simply call that condition which is
easiest to manipulate as the cause.
160
CAUSALITY
Summarizing: Our concept of a cause has one objective and
subjective component. The objective content of the concept of a
cause is expressed by its being an INUS condition. The
subjective part is that our choice of one factor as the cause
among the necessary parts in the complex is a matter of
interest.
161
CAUSE AND CORRELATION
Instead of saying that the same cause always is followed by the
same effect it is said that the occurrence of a particular cause
increases the probability for the associated effect, i.e., that the
cause sometimes but not always are followed by the effect.
Hence cause and effect are statistically correlated.
162
CAUSE AND CORRELATION
X and Y are correlated if and only if:
P(X/Y) > P(X)
and
P(Y/X) > P(Y)
[The events X and Y are positively correlated if the conditional
probability for X, if Y has happened, is higher than the
unconditioned probability, and vice versa.]
163
CAUSE AND CORRELATION
Reichenbach's principle:
If events of type A and type B are positively correlated, then one of
the following possibilities must obtain:
i)
A is a cause of B
B, or
ii)
B is a cause of A, or
iii)
A and B have a common cause.
164
CAUSE AND CORRELATION
The idea behind Reichenbach’s principle is:
Every real correlation must have an explanation in terms of
causes. It just can’t happen that as a matter of mere coincidence
a correlation obtains.
165
CAUSE AND CORRELATION
We and other animals notice what goes on around us. This helps
us by suggesting what we might expect and even how to prevent
it, and thus fosters survival. However, the expedient works only
imperfectly. There are surprises, and they are unsettling. How
can we tell when we are right? We are faced with the problem of
error.
W.V.
W
V Quine,
Quine 'From Stimulus To Science',
Science' Harvard University Press,
Press
Cambridge, MA, 1995.
166
The Classical (Ideal) Model of Science
The Classical Model of Science is a system S of propositions
and concepts satisfying the following conditions:
•
All propositions
iti
and
d allll concepts
t (or
( terms)
t
) off S concern a
specific set of objects or are about a certain domain of being(s).
•
There are in S a number of so-called fundamental concepts
p ((or
terms).
•
All other concepts (or terms) occurring in S are composed of (or
are definable
f
from)
f
) these ffundamental concepts (or
( terms).
)
167
The Classical (Ideal) Model of Science
•
There are in S a number of so-called fundamental propositions.
•
All other propositions of S follow from or are grounded in (or are
provable
bl or demonstrable
d
t bl ffrom)) th
these ffundamental
d
t l propositions.
iti
•
All propositions of S are true.
•
All propositions of S are universal and necessary in some sense
or another.
168
The Classical (Ideal) Model of Science
•
All concepts or terms of S are adequately known. A nonfundamental concept is adequately known through its
composition (or definition).
•
The Classical Model of Science is a reconstruction a posteriori
and sums up the historical philosopher’s ideal of scientific
explanation.
explanation
•
The fundamental is that “All propositions and all concepts (or
terms) of S concern a specific set of objects or are about a
certain domain of being(s).”
Betti A & De Jong W. R., Guest Editors, The Classical Model of Science I: A MillenniaOld Model of Scientific Rationality, Forthcoming in Synthese, Special Issue
169
SCIENCE, KNOWLEDGE, TRUTH, MEANING.
FORMAL LOGICAL SYSTEMS
AND THEIR LIMITATIONS
The science is not about the search for truth (“absolute truth”)
but the search for meaning in the form of explanations/models/
simulations that work:
“No such (scientific) model, however comprehensive, coherent
or well entrenched it might be
be, can lay an automatic claim to
objective truth, even though contextually it may provide a
reliable and successful explanatory tool for making sense of
what is going on around us
us.” Edo Pivčević,
Pivčević The Reason Why: A
Theory of Philosophical Explanation, KruZak, 2007
Knowledge networks in communities of practice - Language
170
Classical Sciences in their Cultural Context –
A Language Based Scheme
Logic
&
Mathematics
1
Natural Sciences
(Physics
(Physics,
Chemistry,
Biology, …)
2
Social Sciences
(Economics,
Sociology,
Anthropology, …)
3
Culture
l
(Religion, Art,
…)
5
The Humanities
(Philosophy, History,
Linguistics …)
4
171
CRITICAL THINKING (1)
Critical thinking is rationally deciding what to believe or do.
To rationally decide something is to evaluate claims to see
whether they make sense, whether they are coherent, and
whether they are well-founded on evidence, through inquiry
and the use of criteria developed for this purpose.
Critical Thinking http://en.wikipedia.org/wiki/Critical_thinking
172
CRITICAL THINKING (2)
How Do We Think Critically?
A. Question
First we ask a question about the issue that we are wondering
First,
about.
For example, "Is there right and wrong?"
B. Answer (hypothesis)
Next, we propose an answer or hypothesis for the question raised.
A hypothesis is a "tentative theory provisionally adopted to explain
certain facts." We suggest a possible hypothesis, or answer, to
the question posed.
F example,
For
l "N
"No, there
h
iis no right
i h and
d wrong.""
173
CRITICAL THINKING (3)
C. Test
Testing the hypothesis is the next step. With testing, we draw out
the implications of the hypothesis by deducing its consequences
(deduction). We then think of a case which contradicts the
claims and implications of the hypothesis (inference).
For example, "So if there is no right or wrong, then everything has
equal moral value (deduction); so would the actions of Hitler be
of equal moral value to the actions of Mother Theresa
(inference)? as Value nihilism ethics claims"
174
CRITICAL THINKING (4)
1. Criteria for truth
Criteria are used for testing the truth of a hypothesis. The criteria
may be used singly or in combination.
a. Consistent with a precondition
Is the hypothesis consistent with a precondition necessary for its
own assertion?
For example, is the assertion "there is no right or wrong" made
possible only by assuming a concept of right or wrong - namely,
that it is right that there is no right or wrong and that it is wrong
that there is right or wrong?
175
CRITICAL THINKING (5)
b. Consistent with itself (self-consistent)
Is the hypothesis consistent with itself?
For example, is the assertion that "there is no right or wrong" itself
an assertion of right or wrong?
c. Consistent with language
Is the hypothesis consistent with the usage and meaning of
ordinary language?
For example, do we use the words "right" or "wrong" in our
language and do the words refer to concepts and meanings
which we consider "right" and "wrong"?
176
CRITICAL THINKING (6)
d. Consistent with experience
Is the hypothesis consistent with experience?
For example,
p , do people
p p really
y live as if there is no right
g or wrong?
g
e. Consistent with the consequences
Is the hypothesis consistent with its own consequences, can it
actually bear the burden of being lived?
For example, what would the consequences be if everyone lived as
if there was no right or wrong?
177