ARITHMETIC AND REALITY: A DEVELOPMENT OF POPPER'S IDEAS
Frank H. Gregory
Logicalfrank(at)gmail.com
Most computerized information systems operate by means of rules that are incorrigible within
the system. They have the same status as necessary, or logical, truths. There is a problem here
that dates back to the beginning of British Empiricism. According to David Hume "there is no
necessity in the object". In other words the rules that govern the behaviour of the physical world
are not necessary but contingent truths. We are, therefore, faced with the problem of explaining
how a systems of necessary truths can tell us anything about, or be useful in dealing with, a
contingent world.
The problem is not unique to computer systems. Prima facie it seems that mathematical formulae
are logically true. The question of how, given this, they can apply to reality has been the subject
of lengthy debate in the philosophy of mathematics. The present paper recounts how the problem
has been structured and offers a new solution.
The downstream relevance to information system design should be obvious. Whatever, principles
underlie the application of arithmetic to reality will also need to underlie the design of any
information system intended to be informative about the real world.
INTRODUCTION
There are two basic questions that can be asked in respect of mathematical propositions. One is
"what are they about?" the other is "how are they justified?".
Korner [1968] makes a distinction between, what he calls, pure and applied mathematics. A pure
mathematical proposition is of the form "1 + 1 = 2" while a proposition of the form "one apple
and one apple makes two apples" is a proposition of applied mathematics. This distinction opens
the door to the possibility that there are two different types of mathematical proposition and that
these are about different things. Prima facie the propositions of applied mathematics appear to be
about objects and events in the real world while those of pure mathematics do not.
If it were true that propositions of applied mathematics are about real world objects then this
would suggest that they are justified empirically. Here we can identify two broad schools of
thought. Following Tymoczko , accounts of the nature of arithmetic and mathematics can be
described as realist or constructivist. "Realism assumes the reality of a mathematical universe
which is independent of mathematicians who discover truths about this reality. Constructivism
insists that any mathematical reality is conditioned by the actual and potential constructions of
mathematicians who invent mathematics." [p xiv, 1985]
Mathematics is a large subject and it not obvious that it is completely homogenous. The idea that
parts of mathematics are invented and parts discovered should not discounted out of hand.
However, if the inquiry is limited to basic propositions of arithmetic i.e. the addition or
subtraction of finite numbers, as will be the case in this paper, then the realist and constructivist
accounts have the appearance of contradictories. It might be assumed that arguments against one
would count in favour of the other. However, if the realist/constructivist distinction is combined
with the pure/applied distinction then there are four permutations and in two of these realism and
constructivism are not even contraries let alone contradictories. These are:
First: A realist account of pure and applied arithmetic.
Second: A constructivist account of pure and applied arithmetic.
Third: A constructivist account of pure arithmetic and a realist account of applied arithmetic.
Fourth: A realist account of pure mathematics and a constructivist account of applied
mathematics.
All four permutations are open to immediate difficulties. The first needs to explain why
arithmetic propositions do not appear to be falsifiable by experience. The second needs to
explain why a mental construct, such as arithmetic, can be informative about reality while other
analogous mental constructs, such as chess, are not. The third has similar difficulties, it needs to
explain how a mental construct relates to a real world discovery. The fourth appears to combine
the worst aspects of the other three permutations, it needs to explain why pure arithmetical
propositions are falsifiable while the apparently contingent applied arithmetical propositions are
not.
Popper put forward a version of the third permutation. However, he did not regard arithmetic as
comprising two distinct types of statement, statements of pure arithmetic and statements of
applied arithmetic. Rather his idea was that a number statement such as "2 apples + 2 apples = 4
apples" can be taken in two senses. In one sense it is irrefutable and logically true in the second
sense it is factually true and falsifiable. Another way of putting this is to say that a single number
statement can express two proposition one of which can be explained on constructivist lines the
other on realist lines.
Popper's argument is not tenable as it stands. This is because it functions at a psychological level
rather than at a logical level. However, a similar but tenable, logical, argument can be
formulated. This is undertaken in Part I of the present paper. Here it will be argued that there
cannot be a meaningful system that consists only of logically true universals and factual
particulars. Factual universals must be introduced into the system to make it workable. Part I
argues the case for a realist element in any number system.
Part II makes the much stronger claim that there cannot be a meaningful system that consists
only of factually universals and factual particulars. Here logical universals must be introduced
into the system to make it workable. Part II argues for a constructivist element in any number
system.
The main thrust of the paper is to develop a tenable version of the third permutation. Somewhat
surprisingly the consequences of the Part II arguments show that the fourth permutation, while
not necessarily a practical perspective, is also logically tenable.
PART I
Popper's account
The question "Why are the calculi of logic and arithmetic applicable to reality?" was the subject
of a symposium at which Gilbert Ryle, Karl Popper, and C. Lewy presented papers. Both Ryle
[1946] and Lewy [1946] limited their papers to a discussion of logic, but Popper directly
addressed the issue of how arithmetic applies to reality.
Ryle contended that the rules of logic are rules of procedure and therefore do not apply to reality
at all. In the earlier sections of his paper Popper [1946] agreed with Ryle that the rules of logic
(or of inference) are rules of procedure and as such they are not meant to fit the facts of the
world. Thus the problem disappears. But Popper felt that there was an underlying problem that
had not been solved. This was the question of how the rules of logic can be useful in dealing with
the world: "Why are the rules of logic good, or useful, or helpful rules of procedure?"
Popper thought that this could be answered rather easily. A man will find "the procedure useful
because he finds that, whenever he observes the rules of logic, whether consciously or
intuitively, the conclusion will be true, provided the premises were true". Here we would expect
the argument to move into a discussion of theories of truth, but Popper does not do this. Instead
he says "... a "good" or "valid" rule of inference is useful because no counter example can be
found," and continues
... since we can say of a true description that it fits the facts ... we can say that rules of inference
apply to facts in so far as every observance of them which starts with a fitting description can be
relied on to lead to a description which likewise fits the facts. [Popper, 1946, p48]
The key point here is what counts as a counter example. Popper could be making the point that a
rule of inference will only be valid if its use in an axiomatic system will not lead that system into
inconsistency. That is, the use of a rule of inference will not lead to the production of any
theorem and its contradictory. On this interpretation the theorem and its contradictory would be
the counter example. But it seems unlikely that this is what Popper had in mind as this would not
go far towards solving the usefulness problem. There are many consistent systems that have no
relation to and no use in the real world.
A more likely candidate is that he was saying that rules of inference are open to falsification by
facts. I.e. that if "All men are mortal" is a description that fits the world and "Socrates is a man"
is a description that fits world, but "Socrates is mortal" is a description that does not fit the
world, then it would be shown that modus ponens is not valid. In this case it must be at least
logically possible for modus ponens to be false, therefore, modus ponens is contingent. This is
effectively an inductive account of deduction. However, this was not Popper's position either.
This becomes clear when he extends his ideas on logic to arithmetic:
In so far as a calculus is applied to reality, it loses its character as a logical calculus and becomes
a descriptive theory which may be empirical refutable; and in so far as it is treated as irrefutable,
i.e., as a system of logically true formulae, rather than a descriptive scientific theory, it is not
applied to reality. [Popper, 1946, p 54]
So, it would appear, that a calculus is only useful when it becomes a descriptive theory and
therefore falsifiable. Two questions now need to be answered: firstly, how does a calculus
become a descriptive theory, and, secondly, which calculi can become descriptive theories? (it is
not clear that all calculi can become descriptive theories, some calculi have been developed
merely to explore the properties of formal systems, for example the MIU-system Post Production
System in Hofstadter [1980].
Popper attempts to answer the second question as follows:
...if we consider a proposition such as "2 + 2 = 4", then it may be applied - for example to apples
- in two different senses... In the first of these senses, the statement "2 apples + 2 apples = 4
apples" is taken to be irrefutable and logically true. But it does not describe any fact involving
apples - any more than "All apples are apples" does. ...it is based ... on certain definitions of the
signs "2", "4", "+" and "=".
More important is the application in the second sense. In this sense, "2 + 2 = 4" may be taken to
mean that, if somebody has put two apples in a basket, and then again two, and has not taken any
apples out of the basket, there will be four in it. In this interpretation "2 + 2 = 4" helps us to
calculate, i.e., to describe certain physical facts, and the symbol "+" stands for a physical
manipulation - for physically adding certain things to other things. ...But in this interpretation "2
+ 2 = 4" becomes a physical theory, rather than a logical one; and as a consequence, we cannot
be sure whether it remains universally true. As a matter of fact, it does not. ...It may hold for
apples, but it hardly holds for rabbits. If you put 2 + 2 rabbits in a basket you may soon find 7 or
8 in it. [Popper, 1946, p 55].
The key question here is when is "2 + 2 = 4" operative in the logical and when is it operative in
the physical, factual and contingent sense. Popper seems to be giving a psychological account
here. He could be saying that people do, as a matter of fact, interpret "2 + 2 = 4" in two ways.
That, as a matter of fact, there is an oscillation of "2 + 2 = 4" between being a logical truth and a
physical truth in every person's thinking. As a psychological account it has a lot to commend it. It
can help to explain why the problem is such an intractable problem and why it has a now you see
it, now you don't quality. "2 + 2 = 4" taken as purely logical throughout a system or narrative,
will not be a problem; nor will it be a problem if it is taken as purely physical throughout a
system or narrative. The errors that undoubtedly occur in this area are when a given instance of
"2 + 2 = 4" is taken to be logical and physical in the same system or narrative. We then have the
situation were people claim that there must, as a matter of logic, be four rabbits in a basket; and
the opposite error where people claim that arithmetic is a branch of physics. The problem is how
to deal with "2 + 2 = 4" in such a way that it has logical and physical implications in the same
system or narrative.
A psychological account will not solve this problem because we require a logical account of
when, where and how logical systems apply reality. If Popper's account is taken as purely
psychological then he will not have explained how and why "2 + 2 = 4" taken as logical and a
calculus can determine, or help to determine, what the physical state of affairs is with regard to
apples. The psychological account says only that arithmetic is logical and it can work in the real
world and people have learned to use it. It does not explained which calculi can become
descriptive theories. It does not say why arithmetic can work in the real world; therefore it
cannot explain how people have learned that it can work in the real world. Briefly, arithmetic can
work in the real world but we don't know how, and people have learned that it can work in the
real world but we don't know how they have done that either; however, we do that they have
learned to use it. But this says no more than that people have learned to use arithmetic and this, I
think, we knew already.
A logical reformulation
The apples example can be reformulated as an experiment. Take a basket that contains a pair of
apples and nothing else. Take a bucket that contains a pair of apples and nothing else. Empty the
entire contents of the basket into the bucket taking care to make sure that everything that is in the
basket goes into the bucket. Now how can we determine how many apples are in the bucket?
One way is to use the calculus of arithmetic. We can take the contents of the basket as an
instantiation of the arithmetical notion "2". We can take the contents of the bucket as another
instantiation of the arithmetical notion "2". We can take the act of emptying the entire contents of
the basket into the bucket as an instantiation of the arithmetical notion of "+". Given this we can
describe our experiment arithmetically as "2 + 2". We can use to the calculus of arithmetic to
show "2 + 2 = 4" and from this we can conclude that there are four apples in the bucket. Let us
call this the "calculation method". There is another way to determine the number of apples in the
bucket and this is by counting them. We can take an apple out of the bucket and say "one", then
we can then take another apple out and say "two" and so forth. When there are no more apples
left in the bucket we know we have counted them all. Let us call this the "counting method". The
contention that arithmetic, understood in the constructivist sense, applies to reality is the
contention that the calculation and counting methods will always give the same results.
Popper's mistake was to take "2 + 2 = 4" as being at one time (the time depending on
psychological factors) logically true and at another factually, and therefore contingently, true. A
better account is that "2 + 2 = 4" is always logically true. What is only contingently true is that
objects and events in the world are instantiations of its components: "2", "+", "=", "4".
If two apples are taken as being a contingent instantiation of the arithmetic "2", four apples as
being a contingent instantiation of the arithmetic "4" and emptying the contents of a basket into a
bucket as a contingent instantiation of the arithmetic "+", then the problem is on the way to being
solved. We can say that it is true as a matter of logic that any instantiation of "2" combined with
an instantiation of plus and another instantiation of "2", is an instantiation of "4" while it remains
contingent whether apples are an instantiation.
This can be set up as follows:
Apple System 1
(1) Apples when counted as two are an instantiation of "2 apples". (factual hypothesis).
(2) The apples in the basket have been counted as "2 apples" (factual particular)
(3) The apples in the bucket have been counted as "2 apples" (factual particular)
(4) Emptying a basket into a bucket is an instantiation of "+" for the things in the bucket (factual
hypothesis)
(5) Any instantiation of "2x" combined with an instantiation of "+" and another instantiation of
"2x", is an instantiation of "4x". (definition)
(6) An instantiation of "4 apples" when counted will be counted as four apples. (factual
hypothesis)
Suppose we count the apples in the basket as two, count the apples in the bucket as two, empty
the basket into the bucket and then count the apples in the bucket. Further suppose that the count
results in three apples. Then we could assume that the count has gone wrong somewhere. But we
could repeat the count using other methods of counting. If we are satisfied that our counting is
correct then we might think that (4) is false or we might think that (1) or (6) is false. Whatever
the circumstances we would never have to conclude that (5) was false.
This gives us necessity and falsifiability in all the places where we want it. In fact (4) is false as
it stands, as Popper points out two rabbits plus two more rabbits may produce seven or eight
rabbits. In order to avoid completely abandoning (4) the universe of discourse will need to
exclude rabbits, we could perhaps limit it to inanimate objects. But this limitation placed on the
universe of discourse only effects (4), (1) and (2), it has no effect what so ever on (5) we do not
need to posit a limited universe of discourse for arithmetic. It can be understood as a set of
logical truths that apply to any universe of discourse.
Realist objections
Apple System 1 shows how non-falsifiable statements such as (5) can play a role in our
calculation of quantities in the real world. Unfortunately it does not show that such statements
are necessary for our calculation of real world quantities. This is because all six statements in
Apple System 1 could be replaced by a single factual hypothesis: When the apples in a basket
are counted as "2" and the apples in a bucket are counted as "2" and the contents of the basket
are emptied into the bucket then the contents of the bucket will be counted as "4". At first glance
it might be thought that a non-falsifiable system of arithmetic is necessary in order to extrapolate.
Inductively one would not be able to say that 67 apples in the basket and 95 apples in the bucket
would result in 162 unless one had observed these quantities being put together before. To make
the extrapolation requires the abstract, i.e. definitional, notion of arithmetic.
But this argument does not stand up to a more subtle version of the realism. We could adopt a
similar strategy to that adopted by Field [p 274, 1989] in his version of logicism "What ... is the
value of the search for modal translations (or any other sort of translations of mathematics into
acceptable nominalistic terms)? Why not instead adopt the easier course of simply trying to
translate each of the applications of mathematics?" In order to give the empirical/realist account
we need not say that every statement of arithmetic is induced from observations of real world
quantities. Nor need we say that the system of arithmetic is open to falsification, but is not in fact
never falsified by the observation of real world quantities. All we need to say is that in any
system for the calculation of real world quantities that employs logically true and non-falsifiable
statements of arithmetic these statements can be replaced by a statement or statements that are
not non-falsifiable statements of arithmetic.
This opens the door for the contention that non-falsifiable arithmetic is just a useful but nonessential tool, rather like a typist's shorthand, or that it is a useful fiction. This is a position that is
counter to our intuition and today few would advocate it. Ayer, in what Lakatos [p 30, 1985]
described as logical empiricist orthodoxy, came close to it when he claimed that truths of
mathematics are analytic and a priori, that there can be no a priori knowledge of reality, and that
if a proposition is true a priori it is a tautology. For Ayer "tautologies [such as the propositions of
mathematics], though they may serve to guide us in our empirical search for knowledge, do not
in themselves contain any information about any matter of fact." [1946, p 87].
Gaskin produced an argument that counts against this sort of realist account. Gaskin argues that
an arithmetic formula such as "7 + 5 = 12" cannot mean the same as an empirical proposition
such as would be obtained from counting groups of objects. He argues that in order to explain
mistakes in counting we need to invoke the notion of counting correctly. But Gaskin argues the
meaning of correct counting is dependent on logically true propositions of arithmetic. Therefore,
empirical propositions based on counting do not have equivalent meaning, nor can they be used
as equivalent substitutions for, arithmetical propositions.
"... what is the criterion for correctness in counting? ... "Correctness" has no meaning in this
context, independent of the mathematical proposition. So our suggested analysis of the meaning
of "7 + 5 = 12" runs when suitably expanded: "7 + 5 = 12" means "If you count objects correctly
(i.e. in such a way as to get 12 on adding 7 and 5) you will, on adding 7 to 5, get 12.""[Gaskin,
1940]
If Gaskin's argument were correct this paper could be rapidly brought to a close because it would
show the necessity of logically true statements, such as (5) in Apple System 1, in every system of
applied arithmetic. It would show that the substitution of a single factual hypothesis, such as the
one suggested at the beginning of this section, was inadequate. It would, along with earlier
arguments, establish the main point of the present essay which is that every system, that is
informative about reality, must contain factual particulars, factual universals and logically true
universals.
However, Gaskin's argument is not, as it stands, sufficient to prove the point.
Mistakes in counting
Gaskin's idea that there must some form of logical truth underlying our notion of "correctness" in
the assignment of number, is as I shall argue, quite right. However, he says that the notion of
incorrect counting would be meaningless without mathematical propositions such as "7 + 5 =
12". This suggests that mistakes in counting cannot be identified without an arithmetic calculus
and this is plainly not true. Four types of mistakes in counting can occur:
Case C1. A child counting apples in a bucket says "one apple, two apples, four apples, five
apples" and concludes that there are five apples in the bucket when there are in fact only four. In
this case it is clear that the child has not learned how to count. The mistake can be identified and
corrected by a parent or teacher.
Case C2. A person who has learned to count correctly makes a mistake through inattention. This
mistake can be identified and corrected by subsequent counts by the same person or by other
people. If a second, third and fourth count all agree then we will conclude that the first count was
incorrect.
Case C3. Most people counting by means of saying aloud or in silently soliloquy "one, two,
three" etc. will make mistakes when counting large numbers. These mistakes can be identified
and corrected by other methods of counting. There are many other ways of counting apples: i)
writing the count down by taking an apple out and writing down "1" taking out another and
writing down "2" etc. ii) using a tally board and crossing off "1" then "2" then "3" as the apples
are removed, iii) using a machine, banks have bank note counting machines and, no doubt,
somewhere in some packing factory or cannery there is a machine that counts apples.
In none of these three different ways of identifying mistakes is there any need to use the calculus
of arithmetic.
Gaskin's contention that the notion of correct counting is, on the basis of the arguments so far
considered, rather implausible. The situation is made worse when we consider that people make
mistakes in arithmetic and these mistakes in arithmetic can be identified and corrected by
counting. A person might use addition to determine the sum of a bucket containing seven apples
and a basket containing five. He might come up with the answer "eleven". This mistake could be
identified and corrected by a continuous count of the total. We could reverse Gaskin's argument
and argue that the notion of correctness in arithmetic is meaningless without propositions
resulting from counting.
That some form of logical truth underlining our notion of "correctness" in the assignment of
number requires a more powerful and more general argument than Gaskin's simple counting
example.
PART II
The need for logical truth
A comprehensive account of the distinction between logical and factual truth would involve a
discussion of the terms: a priori, a posteriori, empirical, analytic, synthetic, necessary and
contingent. Such a massive digression into philosophical logic can, for present purposes, be
circumvented if the distinction between logical and factual truth is based on the key terms used
to describe the difference between realism and constructivism. That is, logically true statements
are those that are invented and what follows from them, factually true statements are those that
are discovered to be true and what follows from them.
Following Popper we can say that all factually true universals are open to falsification. Therefore
they are contingent. Logically true statements by contrast are not open to falsification, they are
necessarily true. The relations between the two types of statement can be seen in axiomatic
systems. The axioms, definitions and rules of production are inventions of the person or persons
developing the system and are, therefore, logically true. Any theorems that follow from the
axioms and definitions by means of the rules of production will also be logically true. Factually
true premises can be introduced into an axiomatic system and theorems that follow from axioms
and factual premises by means of the rules of production will inherit the contingency of the
premises and be factually true. The problem is to determine why we need the logical truths.
Axioms and definitions could be replaced by factual premises and factual theorems generated by
the rules of production, and, as was suggested above, the rules of production could themselves be
open to falsification and therefore be factual.
However, a comprehensive system that comprises only factual statements, that is a system that is
not underpinned by any logical statement, is not possible. The later Wittgenstein argued that all
languages are rule based. Rules may change but they not falsifiable. As they are not falsifiable
they have a very similar status to logically true statements.
If an informative system consisted entirely of falsifiable statements, then in the case that two
statements contradicted each other we would not know which had been falsified. Suppose we
take "all swans are white" to be a factual statement. Then this can be falsified by "Donald is a
swan and Donald is white". However, in order to know that Donald is a swan we must have a
criterion for including Donald in the class of swans that is independent of Donald's colour. This
criterion might be "being a water-fowl with a long neck". However, if we are going to say that,
on the basis of Donald being a black water-fowl with a long neck, that "all swans are white" is
false then we have taken "being a water-fowl with a long neck" as being a defining criterion for
swans. That is we will have taken it to be logically true. A fixed pivotal point is needed if we are
going to operate the lever of falsification.
With Donald, the newly discovered black water-fowl with a long neck. The crucial point is that
before you can say "Donald is a swan" or "Donald is not a swan" you must have decided if white
is a logical or a contingent identifying criterion for swans. The need for both factually true and
logically true statements in any informative system can be seen clearly when we consider how
the two forms of definition, intensive and extensive, can be useful.
Intensive and extensive definition
We can use the notions of "conjunction" and "disjunction" to make a distinction between
intensive and extensive definition. An extensive definition, where it consists of more than one
term, will be characterized by the disjunction of the terms. Extensive definition gives the
reference (denotation) of the definiendum. An extension specifies members of a class, in the case
of extensive definition we need to specify all the members of the class i.e. all the extensions.
Where a class F has three members, G, H, I, we can express its extension as (A x) (Fx -> (Gx v
Hx v Ix)). If this class has, as a matter of logic, only these three members we can formulate an
extensive definition:
L (A x) (Fx <-> (Gx v Hx v Ix)).
Extensive definitions can be useful. Given that we have fixed members of a class we can
generate factual hypotheses about them. Suppose we define "cat" in terms of its member species.
E.g. every cat is a lion or a tiger or a leopard or a puma etc. On the basis of this we might
formulate various factual hypotheses, i.e. that only cats have claws and that all cats have sharp
teeth. These could lead to other factual universals e.g. that anything that has claws also has sharp
teeth. Thus, extensive definitions can be instrumental in formulation of factual universals.
An intensive definition, where it consists of more than one term, will be characterized by the
conjunction of the defining terms.
Intensive definitions give the sense (connotation) of the definiendum. An intension will give a
criterion for class inclusion, in the case of intensive definition we need to specify all the criteria.
Where a member of a class J must meet three criteria, K, L, M, we can express these as (A x)
((Kx & Lx & Mx) -> Jx)). If as a matter of logic there only three criteria, we can formulate an
intensive definition: L (A x) ((Kx & Lx & Mx) <-> Jx).
Intensive definitions can be useful. A fixed criteria for class membership will enable us to
identify members of the class. If we define a tiger as a cat with stripes then we can say that if X
is a cat and X has stripes then X is a Tiger. It might also be true as a matter of fact that all wild
Tigers live in Bengal or Assam. In this case if we find an animal that is a cat and has strips and
lives in Africa we will know that it is not wild. Thus, intensive definitions can be useful in the
formulation of particular factual conclusions.
In these examples there has been a logical extension with a factual intension or a logical
intension with a factual extension. Now let us consider the case where a term has a logical
extension and a logical intension. Surely such a formulation is useless. If the extension is fixed
then intension can play no part in helping us identify members of the class, nor is it factual. In
this case, therefore, the intension is useless.
The situation is hardly better where a term has a factual intension and a factual extension. As
neither are fixed both are open to revision. But if one is to be revised it must surely be revised in
the light of the other. We can discover that the a putative intension is false based upon the
extension. Or we can discover that a putative extension is false based upon the intension. But we
cannot make any discoveries about one without taking the other as fixed. If we are to determine
that something is a member of a class there must be some criterion for class inclusion that we use
to make the determination. Alternatively if we are to determine a criterion for class inclusion
then that criterion must be true of all members of the class, therefore, in order to make the
determination we must have identified the members of the class. It can be concluded that any
useful term or class that has a logical extension must have a factual and contingent intension; and
any term or class that has a logical intension must have a factual and contingent extension. An
example will make this clear.
The intention of "a snake" could be "any reptile that does not have legs and does not have
eyelids", an extension could be "any member of the viper family or cobra family or boa family
or colubrid family or hydrophida family". Suppose we adopt this extention as a definition of
"snake", and suppose we give "viper" the intensive definition of "any reptile with retractable
fangs" Then, if we find a reptile that has retractable fangs and eyelids, then we will have
discovered that some snakes have eyelids. We will have discovered that the putative intension of
"snake" as "any reptile that does not have legs and does not have eyelids" is false.
Alternatively we could take the intension of snake as definitional. In this case our discovery of
the reptile with retractable fangs and eyelids would be the discovery of a viper (because of the
intensive definition of "viper") but it would also be a discovery that not all vipers are snakes. The
putative extension of "snake" that included all members of the viper family would have been
discovered to be false.
An important point here is that as things currently stand in the world both the intension and the
extension given above are sufficient for the identification of snakes. This means that for the
practical purpose of identifying a snake we do not have to know, or do not have to decide,
whether it is the intension or the extension that is definitional. It is only when something like the
reptile with retractable fangs and eyelids is discovered that we have to make a decision. These
are situations which offer no precedence. A situation where the existing rules of language will
not provide a decision procedure. They require that a new rule be made but this will not
necessarily be the product of existing rules. There may be a host of psychological factors that go
into the decision but logically it will be arbitrary. In these situation a stipulation is required in
order to proceed. We need to make a stipulative definition.
Take the following:
i) An animal is a snake if and only if it is a reptile that does not have legs and does not have
eyelids.
ii) An animal is a snake if and only if it is a member of the viper family or cobra family or boa
family or colubrid family or hydrophida family.
The players in a language game might assent to both statements without having decided which is
a definition. Both are sufficient identification criteria. They therefore inhabit a logical limbo
which will not be resolved until a particular fact forces the issue. Lets imagine a animal called
"Olga" and the following:
iii) Olga is a reptile without eyelids or legs.
iv) Olga is a snake. From i) and iii) by modus ponens.
As iii) is factual iv) is bound to be factual whether or not i) is factual. However, the truth of iv)
may depend on whether
i) is factual or not.
v) By definition a viper is any reptile with retractable fangs.
vi) Karl is a reptile with retractable fangs and eyelids.
This forces a decision about whether i) or ii) is false. And this decision is not factual, it is solely
about whether the language game player choose to take one or other as a definition. If i) is taken
to be definitional then ii) will be false - it will not be true that all vipers are reptiles. However, if
ii) is taken as definitional then i) will be false and we will have insufficient grounds for asserting
that Karl is a snake. The truth value of particulars is therefore dependent on definitions. Though
the definitions may not have been accepted as definitions as yet. One might say that the truth
value of particulars is dependent on definitions present or future.
Two systems of making a tally
A case can now be constructed to show the relation between the calculus of arithmetic and
systems of counting are of the same order as that between intensions and extensions. However,
the word "counting" will be dropped because this is sometimes and sometimes not, used, like
"knowledge" as a success word. It could be argued, one way or the other, that a correspondence
with the arithmetic calculus is built into the concept of counting. The word "tally" will be used in
its place. The word "tally" will imply nothing more than a system or ritual for producing totals.
Tally System 1
There is a tribe of goat-herds who live in an enclosed valley from which no goat can escape.
Each member of the tribe has a tally stick onto which beads are threaded. When a tribe member
is given a goat, or when one of his goats gives birth, a new bead is threaded on to the owners
tally stick. When one of his goats dies a bead is taken off the owner's tally stick. We can imagine
that in the tribe social prestige and privilege is the determined by the number of goats that a
person owns. Given this the tally system will be useful. It can be determined who has the most
goats by placing different owners' tally sticks side by side.
Tally System 2
In a second tribe the system beads are added as follows:
first goat 0
second goat 00
third goat 0000
fourth goat 00000000
This form of tally system differentiates the social ranking more clearly that in Tally System 1,
therefore one might argue, it is more useful. However, in this system when a goat dies only one
bead is removed from the tally stick. Therefore, we can assume that the number of beads on the
tally stick will not normally correspond to the number of goats that a goat herd owns. The
number of beads on the tally stick will not normally even correspond to the number of goats that
a goat herd has owned. A goat herd who has had four goats and four have died and a goat herd
who has had three goats and none have died will both have 0000 on their tally sticks. But we
need not assume that this system is any less useful that system 1. Perhaps goats require skill to
breed but die largely by accident. It is, we might imagine, quite right that a man who has had
four goats, but been unlucky and lost them all, should be given the same respect as a man who
has only ever had three.
We need not assume that the goat-herds using either Tally System 1 or Tally System 2 have any
knowledge of arithmetic. Nor need we assume that they can count in any way independently of
their tally sticks. Beads are threaded and taken off as part of a public semi-religious ritual.
Everybody in the tribe can agree when this ritual is properly performed. Children are taught the
ritual along with various occult rituals.
The definition of number
The way in which number is defined can now be considered. One possibility is to define number
in terms of arithmetic formulae. An intensive definition of the number "three" can be as follows:
"(1 + 1 + 1) & (1 + 2) & (2 + 1)". Given this logical intension the contingent extension would be
the total returned by a system or ritual that produced a corresponding result in appropriate
circumstances, that is "a total from System 1 or a total from System 2 or a total from System 3
etc.". Any system that produced a total would be a candidate for inclusion. Tally System 1 and
Tally System 2 would both be candidates and, as a matter of fact, totals from Tally System 1
would be part of the extension but those from Tally System 2 would not.
It is fortuitous that the bead threading ritual of the tribe using Tally System 1 corresponds to
numbers defined by arithmetic. Tally System 1 is meaningful quite independently of arithmetic
formulas. However, as such correspondence does exist we are entitled to call it a system of
counting. It is not that a logically true arithmetic is required in order to determine what correct
counting is, as Gaskin suggested, but that a logically true arithmetic will enable us to determine
what systems are to count as counting systems. Given arithmetic formulae as the logical
intension of number it will be a matter of empirical inquiry and discovery which systems and
rituals can be part of the extension i.e. which systems and rituals are counting systems where
counting is defined in terms of the calculus of arithmetic.
As Tally System 1 is contingent with regard to number there is no logical problem in explaining
how it applies to reality (our problem was explaining how logically true number systems could
apply to reality). It can now be explained, which Popper failed to do, why a logically true system
of arithmetic can be useful.
Given that tally systems are contingent with regard to number, the logically true system of
arithmetic is not only useful but essential. This might not be immediately apparent because it
might seem that tally systems can be identified as systems of counting by the fact that the totals
they produce stand in a one-to-one relation with objects in the real world. But it is difficult to see
how "one" can be given meaning independently of some stipulative and logically true definition.
It is only because arithmetic provides such a definition, i.e. "(3 - 2) & (4 - 3) & (5 - 4)" that one
can identify the "one" in a one-to-one correspondence. A second point is why do we have to say
that a counting system must produce one-to-one totals unless we have stipulated this by defining
number in terms of arithmetic. Tally System 2, which by the present account is not a system of
counting, is just as dependent on real world objects as Tally System 1, which is. Tally system 2 is
also in some mathematical relation to real world objects. If number were defined in some other
way it would open up the possibility of a system of "counting" in which the total were not in a
one-to-one relation with real world objects.
This answers the question of "why, if arithmetic is logically true, is it necessary in an account of
real world quantities?". Neither Popper nor Gaskin offered an adequate answer to this question.
However, the arguments used above can also be used to challenge the contention that the
formulae of arithmetic are logically true. This possibility must now be briefly examined.
It is logically tenable to present a case for a definition of number opposite to that given in the
previous section. Number and counting could be defined in terms of Tally System 1. Numbers
could be determined by placing tally sticks upright next to each other. A "one" tally stick is
higher than an empty tally stick but shorter than a "two" tally stick. A "two" tally stick is higher
than a "one" tally stick but shorter than a "three" tally stick. Counting is the act of assigning the
number to a tally stick. Given that numbers are defined in this way it will be contingent and a
matter of empirical discovery that arithmetic formulas correspond to them.
As an account of the historical development of numbers it seems probable that tally systems
existed before arithmetic. In this case originally it must have been that arithmetic was in fact
contingent and a discovery.
CONCLUSION
There are two tenable accounts of number. One is to regard numbers as defined by the
propositions of arithmetic in which case these propositions are, incorrigible and logically true,
while the propositions based on tally systems are falsifiable and factually true. The other is to
regard number as defined by a tally system in which case propositions based on the system will
be incorrigible and logically true while the propositions of arithmetic will be falsifiable and
factually true.
The most plausible historical account is that arithmetic was a series of discoveries based on tally
systems used in different cultures. As arithmetic knowledge spread and the notation for
expressing it because increasingly uniform more people began to regard it as logically true. This
trend continued to the present day when most people would take any tally system and possibly
every tally system as falsifiable rather than regard arithmetic as falsifiable.
Most philosophers of mathematics have assumed that there is only one tenable account of
formulas such as "2 + 2 = 4", they have assumed that they are either logically true or contingent
in an absolute sense. They have assumed that some arguments would be produced that would
show conclusively that they are one or the other. Popper is, I think, the only one to have come up
with the idea that "2 + 2 = 4" can at one time be logically true and at another be factually true,
that the formula can have two different senses.
Popper's mistake was to think that it was the formula "2 + 2 = 4" rather than the number "4" that
could be taken in two sense. As we have seen "4" is in one sense the product of an arithmetic
formula in another sense it is the product of a tally system. Popper also regarded arithmetic as
being logically true. This might be true, as far as most people, as a matter of fact but it is not true
as a matter of logic. As has been shown, there need not be any self contradiction involved in
taking arithmetic as factually true provided the products of a given tally system are taken as
logically true.
Although there are two logically tenable accounts of number, it is a legitimate question to ask
which is more practical. There could be logical problems in defining numbers in terms of more
than one tally system. As things currently stand it would be quite impractical to define numbers
in terms of a single tally system. It would be an enterprise similar to basing linear measurement
on the standard meter in Paris. This was possible for Napoleon but would be difficult to arrange
today.
REFERENCES
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HOFSTADTER, D.R. [1980]: Gödel, Escher, Bach: an Eternal Golden Braid. London: Penguin
Books.
KORNER, S. [1968]: The Philosophy of Mathematics. New York: Dover Publications.
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