Arithmetic and Reality: A Development of Popper's Ideas

Publishing history: This paper was first published as Working Paper Series No. WP96/01, June 1996. Dept. of Information Systems, City University of Hong Kong. Editor: Dr. Matthew Lee. It was republished in Philosophy of Mathematics Education Journal No. 26 (December 2011).

INTRODUCTION FOR THE INFORMATION SYSTEMS READER
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/contructivist 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.

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 [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 instatiation of the arithmetic "4" and emptying the contents of a basket into a bucket as a contingent instantation 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 instatiation of "2" combined with an instatiation of plus and another intantiation of "2", is an instatiation of "4" while it remains contingent whether apples are an instatiation.

This can be set up as follows:

Apple System 1

(1) Apples when counted as two are an intantiation 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 instatiation of "2x" combined with an instatiation of "+" and another intantiation of "2x", is an instatiation of "4x". (definition)

(6) An intantiation 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 falsifibility 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 non-essential 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 soliloqy "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.

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 contructivism. 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 $$ (\forall x) (Fx \rightarrow (Gx \lor Hx \lor Ix))$$.

If this class has, as a matter of logic, only these three members we can formulate an extensive definition:

$$L (\forall x) (Fx \leftrightarrow (Gx \lor Hx \lor Ix))$$

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 $$ (\forall x) ((Kx \And Lx \And Mx)\rightarrow Jx)$$.

If as a matter of logic there only three criteria, we can formulate an intensive definition:

$$ L (\forall x) ((Kx \And Lx \And Mx)\leftrightarrow Jx)$$

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 extension 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 issometimes and somtimes 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 that 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 fasifiable 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.