CONTENTS

1   INTRODUCTION

2   SCIENCE OF SENSE

3   APPLIED MATHEMATICS

4   AXIOMS AS DEFINITIONS

5   ALTERNATIVE THEORIES

6   CONCLUSIONS

4   AXIOMS AS DEFINITIONS

As discussed above, the SS thesis is a development of the Hilbert account of pure mathematics. To explore this linkage further it is useful to quote more of the exposition of Hilbert's approach given by Kneale and Kneale. They trace the connection between Frege's polemic against formalism and the dispute between Hilbert and Brouwer.

"Sometimes Hilbert's philosophy of mathematics is called formalism, and just as nineteenth-century formalists were attacked by Frege, so Hilbert is attacked by Brouwer for failure to appreciate the real content of mathematics. Frege's opponents maintained that new kinds of numbers could be introduced by creative definitions, and that if the definitions led to no self-contradictions it was pointless to ask for any further proof of the existence of the numbers. To this Frege replied that there could be no proof of freedom from contradiction except by the production of an example, and that the converse inference was a fallacy. Hilbert now argues that the consistency of a postulate system may be established without the production of a model. To this Brouwer replies that intuitionism does not assume an inconsistency in the application of the principle of excluded middle to infinite sets, and that even if Hilbert succeeded in his enterprise he would have done nothing to prove the truth of his 'ideal' statements, because there is a vicious circle in the attempt to justify formalist mathematics through a proof of its consistency. We are asked to accept the correctness of a system of postulates when it has been shown that they do not lead to self-contradiction; but according to Brouwer this justification is of no value unless we can already accept the principle that the correctness of a proposition follows from the absurdity of its absurdity, and that is just another version of the law of excluded middle.

... This is a very curious dispute in which the parties seem to be arguing at cross purposes. Brouwer talks as though Hilbert were trying to prove the truth of his axioms, and so of what follows from them, by proving that they are self-consistent. But when Hilbert undertakes to axiomatize all of mathematics he presumably thinks of his axioms as postulates which determine the sense [my italics] of the otherwise undefined symbols which he uses".

The presumption Kneale and Kneale make about Hilbert effectively attributes to him a belief in the SS thesis. The answer to Brouwer and Frege begins by conceding that postulation by itself can not assure reference in general, and truth in particular, merely by virtue of self-consistency. On the other hand, sense is precisely the thing that can be so assured. (One could even say that inconsistent postulates determine a sense, but by virtue of the inconsistency it is an analytically non-referential sense.) The parties are at cross-purposes because Frege and Brouwer, in their different ways, see mathematics as a Science of Necessary Reference, while Hilbert sees it as a Science of Sense.

To establish the SS thesis as a viable candidate for being a theory of mathematics, we have show that stipulation, by means of axioms, can do what we claim for it, even at the level of sense. This is a deep problem, and here only the beginnings of an approach can be sketched. The intuitive picture of what stipulation does can be presented informally as follows.

NORMAL USE OF LANGUAGE - starting from the senses of the components of a sentence we build up the sense of the whole, and compare the result with what goes on the the world to determine whether the sentence is true or not.

STIPULATION - starting from the stipulation that the sentence is true, we run the machinery in reverse to determine the sense of one of the components of the sentence.

For use in the SS context, this account has to be modified. The appearance of the notion of truth in the account of stipulation takes us wrongly into the realm of reference. What we stipulate is the analyticity of the thought. This defines the sense of the component which previously lacked one. If it, and the other components, have reference then the stipulated analyticity is turned into a stipulated truth. Even with this modification, the fundamental problem remains: can the machinery of sense be run in reverse as claimed, and if so, does it yield the advertised result?

The first step is to acknowledge that there are limits to what can be done with stipulation. To take a simple example, suppose we stipulated that "Q is the largest natural number", where the sense of "x is a natural number" has previously been established using the Peano axioms. The sense of "Q" thus established is non-referential, and since it follows directly from the Peano axioms that there is no largest natural number, it is analytically non-referential. The sentence might still be said to be analytic, by stipulation, but it is analytic in a way which will never allow it to be true.

A classic example of the limitations of stipulation is Prior's "tonk". (This is different in that the stipulation is of the validity of an argument rather than the analyticity of a single sentence, but the difference is not so great, given that analyticity of P can be seen as the validity of the argument, "P", with no premisses and P as the conclusion.) In his paper on tonk, Belnap concludes by saying,

"one can define connectives in terms of deducibility, but one bears the onus of proving at least consistency (existence); and if one wishes further to talk about the connective, it is necessary to prove uniqueness as well."

This takes us back to Hilbert's point that the one constraint on stipulation by axioms is the requirement of self-consistency.

To look more closely at the stipulation process, let us look at two examples. As a simple example, take first the partial ordering axioms:

One might say that these constitute the definition of the new symbol " ³", in terms of all the others in the axioms, which we have already from first order logic with identity.

We are not, however, justified in talking about the relation " ³". Perhaps we can prove the existence condition, which while we are doing the Science of Sense, means consistency, but we are short of uniqueness. We know that our discourse is full of quite different partial orderings. Instead we should extract the symbol " ³" from the axioms, leaving behind a relational expression argument place. The conjunction of the three axioms is then a 1S2SNN predicate, which in regimented English would be called:

"to say F(x,y) is to say x and y are related by a partial ordering".

This predicate is true of all the relations we happen to come across which are partial orderings. Written out explicitly this expression is:

We are no longer studying a relation, whose properties have been defined by stipulating the axioms. Instead we have shifted our attention one level up the Fregean hierarchy and have singled out for study an interesting property which may be true of some relations (which is in turn the conjunction of three simpler 1S2NN properties: reflexivity, antisymmetry and transitivity.) In doing so we have also replaced a contextual definition by a direct one.

Another consequence of this move is that not only do we attribute to PO a sense in a direct way, we also guarantee its reference (relative to the thesis that the sentential connective, the quantifiers and the identity relation necessarily refer). We have stripped out all non-logical ingredients, and are left only with what is logical. This transition to a necessary reference is discussed further in Section 5.1 below.

As a second, more complicated, example, let us consider the Peano axioms in the form given by Tennant:

The partial ordering axioms are particularly simple because only one new symbol is introduced. The Peano axioms in this form contain two new symbols, the name "0" and the functional expression "s(x)". If we regard the axioms as defining them both, then the definitions are intertwined.

Entangled definitions can, however, be seen as defining relations (of a suitably high level) by the removal of both items. They are entangled when seen as being contextual definitions of the undefined terms, but they become simple when the terms are removed leaving behind an incomplete expression with the appropriate number and type of argument places. The Peano axioms become an explicit definition of the 2S1NNN expression:

Here the third conjunct is the first example of the axiom schema, and the "...." indicates the extension to all the other examples, treated as additional conjuncts. The argument places are x for the name and f for the function. In regimented English, this expression could be written as:

"the function mapping x onto f(x) is a successor function of which x is the first member" .

In making this move towards incomplete expressions (Quine's "semantic ascent") we have come along way from the original idea of defining entities by the stipulation that certain sentences referring to them are true. The Peano axioms have been introduced here in a form which says nothing explicit about natural number, but Peano's initial intention was to use them to provide a complete description of what is meant by natural number. Instead we end up here studying a higher order relation which is found as a feature of the natural numbers, but could also be found elsewhere. Frege's polemic against formalism was based on the idea that we should start from reference (in the case of mathematics, a necessary reference) and then derive what properties are true of the entities, rather than collecting together some properties and stipulating that the entities we want are those of which the properties are true. Here we have moved to shift the focus from the entities to the properties themselves, defining their sense, and, if all non-logical items have been removed and if the logical items are seen as having guaranteed reference, guaranteeing them a reference.

If there is only one argument place in the incomplete expression, then we can use its sense, in conjunction with the definite description operator of the appropriate level, to define the sense of the expression referring, if it refers at all, to the (existing and unique) bearer of the property. (This cannot be done with the multiple-place expressions derived from entangled definitions.) This only works at the level of sense - the definite description operator is a sense-function not a reference-function. In as much as Frege was looking to mathematics to be a science of part of reference, his polemic against the formalists was entirely justified. Within the realm of sense, a realm of which Frege himself was the discoverer, we can make sense of what the formalists were trying to do.

Another question about stipulation is the extent to which purely existential axioms (those where the leftmost quantifier is existential, and which can not be true in an empty universe) can be a part of stipulation. Dummett hints at a problem here, by saying the following about stipulation

" In number theory, for example, the recursion equations for addition and multiplication may be regarded as stipulating the meanings of these function symbols; but the third and fourth Peano axioms cannot be viewed as merely part of a stipulation of the meaning of 'natural number', since they require the prior assumption of the existence of an infinite totality."

It is not directly clear from this whether Dummett sees the problem with an existential assumption per se or with the assumption of the existence of something infinite. Let us concentrate on the former alternative. The way the Peano axioms were introduced above avoided saying that 0 exists. But if we wanted to turn the two-place 2S2SNNN expression into a one-place 1S2SNN expression which simply refers to the property of a relation that it is a successor relation, then the obvious way to do so is with an existential claim:

The question is whether it is legitimate to make the sense of an expression dependent on a claim about the existence of an object. The intuitive source of concern is that this mixes in an intrinsically referential statement with something which should be done entirely within the realm of sense.

Click here to go to Section 5