Validity, Necessity and Mathematics

Few mathematicians doubt that the truths of mathematics are independent of the particular matters of fact which constitute the universe we live in. Mathematics is not about the world, it is about abstract ideas. This is reflected in the character of the logical foundations which have been preferred by the main body of mathematicians for most of this century. ZFC (Zermelo-Fraenkel with Choice) and NBG (von-Neumann, Bernays and Gödel), the most popular first order axiomatisations of set theory, are both pure set theories, in which there is no place for anything of the material world. Nothing but sets or classes.

The notion of logical necessity is relevant in this context. A proposition is logically necessary, it is sometimes said, if it is true in every possible world. Technically, to explain necessity in terms of possibility merely establishes a relationship between the two concepts, and without further explanation may leave both adrift. Nevertheless this move seems to me to be of some benefit, and we will shortly discuss how different ideas of what constitutes a possible world determine distinct ideas of logical necessity.

Despite the instincts of mathematicians, and the efforts of philosophers Frege and Russell to establish as a philosophical theory that mathematics is logically necessary, it is now widely held by philosophers that the truths of mathematics are not logically necessary.

To understand the grounds of this view we must consider a concept closely related to logical necessity, that of validity. Validity is a technical term used in giving an account of the semantics of logical systems, most closely associated with first order logic, which specific notion of validity may be called first-order validity. A statement is valid if it is true in all interpretations. We are naturally tempted to connect the idea of an interpretation of a first order language with the idea of a possible world, thinking of the one as a precise idea corresponding to an informal and vague idea.

We will succumb to this temptation for a while, and explain briefly how this leads us to the conclusion that mathematics cannot consist of necessary truths.

For a proposition to be first-order valid it is necessary that it be true under all interpretations. An interpretation consist of an identified domain of discoursewhich is subject only to the constraint that it be non-empty, together with n-ary relations and n-ary functions corresponding to each of the n-ary relation and function symbols employed in the proposition.

Considering the language of arithmetic alone, for the true p

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