Truth and objectivity from a verificationist point of view

Abstract

Abstract Truth in mathematics is a very appropriate theme for a conference on what mathematics is about. Questions of truth often work as a catalyst in bringing out contrasts between different positions concerning the nature of a given field. For instance, the question of the applicability of truth is a main issue in discussions on the position known as mathematical formalism, a position that occurs in many versions. Hilbert’s variant of formalism is based on the idea that so-called real sentences are true or false while all other sentences lack truth values. In his chapter in this volume, Dales (1998) seems to advocate a more radical formalism. Challenging the realist conception of mathematics by saying that it takes truth as a key notion but leaves unanswered how truth is established, he takes the merit of formalism to be that it accounts for the nature of mathematics without relying upon any notion of truth. The critics of such a strict formalism typically claim that in the end even the formalist account will depend on some notion of truth. I think that such a criticism can be rightly levelled also against Dales’s formalism. He proposes that a mathematical theorem is to be understood as asserting that a certain formula follows logically from certain axioms, but if fallows logically is understood as follows according to the rules of predicate calculus, then the proposal reduces mathematics to a body of truths about a certain calculus, a certain formal game if you want. This makes mathematics a science that still pursues truths, although truths of a special kind, and we are left with explaining the nature of such truths.