Poincaré’s Theory of Predicativity

Abstract

Poincaré’s theory of predicativity is a central and exciting component of his general philosophical position. As is well known, his philosophy of mathematics was foundational for intuitionism. It is also well known that he was concerned about the set-theoretic paradoxes, and that he was one of the first to write about the ‘Vicious Circle Principle’ (VCP). Just what constitutes Poincaré’s version of the VCP, the theory of predicativity which underlies it, and his contribution to the solution of the contradictions of classical mathematics, is much more obscure. To be sure, his work in this area ought to be regarded as ancestrally related to modem programmes in predicative analysis and predicative set theory. However, just as it is wrong to consider a modern formalised intuitionism as a natural extension of his general philosophical views, so is it a mistake to consider a predicative version of axiomatised set theory as a programme he would have unequivocally endorsed. In fact, in view of the formality of both of these programmes, he probably would have opposed them.