Poincaré on logic and foundations

Abstract

Poincaré's intervention in the philosophical debate on mathematics of the beginning of the twentieth century is circumscribed but had lasting effects. The important aspects considered here are his defence of intuition against all forms of logicism and his fierce condemnation of the use of symbolic languages; the criticism of the vicious circle as responsible for all the antinomies, and the influence both on Russell, who was led to adopt the theory of types, and on Weyl, who tried a predicativistic reconstruction of mathematical analysis; the fight with Hilbert for the circular use of induction of the proof of the consistency of arithmetic; his discussions with Peano and Zermelo on the Cantor-Bernstein theorem and on predicative proofs in general.