Kronecker, the semi-intuitionists, Poincaré

Abstract

Modern foundational research got its full start with Cantor’s publications on set theory from 1874 onwards. The tradition which developed in this style was to include as its contributors some of the most famous mathematicians of that time. With the publication of Was sind and was sollen die Zahlen? (‘What are numbers and what should they be?’) in 1887, Dedekind was considered to have given a secure foundation for the theory of natural numbers, based on the laws of logic. Hilbert gave a strictly axiomatic foundation to geometry in his Grundlagen der Geometrie (‘Foundations of geometry’) in 1899, and later extended this to his general proof theory. Zermelo axiomatised set theory. It was a tradition of rigorous proofs with an appeal to logic rather than to intuition. And it was against this tradition that several mathematicians protested, who, by so doing, held views more or less similar to the ones Brouwer was to take later on. This chapter is about these mathematicians. In chronological order, they are: Kronecker, the French semi-intuitionists Borel, Baire and Lebesgue, and Poincaré.