Poincaré against foundationalists old and new

Abstract

The early 20th century witnessed concerted research in foundationalism in mathematics. Those pursuing a basis for mathematics included Hilbert, Russell, Zermelo, Frege, and Dedekind. They found a vocal opponent in Poincaré, whose attacks were numerous, vituperative, and often indiscriminate. One of the objections was the petitio argument that claimed a circularity in foundationalist arguments. Any derivation of mathematical axioms from a supposedly simpler system would employ induction, one of the very axioms purportedly derived. Historically, these attacks became somewhat moot as both Frege and Hilbert had their programs devastated-Frege's by Russell's paradox and Hilbert's by Godel's incompleteness result. However, the publication of Frege's Conception of Numbers as Objects by Crispin Wright began the neo-logicist program of reviving Frege's project while avoiding Russell's paradox. The neo-logicist holds that Frege's theorem-the derivation of mathematical axioms from Hume's Principle(HP) and second-order logic-combined with the transparency of logic and the analyticity of HP guarantees knowledge of numbers. Moreover, the neo-logicist conception of language and reality as inextricably intertwined guarantees the objective existence of numbers. In this context, whether or not a revived version of the petitio objection can be made against the revived logicist project. The current project investigates Poincaré's philosophy of arithmetic-his psychologism, conception of intuition, and understanding of induction, and then evaluates the effectiveness of his petitio objection against three foundationalist groups: Hilbert's early and late programs, the logicists, and the neo-logicists.