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.
Type
Thesis, MPhil Master of Philosophy
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