Introduction foundations of mathematics

pp. 1-21

in: Yvon Gauthier, Internal logic, Berlin, Springer, 2002

Abstract

Foundational questions in mathematics were born with Hilbert, but foundational programs existed before him. Arithmetization of analysis and arithmetization of algebra (for Kronecker) antidate Hilbert's idea of axiomatization. While Frege was struggling with the logical concept of number as the extension of a concept and while Cantor (and Dedekind) imagined infinite (transfinite) extensions of the ordinary number concept. Kronecker was busy devising a general arithmetic that would arithmetize mathematics without transcending the realm of the algebraic. The so-called foundational crisis did affect only the logicist program, comforting in a sense the arithmetical program. It is that program that Hilbert wanted to pursue with other means in order to rescue set theory from its logico-paradoxical consequences.