Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme

Abstract

Kronecker called his programme of arithmetization “General Arithmetic” (Allgemeine Arithmetik). In his view, arithmetic is the building block of the whole edifice of mathematics. The aim of this paper is to show that Kronecker’s arithmetical philosophy and mathematical practice have exerted a permanent influence on a long tradition of mathematicians from Hilbert to Weil, Grothendieck and Langlands. The conclusion hints at a constructivist finitist stance in contemporary mathematical logic, especially proof theory, beyond Hilbert’s programme of finitist foundations which can be seen as the continuation of Kronecker’s arithmetization programme by metamathematical or logical means. It is finally argued that the introduction of higher-degree polynomials by Kronecker inspired Hilbert’s notion of functionals, which in turn influenced G¨odel’s functional Dialectica interpretation for his intuitionistic proof of the consistency of arithmetic.