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10 - Foundations of mathematics

from 3 - Logic, mathematics, and judgement

Published online by Cambridge University Press:  28 March 2008

By
Michael Hallett
Edited by
Thomas Baldwin
Michael Hallett
Affiliation:
McGill University
Thomas Baldwin
Affiliation:
University of York
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Summary

INTRODUCTION

It is uncontroversial to say that the period in question saw more important changes in the philosophy of mathematics than any previous period of similar length in the history of philosophy. Above all, it is in this period that the study of the foundations of mathematics became partly a mathematical investigation itself. So rich a period is it, that this survey article is only the merest sketch; inevitably, some subjects and figures will be inadequately treated (the most notable omission being discussion of Peano and the Italian schools of geometry and logic). Of prime importance in understanding the period are the changes in mathematics itself that the nineteenth century brought, for much foundational work is a reaction to these, resulting either in an expansion of the philosophical horizon to incorporate and systematise these changes, or in articulated opposition. What, in broad outline, were the changes?

First, traditional subjects were treated in entirely new ways. This applies to arithmetic, the theory of real and complex numbers and functions, algebra, and geometry. (a) Some central concepts were characterised differently, or properly characterised for the first time, for example, from analysis, those of continuity (Weierstrass, Cantor, Dedekind) and integrability (Jordan, Lebesgue, Young), from geometry, that of congruence (Pasch, Hilbert), and geometry itself was recast as a purely synthetic theory (von Staudt, Pasch, Hilbert). (b) Theories were treated in entirely new ways, for example, as axiomatic systems (Pasch, Peano and the Italian School, Hilbert), as structures (Dedekind, Hilbert), or with entirely different primitives (Riemann, Cantor, Frege, Russell).

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The Cambridge History of Philosophy 1870–1945 , pp. 128 - 156
Publisher: Cambridge University Press
Print publication year: 2003

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