Logicism

In philosophy of mathematics, logicism is a school of thought comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

Overview

Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.

The philosophical impetus behind Frege's logicist programme from Die Grundlagen der Arithmetik onward was in part his dissatisfaction with the epistemological and ontological commitments of then-extant accounts of the natural numbers, and his conviction that Kant's use of truths about the natural numbers as examples of synthetic a priori truth was incorrect.

This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor's 1896, Zermelo and Russell's 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an inconsistency in Frege's system set out in Grundgesetze der Arithmetik.