Overview

Different philosophical views of the nature of mathematics and its foundations came to a head in the early twentieth century. Among the different schools of thought were the logicism of Gottlob Frege (1848-1925), the formalism of David Hilbert (1862-1943), and the intuitionism of Luitzen Egbertus Jan Brouwer (1881-1966).

Background

Frege, who founded modern mathematical logic in the 1870s, claimed that mathematics is reducible to logic. That is, if all logic were understood perfectly, then all mathematics could be derived from it, or considered part of logic. This view, logicism, was always controversial, but it started important lines of inquiry in philosophy, logic, and mathematics. Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) adopted a weaker version of logicism than Frege's. Ludwig Wittgenstein (1889-...