Formalism

Formalism is a philosophical theory of the foundations of mathematics that had a spectacular but brief heyday in the 1920s. After a long preparation in the work of several mathematicians and philosophers, it was brought to its mature form and prominence by David Hilbert and co-workers as an answer to both the uncertainties created by antinomies at the basis of mathematics and the criticisms of traditional mathematics posed by intuitionism. In this prominent form it was decisively refuted by Gödel’s incompleteness theorems, but aspects of its methods and outlook survived and have come to inform the mathematical mainstream. This article traces the gradual assembly of its components and its rapid downfall.