Formalism and Hilbert's Program

Formalism treats mathematics as, at bottom, a system of formal symbol manipulation according to explicit, stipulated rules, without reference to any independent mathematical reality: on a strict game formalist reading, mathematical statements are true or false only relative to a chosen axiomatic system, much as a chess move is "correct" only relative to chess's rules, and the symbols themselves carry no meaning outside the game. David Hilbert, the movement's towering figure, did not intend to strip mathematics of all content; rather, he sought to secure the whole of classical mathematics, including Georg Cantor's transfinite set theory, against the strictures L.E.J. Brouwer's intuitionism threatened to impose, by finding an uncontroversial foundation acceptable even to a skeptic.