A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. Hilbert aimed to show the consistency of mathematical systems from the assumption that the “finitary arithmetic” (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent (i.e. no contradictions can be derived from the system).

Godel’s conclusion in his incompleteness theorems was that you cannot prove consistency within any axiomatic system rich enough to include classical arithmetic.

— Wikipedia on Formalism (mathematics)

2012.04.05 Thursday ACHK