The premiere text on game formalism is not a defense made by an ardent supporter, but rather an attempt to refute it, written by Frege, where he lays into Heine and Thomae. Philosophers of mathematics claim that the position is still held by mathematicians, however, formalism in the sense that Frege set out to dispute, should be distinguished from what some argue is the more refined formalism of Hilbert, and his program.

The mathematician David Hilbert was promoting a formalized mathematics, that treats a number with an infinite amount of digits as a real physical object.

While acceptable in modern times, the advent of infinite numbers after decimal points was not warmly received when the idea was first put forth. David Hilbert was in support of the idea, where as LEJ Brouwer felt Mathematics was a construct, and that numbers are finite, their digits being the result of calculation or random selection, one at a time.

These papers, with their mathematical precision, had been preceded by a paper by Hilbert, Neumann and Nordheim.