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(Game) Formalism in the Philosophy of Mathematics

One common presentation of the formalism is that it holds that mathematics is not representative of an abstract sector of reality, but rather that is somewhat of a game, being no more wed to an ontology than chess. There is some intuitive support for this, when one considers the novice working on calculations or the application of some method or technique for solving problems. It also refers to the nature of the work done by advanced mathematicians, such as the handling of complex numbers after their introduction, or the opinion of mathematicians on higher levels of set theory. Lastly, it is the position to which many will gesture, when questioned about the nature of mathematics.

I. Introduction to Formalism

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.