Is Mathematics Like a Game?

Abstract We re-examine the old question to what extent mathematics may be compared with a game. Mainly inspired by Hilbert and Wittgenstein, our answer is that mathematics is something like a “rhododendron of language games”, where the rules are inferential. The pure side of mathematics is essentially formalist, where we propose that truth is not carried by theorems correspondencing to whatever independent reality and arrived at through proof, but is defined by correctness of rule-following (and as such is objective given these rules). Gödel’s theorems, which are often seen as a threat to formalist philosophies of mathematics, actually strengthen our concept of truth. The applied side of mathematics arises from two practices: first, the dual nature of axiomatization as taking from heuristic practices like physics and informal mathematics whilst giving proofs and logical analysis; and second, the ability of using the inferential role of theorems to make “surrogative” inferences about natural phenomena. Our framework is pluralist, combining various (non-referential) philosophies of mathematics.