Is mathematics a game?

Abstract: We re-examine the old question to what extent mathematics may be compared with a game, realizing that a "motley of language games" is the more refined object of comparison. Our analysis owes at least as much to (late) Wittgenstein, updated by a Brandomian emphasis that the rules governing language games be inferential, as to Hilbert, supplemented with some empiricism taken from van Fraassen. Our "motley" provides a coat rack onto which various philosophies of mathematics may be attached and may even peacefully support each other. Pure mathematics is a language game incorporating aspects of formalism and structuralism, modified however by a different notion of truth. Applied mathematics corresponds to a different language game in which mathematical theories provide generalized yardsticks that "measure" (as opposed to: represent) natural phenomena. Thus our framework is non-referential for both pure and applied mathematics. The certainty of pure mathematics in principle resides in proofs, but in practice these must be surveyable'. Hilbert and Wittgenstein proposed almost opposite criteria for surveyability; we try to overcome their difference.