  We re-examine the old question to what extent mathematics may be compared to
a game. Under the spell of Wittgenstein, we propose that the more refined
object of comparison is a "motley of language games", the nature of which was
(implicitly) clarified by Hilbert: via different language games, axiomatization
lies at the basis of both the rigour and the applicability of mathematics. In
the "formalist" game, mathematics resembles chess via a clear conceptual
dictionary. Accepting this resemblance: like positions in chess, mathematical
sentences cannot be true or false; true statements in mathematics are about
sentences, namely that they are theorems (if they are). In principle, the
certainty of mathematics resides in proofs, but to this end, in practice these
must be "surveyable". Hilbert and Wittgenstein proposed almost oppositie
criteria for surveyability; we try to overcome their difference by invoking
computer-verified proofs. The "applied"' language game is based on Hilbert's
axiomatization program for physics (and other scientific disciplines), refined
by Wittgenstein's idea that theorems are yardsticks to which empirical
phenomena may be compared, and further improved by invoking elements of van
Fraassen's constructive empiricism. From this perspective, in an appendix we
also briefly review the varying roles and structures of axioms, definitions,
and proofs in mathematics. Our view is not meant as a philosophy of mathematics
by itself, but as a coat rack analogous to category theory, onto which various
(traditional and new) philosophies of mathematics (such as formalism,
intuitionism, structuralism, deductivism, and the philosophy of mathematical
practice) may be attached and may even peacefully support each other.
