Mathematical pluralism

There is a plurality of mathematical investigations. These cannot all be reduced to proofs within the framework of Zermelo Fraenkel set theory, if only because some of them use non-classical logic (such as the various branches of intuitionist mathematics). How is one to understand this situation? In this article, I suggest that one should see this plurality as analogous to the plurality of games, any of which may be played. Various objections are considered and rejected, including the charge that the picture engenders a pernicious relativism.