ABSTRACT
A succinct characterization of the family of Platonist and nominalist positions is given. Nominalist motivations are described: epistemic and metaphysical ones. That objects that exist should be observable, or accessible epistemically in some way; that objects that exist should be concreta, or in space and time, or whatnot. A Platonist motivation is described – the indispensability thesis: that the application of mathematics to the empirical sciences requires the truth of said mathematics and that these indispensable truths in turn are committed to various apparent mathematical objects by, for example, Quine's criterion. Nominalist joinder-strategies to the indispensability thesis are described: finding nominalist ersatz that can play the semantic/epistemic role that Platonist objects play, in the space-time continuum itself, for example; rejiggering the logic taken to be presupposed in play so that the apparent commitments of mathematical truths to mathematical objects vanish. The current state of these various philosophical programs is evaluated.
