The quest for purity : another look at the new Wittgenstein

In his Realism, Mathematics, and Modality, Hartry Field attempted to revitalize the epistemological case against mathematical platontism by challenging mathematical platonists to explain how we could be epistemically reliable with regard to the abstract objects of mathematics. Field suggested that the seeming impossibility of providing such an explanation tends to undermine belief in the existence of abstract mathematical objects regardless of whatever reason we have for believing in their existence. After more than two decades, Field’s explanatory challenge remains among the best available motivations for mathematical nominalism. This paper argues that Field’s explanatory challenge misidentifies the central epistemological problem facing mathematical platonism. Contrary to Field’s suggestion, inexplicability of epistemic reliability does not act as an epistemic defeater. The failure to explain our epistemic reliability with respect to the existence and properties of abstract mathematical objects is simply one aspect of a broader failure to establish that we are epistemically reliable with respect to abstract mathematical objects in the first place. Ultimately, it is this broader failure that is the source of mathematical platonism’s real epistemological problems.