Potentialist set theory and the nominalist’s dilemma

Mathematicalnominalists have argued that we can reformulate scientific theories without quantifying over mathematical objects.However, worries about the nature and meaningfulness of these nominalistic reformulations have been raised, like Burgess and Rosen’s dilemma. In this paper, I’ll review (what I take to be) a kind of emerging consensus response to this dilemma: appeal to the idea of different levels of analysis and explanation, with philosophy providing an extra layer of analysis “below” physics, much as physics does below chemistry. I’ll argue that one can address certain lingering worries for this approach by appeal to the apparent usefulness of a distinction between foundational and non-foundational contexts within mathematics and certain (admittedly controversial) arguments for Potentialism about set theory.