Abstract
In Defending the Axioms: On the Philosophical Foundations of Set Theory, Penelope Maddy introduces two methodologically equivalent but philosophically distinct positions, termed Thin Realism and Arealism, which presumably respect set-theoretic practice. Further, Maddy concludes that for her idealized naturalistic inquirer, there is no substantive difference between the two positions. However, I argue that Thin Realism loses its tenability due to the presence of foundational pluralism in broader mathematical practice. In turn, this presents a naturalistic way to undermine Maddy’s conclusion that these are two equally admissible ways of describing the underlying constraints of mathematical practice for the philosophical naturalist.
