  One of the implications of this view is that the kinds of maxims Maddy sees as purely internal to mathematics, such as MAXIMIZE or UNIFY, are never themselves underlain by philosophical beliefs. UNIFY is a methodological consequence of set theory's foundational ambition of providing ‘a single system in which all objects and structures of mathematics can be modelled or instantiated’ (1997, 208–9). Yet perhaps UNIFY and set theory's foundational ambition—the fact that as Maddy correctly observes, “set theory is (at least partly) designed to provide a foundation for classical mathematics” (2007, 355)—are themselves in some small way, perhaps to some slight degree, underpinned by set-theoretic realism, i.e., the view that set theory is about a single universe of sets. Likewise for the Axiom of Choice, which on the surface owes its place in the set-theoretic canon partly—perhaps only to a small degree—to an ingrained realism about set theory. Several set theorists are on record as making claims to that effect, so the burden of proof is on Maddy to explain away these remarks.