The Structural Debt of Nominalism: Why Azzouni’s use of PFL fails to eliminate mathematical commitment.
I’ve been diving into Jody Azzouni’s "Deflationary Nominalism," specifically his use of Predicate Functor Logic (PFL) to dodge ontological commitment to mathematical objects. The idea is that by stripping away variables/quantifiers, we can do science without "committing" to the existence of the objects the math describes.
However, I think there is a massive structural flaw here that often gets overlooked: The "Variable-Free" shell game.
Azzouni argues that PFL allows us to avoid "objects," but he fails to account for the fact that the Functors themselves (the operators) are embedded with the very relations he’s trying to deflate. To even run a PFL system, you have to presuppose the Type-Theoretic relations of distinction, identity, reflexivity, composition, and transitivity.
Even if you adopt quantifier variance or a deflationary theory of truth (where truth is just "warranted assertibility"), you are still trapped. If "truth" is grounded in logical implication, and logical implication is a structural/type-theoretic relation, then you haven't eliminated the math; you’ve just moved it from the "nouns" (variables) into the "verbs" (functors).
You can't have a "variable-free" logic if your operators rely on the rigid, non-negotiable architecture of Type Formation. Mathematically, logic is just a shadow cast by these deeper structural relations. Azzouni wants to have the "pragmatic cash value" of the assertion without paying the "structural debt" of the relations that make the assertion possible.
Essentially, nominalism in this form isn't an elimination; it’s just a rebranding of structural realism. Thoughts?