Benacerraf identification problem

The Benacerraf identification problem is a challenge to the Platonist and structuralist views of mathematics, posed by Paul Benacerraf in his 1965 paper "What Numbers Could Not Be." The problem: the set-theoretic reduction of the natural numbers is not unique. Von Neumann defines 2 as {∅, {∅}}; Zermelo defines 2 as Template:∅. Both definitions are adequate — they make the Peano axioms true and enable all standard arithmetic. If numbers are really set-theoretic objects, which sets are they?

The answer cannot be "both, depending on context," because 2 is not the kind of thing that has multiple identities. And it cannot be "the one that is more natural," because the choice between competing set-theoretic reductions is arbitrary. Benacerraf's conclusion: numbers are not objects at all. Mathematical truth is not about referential relations between mathematical terms and independently existing objects. Whatever mathematics is about, it is not a fixed domain of things-in-themselves.

This argument cuts against Mathematical Platonism (which needs the objects to exist independently) and creates the central challenge for Mathematical Structuralism (which must explain what "same structure" means without appealing to object-identity). The empiricist moral: any foundational program that begins by asking "what are mathematical objects?" may be asking a question with no determinate answer. The right question may be structural: what roles do mathematical expressions play in our inferential practices?