Structuralism in the Philosophy of Mathematics

Structuralism holds that mathematics is not fundamentally about a fixed domain of individual objects, such as specific numbers or specific sets, but about abstract structures or patterns, and that what matters about any mathematical object is solely its relations to other objects within the structure it occupies, not any further "intrinsic" identity or nature it might otherwise be thought to have. Paul Benacerraf's 1965 paper "What Numbers Could Not Be" (Philosophical Review) supplied the classic motivating argument: since natural numbers can be modeled equally well by Ernst Zermelo's construction, where 0 is the empty set and each successor is the singleton of its predecessor, or by John von Neumann's construction, where each number is the set of all smaller numbers, and no principled mathematical or metaphysical fact settles which construction the numbers "really are," Benacerraf concluded that numbers are not objects with any intrinsic identity at all, but merely positions in the natural-number structure, the abstract pattern common to any such sequence.