Benacerraf's Identification Problem

Benacerraf's identification problem, set out by Paul Benacerraf in his 1965 paper "What Numbers Could Not Be," is an argument against the view that numbers simply are particular sets, a form of mathematical Platonism sometimes called set-theoretic reductionism. Benacerraf observes that there are multiple equally adequate ways of defining the natural numbers as sets, such as the systems proposed by Ernst Zermelo and John von Neumann, and that nothing in mathematical practice picks out one of these systems as revealing what a number truly is rather than merely providing a structure that behaves like the numbers. Since no principled way exists to say which reduction is correct, Benacerraf concludes that numbers cannot simply be identical to any particular sets, and more broadly that questions about what mathematical objects intrinsically are, apart from the structural relations they bear to one another, may not have determinate answers, a conclusion that helped motivate mathematical structuralism as an alternative to Platonism.