Benacerraf's identification problem

In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the development of mathematical structuralism. The identification problem argues that there exists a fundamental problem in reducing natural numbers to pure sets. Since there exists an infinite number of ways of identifying the natural numbers with pure sets, no particular set-theoretic method can be determined as the "true" reduction. Benacerraf infers that any attempt to make such a choice of reduction immediately results in generating a meta-level, set-theoretic falsehood, namely in relation to other elementarily-equivalent set-theories not identical to the one chosen. The identification problem argues that this creates a fundamental problem for Platonism, which maintains that mathematical objects have a real, abstract existence. Benacerraf's dilemma to Platonic set-theory is arguing that the Platonic attempt to identify the "true" reduction of natural numbers to pure sets, as revealing the intrinsic properties of these abstract mathematical objects, is impossible. As a result, the identification problem ultimately argues that the relation of set theory to natural numbers cannot have an ontologically Platonic nature. The historical motivation for the development of Benacerraf's identification problem derives from a fundamental problem of ontology. Since Medieval times, philosophers have argued as to whether the ontology of mathematics contains abstract objects. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that: (1) exists independent of the mind; (2) exists independent of the empirical world; and (3) has eternal, unchangeable properties. Traditional mathematical Platonism maintains that some set of mathematical elements–natural numbers, real numbers, functions, relations, systems–are such abstract objects.