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Benacerraf's identification problem
Benacerraf's identification problem is a philosophical argument, developed by Paul Benacerraf, against set-theoretic Platonism.〔Paul Benacerraf (1965), “What Numbers Could Not Be”, ''Philosophical Review'' Vol. 74, pp. 47–73.〕 In 1965, Benacerraf published a paradigm changing article entitled "What Numbers Could Not Be".〔〔Bob Hale and Crispin Wright (2002) ("Benacerraf's Dilemma Revisited" ) ''European Journal of Philosophy'', Issue 10:1.〕 Historically, the work became a significant catalyst in motivating the development of structuralism in the philosophy of mathematics.〔Stewart Shapiro (1997) ''Philosophy of Mathematics: Structure and Ontology'' New York: Oxford University Press, p. 37. ISBN 0195139305〕
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.〔
== Historical motivations ==
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.〔Michael Loux (2006) Metaphysics: A Contemporary Introduction (Routledge Contemporary Introductions to Philosophy), London: Routledge. ISBN 0415401348〕 Traditional mathematical Platonism maintains that some set of mathematical elements–natural numbers, real numbers, functions, relations, systems–are such abstract objects. Contrarily, mathematical nominalism denies the existence of any such abstract objects in the ontology of mathematics.
In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included intuitionism, formalism, and predicativism. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for the identification problem developed.
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