The ante rem structuralist holds that places in ante rem structures are objects with determinate identity conditions, but he cannot justify this view by providing places with criteria of identity. The latest response to this problem holds that no criteria of identity are required because mathematical practice presupposes a primitive identity relation. This paper criticizes this appeal to mathematical practice. Ante rem structuralism interprets mathematics within the theory of universals, holding that mathematical objects are places in universals. The identity problem should be read as challenging this claim about universals. However, what mathematical practice presupposes is only relevant to what is true according to the theory of universals, if one takes it for granted such a theory offers the best interpretation of mathematics. In the current context, taking this for granted begs the question. Therefore, the appeal to mathematical practice in response to the identity problem is illegitimate.
Ante REM Structuralism Identity problem Primitive identity Mathematical practice
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Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 47, 43–47.Google Scholar
Burgess, J. P. (1999). Review of Stewart Shapiro, philosophy of mathematics: Structure and ontology. Notre Dame Journal of Formal Logic, 40(2), 283–291.CrossRefGoogle Scholar
Button, T. (2006). Realistic structuralism’s identity crisis: A hybrid solution. Analysis, 66(3), 216–222.CrossRefGoogle Scholar