Criteria of identity and the hermeneutic goal of ante rem structuralism

Synthese 195 (5):2141-2153 (2018)
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Abstract

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.

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Citations of this work

Collective Abstraction.Jon Litland - forthcoming - Philosophical Review.

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References found in this work

What Numbers Could Not Be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
Mathematics as a Science of Patterns.Michael David Resnik - 1997 - Oxford, England: New York ;Oxford University Press.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press USA.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.

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