The Identity Problem for Realist Structuralism

Philosophia Mathematica 9 (3):308--330 (2001)

Abstract

According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = − 1 in the group (Z, +). We call this the identity problem and conclude that nominalism is presently the safest route for the structuralist

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References found in this work

Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.
The Philosophy of Mathematics.Wilbur Dyre Hart (ed.) - 1996 - Oxford, England: Oxford University Press.

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Citations of this work

What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
Criteria of Identity and Structuralist Ontology.Hannes Leitgeb & James Ladyman - 2008 - Philosophia Mathematica 16 (3):388-396.
Structuralism and the Notion of Dependence.Øystein Linnebo - 2008 - Philosophical Quarterly 58 (230):59-79.
A Proposal for a Minimalist Ontology.Michael Esfeld - 2020 - Synthese 197 (5):1889-1905.

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