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The Oxford Handbook of Philosophy of Mathematics and Logic$
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Stewart Shapiro

Print publication date: 2005

Print ISBN-13: 9780195148770

Published to Oxford Scholarship Online: July 2005

DOI: 10.1093/0195148770.001.0001

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Structuralism Reconsidered

Structuralism Reconsidered

(p.563) Structuralism Reconsidered
The Oxford Handbook of Philosophy of Mathematics and Logic

Fraser MacBride

Oxford University Press

The basic relations and functions that mathematicians use to identify mathematical objects fail to settle whether mathematical objects of one kind are identical to or distinct from objects of an

apparently different kind, and what, if any, intrinsic properties mathematical objects possess. According to one influential interpretation of mathematical discourse, this is because the objects under study are themselves incomplete; they are positions or akin to positions in patterns or structures. Two versions of this idea are examined (Resnik, Shapiro). It is argued that the evidence adduced in favor of the incompleteness of mathematical objects underdetermines whether it is the objects themselves or our knowledge of them that is incomplete. Also, holding that mathematical objects are incomplete conflicts with the practice of mathematics. The objection that structuralism is committed to the identity of indiscernibles is evaluated and it is also argued that the identification of objects with positions is metaphysically suspect.

Keywords:   mathematical objects, identity, positions, structures, Resnik, Shapiro, incompleteness, structuralism, identity of indiscernibles

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