Philosophia Mathematica 16 (3):285-309 (2008)

Authors
Stewart Shapiro
Ohio State University
Abstract
Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike    Connotea    Del.icio.us    What's this?
Keywords structuralism   identity
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Reprint years 2007, 2008
DOI 10.1093/philmat/nkm042
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References found in this work BETA

On Denoting.Bertrand Russell - 1905 - Mind 14 (56):479-493.
Mathematics as a Science of Patterns.Michael D. Resnik - 1997 - New York ;Oxford University Press.
Methods of Logic.W. V. O. Quine - 1950 - Harvard University Press.

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Citations of this work BETA

What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
Is Identity Really so Fundamental?Décio Krause & Jonas R. Becker Arenhart - 2019 - Foundations of Science 24 (1):51-71.
Speaking of Essence.Alessandro Torza - 2015 - Philosophical Quarterly:754-771.
How to Be a Minimalist About Sets.Luca Incurvati - 2012 - Philosophical Studies 159 (1):69-87.

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