Structure and Identity
Shapiro (Stewart)
Source: MacBride - Identity and Modality, 2006, Chapter 5
Paper - Abstract

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Editor’s Introduction1

  1. In his 'Structure and Identity' Stewart Shapiro reflects upon the doctrine (advanced in his Philosophy of Mathematics: Structure and Ontology (Oxford: OUP, 1997)) that mathematical objects are places in structures where the latter are conceived as ante rem universals2.
  2. This doctrine — that Shapiro dubs 'ante rem structuralism' — suggests that there is no more to a mathematical object than the (structural) relations it bears to the other objects within the structure to which it belongs.
  3. However, as Shapiro recognizes, when conceived in this way ante rem structuralism is open to a variety of criticisms. This is because there appears to be more to a mathematical object than the relations it bears to other objects within its parent structure.
  4. Mathematical objects enjoy relations to
    • (i) items outside the mathematical realm (e.g. the concrete objects they are used to measure or count) and
    • (ii) objects that belong to other structures inside the mathematical realm.
    • Moreover, (iii) there are mathematical objects (e.g. points in a Euclidean plane) that are indiscernible with respect to their (structural) relations but nevertheless distinct.
    This makes it appear that ante rem structuralism is committed to the absurdity of identifying these objects.
  5. Shapiro seeks to overcome these difficulties by a series of interlocking manoeuvres.
    • First, he seeks to overturn the metaphysical tradition about numbers, suggesting that it may be contingent whether a given mathematical object is abstract or concrete.
    • Second, Shapiro questions whether mathematical discourse is semantically determinate.
    • Finally, Shapiro rejects the requirement that ante rem structuralism provide for the non-trivial individuation3 of mathematical objects.

Author’s Abstract
  1. The purpose of this paper is to further articulate my preferred version of mathematical structuralism … ante rem structuralism, the thesis that mathematical structures exist prior to, and independent of, any exemplifications they may have in the non-mathematical world.
  2. The contrast is with an approach that either adopts an Aristotelian in re view that a given structure exists only in the systems that exemplify it or the more common eliminative thesis that structures do not exist at all – talk of structures is to be paraphrased away.

  1. What is (Ante Rem) Structuralism?
  2. Cross-Structural Identity
  3. Identity and Indiscernibility

In-Page Footnotes

Footnote 1:

Text Colour Conventions (see disclaimer)

  1. Blue: Text by me; © Theo Todman, 2019
  2. Mauve: Text by correspondent(s) or other author(s); © the author(s)

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