- 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.
- 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.
- 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.
- Mathematical objects enjoy relations to
This makes it appear that ante rem structuralism is committed to the absurdity of identifying these objects.
- (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.
- 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.
- 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.
- 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.
- What is (Ante Rem) Structuralism?
- Cross-Structural Identity
- Identity and Indiscernibility
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