- In 'The Identity Problem for Realist Stucturalism II: A Reply to Shapiro' Jukka Keranen argues that Shapiro2 nevertheless fails to provide an adequate account of the identity of numbers conceived as places in structures.
- According to Keranen, it is an adequacy constraint upon the introduction of a type of object that some account be given of the kinds of fact that metaphysically underwrite the sameness and difference of objects of this type. More specifically, Keranen favours the view that facts about the sameness and difference of objects must be underwritten by facts about the properties they possess or relations they stand in.
- He holds up set theory as an exemplar of a theory that meets this adequacy constraint, grounding the identity of sets — via the Axiom of Extensionality — in facts about their members.
- Keranen doubts, however, whether ante rem structuralism can meet this adequacy constraint because there are no structural properties or relations that can be used to distinguish between (e.g.) the structurally indiscernible points in a Euclidean plane.
- Of course, the structuralist can meet the constraint by force majeure, positing a supply of haecceitistic properties to distinguish between structurally indiscernible objects. But, as Keranen reflects, the positing of haecceities opens up the possibility of indiscernible structures that differ only haecceitisticallv.
- Since mathematical discourse lacks the descriptive resources to distinguish between these structures, this manoeuvre on the part of the structuralist threatens to render reference to mathematical objects deeply inscrutable.
- Keranen concludes that the particular difficulties encountered by ante rem structuralism in particular reflect deep difficulties for ontological realism in general.
- Identity and Individuation3
- The Trivializing Objection
- The Leibniz Principle
- The ‘Trivial’ Accounts of Identity
- Realist Structuralism Reconsidered
- Conclusion: The Identity Problem for Realism
Footnote 1: Footnote 2: In "Shapiro (Stewart) - Structure and Identity".
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