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Mathematical Structuralism and the Third Man

Published online by Cambridge University Press:  01 January 2020

Michael Hand
Michael Hand*
Affiliation:
Texas A & M University, College Station, TX, 77843-4237, USA
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Extract

Plato himself would be pleased at the recent emergence of a certain highly Platonic variety of platonism concerning mathematics, viz., the structuralism of Michael Resnik and Stewart Shapiro. In fact, this species of platonism is so Platonic that it is susceptible to an objection closely related to one raised against Plato by Parmenides in the dialogue of that name. This is the Third Man Argument (TMA) against a view about the relation of Forms to particulars. My objection is not a TMA against structuralism; the position avoids that objection, but is vulnerable to a different one precisely at the point where it avoids the TMA. The way structuralism avoids the TMA has in fact been considered, as one of Plato’s options, by at least one commentator on the Parmenides, Colin Strang, who explicitly rejects it on logical grounds. In the course of the discussion, I shall clarify the reason that I believe led Strang to reject this option, and shall modify his own statement of that reason.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 23 , Issue 2 , June 1993 , pp. 179 - 192
Copyright
Copyright © The Authors 1993

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References

1 See, in particular, his ‘Mathematics as a Science of Patterns: Ontology and Reference,’ Nous 15 (1981) 529-50; ‘Mathematics as a Science of Patterns: Epistemology,’ Nous 16 (1982) 95-105; ‘Mathematics from a Structuralist Point of View,’ Revue Internationale de Philosophie 42 (1988) 400-24.

2 See his ‘Mathematics and Reality,’ Philosophy and Science 50 (1983) 523-48; ‘Structure and Ontology,’ Philosophical Topics 17 (1989) 145-71.

3 ‘Plato and the Third Man,’ in Gregory Vlastos, ed., Plato I: Metaphysics and Epistemology (Garden City, NY: Doubleday 1971) 184-200

4 See Hellman, Geoffrey Mathematics without Numbers (Cambridge: Oxford University Press 1989)Google Scholar.

5 ‘Plato’s Third Man Argument (Parm. 132A1-B2): Text and Logic,’ in his Platonic Studies, 2nd ed. (Princeton: Princeton University Press 1981) 342-65

6 ‘The Philosophical Economy of the Theory of Ideas,’ in Vlastos, ed., Plato I: Metaphysics and Epistemology 16-27Google Scholar

7 Universals and Scientific Realism (Cambridge: Cambridge University Press 1978)

8 I am indebted to Hugh Benson, Jon K vanvig, Chris Menzel, Michael Resnik, Michael Silberstein, Chris Swoyer, a referee of this journal, and especially William Lycan for discussion and for comments on earlier drafts of this paper.