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Structuralism and Conceptual Change in Mathematics

Published online by Cambridge University Press:  31 January 2023

Christopher Menzel
Christopher Menzel*
Affiliation:
Texas A&M University
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Professor Grosholz packs a lot into her interesting and suggestive paper “Formal Unities and Real Individuals” (Grosholz 1990b). In the limited space available I can comment briefly on its several parts, or direct more substantive comments at a single issue. I will opt for the latter; specifically, I want to address her critique of mathematical structuralism, as found especially in the writings of Michael Resnik.

I begin with a brief, hence necessarily caricatured, summary of Resnik’s influential view. According to structuralism, the subject matter of a mathematical theory is a given pattern, or structure, and the objects of the theory are intrinsically unstructured points, or positions, within that pattern. Mathematical objects thus have no identity, and no intrinsic features, outside of the patterns in which they occur. Hence, they cannot be given in isolation but only in their role within an antecedently given pattern, and are distinguishable from one another only in virtue of the relations they bear to one another in the pattern (see, e.g., Resnik (1981)).

Type
Part IX. Mathematical and Physical Objects
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1990 , Issue 2: Volume Two: Symposia and Invited Papers 1990 , 1990 , pp. 397 - 401
Copyright
Copyright © Philosophy of Science Association 1991

References

Grosholz, E. (1990a), Cartesian Method and the Problem of Reduction. Oxford: Oxford University Press.Google Scholar
Grosholz, E. (1990b), “Formal Unities and Real Individuals,” in PSA 1990, Volume Two, Fine, A., Forbes, M., and Wessels, L. (eds.). East Lansing: Philosophy of Science Association.Google Scholar
Grosholz, E. (1985), “Two Episodes in the Unification of Logic and Topology, British Journal for the Philosophy of Science 36: 147-57.CrossRefGoogle Scholar
Grosholz, E. (forthcoming), “Was Leibniz a Mathematical Revolutionary?”Google Scholar
Resnik, M. (1981), “Mathematics as a Science of Patterns: Ontology and Reference,” Nous 15: 529–550.CrossRefGoogle Scholar
Resnik, M. (1988), “Mathematics from the Structural Point of View,” Revue Internationale de Philosophic 42: 400–424.Google Scholar