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NEITHER CATEGORICAL NOR SET-THEORETIC FOUNDATIONS

  • GEOFFREY HELLMAN (a1)

Abstract

First we review highlights of the ongoing debate about foundations of category theory, beginning with Feferman’s important article of 1977, then moving to our own paper of 2003, contrasting replies by McLarty and Awodey, and our own rejoinders to them. Then we offer a modest proposal for reformulating a theory of category of categories that would actually meet the objections of Feferman and Hellman and provide a genuine alternative to set theory as a foundation for mathematics. This proposal is more modest than that of our (2003) in omitting modal logic and in permitting a more “top-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

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Corresponding author

*UNIVERSITY OF MINNESOTA, 831 HELLER HALL, 271-19TH AVENUE SOUTH, MINNEAPOLIS, MN 55455 E-mail:hellm001@umn.edu

References

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Awodey, S. (2004). An answer to Hellman’s question: ‘Does category theory provide a framework for mathematical structuralism? ’. Philosophia Mathematica, 12, 5464.
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Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica, 11, 129157.
Hellman, G. (2006). What is categorical structuralism? In van Benthem, J., Heinzmann, G., Rebuschi, M., and Visser, H., editors. The Age of Alternative Logics. Assessing Philosophy of Logic and Mathematics Today. Dordrecht, The Netherlands: Springer.
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McLarty, C. (2004). Exploring categorical structuralism. Philosophia Mathematica, 12, 3753.
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Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64, 1.
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Zermelo, E. (1930). On boundary numbers and set domains: New investigations in the foundations of set theory. In Ewald, W., editor. From Kant to Hilbert: Readings in the Foundations of Mathematics. Oxford, UK: Oxford University Press, 1996, pp. 12081233. Translated: M. Hallett, from the German original, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16, 29–47.

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