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On three arguments against categorical structuralism

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Abstract

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely autonomous from set theory.

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References

  • Awodey S. (1996). Structure in mathematics and logic: A categorical perspective. Philosophia Mathematica 4(3): 209–237

    Google Scholar 

  • Awodey S. (2004). An answer to G. Hellman’s question: Does category theory provide a framework for mathematical structuralism?. Philosophia Mathematica 12(1): 54–64

    Article  Google Scholar 

  • Awodey S. (2006). Category theory. Clarendon Press, Oxford

    Book  Google Scholar 

  • Bell J. (1986). From absolute to local mathematics. Synthese 69(3): 409–426

    Article  Google Scholar 

  • Benacerraf P. (1965). What numbers could not be. Philosophical Review 74(1): 47–73

    Article  Google Scholar 

  • Enderton H. (1977). Elements of set theory. Academic Press, New York

    Google Scholar 

  • Feferman S. (1977). Categorical foundations and foundations of category theory. In: Butts, R. and Hintikka, J. (eds) Foundations of mathematics and computability theory. Reidel, Dordrecht

    Google Scholar 

  • Hellman G. (2003). Does category theory provide a framework for mathematical structuralism. Philosophia Mathematica 11(2): 129–157

    Google Scholar 

  • Heijenoort J. (1967). Logic as calculus and logic as language. Synthese 17(1): 324–330

    Article  Google Scholar 

  • Johnstone P. (1977). Topos theory. London Mathematical Society Monographs (Vol. 10). Academic Press, London

    Google Scholar 

  • Landry E. and Marquis J.-P. (2005). Categories in Context: Historical, foundational and philosophical. Philosophia Mathematica 13(1): 1–43

    Article  Google Scholar 

  • Lawvere F.W. (1964). An elementary theory of the category of sets. Proceedings of the National Academy of Science of the USA 52: 1506–1511

    Article  Google Scholar 

  • Lawvere, F. W. (1965). The category of categories as a foundation for mathematics. In S. Eilenberg et al. (Eds.), Proceedings of the Conference on Categorical Algebra in La Jolla, pp 1–21.

  • Levy A. (1979). Basic set theory. Dover Publications, New York

    Google Scholar 

  • Mac Lane, S. (1986). Mathematics: Form and function. Springer-Verlag.

  • Mac Lane S. (1996). Structure in mathematics. Philosophia Mathematica 3: 174–183

    Google Scholar 

  • Mac Lane S. (1998). Categories for the working mathematician (2nd ed). Springer-Verlag, New York

    Google Scholar 

  • Maddy P. (1997). Naturalism in mathematics. Oxford University Press, Oxford

    Google Scholar 

  • McLarty C. (1992). Elementary categories, elementary toposes. Oxford University Press, Oxford

    Google Scholar 

  • McLarty C. (1993). Numbers can be just what they have to. Nous 27: 487–498

    Google Scholar 

  • McLarty C. (2004). Exploring categorical structualism. Philosophia Mathematica 12(3): 37–53

    Article  Google Scholar 

  • McLarty C. (2005). Learning from questions on categorical foundations. Philosophia Mathematica 13(1): 44–60

    Article  Google Scholar 

  • Moschovakis Y. (2005). Notes on set theory (2nd ed). Springer-Verlag, New York

    Google Scholar 

  • Osius G. (1974). Categorical set theory: A characterization of the category of sets. Journal of Pure and Applied Algebra 4: 79–119

    Article  Google Scholar 

  • Reck E. and Price M. (2005). Structures and structuralism in contemporary philosophy of mathematics. Synthese 125: 341–383

    Article  Google Scholar 

  • Shapiro S. (2005). Categories, structures and the Frege–Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica 13(1): 61–77

    Article  Google Scholar 

  • Zach R. (2007). Hilbert’s program then and now. In: Jacquette, D. (eds) Philosophy of logic. Elsevier, Amsterdam

    Google Scholar 

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Correspondence to Makmiller Pedroso.

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Pedroso, M. On three arguments against categorical structuralism. Synthese 170, 21–31 (2009). https://doi.org/10.1007/s11229-008-9346-2

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  • DOI: https://doi.org/10.1007/s11229-008-9346-2

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