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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Awodey S. (1996). Structure in mathematics and logic: A categorical perspective. Philosophia Mathematica 4(3): 209–237
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
Awodey S. (2006). Category theory. Clarendon Press, Oxford
Bell J. (1986). From absolute to local mathematics. Synthese 69(3): 409–426
Benacerraf P. (1965). What numbers could not be. Philosophical Review 74(1): 47–73
Enderton H. (1977). Elements of set theory. Academic Press, New York
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
Hellman G. (2003). Does category theory provide a framework for mathematical structuralism. Philosophia Mathematica 11(2): 129–157
Heijenoort J. (1967). Logic as calculus and logic as language. Synthese 17(1): 324–330
Johnstone P. (1977). Topos theory. London Mathematical Society Monographs (Vol. 10). Academic Press, London
Landry E. and Marquis J.-P. (2005). Categories in Context: Historical, foundational and philosophical. Philosophia Mathematica 13(1): 1–43
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
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
Mac Lane, S. (1986). Mathematics: Form and function. Springer-Verlag.
Mac Lane S. (1996). Structure in mathematics. Philosophia Mathematica 3: 174–183
Mac Lane S. (1998). Categories for the working mathematician (2nd ed). Springer-Verlag, New York
Maddy P. (1997). Naturalism in mathematics. Oxford University Press, Oxford
McLarty C. (1992). Elementary categories, elementary toposes. Oxford University Press, Oxford
McLarty C. (1993). Numbers can be just what they have to. Nous 27: 487–498
McLarty C. (2004). Exploring categorical structualism. Philosophia Mathematica 12(3): 37–53
McLarty C. (2005). Learning from questions on categorical foundations. Philosophia Mathematica 13(1): 44–60
Moschovakis Y. (2005). Notes on set theory (2nd ed). Springer-Verlag, New York
Osius G. (1974). Categorical set theory: A characterization of the category of sets. Journal of Pure and Applied Algebra 4: 79–119
Reck E. and Price M. (2005). Structures and structuralism in contemporary philosophy of mathematics. Synthese 125: 341–383
Shapiro S. (2005). Categories, structures and the Frege–Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica 13(1): 61–77
Zach R. (2007). Hilbert’s program then and now. In: Jacquette, D. (eds) Philosophy of logic. Elsevier, Amsterdam
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pedroso, M. On three arguments against categorical structuralism. Synthese 170, 21–31 (2009). https://doi.org/10.1007/s11229-008-9346-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-008-9346-2