Abstract
The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression ‘foundations of’. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barr, M. and Wells, C.: 1985,Toposes, Triples and Theories, Ergenbnisse Math. Wiss. 278, Springer-Verlag, New York.
Barr, M. and Wells, C.: 1988, ‘The Formal Description of Data Types Using Sketches’, inMathematical Foundations of Programming Language Semantics, LNCS 298, Springer-Verlag, New York, pp. 490–527.
Barr, M. and Wells, C.: 1990,Category Theory for Computing Science, Prentice Hall, New York.
Beeson, M. J.: 1985,Foundations of Constructive Mathematics, Springer-Verlag, New York.
Bell, J. L.: 1981, ‘Category Theory and the Foundations of Mathematics’,British Journal for the Philosophy of Science 32, 349–58.
Bell, J. L.: 1986, ‘From Absolute to Local Mathematics’,Synthese 69, 409–26.
Bell, J. L.: 1988,Toposes and Local Set Theories, Oxford University Press, New York.
Bénabou, J.: 1985, ‘Fibered Categories and the Foundations of Naive Category Theory’,Journal of Symbolic Logic 50(1), 10–37.
Blass, A.: 1984, ‘The Interaction Between Category Theory and Set Theory’,Contemporary Mathematics 30, 5–29.
Boileau, A. and Joyal, A.: 1981, ‘La logique des Topos’,Journal of Symbolic Logic 46(1), 6–16.
Boolos, G.: 1971, ‘The Iterative Conception of Set’,Journal of Philosophy 68, 215–32.
Bunge, M.: 1979,A World of Systems, D. Reidel, Dordrecht.
Bunge, M.: 1985,Philosophy of Science and Technology, D. Reidel, Dordrecht.
Chapman, J. and Rowbottom, F.: 1992,Relative Category Theory and Geometric Morphisms, Oxford University Press, Oxford.
Cole, J. C.: 1973, ‘Categories of Sets and Models of Set Theory’,Proceedings of the Bertrand Russell Memorial Logic Conference, 351–99.
Corry, L.: 1992, ‘Nicolas Bourbaki and the Concept of Mathematical Structure’,Synthese 92, 315–48.
Couture, J. and Lambek, J.: 1991, ‘Philosophical Reflections on the Foundations of Mathematics’,Erkenntnis 34, 187–209.
Dedekind, R.: 1901,Essays on the Theory of Numbers, W. W. Beman, trans., 2nd ed. 1909, Open Court, Chicago.
Faith, C.: 1973,Algebra Rings, Modules, and Categories, Vol. I, Springer-Verlag, New York.
Feferman, S.: 1969, ‘Set-Theoretical Foundations of Category Theory’,LNM 106, Springer-Verlag, 201–47.
Feferman, S.: 1977, ‘Categorical Foundations and Foundations of Category Theory’, in R. Butts (ed.),Logic, Foundations of Mathematics and Computability, D. Reidel, pp. 149–169.
Feferman, S.: 1985, ‘Working Foundations’,Synthese 62, 229–54.
Field, H.: 1989,Realism, Mathematics, and Modality, Basil Blackwell, Oxford.
Fourman, M.: 1980, ‘Sheaf Models for Set Theory,’Journal of Pure and Applied Algebra 19, 91–101.
Goguen, J. A.: 1991, ‘A Categorical Manifesto’,Mathematical Structures in Computer Science 1, 49–68.
Goguen, J. and Burstall, R.: 1984, ‘Introducing Institutions’, in E. Clarke and D. Kozen (eds.),Logics of Programs, LNCS, 164, New York: Springer-Verlag, pp. 221–56.
Goguen, J. and Burstall, R.: 1986, ‘A Study in the Foundations of Programming Methodology: Specifications, Institutions, Charters and Parchments’, in D. Pitt, S. Abramsky, A. Poigne and D. Rydeheard (eds.),Proceedings, Conference on Category Theory and Computer Programming, LNCS, 240, Springer-Verlag, New York, 313–33.
Goldblatt, R.: 1984,Topoi, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 98, North-Holland.
Goldfarb, W.: 1988, ‘Poincaré Against the Logicists’, in W. Aspray and P. Kitcher (eds.),History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, pp. 61–81.
Griffiths, P. and Harris, J.: 1978,Principles of Algebraic Geometry, John Wiley and Sons, New York.
Hallett, M.: 1984,Cantorian Set Theory, Oxford University Press, Oxford.
Hartshorne, R.: 1977,Algebraic Geometry, Springer-Verlag, New York.
Hatcher, W. S.: 1982,The Logical Foundations of Mathematics, Pergamon Press, New York.
Heinzmann, G.: (ed.), 1986,Poincaré, Russell, Zermelo et Peano, Alben Blanchard, Paris.
Hilbert, D.: 1927, ‘The Foundations of Mathematics’, in J. van Heijennort (ed.),From Frege to Gödel, 1967, Harvard University Press, Cambridge, MA.
Hylton, P.: 1980, ‘Russell's Substitutional Theory’,Synthese 45, 1–31.
Hylton, P.: 1990,Russell, Idealism, and the Emergence of Analytic Philosophy, Oxford University Press, New York.
Irvine, A.: 1989, ‘Epistemic Logicism and Russell's Regressive Method,Philosophical Studies 55, 303–27.
Isbell, J.: 1967, ‘Review of Lawvere 1966,Mathematical Reviews,34(7332), 1354–5.
Johnstone, P. T.: 1977,Topos Theory, Academic Press, London.
Johnstone, P. T.: 1987,Notes on Logic and Set Theory, Cambridge University Press, Cambridge.
Kitcher, P.: 1983,The Nature of Mathematical Knowledge, Oxford University Press, Oxford.
Kock, A. and Reyes, G. E.: 1977, ‘Doctrines of Categorical Logic’, in J. Barwise (ed.),Handbook of Mathematical Logic, North-Holland, Amsterdam.
Kuyk, W.: 1977,Complementarity in Mathematics, D. Reidel, Boston.
Lambek, J.: 1992, ‘Are the Traditional Philosophies of Mathematics really Incompatible?’, Preprint, McGill University.
Lambek, J. and Couture, J.: 1991, ‘Philosophical Reflections on the Foundations of Mathematics’,Erkenntnis 34, 187–209. Lambek, J. and Scott, P. J.: 1986,Introduction to Higher Order Categorical Logic, Cambridge University Press.
Lawvere, W.: 1966, ‘The Category of Categories as a Foundation for Mathematics’,Proceedings of La Jolla Conference on Categorical Algebra, Springer, New York, pp. 1–20.
Lawvere, W.: 1969, ‘Adjointness in Foundations’,Dialectica 23, 281–96.
Lawvere, W.: 1975, Introduction ofToposes, Algebraic Geometry and Logic, LNM 274, Springer-Verlag, New York, pp. 1–12.
Levy, I.: 1979,Basic Set Theory, Springer Verlag, New York.
Mac Lane, S.: 1969, ‘One Universe as a Foundation for Category Theory’,LNM 106, Springer-Verlag, New York, pp. 192–9.
Mac Lane, S.: 1971, Categories for the Working Mathematician, Springer-Verlag.
McLarty, C.: 1990, ‘The Uses and Abuses of the History of Topos Theory’,British Journal for the Philosophy of Science 41(3), 351–76.
Maddy, P.: 1990,Realism in Mathematics, Clarendon Press, Oxford.
Makkai, M. and Paré, R.: 1989, Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics 104, American Mathematical Society, Providence.
Makkai, M. and Reyes, G.: 1977,First-Order Categorical Logic, LNM 611, Springer-Verlag, New York.
Marquis, J.-P.: 1993, ‘Russell's Logicism and Categorical Logicisms’, inRussell and Analytic Philosophy, A. Irvine and G. Weddekind (eds.), University of Toronto Press, Toronto, pp. 293–324.
Mathias, A. R. D.: 1992, ‘The Ignorance of Bourbaki’,The Mathematical Intelligencer 14(3), 4–13.
Mayberry, J.: 1977, ‘On the Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Classical Mathematics (I)’,The British Journal for the Philosophy of Science 28, 1–34.
Meseguer, J.: 1989, ‘General Logics’,Logic Colloquium 87, in H.-D. Ebbinghauset al. (eds.), North-Holland, pp. 275–329.
Mitchell, W.: 1972, ‘Boolean Topoi and the Theory of Sets’,Journal of Pure and Applied Algebra 2, 261–74.
Moerdijk, I. and Reyes, G.: 1991,Models for Smooth Infinitesimal Analysis, 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.
Parsons, C.: 1983,Mathematics in Philosophy, Cornell University Press.
Piaget, J.: 1977, ‘General Conclusions’,Epistemology and Psychology of Functions, by J. Piaget, Jean-Blaise Grize, Alina Szeminska and Vinh Bang, transl. by, F. X. Castellanos and V. D. Anderson, D. Reidel, Boston, pp. 167–96.
Poincaré, H.: 1909, ‘La Logique de l'Infini’, In. G. Heinzmann (ed.),Poincaré, Russell, Zermelo et Peano, 1986, Blanchard, Paris.
Resnik, M.: 1981, ‘Mathematics as a Science of Patterns: Ontology and Reference’,Noûs 15, 529–50.
Resnik, M.: 1982, ‘Mathematics as a Science of Patterns: Epistemology’,Noûs 16, 95–105.
Resnik, M.: 1988, ‘Mathematics from the Structural Point of View’,Revue Internationale de Philosophie 42(167), 400–24.
Russell, B.: 1897,An Essay in the Foundations of Geometry, 2nd ed. 1956, Dover, New York.
Russell, B.: 1903,The Principles of Mathematics, W. W. Norton and Co, New York.
Russell, B.: 1907, ‘The Regressive Method of Discovering the Premises of Mathematics’, in D. Lackey (ed.),Essays in Analysis, 1973, George Allen and Unwin, London, pp. 272–83.
Shafarevich, I. R.: 1977,Basic Algebraic Geometry, Springer-Verlag, New York.
Shapiro, S.: 1989, ‘Logic, Ontology and Mathematical Practice’,Synthese 79, 13–50.
Shoenfield, J.: 1977, ‘Axioms of Set Theory’, in J. Barwise (ed.),Handbook of Mathematical Logic, North-Holland, pp. 322–44.
Steiner, M.: 1989, ‘The Applications of Mathematics to Natural Sciences’,The Journal of Philosophy LXXXVI(9), 449–80.
Tieszen, R. L.: 1989,Mathematical Intuition, Kluwer Academic Publishers, Dordrecht.
Wang, H.: 1974,From Mathematics to Philosophy, Routledge and Kegan Paul, London.
Wang, H.: 1983, ‘The Concept of Set’, in P. Benacerraf and H. Putnam (eds.),Philosophy of Mathematics, 2nd ed., Cambridge: Cambridge University Press, pp. 520–70.
Author information
Authors and Affiliations
Additional information
Various versions of this paper have been read by many people, many of whom have made crucial comments. Needless to say, I am entirely responsible for the claims made in this paper. I would particularly like to thank, in alphabetical order, Mario Bunge, Marta Bunge, Michael Hallett, Andrew Irvine, Saunders Mac Lane, Collin McLarty, Peneloppe Maddy and Mihaly Makkai. Part of the work was done while the author was a visiting fellow at REHSEIS in Paris and at the Center for Philosophy of Science in Pittsburgh. I would like to thank everyone for his or her help and support. I gratefully acknowledge the financial support received from the SSHRC of Canada while this work was done.
Rights and permissions
About this article
Cite this article
Marquis, JP. Category theory and the foundations of mathematics: Philosophical excavations. Synthese 103, 421–447 (1995). https://doi.org/10.1007/BF01089735
Issue Date:
DOI: https://doi.org/10.1007/BF01089735