Abstract
This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.
Article PDF
Similar content being viewed by others
References
Antonelli A., May R. (2002) Frege’s new science. Notre Dame Journal of Formal Logic 41(3): 242–270
Awodey S. (2004) An answer to Hellman’s question: “Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica 13(1): 54–64
Bernays P. (1967) Hilbert, David. In: Edwards P. (eds) The encyclopedia of philosophy (Vol. 3). Macmillan, New York, pp 496–504
Ewald W. (1999) From Kant to Hilbert: A source book in the foundations of mathematics (Vol. II). Oxford University Press, Oxford
Goldfarb W. D. (1979) Logic in the twenties: The nature of the quantifier. The Journal of Symbolic Logic 44(3): 351–368
Hallett M. (1990) Physicalism, reductionism & Hilbert. In: Irvine A. D. (eds) Physicalism in mathematics. Kluwer Academic Publishers, Dordretch
Hallett M. (1994) Hilbert’s axiomatic method and the laws of thought. In: George A. (eds) Mathematics and Mind. Oxford University Press, Oxford, pp 158–200
Hallett, M. (2007). Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie, manuscript.
Hallett M., Majer U. (2004) David Hilbert’s lectures on the foundations of geometry, 1891–1902. Springer Verlag, New York
Hellman G. (2003) Does category theory provide a framework for mathematical structuralism. Philosophia Mathematica 11(2): 129–157
Hilbert, D. (1899). Grundlagen der Geometrie, Leipzig, Teuber; Foundations of geometry (E. Townsend, 1959, Trans.). La salle, IL: Open Court.
Johnstone P.T. (2002). Sketches of an elephant: A Topos theory compendium (Vols 1, 2). Oxford, Oxford University Press.
Kreisel G. (1960) Ordinal logics and the characterization of informal notions of proof. In: Todd J.A. (eds) Proceedings of the international congress of mathematicians. Cambridge University Press, Cambridge, pp 289–299
Lambek J., Scott P. J. (1986) Introduction to higher order categorical logic. Cambridge University Press, Cambridge
Landry E., Marquis J.-P. (2005) Categories in context: Historical, foundational and philosophical (co-author Jean-Pierre Marquis). Philosophia Mathematica 13(1): 1–43
Lawvere F. W. (1964) An elementary theory of the category of sets. Proceedings of the National Academy of Sciences USA 52: 1506–1511
Lawvere, F. W. (1966). The category of categories as a foundation for mathematics. In Proceedings of the conference on categorical algebra (pp. 1–21). La Jolla, New York: Springer-Verlag.
Mac Lane, S. (1968). Foundations of mathematics: Category theory. In R. Klibansky (Ed.), Contemporary philosophy (Vol. I, pp 286–294). Firenze: La Nuova Italia Editrice.
Mac Lane S. (1986) Mathematics, form and function. Springer-Verlag, New York
Mac Lane S. (1996) Structure in mathematics. Philosophia Mathematica 3(4): 174–183
Makkai, M., & Reyes, G. E. (1977). First order categorical logic, Lecture Notes in Math. Vol. 611. Berlin: Springer-Verlag
Marquis, J.-P. (2007). Category theory. In Stanford encyclopedia of philosophy (SEP). http://plato.stanford.edu/entries/category-theory/
McLarty C. (1991) Axiomatizing a category of categories. Journal of Symbolic Logic 56(4): 1243–1260
McLarty C. (1992) Elementary categories, elementary toposes. Oxford University Press, Oxford
McLarty C. (2004) Exploring categorical structuralism. Philosophia Mathematica 3(1): 37–53
McLarty C. (2005) Learning from questions on categorical foundations. Philosophia Mathematica 13(1): 61–77
McLarty, C. (2007). The central insight of categorical mathematics. In C. Glymour, W. Wang, & D. Westerstahl (Eds.), Invited talks of the thirteenth international congress of logic, methodology and philosophy of science, Beijing, 2007. London, King’s College Publications (submitted for Studies in Logic and the Foundations of Mathematics)
Parsons C. (1998) Finitism and intuitive knowledge. In: Schirn M. (eds) The philosophy of mathematics today. Oxford University Press, Oxford, pp 249–270
Shapiro S. (1991) Foundations without foundationalism. Oxford University Press, Oxford
Shapiro S. (1997) Philosophy of Mathematics: structure and ontology. Oxford University Press, Oxford
Shapiro S. (2005) Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica 13(1): 61–77
Tait W. (1981) Finitism. Journal of Philosophy 78: 524–546
Zach R. (2006) Hilbert’s program then and now. In: Jacquette D. (eds) Handbook of the philosophy of science, Volume 5: Philosophy of Logic. Elsevier, Amsterdam
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
For John, for Mimi.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Landry, E. How to be a structuralist all the way down. Synthese 179, 435–454 (2011). https://doi.org/10.1007/s11229-009-9691-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-009-9691-9