, Volume 190, Issue 12, pp 2141–2164 | Cite as

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

  • Jean-Pierre MarquisEmail author


In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.


Philosophy of mathematics Algebraic geometry Category theory Homotopy theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aguilar M., Gitler S., Prieto C. (2002) Algebraic topology from a homotopical viewpoint. Springer, New YorkGoogle Scholar
  2. Alexandroff, P. (1962). Commemoration of Eduard C̆ech (pp. 29–30). Czech Republic: Academia Publishing House of Czechoslovak Academy of Sciences.Google Scholar
  3. Awodey S., Warren M. A. (2009) Homotopy theoretic models of identity type. Math. Proc. Cambrdige Philos. Soc. 146(1): 45–55CrossRefGoogle Scholar
  4. Baez, J. C., Shulman, M. (2006). Lectures on n-categories and cohomology. arXiv. math.CT.Google Scholar
  5. Baues H.-J. (1995) Homotopy types. In: James I. M. (Ed.) Handbook of algebraic topology. Elsevier, New York, pp 1–72CrossRefGoogle Scholar
  6. Baues H.-J. (1996) Homotopy type and homology. Clarendon Press. Oxford Science Publications, OxfordGoogle Scholar
  7. Baues H.-J. (2002) Atoms of topology. Jahresber. Deutsch. Math.-Verein 104(4): 147–164Google Scholar
  8. Carter J. (2005) Individuation of objects—a problem for structuralism?. Synthese 143(3): 291–307CrossRefGoogle Scholar
  9. Carter J. (2008) Structuralism as a philosophy of mathematical practice. Synthese 163(2): 119–131CrossRefGoogle Scholar
  10. Dieudonné, J. (1985). History of algebraic geometry: An outline of the history and development of algebraic geometry Wadsworth mathematics series. Belmont, CA: Wadsworth International Group (J. D. Sally, Trans. from French).Google Scholar
  11. Dieudonné J. (1989) A history of algebraic and differential topology, 1900–1960. Birkhäuser, BostonGoogle Scholar
  12. Dwyer, W. G., Hirschhorn, P. S., Kan, D. M., & Smith, J. H. (2004). Homotopy limit functors on model categories and homotopical categories. Mathematical surveys and monographs (Vol. 113). Providence, Rhodes Island: American Mathematical Society.Google Scholar
  13. Eilenberg S., Mac Lane S. (1945) A general theory of natural equivalences. Transactions of the American Mathematical Society 58: 231–294Google Scholar
  14. Grassi A. (2009) Birational geometry old and new. Bulletin of the American Mathematical Society (N.S.) 46(1): 99–123CrossRefGoogle Scholar
  15. Hartshorne R. (1977) Algebraic geometry. Graduate texts in mathematics, No. 52. Springer, New YorkCrossRefGoogle Scholar
  16. Hatcher A. (2002) Algebraic topology. Cambridge University Press, CambridgeGoogle Scholar
  17. Krömer, R. (2007). Tool and object: A history and philosophy of category theory. Science networks. Historical studies (Vol. 32). Basel: Birkhäuser.Google Scholar
  18. Mac Lane, S. (1998). Categories for the working mathematician. Graduate texts in mathematics (2nd ed., Vol. 5). New York: Springer.Google Scholar
  19. Mac Lane S., Moerdijk I. (1994) Sheaves in geometry and logic: A first introduction to topos theory. Springer, Universitext. New YorkGoogle Scholar
  20. Makkai, M. (1998). Towards a categorical foundation of mathematics. In: Logic colloquium ’95 (Haifa). Lecture notes logic (Vol. 11, pp. 153–190). Berlin: Springer.Google Scholar
  21. Marquis J.-P. (2006) A path to the epistemology of mathematics: Homotopy theory. In: Ferreirós J., Gray J. J. (eds) The architecture of modern mathematics. Oxford University Press, Oxford, pp 239–260Google Scholar
  22. Rotman, J. J. (1988). An introduction to algebraic topology. Graduate texts in mathematics (Vol. 119). New York: Springer.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Département de philosophieUniversité de MontréalMontrealCanada

Personalised recommendations