Advertisement

For a Continued Revival of the Philosophy of Mathematics

  • Jean-Jacques SzczeciniarzEmail author
Chapter
  • 199 Downloads
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 43)

Abstract

This paper argues in favor of a nonreductionist and nonlocal approach to the philosophy of mathematics. Understanding of mathematics can be achieved neither by studying each of its parts separately, nor by trying to reduce them to a unique common ground which would flatten their own specificities. Different parts are inextricably interwined, as emerges in particular from the practice of working mathematicians. The paper has two topics. The first one concerns the conundrum of the unity of mathematics. We present six concepts of unity. The second topic focuses on the question of reflexivity in mathematics. The thesis we want to defend is that an essential motor of the unity of the mathematical body is this notion of reflexivity we are promoting. We propose four kinds of reflexivity. Our last argument deals with the unity of both of the above topics, unity and reflexivity. We try to show that the concept of topos is a very powerful expression of reflexivity, and therefore of unity.

References

  1. Belanger, B. (2010). La vision unificatrice de Grothendieck : au-delà de l’unité (méthodologique?) de Lautman. Philosophiques, 37(1).Google Scholar
  2. Bell, J. L. (1988). Toposes and local sets theories. Oxford: Oxford Science Publications.Google Scholar
  3. Bernays, P. (1934). Hilberts Untersuchungen über die Grundlagen der Arithmetik. Berlin: Springer.Google Scholar
  4. Boileau, A. (1975). Types versus topos (Thèse de Philosophie Doctor Université de Montréal).Google Scholar
  5. Caramello, O. (2014). Topos-theoretic background. France: IHES.Google Scholar
  6. Connes, A. (2008). A view of mathematics Mathematics: Concepts and Foundations (Vol. 1). www.colss.net/Eolss.
  7. Fourman, M. P. (1974). Connections between category theory and logic. D. Phil: Thesis, Oxford University.Google Scholar
  8. Goldblatt, W. (1979). Topoi the categorical analysis of logic. Amsterdam, New York, Oxford: North-Holland.Google Scholar
  9. Granger, G. G. (1994). Formes opérations, objets. Paris: Vrin.Google Scholar
  10. Grothendieck, A. et al.Théorie des topos et cohomologie étale des schémas. Lectures Notes, Vols. (269, 270, 305).Google Scholar
  11. Grothendieck, A. (1985). Reaping and sowing 1985. Récoltes et Semailles.Google Scholar
  12. Grothendieck, A. (1960). EGA I Le langage des schémas, (Vol. 4). France: Publications Mathamatiques de IHES.Google Scholar
  13. Hartshorne, R. (1977). Algebraic geometry (p. 1977). New York: Springer.CrossRefGoogle Scholar
  14. Hilbert, D. (1932). Gesammelte Abhandlungen. Berlin: Springer.CrossRefGoogle Scholar
  15. Kant, I. (1929). Kritik der reinen Vernunft In J. F. Hartnoch (transl.) N. Kemp Smith as Critique of Pure Reason.Google Scholar
  16. Larvor, B. (2010). Albert Lautman: Dialectics in mathematics. In Foundations of the Formal Science.Google Scholar
  17. Lautman, A. (2006). Les mathématiques, les idées et le réel physique. In J. Lautman, J. Dieudonné, & F. Zalamea (Eds.), Introduction and biography; introductory essay; Preface to the (1977th ed.). Paris: Vrin.Google Scholar
  18. Lawvere, W. (1966). The category of categories as a foundation of mathematics. In Proceedings of the Conference on Categorical Algebra La Jollia Californa, 1965 (pp. 1–20). New York: Springer.Google Scholar
  19. Mac Lane, S., & Moerbijk, I. J. (1992). Shaeves and geometry. New York: Springer.Google Scholar
  20. Panza, M. (2005). Newton et les origines de l’analyse: 1664–1666 Paris: Blanchard.Google Scholar
  21. Proutè, A. (2010). Introduction à la Logique Catégorique. Paris-Diderot: Cours Université.Google Scholar
  22. Szczeciniarz, J.-J. (2013). COED. Fabio Maia Bertato, Jose Carlos Cifuentes: In the steps of Galois. Hermann Cle Paris Campinas.Google Scholar
  23. Weyl, H. (1912). The concept of a Riemann Surface Addison and Wesley 1964 First edition die Idee der Riemannschen Fläche. Berlin: Teubner.Google Scholar
  24. Weyl, H. (1928). Gruppentheorie and Quantenmechnik. Leipzig: Hirzel.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS SPhERE University of Paris DiderotParisFrance

Personalised recommendations