, Volume 92, Issue 3, pp 315–348 | Cite as

Nicolas Bourbaki and the concept of mathematical structure

  • Leo Corry


In the present article two possible meanings of the term “mathematical structure” are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.


Present Article Formal Theory Mathematical Structure Contemporary Mathematics Category Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


I. Primary Works and Book Reviews

  1. Artin, E.: 1953, Review of Bourbaki (1939– ), Book II, Chaps. 1–7, (1942–52), Bull. AMS 59, 474–79.Google Scholar
  2. Bagemihl, R.: 1958, Review of Bourbaki (1939– ), Book I, chap. 3, Bull. AMS 64, 390.Google Scholar
  3. Bourbaki, N.: (1939– ), Éléments de Mathématique, 10 vols., Hermann, Paris.Google Scholar
  4. Bourbaki, N.: 1949, ‘The Foundations of Mathematics’, J. of Symbolic Logic 14, 1–8.Google Scholar
  5. Bourbaki, N.: 1950, ‘The Architecture of Mathematics’, Amer. Math. Monthly 57, 221–32.Google Scholar
  6. Bourbaki, N.: 1960, Éléments d'Histoire des Mathématiques, Hermann, Paris.Google Scholar
  7. Bourbaki, N.: 1966, General Topology, 2 vols., Hermann, Paris.Google Scholar
  8. Bourbaki, N.: 1968, Theory of Sets, Hermann, Paris.Google Scholar
  9. Bourbaki, N.: 1972, Commutative Algebra, Hermann, Paris.Google Scholar
  10. Bourbaki, N.: 1972a, ‘Univers’, in M. Artin, A. Grothendieck, and J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas, (SGA 4), Springer Verlag, Heidelberg, pp. 23–72.Google Scholar
  11. Bourbaki, N.: 1973, Algebra, Hermann, Paris.Google Scholar
  12. Bourbaki, N.: 1980, Homological Algebra, Hermann, Paris.Google Scholar
  13. Cartan, H.: 1959, ‘Nicolas Bourbaki und die heutige Mathematik’, Arbeitsgemeinschaft für Forsch. des Landes Nord Westf., Heft 76, pp. 5–18.Google Scholar
  14. Dieudonné, J.: 1970, ‘The Work of Nicolas Bourbaki’, Amer. Math. Monthly 77, 134–45.Google Scholar
  15. Dieudonné, J.: 1979, ‘The Difficult Birth of Mathematical Structures, (1840–1940)’, in U. Mathieu and P. Rossi (eds.), Scientific Culture in Contemporary World, Scientia, Milan, pp. 7–23.Google Scholar
  16. Dieudonné, J.: 1982, ‘The Work of Bourbaki in the Last Thirty Years’, Notices AMS 29 618–23.Google Scholar
  17. Eilenberg, S.: 1942, Review of Bourbaki (1939– ), Book I, Fascicule de Résultats, 1st ed. (1939), MR 3, 55–56.Google Scholar
  18. Eilenberg, S.: 1945, Review of Bourbaki (1939– ), Book II, Chap. I, MR 6, #113.Google Scholar
  19. Fang, J.: 1970, Bourbaki: Towards a Philosophy of Modern Mathematics - I, Paideia Press, New York.Google Scholar
  20. Friedman, J.: 1975, L'Origine et le Développement de Bourbaki, unpublished Ph.D. Dissertation, Centre Alexandre Koyré, École Practique des hautes Études en Sciences Sociales, Paris.Google Scholar
  21. Frink, O.: 1950, Review of Bourbaki (1949), MR 11, 73.Google Scholar
  22. Gandy, R. O.: 1959, Review of Bourbaki (1949), J. Symbolic Logic 24, 71–73.Google Scholar
  23. Gauthier, Y.: 1969, ‘La Notion Theoretique de Structure’, Dialectica 23, 217–27.Google Scholar
  24. Gauthier, Y.: 1972, Review of Bourbaki (1939– ), Book I, Canadian Mathematical Bulletin 15, 623–26.Google Scholar
  25. Gillman, L.: 1964, Review of Bourbaki (1939– ), Book II, Fascicule de Résultats, 3rd ed. (1958), MR 27, #4757.Google Scholar
  26. Guedj, D.: 1985, ‘Nicolas Bourbaki, Collective Mathematician, An Interview with Claude Chevalley’, Math. Intelligencer 7, 18–22 (originally appeared in Dédales, Nov. 1981).Google Scholar
  27. Halmos, P.: 1953, Review of Bourbaki (1939– ), Book VI, Chaps. 1–4 (1952), Bull. AMS 59, 249–55.Google Scholar
  28. Halmos, P.: 1955, Review of Bourbaki (1939– ), Book I, Chaps. 1–2 (1954), MR 16, #454.Google Scholar
  29. Halmos, P.: 1956, Review of Bourbaki (1939– ), Book I, Chaps. 3 (1956), MR 17, #1062.Google Scholar
  30. Halmos, P.: 1957, ‘Nicolas Bourbaki’, Scientific American 196(5), 88–99.Google Scholar
  31. Halmos, P.: 1970, Review of Fang (1970), MR 40, #7066.Google Scholar
  32. Hermann, R.: 1986, ‘Mathematics and Bourbaki’, Math. Intelligencer 8, 32–33.Google Scholar
  33. Hewitt, E.: 1956, Review of Bourbaki (1939– ), Book IV (1953–55), Bull. AMS 62, 507–08.Google Scholar
  34. Hewitt,E.: 1966, Review of Bourbaki (1939– ), Book VI, Chaps. 7–8, MR 31, #3539.Google Scholar
  35. Israel, G.: 1977, ‘Un Aspetto Ideologico della Mathematica Contemporanea: Il “Bourbakismo”’, in E. Donini et al. (eds.), Mathematica e Fisica: Cultura e Ideologia, De Donato Editore, Bari, pp. 35–70.Google Scholar
  36. Jónsson, B.: 1959, Review of Bourbaki (1939– ), Book I (1957), MR, 20, #3804.Google Scholar
  37. Kaplanski, I.: 1953, Review of Bourbaki (1939– ), Book II, Chaps. 6–7, MR 14, #237.Google Scholar
  38. Kelley, J. L.: 1956, Review of Bourbaki (1939– ), Book IV, MR 17, #1109.Google Scholar
  39. Kolchin, E. R.: 1951, Review of Bourbaki (1939– ), Book II, Chaps. 4–5, MR 12, #6.Google Scholar
  40. Mac Lane, S.: 1948, Review of Bourbaki (1939– ), Book II, Chap. 2, MR 9, #406.Google Scholar
  41. Mac Lane, S.: 1986, Review of Bourbaki (1960/1984 English ed.), MR 86h, 01:005.Google Scholar
  42. Michael, E.: 1963, Review of Bourbaki (1939– ), Book III, Chaps. 1–2, MR 25, #4480.Google Scholar
  43. Mostowski, A.: 1967, Review of Bourbaki (1939– ), Book I, Chap. 4, MR 34, #2425.Google Scholar
  44. Munroe, H. E.: 1958, Review of Bourbaki (1939– ), Book V, Chap. 5 (1956), Bull. AMS 64, 105–06.Google Scholar
  45. Queneau, R.: 1962, ‘Bourbaki et les Mathématiques de Demain’, Critique 18, 3–18.Google Scholar
  46. Rosenberg, A.: 1960, Review of Bourbaki (1939– ), Book II, Chap. 8, Bull. AMS 66, 16–19.Google Scholar
  47. Samuel, P.: 1948, ‘On Universal Mappings and Free Topological Groups’, Bull. AMS 54, 591–98.Google Scholar
  48. Samuel, P.: 1972, Review of Bourbaki (1939– ), Book II, Chaps. 1–3 (1970), MR 43, #2.Google Scholar
  49. Stegmüller, W.: 1979, The Structuralist View of Theories: A Possible Analogue to the Bourbaki Programme in Physical Sciences, Springer Verlag, Berlin-New York.Google Scholar
  50. Toth, I.: 1980, ‘Nicolas Bourbaki. S.A.’, Princeton (Preprint).Google Scholar

II. General Works

  1. Aspray, W. and P. Kitcher (eds.): 1988, History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, Minneapolis.Google Scholar
  2. Bell, E. T.: 1945, The Development of Mathematics, 2nd ed., McGraw-Hill, New York.Google Scholar
  3. Birkhoff, G. H.: 1948, Lattice Theory, 2nd ed., AMS Colloquium Publication, Providence.Google Scholar
  4. Buchsbaum, D.: 1955, ‘Exact Categories and Duality’, Trans. AMS 80, 1–34.Google Scholar
  5. Cartan, H. and S. Eilenberg: 1956, Homological Algebra, Princeton University Press, Princeton.Google Scholar
  6. Corry, L.: 1990, ‘Linearity and Reflexivity in the Growth of Mathematical Knowledge’, Science in Context 3, 409–40.Google Scholar
  7. Corry, L.: 1991, ‘Libros de Texto e Imágenes del Algebra en el Siglo XIX’, Llull 14, 7–30.Google Scholar
  8. Ehresmann, C.: 1957, ‘Gattungen von lokalen Strukturen’, Jber. DMV 60, 49–77 (also in Ehresmann (1981), II–1, pp. 125–53).Google Scholar
  9. Ehresmann, C.: 1965, Categories et Structures, Dunod, Paris.Google Scholar
  10. Ehresmann, C.: 1981, Ouevres Complètes et Comentées, ed. by Andrée Charles Ehresmann, Amiens.Google Scholar
  11. Eilenberg, S. and S. Mac Lane: 1942, ‘Natural Isomorphisms in Group Theory’, Proc. Acad. Sci. 28, 537–43.Google Scholar
  12. Eilenberg, S. and S. Mac Lane: 1945, ‘General Theory of Natural Equivalences’, Trans. AMS 28, 231–94.Google Scholar
  13. Eilenberg, S. and N. Steenrod: 1952, Foundations of Algebraic Topology, Princeton University Press, Princeton.Google Scholar
  14. Freudenthal, H.: 1973, Mathematics as an Educational Task, D. Reidel, Dordrecht.Google Scholar
  15. Gandillac, H., L. Goldmann and J. Piaget (eds.): 1965, Entretiens sur les Notions de Genèse et de Structure, Mouton & Co., Paris.Google Scholar
  16. Gauthier, Y.: 1976, Fondements des Mathématiques, Les Presses de l'Université de Montréal, Montreal.Google Scholar
  17. George, E.: 1939, ‘Über den Satz von Jordan-Hölder’, J. reine und ang. Math. 180, 110–20.Google Scholar
  18. Gillois, M.: 1965, ‘Relation d'identité en Génétique’, Annales de l'Institut H. Poincaré 2, 1–94.Google Scholar
  19. Godement, R.: 1958, Théorie de Faisceaux, Hermann, Paris.Google Scholar
  20. Grothendieck, A.: 1957, ‘Sur quelques Points d'Algebre Homologique’, Tôhoku Math. Journal 9, 119–221.Google Scholar
  21. Hasse, H.: 1930, ‘Die moderne algebraische Methode’, Jber. DMV 39, 22–34 (reprinted in: 1986, Math. Intellingencer 8, 18–23, English trans. by Abe Shenitzer).Google Scholar
  22. Kakutani, N.: 1944, ‘Free Topological Groups and Infinite Direct Product of Topological Groups’, Proc. Imp. Acad. Tokyo 20, 595–98.Google Scholar
  23. Kan, D.: 1958, ‘Adjoint Functors’, Trans. AMS 87, 294–329.Google Scholar
  24. Lane, M. (ed.): 1970, Introduction to Structuralism, Basic Books, New York.Google Scholar
  25. Macaulay, F. S.: 1933, ‘Modern Algebra and Polynomial Ideals’, Proc. Cambridge Phil. Soc. 30, 27–46.Google Scholar
  26. Mac Lane, S.: 1939, ‘Some Recent Advances in Algebra’, Amer. Math. Monthly 46, 3–19.Google Scholar
  27. Mac Lane, S.: 1948a, ‘Groups, Categories and Duality’, Proc. Nat. Acad. Sci. 34, 263–67.Google Scholar
  28. Mac Lane, S.: 1948b, Review of Samuel (1948), MR 9, 605.Google Scholar
  29. Mac Lane, S.: 1950, ‘Duality for Groups’, Bull. AMS 56, 485–516.Google Scholar
  30. Mac Lane, S.: 1963, ‘Some Additional Advances in Algebra’, in A. A. Albert (ed.), Studies in Modern Algebra, Prentice-Hall, Englewood Cliffs, pp. 35–58.Google Scholar
  31. Mac Lane, S.: 1971, Categories for the Working Mathematician, Springer Verlag, New York.Google Scholar
  32. Maddy, P.: 1980, ‘Perception and Mathematical Induction’, The Philosophical Review 89, 163–96.Google Scholar
  33. Markoff, A.: 1942, ‘On Free Topological Groups’, Bull. Acad. Sci. USSR (Ser Math.) 9, 3–64.Google Scholar
  34. Nakayama, T.: 1943, ‘Note on Free Topological Groups’, Proc. Imp. Acad. Tokio 19, 471–75.Google Scholar
  35. Novy, L.: 1973, Origins of Modern Algebra, English trans. by J. Taver, Noordhof International Publications, Leyden.Google Scholar
  36. Ore, O.: 1931, ‘Some Recent Developments in Algebra’, Bull. AMS 39, 728–74.Google Scholar
  37. Ore, O.: 1935, ‘On the Foundations of Abstract Algebra, I’, Annals of Maths. 36, 406–37.Google Scholar
  38. Ore, O.: 1936, ‘On the Foundations of Abstract Algebra’, II, Annals of Maths. 37, 265–92.Google Scholar
  39. Ore, O.: 1937, ‘On the Theorem of Jordan-Hölder’, Trans. AMS 41, 247–69.Google Scholar
  40. Piaget, J.: 1968, Le Structuralisme, Presses Universitaire de France, Paris.Google Scholar
  41. Purkert, W.: 1971, ‘Zur Genesis des abstrakten Körperbegriffs (1)’, Schriftenreihe für Geschichte der Naturwissenschaft, Technik, und Medizin 8, 23–37.Google Scholar
  42. Resnik, M.: 1981, ‘Mathematics as a Science of Patterns: Ontology and Reference’, Noûs 15, 529–50.Google Scholar
  43. Resnik, M.: 1982, ‘Mathematics as a Science of Patterns: Epistemology’, Noûs 16, 95–105.Google Scholar
  44. Shapiro, S.: 1983, ‘Mathematics and Reality’, Philosophy of Science 50, 523–48.Google Scholar
  45. Waerden, B. L. van der: 1930, Moderne Algebra, 2 vols., Springer Verlag, Berlin (English trans. of 2nd ed. — Vol. I by Fred Blum (1949), Vol. II by T. J. Benac (1950) — Frederic Ungar Publishing, New York).Google Scholar
  46. Wussing, H.: 1984, The Genesis of the Abstract Group Concept. A Contribution to the History of the Origin of Abstract Group Theory, MIT Press, Cambridge, MA (English trans. by Abe Shenitzer of: 1969, Die Genesis des abstrakten Gruppenbegriffs, Berlin).Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Leo Corry
    • 1
  1. 1.The Cohn Institute of the History and Philosophy of ScienceTel-Aviv UniversityRamat AvivIsrael

Personalised recommendations