Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-20T04:10:22.018Z Has data issue: false hasContentIssue false
  • English
  • Français

The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond1

Published online by Cambridge University Press:  26 September 2008

Leo Corry
Leo Corry
Affiliation:
Cohn Institute for the History and Philosophy of Science and Ideas Tel Aviv University
Get access Rights & Permissions [Opens in a new window]

Abstract

The belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.

Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs.

Type
Article
Information
Science in Context , Volume 10 , Issue 2 , Summer 1997 , pp. 253 - 296
Copyright
Copyright © Cambridge University Press 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

I wish to thank David Rowe for helpful editorial comments.

References

Andrews, G. E. 1994. “The Death of Proof? Semi-Rigorous Mathematics? You've Got to Be Kidding!” Math. Int. 16(4): 16–18.Google Scholar
Atiyah, M. et al. 1994. Responses to Jaffe and Quinn 1994. Bull. AMS 30:178–207.Google Scholar
Beaulieu, L. 1993. “A Parisian Café and Ten Proto-Bourbaki Meetings (1934–35).” Math. Int. 15:27–35.Google Scholar
Beaulieu, L. 1994. “Dispelling the Myth: Questions and Answers about Bourbaki's Early Work, 1934–1944.” In The Intersection of History and Mathematics, edited by Sasaki, C. et al. , 241–52. Basel: Birkhäuser.CrossRefGoogle Scholar
Bernays, P. 1922. “Die Bedeutung Hilberts für die Philosophic der Mathematik.” Die Naturwissenschaften 10:93–99.CrossRefGoogle Scholar
Boos, W. 1985. “'The True' in Gottlob Frege's ”Über die Grundlagen der Geometrie” Arch. Hist. Ex. Sci. 34:141–92.CrossRefGoogle Scholar
Born, M. 1922. “Hilbert und die Physik.” Die Naturwissenschaften 10:88–93. Reprinted in Ausgwählte Abhandlungen 2(1963): 584–98. Göttingen: Vanden-hoek and Ruprecht.CrossRefGoogle Scholar
Borwein, J. et al. 1996. “Making Sense of Experimental Mathematics.” Math. Int. 18(4): 12–18.Google Scholar
Bourbaki, N. 1950. “The Architecture of Mathematics.” Am. Math. Mo. 67:221–32.CrossRefGoogle Scholar
Bourbaki, N. 1969. Eléments d'histoire des mathématiques, 2nd ed. Paris: Hermann.Google Scholar
Cartan, H. [1958] 1980. “Nicolas Bourbaki and Contemporary Mathematics.” Translated by Kevin Lenzen. MI2:175–80. Originally published as “Nicolas Bourbaki und die heutige Mathematik,” Arbeitsg. Forsch. Landes Nordheim 76:5–18.CrossRefGoogle Scholar
Corcoran, J. 1980. Review of Kline 1980. Math. Rev. 82e:03013.Google Scholar
Corry, L. 1989. “Linearity and Reflexivity in the Growth of Mathematical Knowledge.” Science in Context 3:409–40.CrossRefGoogle Scholar
Corry, L. 1996. Modern Algebra and the Rise of Mathematical Structures. Science Network, vol. 17. Boston: Birkhäuser.Google Scholar
Corry, L. 1997a. “David Hilbert and the Axiomatization of Physics (1894–1905).” Arch. Hist. Ex. Sci. 51:83–198.CrossRefGoogle Scholar
Corry, L. 1997b “Hermann Minkowksi and the Principle of Relativity.” Arch. Hist. Ex.Sci. 51:273–314.CrossRefGoogle Scholar
Corry, L. — Forthcoming. “David Hilbert and Physics.” In The Symbolic Universe:Geometry and Physics, 1890–1930, edited by Gray, J. J.. Oxford: Oxford University Press.Google Scholar
Dedekind, R., Weber, H.. 1882. “Theorie der algebraischen Funktionen einer Veränderlichen.” Jour. r. ang. Math. 92:181–290.Google Scholar
Detlefsen, M., Luker, M.. 1980. “The Four-Color Problem and Mathematical Proof.” Journal of Philosophy 77:803–24.CrossRefGoogle Scholar
Dieudonné, J. 1962. “Les méthodes axiomatiques modernes et les fondements des mathématiques.” In Lesgrands Courantsde la Pensée Mathématique, 2nd enl. ed., edited by Lionnais, F. Le, 443–555. Paris: Blanchard.Google Scholar
Dieudonné, J. — [1977] 1982. A Panorama of Pure Mathematics: As Seen by Nicolas Bourbaki. Translated by MacDonald, I. G.. New York: Academic Press. Originally published as Panorama des mathematiques pures. Le choix bourbachique.Google Scholar
Dieudonné, J. 1982. “The Work of Bourbaki in the Last Thirty Years.” Notices AMS 29:618–23.Google Scholar
Dieudonné, J. 1985. History of Algebraic Geometry. An Outline of the Historical Development of Algebraic Geometry. Monterey, Calif.: Wadsworth.Google Scholar
Fang, J. 1970. Bourbaki: Towards a Philosophy of Modern Mathematics I. New York: Paideia Press.Google Scholar
Gabriel, G. 1980. et al., eds. Gottlob Frege—Philosophical and Mathematical Correspondence. Chicago: The University of Chicago Press.Google Scholar
Grattan-Guinness, I. 1979. Review of Dieudonné, ed., Abrégé d'histoire des mathématiques, 1700–1900. Annals of Science 36:653–55.Google Scholar
Hertz, H., [1894] 1956. The Principles of Mechanics Presented in a New Form. New York: Dover. Originally published as Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt. Leipzig.Google Scholar
Hilbert, D. 1899. Grundlagen der Geometric (Festschrift zur Feier der Enthüllung des Gauss- Weber-Denkmals in Göttingen). Leipzig: Teubner.Google Scholar
Hilbert, D. 1902. “Mathematical Problems.” Bulletin AMS 8: 437–79.CrossRefGoogle Scholar
Hilbert, D. 1903. Grundlagen der Geometrie, 2nd rev. ed., with five supplements. Leipzig: Teubner.Google Scholar
Hilbert, D. 1916–17. Die Grundlagen der Physik, II, Ms. Vorlesung WS 1916–17. Annotated by R. Bär. Bibliothek des Mathematischen Seminars: Universität Göttingen.Google Scholar
Hilbert, D. 1918. “Axiomatisches Denken.” Math.Ann. 78:405–15. Reprinted in Hilbert 1932–35,3:146–56.CrossRefGoogle Scholar
Hilbert, D. — [1919–20] 1992. Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays. Edited and with an English introduction by Rowe., David E. Basel: Birkhäuser.Google Scholar
Hilbert, D. 1922–23. “Wissen und mathematisches Denken.” WS 1922/23, Ausgearbeitet von W. Ackerman, Mathematisches Institut Universität Göttingen. Revised version.Google Scholar
Hilbert, D. 1932–35. Gesammelte Abhandlungen, vol. 3. Berlin: Springer Verlag.CrossRefGoogle Scholar
Horgan, J. 1993. “The Death of Proof.” Scientific American 269(4):74–103.CrossRefGoogle Scholar
Huntington, E. 1902. “Simplified Definition of a Group.” Bulletin AMS 8: 296–300.CrossRefGoogle Scholar
Jaffe, A. 1997. “Proof and the Evolution of Mathematics.” Synthese 111(2): 133–46.CrossRefGoogle Scholar
Jaffe, A., Quinn, F.. 1993. “Theoretical Mathematics: Towards a Cultural Synthesis of Mathematics and Theoretical Physics.” Bull. AMS 29:1–13.CrossRefGoogle Scholar
Jaffe, A. 1994. Responses to M. Atiyah et al., 1994. Bull. AMS 30:208–11.CrossRefGoogle Scholar
Kitcher, P. 1983. The Nature of Mathematical Knowledge. New York: Oxford University Press.Google Scholar
Kitcher, P. 1988. “Mathematical Naturalism.” In History and Philosophy of Modern Mathematics, Vol. 11 of Minnesota Studies in the Philosophy of Science, edited by Aspray, R. Kitcher, P., 293–323. Minneapolis: University of Minnesota Press.Google Scholar
Kleiner, I., Movshowitz-Hadar, N.. 1997. “Proof: A Many-Splendored Thing,” Mathematical Intelligencer 19(3): 16–32.CrossRefGoogle Scholar
Kline, M. 1980. Mathematics. The Loss of Certainty. New York: Oxford University Press.Google Scholar
Kolata, G. 1976. “Mathematical Proofs: The Genesis of Reasonable Doubt.” Science 192:989–90.CrossRefGoogle ScholarPubMed
Mazur, B. 1997. “Conjecture.” Synthese 111:197–210.CrossRefGoogle Scholar
Moore, E. H. 1902. “A Definition of Abstract Groups.” Transactions AMS 3:485–92.CrossRefGoogle Scholar
Moore, G. H. 1987. “A House Divided against Itself: The Emergence of First-Order Logic as the Basis for Mathematics.” In Studies in the History of Mathematics. MAA Studies in Mathematics, edited by Phillips, E. R., 98–136.Google Scholar
Pasch, M. 1882. Vorlesungen über neuere Geometric. Leipzig: Teubner.Google Scholar
Peckhaus, V. 1990. Hilbertprogramm und Kritische Philosophic. Des Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie. Göttingen: Vandenhoeck & Ruprecht.Google Scholar
Rabin, M. 1976. “Probabilistic Algorithms.” In Algorithms and Complexity: New Directions and Recent Results, edited by Traub, J. F., 21–40. New York: Academic Press.Google Scholar
Rowe, D. Forthcoming “Perspectives on Hilbert—A Review Essay.” In Perspectives on Science.Google Scholar
Scanlan, M. 1991. “Who Were the American Postulate Theorists?” Jour. Symb. Logic 50: 981–1002.CrossRefGoogle Scholar
Spalt, D. 1987. Vom Mythos der mathematischen Vernunft. Darmstadt: Wissen- schaftliche Buchgesellschaft.Google Scholar
Stegmüller, W. 1979. The Structuralist View of Theories: A Possible Analogue to the Bourbaki Programme in Physical Sciences. Berlin: Springer.CrossRefGoogle Scholar
Thurston, W. 1994. “Letter to the Editor.” Scientific American 270(1):9.Google Scholar
Toepell, M. 1986. Über die Entstehung von David Hilberts “Grundlagen der Geometric.” Göttingen: Vandenhoeck & Ruprecht.Google Scholar
Tymoczko, T. 1979. “The Four-Color Problem and Its Philosophical Significance.” Journal of Philosophy 76:67–114.CrossRefGoogle Scholar
Tymoczko, T., — ed. 1985. New Directions in the Philosophy of Mathematics. Boston: Birkhäuser.Google Scholar
vander, Waerden B. L. 1930. Moderne Algebra, 2 vols. Berlin: Springer.Google Scholar
Veblen, O. 1904. “A System of Axioms for Geometry.” Transactions AMS 5: 343–84.CrossRefGoogle Scholar
Weyl, H. 1944. “David Hilbert and His Mathematical Work.” Bulletin AMS 50:612–54.CrossRefGoogle Scholar
Zeilberger, D. 1994. “Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture.” Notices of the AMS 40:978–81 (Reprinted Math Int. 16(4): 11–14).Google Scholar