Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme

Yvon Gauthier

Titles

Abstract

Kronecker called his programme of arithmetization “General Arithmetic” (Allgemeine Arithmetik). In his view, arithmetic is the building block of the whole edifice of mathematics. The aim of this paper is to show that Kronecker’s arithmetical philosophy and mathematical practice have exerted a permanent influence on a long tradition of mathematicians from Hilbert to Weil, Grothendieck and Langlands. The conclusion hints at a constructivist finitist stance in contemporary mathematical logic, especially proof theory, beyond Hilbert’s programme of finitist foundations which can be seen as the continuation of Kronecker’s arithmetization programme by metamathematical or logical means. It is finally argued that the introduction of higher-degree polynomials by Kronecker inspired Hilbert’s notion of functionals, which in turn influenced G¨odel’s functional Dialectica interpretation for his intuitionistic proof of the consistency of arithmetic.

References

[1] E. Bishop [1970], Mathematics as a numerical language, in: Intuitionism and Proof Theory, North-Holland, Amsterdam and New York, 1970, pp. 53–71.

[2] J. Boniface, N. Schappacher [2001], Sur le concept de nombre en math´ematique, Cours in´edit de Leopold Kronecker `a Berlin 1891 (with the integral German text), Revue d’histoire des math´ematiques 7 (2001), pp. 207–275.

[3] J. Dieudonn´e [1974], Cours de g´eom´etrie alg´ebrique 1, PUF, Paris, 1974.

[4] H.M. Edwards [1990], Divisor Theory, Birkh¨auser, Boston-Basel-Stuttgart, 1990.

[5] H.M. Edwards [1992], Kronecker’s Arithmetic Theory of Algebraic Quantities, Jahresberichte der Deutschen Mathematiker Vereinigung 94:3 (1992), pp. 130–139.

[6] G. Faltings, G. W¨ustholz [1984], Rational Points, Vieweg Verlag, BraunschweigWiesbaden,1984.

[7] G. Frege, Grundgesetze der Arithmetik, H. Pohle, Jena, 1893.

[8] Y. Gauthier [1983], Le constructivisme de Herbrand (Abstract), The Journal of Symbolic Logic 48:1230 (1983).

[9] Y. Gauthier [1989], Finite Arithmetic with Inﬁnite Descent, Dialectica 43:4 (1989), pp. 329–337.

[10] Y. Gauthier [1991], De la logique interne. Collection ”Mathesis”, Vrin, Paris, 1991.

[11] Y. Gauthier [1994], Hilbert and the Internal Logic of Mathematics, Synthese101 (1994), pp. 1–14.

[12] Y. Gauthier [2002], Internal Logic. Foundations of Mathematics from Kronecker to Hilbert, Kluwer, Synthese Library, Dordrecht/Boston/London, 2002.

[13] Y. Gauthier [2007], The Notion of Outer Consistency from Hilbert to G¨odel (Abstract), Bulletin of Symbolic Logic 13:1 (2007), pp. 136–137.

[14] Y. Gauthier [2009], Classical Function Theory and Applied Proof Theory, International Journal of Pure and Applied Mathematics 56:2 (2009), pp. 223–233.

[15] Y. Gauthier [2010], Logique arithm´etique. L’arithm´etisation de la logique, Qu´ebec, PUL, 2010.

[16] Y. Gauthier [2011], Hilbert Programme and Applied Proof Theory, Logique et Analyse 54:213 (2011), pp. 49–68.

[17] J. Giraud [1964], M´ethode de la descente, M´emoire 2 de la Soci´et´e Math´ematique de France, 1964.

[18] K. G¨odel [1958], ¨Uber eine noch nicht ben¨utze Erweiterung des ﬁniten Standpunktes, Dialectica 12 (1958), pp. 230–237.

[19] A. Grothendieck [1960], Technique de descente et th´eor`emes d’existence en g´ eom´etrie alg´ebrique I, G´en´eralit´es. Descente par morphismes ﬁd`element plats, S´eminaire Bourbaki, 5 (1958-1960) Exp. No. 90, 29 p.

[20] A. Grothendieck, M. Raynaud [1971], Revˆetements ´etales et Groupe fondamental (SGA1), S´eminaire de G´eom´etrie alg´ebrique du Bois-Marie 1960-1961. Lecture Notes in Mathematics, Springer-Verlag, 1971.

[21] M. Hallett [1995], Hilbert and Logic, in: Qu´ebec Studies in the Philosophy of Science, Part 1, M. Marion and R.S. Cohen, eds., Dordrecht, Kluwer, 1995, pp. 135–187.

[22] J. Herbrand [1968], ´Ecrits logiques, PUF, Paris, 1968.

[23] D. Hilbert [1890], ¨Uber die Theorie der algebraischen Formen, in: Hilbert [1935], vol. III, pp. 199–257.

[24] D. Hilbert [1893], ¨Uber die vollen Invariantensysteme, in: Hilbert [1935], vol. II, pp. 287–365.

[25] D. Hilbert [1905], ¨Uber die Grundlagen der Logik und der Arithmetik, in: Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg, Krazer (ed.) Leipzig : B. G. Teubner, 1905 : 1904, pp. 174–185.

[26] D. Hilbert [1926], ¨Uber das Unendliche, Mathematische Annalen 95 (1926), pp.161–190.

[27] D. Hilbert [1930], Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104 (1930), pp. 485–494.

[28] D. Hilbert [1935], Gesammelte Abhandlungen 3 vols, Chelsea, New-York, 1935.

[29] A. Hurwitz [1895], Mathematische Werke, Vol. 2, pp. 198–207.

[30] U. Kohlenbach [2008], Applied Proof Theory. Proof Interpretations and their Use in Mathematics, Springer-Verlag, Berlin - Heidelberg, 2008.

[31] L. Kronecker [1889], Werke K. Hensel (ed.), 5 vols. Teubner, Leipzig, 1889.

[32] L. Kronecker [1882], Grundz¨uge einer arithmetischen Theorie der algebraischen Gr¨ossen, Werke, vol. II, pp. 245–387.

[33] L. Kronecker [1883], Zur Theorie der Formen h¨oherer Stufen, Werke, vol. II, pp. 419–424.

[34] L. Kronecker [1884], ¨Uber einige Anwendungen der Modulsysteme auf elementare algebraische Fragen, Werke, vol. III, pp. 47–208.

[35] L. Kronecker [1886], Zur Theorie der elliptischen Funktionen, Werke, vol. IV, pp. 309–318.

[36] L. Kronecker [1887a], Ein Fundamentalsatz der allgemeinen Arithmetik, Werke, vol. II, pp. 211–240.

[37] L. Kronecker [1887b], ¨Uber den Zahlbegriﬀ, Werke, Vol. III, pp. 251–274.

[38] L. Kronecker [1901], Vorlesungen ¨uber Zahlentheorie, Vol. I, K. Hensel (ed.), Teubner, Leipzig, 1901.

[39] L. Laﬀorgue [2002], Chtoukas de Drinfeld et correspondance de Langlands, Inventiones Mathematicae, 197:1 (2002), pp. 11–242.

[40] R.P. Langlands [1976], Some Contemporary Problems with Origins in the Jugendtraum, in: Mathematical Developments arising from Hilbert’s problems, American Mathematical Society, Providence, Rhode Island, 1976, pp. 401–418.

[41] S. Lipschitz [1986], Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass, Vieweg Verlag, Braunschweig, 1986.

[42] J. Lurie [2009], Higher Topos Theory, Annals of Mathematics Studies, Princeton University Press, Princeton, 2009.

[43] M. Marion [2009], Kronecker’s Safe Haven of Real Mathematics, in: Quebec Studies in the Philosophy of Science I, M. Marion and R.S. Cohen (eds.) Kluwer, Boston, 2009, pp. 189–215.

[44] J. Molk [1885], Sur une notion qui comprend celle de divisibilit´e et sur la th´eorie g´ en´erale de l’´elimination, Acta Mathematica 6 (1885), pp. 1–166.

[45] H. Poincar´e [1951], Sur les propri´et´es arithm´etiques des courbes alg´ebriques, in: Œuvres, vol. II, pp. 483–550.

[46] B. Russell [1919], Introduction to Mathematical Philosophy, Allen and Unwin, London UK, 1919.

[47] J.-P. Serre [2009], How to use ﬁnite ﬁelds for problems concerning inﬁnite ﬁelds, Proc. Conf. Marseille-Luminy 2007, Contemporary Math. Series, AMS, pp. 1–12.

[48] W. Sieg [1999], Hilbert’s Programs : 1917-1922, Bulletin of Symbolic Logic 5:1 (1999), pp. 1–44.

[49] L. Van den Dries [1988], Alfred Tarski’s Elimination Theory for Real Closed Fields, The Journal of Symbolic Logic 53 (1988), pp. 7–19.

[50] H.S. Vandiver [1936], Constructive Derivation of the Decomposition-Field of a Polynomial, Annals of Mathematics 37:1 (1936), pp. 1–6.

[51] V. Voevodsky [2010], Univalent Foundations Project, (A modiﬁed version of an NSF grant application). October 1, 2010.

[52] A. Weil [1976], Elliptic Functions according to Eisenstein and Kronecker, SpringerVerlag Berlin, 1976.

[53] A. Weil [1979a], Oeuvres scientiﬁques, Collected Papers, Vol. I, Springer-Verlag, New York, 1979.

[54] A. Weil [1979b], Number Theory and Algebraic Geometry, in: Weil [1979a], Vol. III, pp. 442–452.

[55] A. Weil [1979c], L’arithm´etique sur les courbes alg´ebriques, in: Weil [1979a], Vol. I, pp. 11–45.

[56] A. Weil [1984], Number Theory. An Approach through History. From Hammurabi to Legendre, Birkha¨user. Boston-Basel-Berlin, 1984.

[57] H. Weyl [1940], Algebraic Theory of Numbers, Princeton University Press, Princeton, New Jersey, 1940.

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Information: Reports on Mathematical Logic, 2013, Number 48, pp. 37 - 65

DOI: https://doi.org/10.4467/20842589RM.13.002.1254

Article type: Original article

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Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme

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Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme

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Yvon Gauthier

Department of Philosophy University of Montreal C.P. 6128, Succ. Centre-Ville Montreal (Qc), Canada H3C 3J7

Published at: 26.11.2013

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Yvon Gauthier (Author) - 100%

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