Kronecker’s Foundational Programme in Contemporary Mathematics

  • Yvon Gauthier
Part of the Studies in Universal Logic book series (SUL)


A few important mathematicians have emphasized Kronecker’s influence on contemporary mathematics, among them, first and foremost Weil (1976, 1979a) has stressed the fact that Kronecker is the founder of modern algebraic geometry and Edwards (1987a,b, 1992) after Weyl (1940) has insisted on Kronecker’s pioneering work in algebraic number theory (divisor theory). Bishop (1970) has admitted in his work on the computational (or numerical) content of classical analysis that his enterprise was more in line with Kronecker than with Brouwer. Brouwer himself paid tribute to Kronecker—as did Poincaré and Hadamard—for his contribution to the fixed point theorems (see Gauthier 2009a).


Algebraic Geometry Intuitionistic Logic Proof Theory General Arithmetic Euclidean Algorithm 
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  1. W. Ackermann (1940): Zur Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 117(2):162–194.MathSciNetCrossRefGoogle Scholar
  2. E. Bishop (1970): Mathematics as a Numerical Language. In Intuitionism and Proof Theory, 53–71. North-Holland, Amsterdam and New-York.Google Scholar
  3. J. Dieudonné (1974): Cours de Géométrie Algébrique 1. PUF, Paris.zbMATHGoogle Scholar
  4. H. M. Edwards (1987a): Divisor Theory. Birkhaüser, Basel.Google Scholar
  5. H. M. Edwards (1987b): An appreciation of Kronecker. The Mathematical Intelligencer, 9(1):28–35.zbMATHMathSciNetCrossRefGoogle Scholar
  6. H. M. Edwards (1992): Kronecker’s arithmetic theory of algebraic quantities. Jahresbericht der Deutschen Mathematiker Vereinigung, 94(3):130–139.zbMATHMathSciNetGoogle Scholar
  7. G. Faltings and G. Wüstholz (1984): Rational Points. Vieweg Verlag, Braunschweig-Wiesbaden.zbMATHCrossRefGoogle Scholar
  8. G. Frege (1983): Grundgesetze der Arithmetik. H. Pohle, Jena.Google Scholar
  9. Y. Gauthier (1983a): Le constructivisme de Herbrand. Journal of Symbolic Logic, 48(4):1230.MathSciNetGoogle Scholar
  10. Y. Gauthier (1989): Finite Arithmetic with Infinite Descent. Dialectica, 43(4):329–337.zbMATHMathSciNetCrossRefGoogle Scholar
  11. Y. Gauthier (1991): De la logique interne. Vrin, Paris.Google Scholar
  12. Y. Gauthier (1994): Hilbert and the Internal Logic of Mathematics. Synthese, 101:1–14.zbMATHMathSciNetGoogle Scholar
  13. Y. Gauthier (1997a): Logique et fondements des mathématiques. Diderot, Paris.zbMATHGoogle Scholar
  14. Y. Gauthier (2002): Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. Kluwer, Synthese Library, Dordrecht/Boston/London.zbMATHGoogle Scholar
  15. Y. Gauthier (2007): The Notion of Outer Consistency from Hilbert to Gödel (abstract). Bulletin of Symbolic Logic, 13(1):136–137.Google Scholar
  16. Y. Gauthier (2009a): The Construction of Chaos Theory. Found. Sci., 14:153–165.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Y. Gauthier (2010): Logique Arithmétique. L’arithmétisation de la logique. Presses de l’Université Laval, Québec.Google Scholar
  18. Y. Gauthier (2011): Hilbert Programme and Applied Proof Theory. Logique et Analyse, 213:46–68.MathSciNetGoogle Scholar
  19. J. Giraud (1964): Méthode de la descente. Mémoire 2 de la Société Mathématique de France.Google Scholar
  20. K. Gödel (1958): Über eine noch nicht benüzte Erweiterung des finiten Standpunktes. Dialectica, 12:230–237.CrossRefGoogle Scholar
  21. A. Grothendieck (1960): Technique de descente et théorèmes d’existence en géométrie algébrique I, Généralité. Descente par morphisme fidèlement plats. Séminaire Bourbaki, 5 (1958–1960) Exp. No. 90, 29 p.Google Scholar
  22. A. Grothendieck and M. Raynaud (1971): Revêtements étale et Groupe fondamental (SGA1), Séminaire de Géométrie algébrique du Bois-Marie 1960–1961. Lecture Notes in Mathematics, Springer-Verlag.Google Scholar
  23. J. Herbrand (1968): Écrits logiques. J. van Heijenoort, ed. PUF, Paris.Google Scholar
  24. D. Hilbert (1905): Über die Grundlagen der Logik und der Arithmetik. in Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg, Krazer (ed.) Leipzig: B.G. Teubner, 1905, 95:174–185.Google Scholar
  25. D. Hilbert (1926): Über das Unendliche. Mathematische Annalen, 95: 161–190. trad. par A. Weil sous le titre “Sur l’infini”, Acta Mathematica 48: 91–122.Google Scholar
  26. D. Hilbert (1930): Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen, 104:485–494.MathSciNetCrossRefGoogle Scholar
  27. D. Hilbert (1935): Gesammelte Abhandlungen 3 vols. Chelsea, New York.Google Scholar
  28. U. Kohlenbach (2008a): Functional Interpretations and their Use in Current Mathematics. Dialectica, 62:223–267.MathSciNetCrossRefGoogle Scholar
  29. U. Kohlenbach (2008b): Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer, Heidelberg.Google Scholar
  30. G. Kreisel (1981): Extraction of bounds: interpreting some tricks of the trade. In P. Suppes, editor. University-level computer-assisted instruction at Stanford: 1968–1980, Stanford University Institute for Mathematical Studies in the Social Sciences.Google Scholar
  31. S. Kripke (2009): The Collapse of the Hilbert Program: Why a System Cannot Prove its Own 1-consistency. The Bulletin of Symbolic Logic, 15(2):229–230.CrossRefGoogle Scholar
  32. L. Kronecker (1883): Zur Theorie der Formen höherer Stufen. Werke, II:419–424.Google Scholar
  33. L. Kronecker (1884): Über einige Anwendungen der Modulsysteme auf elementare algebraische Fragen. Werke, III:47–208.Google Scholar
  34. L. Kronecker (1889): Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Werke, II:47–208.Google Scholar
  35. L. Kronecker (1901): Vorlesungen über Zahlentheorie I. K. Hensel, ed. Teubner, Leipzig.Google Scholar
  36. L. Kronecker (1968a): Über Systeme von Funktionen mehrerer Variablen. In K. Hensel, editor, Werke, volume I. Teubner, Leipzig.Google Scholar
  37. L. Kronecker (1968b): Werke, éd. K. Hensel, volume III, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, 245–387. Teubner, Leipzig.Google Scholar
  38. L. Lafforgue (2002): Chtoukas de Drinfeld et correspondance de Langlands. Inventiones Mathematicae, 197(1):11–242.Google Scholar
  39. R. P. Langlands (1976): Some Contemporary Problems with Origins in the Jugendtraum, Mathematical Developments arising from Hilbert’s problems. American Mathematical Society, Providence R.I.Google Scholar
  40. J. Lurie (2009): Higher Topos Theory. Annals of Mathematics Studies, Princeton University Press, Princeton.zbMATHGoogle Scholar
  41. J. Molk (1885): Sur une notion qui comprend celle de la divisibilité et sur la théorie générale de l’élimination. Acta Mathematica, 6:1–165.zbMATHMathSciNetCrossRefGoogle Scholar
  42. H. Poincaré (1951): Sur les propriétés arithmétiques des courbes algébriques. Oeuvres, II:483–550.Google Scholar
  43. J.-P. Serre (2009): How to use finite fields for problems concerning infinite fields. Proc. Conf. Marseille-Luminy (2007), Contemporary Math. Series, AMS., 1–12.Google Scholar
  44. W. Sieg (1999): Hilbert’s Program: 1917–1922. Bulletin of Symbolic Logic, 5(1):1–44.zbMATHMathSciNetCrossRefGoogle Scholar
  45. L. van den Dries (1988): Alfred Tarski’s Elimination Theory for Real Closed Fields. Journal of Symbolic Logic, 53:7–19.zbMATHMathSciNetCrossRefGoogle Scholar
  46. H. S. Vandiver (1936): Constructive Derivation of the Decomposition Field of a Polynomial. Annals of Mathematics, 37(1):1–6.MathSciNetCrossRefGoogle Scholar
  47. V. Voevodsky (2010): Univalent Foundation Project. (A modified version of an NSF grant application). October 1.Google Scholar
  48. A. Weil (1976): Elliptic Functions according to Eisentein and Kronecker. Springer, Berlin.CrossRefGoogle Scholar
  49. A. Weil (1979a): Oeuvres scientifiques. Collected Paper, Vol. I, Springer-Verlag, New York.Google Scholar
  50. A. Weil (1979b): Number Theory and Algebraic Geometry. In: Weil, Vol. III:442–452.Google Scholar
  51. A. Weil (1979c): L’arithmétique sur les courbes algébriques. In: Weil, Vol. I:11–45.Google Scholar
  52. A. Weil (1984): Number Theory. An Approach through History. From Hammurabi to Legendre. Birkhäuser. Boston-Basel-Berlin.Google Scholar
  53. H. Weyl (1940): Algebraic Theory of Numbers. Princeton University Press, Princeton, N. J.Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontrealMontrealCanada

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