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Québec Studies in the Philosophy of Science pp 189–215Cite as

Kronecker’s ‘Safe Haven of Real Mathematics’

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Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 177))

Abstract

The mathematical legacy of Kronecker is impressive. The list of mathematicians who took up his problems includes Adolph Hurwitz, David Hilbert, Kurt Hensel, Julius König, Ernst Steinitz, Erich Hecke, Helmut Hasse, Carl Ludwig Siegel, Hermann Weyl and, recently, André Weil, Robert Langlands, Harold Edwards. But, although nobody has ever doubted the brilliance of Kronecker’s results and insights, it is relatively safe to say with Edwards that, more than a century after the publication of his complete works, “there are important passages in Kronecker’s work that no one, ever has fully understood, other than Kronecker himself” (1987, p. 29). This situation is the result of a lack of interest caused by disdain for Kronecker’s well-known foundational stance and for his style of mathematics. Indeed, Kronecker’s mathematical practice features an insistence on providing algorithms which is alien to the abstract, axiomatic approach which has been so dominant in the past hundred years, in particular in Bourbaki’s treatises. From this point of view, algebra consists in the study of (highly) abstract notions — the structures-mères — such as groups, rings, field, and so forth. Dedekind presented and refined his theory of ideals in successive versions of his Supplement XI to Lejeune-Dirichlet’s Vorlesungen über Zahlentheorie (1879). Adopted by Hilbert in his Zahlbericht and studied carefully by emmy Noether and her students, Dedekind’s ideals played a crucial role in the development of modern algebraic number theory.

o̒δòς \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha } \) νω κάτω μία καì ω̒υτή.1

Heraclitus, Fr. 60

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References

  • Arakelov, S. J., 1974a, ‘Intersection Theory of Divisors on an Arithmetic Surface’, Math. USSR Izvestija, 1167–1180.

    Google Scholar 

  • Arakelov, S. J., 1974b, ‘Theory of Intersections on the Arithmetic Surface’, Proceedings of the International Congress of Mathematicians. Vancouver, 1974, vol. 1, 405–408.

    Google Scholar 

  • Arthur, J., 1981, ‘Automorphic Representations and Number Theory’, Canadian Mathematical Society/Société Canadienne de mathématiques. Conference Proceedings, Volume 1: 1980 Seminar on Harmonic Analysis, 3–51.

    Google Scholar 

  • Bell, E. T., 1940, The Development of Mathematics, McGraw-Hill, New York.

    Google Scholar 

  • Birkhoff, G. and MacLane, S., 1941, A Survey of Modern Algebra, Macmillan, New York.

    Google Scholar 

  • Cantor, G., 1878, ‘Ein Beitrag zur Mannigfaltigkeitslehre’, Journal für die reine und angewandte Mathematik 77, 242–258; reprint in E. Zermelo (ed.): 1966, Gesammelte Abhandlungen, reprint, Hildesheim, Georg Olms, 119–133.

    Article  Google Scholar 

  • Cassels, J. W. S. and Fröhlich, J. (eds.), 1967, Algebraic Number Theory, Academic Press, London/New York.

    Google Scholar 

  • Chevalley, C., 1940, ‘La théorie du corps de classes’, Annals of Mathematics 41, 394–417.

    Article  Google Scholar 

  • Couturat, L., 1986, De l’infini mathématique, reprint: 1980, Blanchard, Paris.

    Google Scholar 

  • Dauben, J., 1979, Georg Cantor. His Mathematics and Philosophy of the Infinite, Harvard University Press, Cambridge Mass.

    Google Scholar 

  • Dedekind, R., 1872, Stetigkeit und irrationale Zahlen, Vie weg & Sohn, Brunswick; English translation: ‘Continuity and Irrational Numbers’, in Dedekind (1963), 1–27.

    Google Scholar 

  • Dedekind, R., 1877, ‘Sur la théorie des nombres entiers algébriques’, in Dedekind (1932), vol. 3, 262–296.

    Google Scholar 

  • Dedekind, R., 1888, Was sind und was sollen die Zahlen?, Vieweg & Sohn, Brunswick, second edition: 1893, third edition: 1911; English translation, ‘The Nature and Meaning of Numbers’, in Dedekind (1963), 29–115.

    Google Scholar 

  • Dedekind, R., 1890, ‘Letter to Keferstein’, in J. van Heijenoort (ed.): 1967, From Frege to Gödel. A Sourcebook in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge Mass., 98–103.

    Google Scholar 

  • Dedekind, R., 1892, ‘Über einen arithmetischen Satz von Gauß’, in Dedekind (1932), vol. 2, 28–39.

    Google Scholar 

  • Dedekind, R., 1895, ‘Über die Begründung der Idealtheorie’, in Dedekind (1932), vol. 2, 50–58.

    Google Scholar 

  • Dedekind, R., 1932, Gesammelte mathematische Werke, 3 vols., R. Fricke, E. Noether, and O. Öystein (eds.), Vieweg & Sohn, Brunswick.

    Google Scholar 

  • Dedekind, R., 1963, Essays on the Theory of Numbers, reprint, Dover, New York.

    Google Scholar 

  • Dedekind, R. and Weber, H., 1882, ‘Theorie der algebraischen Funktionen einer Verändlichen’, in Dedekind (1932), vol. 1, 238–349.

    Google Scholar 

  • Dieudonné, J., 1981, ‘La philosophie des mathématiques de Bourbaki’, in Choix d’oeuvres mathématiques, Hermann, Paris, vol. 1, 27–39.

    Google Scholar 

  • Dieudonné, J. (ed.), 1986, Abrégé d’histoire des mathématiques 1700–1900, Hermann, Paris.

    Google Scholar 

  • Dugac, P., 1973, ‘Éléments d’analyse de Karl Weierstrass’, Archives for History of Exact Sciences 10, 41–176.

    Article  Google Scholar 

  • Dugac, P., 1976, Richard Dedekind et les fondements des mathématiques, Librairie philosophique J. Vrin, Paris.

    Google Scholar 

  • Edwards, H. M., 1975, ‘The Background of Kummer’s Proof of Fermat’s Last Theorem for Regular Primes’, Archive for History of Exact Sciences 14, 219–236.

    Article  Google Scholar 

  • Edwards, H. M., 1980, ‘The Genesis of Ideal Theory’, Archive for History of Exact Sciences 23, 322–378.

    Article  Google Scholar 

  • Edwards, H. M., 1983, ‘Dedekind’s Invention of Ideals’, Bulletin of the London Mathematical Society 15, 8–17.

    Article  Google Scholar 

  • Edwards, H. M., 1987, ‘An Appreciation of Kronecker’, The Mathematical Intelligencer 9(1), 28–35.

    Article  Google Scholar 

  • Edwards, H. M., 1989, ‘Kronecker’s Views on the Foundations of Mathematics’, in Rowe, D. E. and J. McCleary (eds.) The History of Modern Mathematics. Volume 1: Ideas and their Reception, Academic Press, London, 67–77.

    Google Scholar 

  • Edwards, H. M., 1990, Divisor Theory, Birkhäuser, Boston-Basel-Berlin.

    Google Scholar 

  • Edwards, H. M., 1992a, ‘Mathematical Ideas, Ideals, and Ideology’, The Mathematical Intelligencer 14(2), 6–19.

    Article  Google Scholar 

  • Edwards, H. M., 1992b, ‘Kronecker’s Arithmetical Theory of Algebraic Quantities’, Jahresbericht der deutschen Mathematiker-Vereinigung 94, 130–139.

    Google Scholar 

  • Edwards, H., Neumann, O., and Purkert, W., 1982, ‘Dedekinds ‘Bunte Bemerkungen’ zu Kroneckers ‘Grundzuge’’, Archive for History of Exact Sciences 27, 49–85.

    Google Scholar 

  • Faltings, G., 1983, ‘Endlichkeitssätze für abelsche Varietäten über Zahlkörpern’, Inventiones Mathematicae 73, 349–366.

    Article  Google Scholar 

  • Faltings, G., 1984, ‘Calculus on Arithmetic Surfaces’, Annals of Mathematics 119, 387–424.

    Article  Google Scholar 

  • Friedman, H. M., Simpson, S. G. and Smith, R. L., 1983, ‘Countable Algebra and Set Existence Axioms’, Annals of Pure and Applied Logic 25, 141–181.

    Article  Google Scholar 

  • Gauthier, Y., 1994, ‘Hilbert and the Internal Logic of Mathematics’, Synthese 101, 1–14.

    Google Scholar 

  • Gelbart, S., 1984, ‘An elementary Introduction to the Langlands Program’, Bulletin of the American Mathematical Society 10, 177–219.

    Article  Google Scholar 

  • Gillet H. and Soulé, C., 1992, ‘An Arithmetic Riemann-Roch Theorem’, Inventiones Mathematicae 110, 473–543.

    Article  Google Scholar 

  • Gordan, P., 1893, ‘Über einem Satz von Hilbert’, Mathematische Annalen 42, 132–142.

    Article  Google Scholar 

  • Hasse, H., 1926–27, Höhere Algebra I, II, Walter de Gruyter, Berlin, second edition: 1933, third edition: 1951; English translation: 1954, Higher Algebra, Frederick Ungar, New York.

    Google Scholar 

  • Hasse, H., 1930, ‘Die moderne algebraische Methode’, Jahresbericht der deutschen Mathematiker-Vereinigung 39, 22–34; English translation: 1986, ‘The Modern Algebraic Method’, The Mathematical Intelligencer 8(2), 18–23.

    Google Scholar 

  • Hasse, H., 1949, Zahlentheorie, Springer, Berlin; English translation: 1980, Number Theory, Springer, Berlin.

    Google Scholar 

  • Hasse, H., 1967, ‘History of Class Field Theory’, in Cassels, J. W. S. and Fröhlich, J. (eds.),  Algebraic Number Theory, Academic Press, London/New York, 266–279

    Google Scholar 

  • Herbrand, J., 1931, ‘Sur la théorie des groupes de décomposition, d’inertie et de ramification’, Journal de mathématiques pures et appliquées 10, 481–498.

    Google Scholar 

  • Herbrand, J., 1932, ‘Théorie arithmétique des corps de nombres de degré infini. I. Extensions algébriques finies de corps infinis’, Mathematische Annalen 106, 473–501.

    Article  Google Scholar 

  • Herbrand, J., 1933, ‘Théorie arithmétique des corps de nombres de degré infini. IL Extensions algébriques de degré infini’, Mathematische Annalen 107, 699–717.

    Article  Google Scholar 

  • Hilbert, D., 1890, ‘Über die Theorie der algebraischen Formen’, Mathematische Annalen 36, 473–534, reprint in Hilbert (1935), vol. 2, 199–257; English translation: ‘On the theory of Algebraic Forms’, in Hilbert (1978), 143–224.

    Article  Google Scholar 

  • Hilbert, D., 1893, ‘Über die vollen Invariantensysteme’, Mathematische Annalen 42, 313–373, reprint in Hilbert (1935), vol. 2, 287–344; English translation: ‘On the Complete systems of Invariants’, in Hilbert (1978), 225–302.

    Article  Google Scholar 

  • Hilbert, D., 1897, ‘Die Theorie der algebraischen Zahlkörper’, Jahresbericht der deutschen Mathematiker-Vereinigung 4, 175–546; reprint in Hilbert (1935), vol. 1, 63–363.

    Google Scholar 

  • Hilbert, D., 1900, ‘Mathematische Probleme’, Göttinger Nachrichten, 253–297; reprint in Hilbert (1935), vol. 3, 290–329; English translation: 1902, ‘Mathematical Problems’ Bulletin of the American Mathematical Society 8, 437–479; reprint in 1976, Proceedings of Symposia in Pure Mathematics 28, 1–34.

    Google Scholar 

  • Hilbert, D., 1922, ‘Neubegründung der Mathematik (Erste Mitteilung)’, reprint in Hilbert (1935), vol. 3, 157–177.

    Google Scholar 

  • Hilbert, D., 1931, ‘Die Grundlegung der elementaren Zahlenlehre’, Mathematische Annalen 104, 485–494.

    Article  Google Scholar 

  • Hilbert, D., 1935, Gesammelte Abhandlungen, Springer, Berlin.

    Google Scholar 

  • Hilbert, D., 1978, Hilberts Invariant Theory Papers, M. Ackerman, R. Hermann (eds.), Math Sci Press, Brookline Mass.

    Google Scholar 

  • Hurwitz, A., 1895, ‘Über einen Fundamentalsatz der arithmetischen Theorie der algebraischen Größen’, in: 1932, Mathematische Werke, Birkhäuser, Basel, vol. 2, 198–207.

    Google Scholar 

  • Jourdain, P. E. B., 1905, ‘On the General Theory of Functions’, Journal für reine und angewandte Mathematik 128, 169–210.

    Article  Google Scholar 

  • Kirk, G. S., Raven, J. E., and Schofield, M., 1983, The Presocratic Philosophers, second edition, Cambridge University Press, Cambridge.

    Google Scholar 

  • Kneebone, G. T., 1963, Mathematical Logic and the Foundations of Mathematics. An Introductory Survey, Van Nostrand, Toronto/New York/Princeton.

    Google Scholar 

  • König, J., 1903, Einleitung in die allgemeine Theorie der algebraischen Grössen, Teubner, Leipzig.

    Google Scholar 

  • Kronecker, L., 1853, ‘Über die algebraisch auflösbaren Gleichungen’, in Kronecker (1899), vol. 4, 27–37.

    Google Scholar 

  • Kronecker, L., 1880, ‘Auszug aus einem Briefe von L. Kronecker and R. Dedekind’, in Kronecker (1899), vol. 5, 453–457.

    Google Scholar 

  • Kronecker, L., 1882, ‘Grundzüge einer arithmetischen Theorie der algebraischen Grössen’, in Kronecker (1899), vol. 2, 287–296.

    Google Scholar 

  • Kronecker, L., 1886a, ‘Über einige Anwedungen der Modulsysteme auf elementare algebraische Fragen’, in Kronecker (1899), vol. 3, 147–208.

    Google Scholar 

  • Kronecker, L., 1886b, ‘Zur Theorie der Elliptischen Funktionen XI’, in Kronecker (1899), vol. 4, 389–470.

    Google Scholar 

  • Kronecker, L., 1887a, ‘Über den Zahlbegriff’, in Kronecker (1899), vol. 3, 251–274.

    Google Scholar 

  • Kronecker, L., 1887b, ‘Einer Fundamentalsatz der allgemeinen Arithmetik’, in Kronecker (1899), vol. 3, 211–240.

    Google Scholar 

  • Kronecker, L., 1899, Werke, 5 vols., K. Hensel (ed.), Teubner, Leipzig.

    Google Scholar 

  • Kronecker, L. 1901, Vorlesungen über Zahlentheorie, K. Hensel (ed.), Teubner, Leipzig.

    Google Scholar 

  • Kummer, E. E., 1851, ‘Mémoire sur la théorie des nombres complexes composés de racines de l’unité et de nombres entiers’, Journal de mathématiques pures et appliquées 15, 377–498, reprint in Kummer (1975), vol. 1, 363–484.

    Google Scholar 

  • Kummer, E. E., 1859, ‘Über die allgemeinen Reciprocitätsgesetze den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist’, Mathematische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, 19–159; reprint in Kummer (1975), vol. 1, 699–838.

    Google Scholar 

  • Kummer, E. E., 1975, Collected Papers, 2 vols., A. Weil (ed.), Springer, Berlin.

    Google Scholar 

  • Lang, S., 1987, Elliptic Functions, second edition, Springer, Berlin.

    Book  Google Scholar 

  • Lang, S., 1988, Introduction to Arakelov Theory, Springer, Berlin.

    Book  Google Scholar 

  • Langlands, R. P., 1976, ‘Some Contemporary Problems with Origins in the Jugendtraum’, Proceedings of Symposia in Mathematics 28, 401–418.

    Google Scholar 

  • Lejeune-Dirichlet, P. G., 1837, ‘Über die Darstellung ganz wilkürlicher Funktionen durch Sinus- und Cosinusreihen’, in P. G. Lejeune-Dirichlet (1969), Werke, L. Kronecker (ed.), reprint, Chelsea, New York, 1969, vol 1, 135–160.

    Google Scholar 

  • Lejeune-Dirichlet, P. G., 1879, Vorlesungen über Zahlentheorie, R. Dedekind (ed.), Vieweg & Sohn, Brunswick.

    Google Scholar 

  • Macaulay, F. S., 1916, The Algebraic Theory of Modular Systems, Cambridge University Press, Cambridge.

    Google Scholar 

  • Marion, M., Forthcoming (a), Wittgenstein, Finitism, and the Foundations of Mathematics, Oxford University Press, Oxford.

    Google Scholar 

  • Marion, M., Forthcoming (b), ‘Wittgenstein and Ramsey on Identity’, to appear in J. Hintikka (ed.)’ 1995, Essays on the Development of the Foundations of Mathematics, Kluwer, Amsterdam.

    Google Scholar 

  • Mazur, B., 1977, ‘Review of E. E. Kummer, Collected papers’, in Bulletin of the American Mathematical Society 83, 976–988.

    Google Scholar 

  • Mazur, B., 1977, ‘Number Theory as Gadfly’, American Mathematical Monthly, 98, 593–610.

    Article  Google Scholar 

  • Meschkowski, H., 1967, Probleme des Unendlichen. Werk und Leben Georg Cantors, Vieweg & Sohn, Brunswick.

    Google Scholar 

  • Molk, J., 1885, ‘Sur une notion qui comprend celle de la divisibilité et sur théorie générale de l’élimination’, Acta Mathematica 6, 1–165.

    Article  Google Scholar 

  • Molk, J., 1909, ‘Principes fondamentaux de la théorie des fonctions’, in Encyclopédie des sciences mathématiques pures et appliquées, Gauthiers-Villars, Paris, vol, II. 1, 1–112.

    Google Scholar 

  • Poincaré, H., 1899, ‘L’oeuvre mathématique de Weierstrass’, Acta Mathematica 22, 1–18.

    Article  Google Scholar 

  • Poincaré, H., 1901, ‘Sur l’arithmétique des courbes algébriques’, in: 1950, Oeuvres de Henri Poincaré, Gauthier-Villars, Paris, vol. 5, 483–548.

    Google Scholar 

  • Poincaré, H., 1906, ‘Les mathématiques et la logique’, Revue de métaphysique et de morale 14, 294–317.

    Google Scholar 

  • Poincaré, H., 1911, ‘The Bolyai Prize. Report on the Works of Hilbert’, Science 34, 753–765 & 801–808.

    Google Scholar 

  • Poincaré, H., 1911, ‘The Bolyai Prize. Report on the Works of Hilbert’, Science 34,  801–808.

    Article  Google Scholar 

  • Pringsheim, A., 1899, ‘Grundlagen der allgemeinen Funktionenlehre’, in Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Teubner, Leipzig, vol. IIa1, 1–41.

    Google Scholar 

  • Russell, B., 1906, ‘On some Difficulties in the Theory of Transfinite Numbers and Order Types’, Proceedings of the London Mathematical Society 4, 29–53.

    Article  Google Scholar 

  • Schoenflies, A., 1927, ‘Die Krisis in Cantor’s Mathematischen Schaffen’, Acta Mathematica 50, 1–23.

    Article  Google Scholar 

  • Shimura, G. and Taniyama, Y., 1961, Complex Multiplication of Abelian Variety and its Applications to Number Theory, Publications of the Mathematical society of Japan 6.

    Google Scholar 

  • Sieg, W., 1990, ‘Relative Consistency and Accessible Domains’, Synthese 84, 259–297.

    Google Scholar 

  • Simpson, S., 1985, ‘Reverse Mathematics’, Proceedings of Symposia in Mathematics 42, 461–471.

    Google Scholar 

  • Simpson, S., 1988, ‘Partial Realizations of Hilbert’s Program’, Journal of Symbolic Logic 53, 359–363.

    Google Scholar 

  • Steinitz, E., 1910, ‘Algebraische Theorie der Körper’, Journal für die reine und angewandte Mathematik 137, 167–309. Book edition: 1930, Algebraische Theorie der Körper, R. Baer & H. Hasse (eds.), Springer, Berlin; reprint: 1950, Chelsea, New York.

    Article  Google Scholar 

  • Tate, J. T., 1967, ‘Global Class Field Theory’, in Cassels, J. W. S. and Fröhlich, J. (eds.), Algebraic Number Theory, Academic Press, London/New York 162–203.

    Google Scholar 

  • van der Waerden, B. L., 1931, Moderne Algebra, Springer, Berlin; English translation: 1949, Modern Algebra, Frederick Ungar, New York.

    Google Scholar 

  • Vladut, S. G., 1991, Kronecker’s Jugendtraum and Modular Functions, Gordon & Breach, New York.

    Google Scholar 

  • Weber, H., 1893, ‘Leopold Kronecker’, Jahresbericht der deutschen Mathematiker-Vereinigung 2, 5–31.

    Google Scholar 

  • Weil, A., 1950, ‘Number-Theory and Algebraic Geometry’, in A. Weil: 1970, Oeuvres Scientifiques. Collected Papers, Springer, Berlin, vol. 1, 442–452.

    Google Scholar 

  • Weil, A., 1976, Elliptic Functions According to Eisenstein and Kronecker, Springer, Berlin.

    Google Scholar 

  • Weyl, H., 1940, The Algebraic Theory of Numbers, Princeton University Press, Princeton.

    Google Scholar 

  • Wittgenstein, L., 1965, Philosophical Remarks, Balckwell, Oxford.

    Google Scholar 

  • Wittgenstein, L., 1974, Philosophical Grammar, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L., 1979, Ludwig Wittgenstein and the Vienna Circle, Blackwell, Oxford.

    Google Scholar 

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Authors and Affiliations

  1. University of Ottawa, Canada

    Mathieu Marion

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  1. Mathieu Marion
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  1. University of Ottawa, Canada

    Mathieu Marion

  2. Boston University, USA

    Robert S. Cohen

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Marion, M. (1996). Kronecker’s ‘Safe Haven of Real Mathematics’. In: Marion, M., Cohen, R.S. (eds) Québec Studies in the Philosophy of Science. Boston Studies in the Philosophy of Science, vol 177. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1575-6_11

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