Leopold Kronecker's conception of the foundations of mathematics

Boniface, Jacqueline

Philosophia Scientiae, Tome 9 (2005) no. S2, pp. 143-156.

On réduit habituellement les idées de Kronecker sur les fondements des mathématiques à quelque boutade ou à quelques principes rétrogrades. Ces idées constituent pourtant une doctrine originale et cohérente, justifiée par des convictions épistémologiques. Cette doctrine apparaît dans un article intitulé ‘Sur le concept de nombre', paru en 1887 dans le Journal de Crelle, et surtout dans le dernier cours professé par Kronecker à Berlin au semestre d'été 1891. Le but de cet article est d'en préciser les principes et les éléments en la comparant aux autres entreprises de fondement, afin d'en souligner l'originalité, et de montrer comment elle a déterminé la pratique mathématique de Kronecker.

Most of the time, Kronecker's views on the foundations of mathematics are reduced to some scattered ideas. However, they constitute an original and coherent doctrine, justified by epistemological convictions. This doctrine appears in the article On the concept of number, published in the Journal of Crelle (1887) and, especially, in the last course taught by Kronecker, which took place in Berlin during the 1891 summer semester. This article would precise the principles and the insights of the Kroneckerian doctrine and then compare it to the other ones in order to underline its originality; and it would highlight how this doctrine determinated Kronecker's mathematical work.

[1] Boniface, Jacqueline 1999.- Kronecker. Sur le concept de nombre, La Gazette des mathématiciens, juillet 1999, 81, 49-70. | MR | Zbl

[2] Boniface, Jacqueline & Schappacher, Norbert 2002.- Sur le concept de nombre dans la mathématique. Cours inédit de Leopold Kronecker à Berlin (1891), Revue d'Histoire des mathématiques, juillet 2002. | Numdam | MR | Zbl

[3] Brouwer, Luitzen. E. J. 1912.- Intuitionisme en Formalisme, Amsterdam: Clausen; engl. trans.: Intuitionism and formalism, Bulletin of American. Mathematical Society, 20, 81-96. | JFM | MR

[4] Brouwer, Luitzen. E. J. 1929.- Mathematik, Wissenschaft, und Sprache, Monatshefte für Mathematik und Physik, 36, 153-64; engl. trans. in [Mancosu 1998], 45-53. | JFM | MR

[5] Cantor, Georg Werke.- Gesammelte Abhandlungen mathematischen und philosophischen Inhhalts, E. Zermelo ed., Berlin: Springer, 1932; reprinted in Hildesheim; G. Olms, 1966. | MR

[6] Dedekind, Richard Werke.- Gesammelte mathematische Werke, vol. 1 1${}^{st}$ publication, Braunschweig, 1930, vol. 2, 1${}^{st}$ publ., Braunschweig, 1931, vol. 3, 1${}^{st}$ publication, New-York, Braunschweig, 1932, reprinted in 2 vol, 1969.

[7] Dedekind, Richard 1871.- Über die Komposition der binären quadratischen Formen423-462; Werke 3, 223-261.

[8] Dedekind, Richard 1876.- Aus Briefen an R. Lipschitz, Werke 3, 464-482.

[9] Dedekind, Richard 1888.- Was sind und was sollen die Zalhen?, Brunswick: Vieweg, Werke 3, 335-391; eng. trans. in W. B. Ewald, From Kant to Hilbert, a source book in the foundations of mathematics, Oxford: Clarendon Press, 2 vols. | JFM | MR

[10] Edwards, Harold M. 1987.- An appreciation of Kronecker, The mathematical Intelligencer, 9, 1, New-York: Springer, 28-35. | MR | Zbl

[11] Edwards, Harold M. 1988.- Kronecker's Place in History, Minnesota Studies in the Philosophy of Science, vol.XI: History and Philosophy of Modern Mathematics, Minneapolis: University of Minnesota Press, 139-144. | MR

[12] Edwards, Harold M. 1989.- Kronecker's Views on the Foundations of Mathematics, History of Modern Mathematics, vol.I: Ideas and their Reception, Academic Press. | MR

[13] Edwards, Harold M. 1995.- Kronecker on the Foundations of Mathematics, From Dedekind to Gödel, Essays on the Development of the Foundations of Mathematics, Boston: J. Hintikka, 45-52. | MR

[14] Frege, Gottlob 1884.- Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: W. Koebner.

[15] Gauss, Carl F. Werke.- Werke 12 vols., Königlichen Gesellschaft der Wissenschaften, Göttingen, 1863-1927.

[16] Gauss, Carl F. 1801.- Disquisitiones Arithmeticae, Leipzig, Werke 1; eng. trans. New Haven, 1966. | MR

[17] Hilbert, David Werke.- Gesammelte Abhandlungen, 3 vols., Berlin: Springer, 1932-1935; 2d edition 1970.

[18] Hilbert, David 1931.- Die Grundlegung der elementaren Zahlentheorie, Mathematische Annalen, 104, 1931, 485-94; eng. trans by W. Ewald, in From Kant to Hilbert, a source book in the foundations of mathematics, Oxford: Clarendon Press, 2 vols., 1996; reprinted in [Mancosu 1998], 266-73. 1876 Vorlesungen über mathematische Physik, Mechanik, Leipzig: Teubner, 1876.

[19] Kronecker, Leopold Werke.- Werke, 5 vols, edited by K. Hensel, Leipzig, 1895-1930, reprinted in New-York, 1968.

[20] Kronecker, Leopold 1887.- Über den Zahlbegriff, Crelle, Journal für die reine und angewandte Mathematik, 101, 337-355; Werke 3, 249-74; introduction and french translation in [Boniface 1999]. | JFM

[21] Kronecker, Leopold 1891.- Lectures of the 1891 summer-semester, Strasbourg, Bibliothèque Université L. Pasteur, publication in [Boniface & Schappacher 2002].

[22] Kronecker, Leopold 1901.- Vorlesungen über Zalhentheorie, Leipzig: Teubner, vol.1. 1998 From Brouwer to Hilbert, The debate on the foundations of mathematics in the 1920s, New-York-Oxford: Oxford University Press, 1998.

[23] Poincaré, Henri 1898.- L'œuvre mathématique de Weierstrass, Acta mathematica, 22. | JFM

[24] Russell, Bertrand 1903.- The principles of mathematics, Cambridge: Cambridge University Press, 1923. | JFM | MR

[25] Russell, Bertrand 1919.- Introduction to mathematical philosophy, London: G. Allen & Unwin.

[26] Weber, Heinrich 1893.- Leopold Kronecker, Mathematische Annalen, 43, 1-25. | JFM | MR