Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T14:55:49.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Between Russell and Hilbert: Behmann on the Foundations of Mathematics

Published online by Cambridge University Press:  15 January 2014

Paolo Mancosu
Paolo Mancosu*
Affiliation:
Department of Philosophy, University of California, Berkeley, Berkeley, CA 94720-2390, E-mail:mancosu@socrates.berkeley.edu, URL: http://socrates.berkeley.edu/~mancosu/
Get access Rights & Permissions [Opens in a new window]

Abstract

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 5 , Issue 3 , September 1999 , pp. 303 - 330
Copyright
Copyright © Association for Symbolic Logic 1999

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.)

References

REFERENCES

[1] Behmann, Heinrich, Über mathematische Logik, unpublished manuscript, dated 12 1, 1914. Behmann Archive, Erlangen.Google Scholar
[2] Behmann, Heinrich, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead, Dissertation, Universität Göttingen, 1918, 352 pp.Google Scholar
[3] Behmann, Heinrich, Entscheidungsproblem und Algebra der Logik, unpublished manuscript, dated 05 10, 1921. Behmann Archive, Erlangen.Google Scholar
[4] Behmann, Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, Mathematische Annalen, vol. 86 (1922), pp. 163–229.Google Scholar
[5] Behmann, Heinrich, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead, Jahrbuch der Mathematisch-Naturwissenschaftlichen Fakultät in Göttingen, vol. 23 (1922), pp. 55–64.Google Scholar
[6] Behmann, Heinrich, Letter to Bertrand Russell, August 8, 1922, Behmann Archive, Erlangen, and Russell Archive, McMaster University, 1368, 1922.Google Scholar
[7] Behmann, Heinrich, Weyls Kritik der Analysis, Hilbert Nachlaß, Niedersächsiche Staats- und Universitätsbibliothek, Göttingen, Cod. Ms. Hilbert 685, Nr. 3, Bl. 13–20, 1917?Google Scholar
[8] Behmann, Heinrich, Beiträge zur axiomatischen Behandlung des Logik-Kalküls, Habilitations schrift, Universität Göttingen, 1918, Bernays Nachlaß, WHS, Bibliothek, ETH Zürich, Hs 973.192.Google Scholar
[9] Bernays, Paul, Letter to Bertrand Russell, 1920,Russell Archive, McMaster University, 110208b.Google Scholar
[10] Bernays, Paul, Über Hilberts Gedanken zur Grundlegung der Arithmetik, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 31 (1922), pp. 10–19, English translation in [24], pp. 215–222.Google Scholar
[11] Bernays, Paul, Erwiderung auf die Note von Herrn Aloys Müller: Über Zahlen als Zeichen, Mathematische Annalen, vol. 90 (1923), pp. 159–63, English translation in [24], pp. 223–226.CrossRefGoogle Scholar
[12] Bernays, Paul, Review of: Aloys Müller, “Der Gegenstand der Mathematik”, Die Naturwissenschaften, vol. 11 (1923), pp. 520–22.Google Scholar
[13] Bernays, Paul and Schönfinkel, Moses, Zum Entscheidungsproblem der mathematischen Logik, Mathematische Annalen, vol. 99 (1928), pp. 342–372.Google Scholar
[14] Ewald, William Bragg (editor), From Kant to Hilbert. A source book in the foundations of mathematics, vol. 2, Oxford University Press, Oxford, 1996.Google Scholar
[15] Haas, Gerrit and Stemmler, Elke, Der Nachlaß Heinrich Behmanns (1891–1970): Gesamtverzeichnis, Aachener Schriften zur Wissenschaftstheorie, Logik und Logikgeschichte, vol. 1 (1981), pp. 1–39.Google Scholar
[16] Hessenberg, Gerhard, Grundbegriffe der Mengenlehre, Abhandlungen der Fries'schen Schule (Neue Folge), vol. 1 (1906), pp. 479–706.Google Scholar
[17] Hilbert, David, Mengenlehre, Lecture notes by Margarethe Loeb. Sommer-Semester 1917. Unpublished manuscript. Bibliothek, Mathematisches Institut, Universität Göttingen.Google Scholar
[18] Hilbert, David, Logische Principien des mathematischen Denkens, Vorlesung, Sommer-Semester 1905. Lecture notes by Ernst Hellinger. Unpublished manuscript. Bibliothek, Mathematisches Institut, Universität Göttingen.Google Scholar
[19] Hilbert, David, Prinzipien der Mathematik, Lecture notes by Paul Bernays. Winter-Semester 1917–1918. Unpublished typescript. Bibliothek, Mathematisches Institut, Universität Göttingen.Google Scholar
[20] Hilbert, David, Axiomatisches Denken, Mathematische Annalen, vol. 78 (1918), pp. 405–15, Lecture given at the Swiss Society of Mathematicians, 11 09 1917. Reprinted in [22], pp. 146–56. English translation in [14], pp. 1105–1115.Google Scholar
[21] Hilbert, David, Neubegründung der Mathematik: Erste Mitteilung, Abhandlungen aus dem Seminar der Hamburgischen Universität, vol. 1 (1922), pp. 157–77, Reprinted with notes by Bernays in [22], pp. 157–177. English translation in [14], pp. 1115–1134.Google Scholar
[22] Hilbert, David, Gesammelte Abhandlungen, vol. 3, Springer, Berlin, 1935.Google Scholar
[23] Hilbert, David, Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen, Birkhäuser, Basel, 1992.Google Scholar
[24] Mancosu, Paolo (editor), From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s, Oxford University Press, Oxford, 1998.Google Scholar
[25] Mancosu, Paolo (editor), Hilbert and Bernays on metamathematics, In From Brouwer to Hilbert [24], pp. 149–188.Google Scholar
[26] Moore, Gregory H., Hilbert and the emergence of modern mathematical logic, Theoria (Segunda Época), vol. 12 (1997), pp. 65–90.Google Scholar
[27] Pasch, Moritz, Die Forderung der Entscheidbarkeit, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 27 (1918), pp. 228–232.Google Scholar
[28] Peckhaus, Volker, Hilbertprogramm und Kritische Philosophie, Vandenhoeck und Ruprecht, Göttingen, 1990.Google Scholar
[29] Schönfinkel, Moses, Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305–316, English translation in [32], pp. 355–366.Google Scholar
[30] Sieg, Wilfried, Hilbert's programs: 1917–1922, this Bulletin, vol. 5 (1999), no. 1, pp. 1–44.Google Scholar
[31] Vaihinger, Hans, Die Philosophie des Als Ob, Felix Meiner, Leipzig, 1911.Google Scholar
[32] Heijenoort, Jean van (editor), From Frege to Gödel. A source book in mathematical logic, 1897–1931, Harvard University Press, Cambridge, Mass., 1967.Google Scholar
[33] Weyl, Hermann, Randbemerkungen zu Hauptproblemen der Mathematik, Mathematische Zeitschrift, vol. 20 (1924), pp. 131–50, Reprinted in: [35], pp. 433–52.Google Scholar
[34] Weyl, Hermann, Die heutige Erkenntnislage in der Mathematik, Symposion, vol. 1 (1925), pp. 1–23, Reprinted in: [35], pp. 511–42. English translation in: [24], pp. 123–42.Google Scholar
[35] Weyl, Hermann, Gesammelte Abhandlungen, vol. 2, Springer, Berlin, 1968.Google Scholar
[36] Whitehead, Alfred North and Russell, Bertrand, Principia mathematica, vol. 1, Cambridge University Press, Cambridge, 1910, Quotes from the 1978 reprint of the second edition, 1927.Google Scholar
[37] Zach, Richard, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Bulletin, vol. 5 (1999), no. 3 (this issue), pp. 331–366.Google Scholar