Advertisement

Logic in Transition: The Logical Calculi of Hilbert (1905) and Zermelo (1908)

  • Volker Peckhaus
Chapter
Part of the Synthese Library book series (SYLI, volume 236)

Abstract

In lecture courses David Hilbert in 1905 and Ernst Zermelo in 1908 presented logical calculi which can be regarded as typical representatives of logical systems at a time when logic was in transition towards forming a new base for the foundations of mathematics. These calculi are the first fruits of discussions in David Hilbert’s circle in Göttingen which were provoked by the publication of the logical paradoxes by Bertrand Russell and Gottlob Frege in 1903. In the course of these discussions the Göttingen mathematicians in Hilbert’s circle reconsidered the interrelations between logic and mathematics, and fully grasped the eminent role of set theory for the foundation of mathematics. In this paper I intend to give a brief presentation of these calculi using hitherto unpublished material from the Nachltsse of Hilbert in Göttingen1 and Zermelo in Freiburg i.Br.2

Keywords

Propositional Logic Logical System Conjunctive Normal Form Logical Calculus Summer Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Blumenthal, Otto 1935 “Lebensgeschichte, in: David Hilbert, Gesammelte Abhandlungen, vol. 3: Analysis, Grundlagen der Mathematik, Physik, Verschiedenes, Lebensgeschichte, Springer: Berlin/Heidelberg, 2nd ed. Springer: Berlin/Heidelberg/New York 1970.Google Scholar
  2. Frege, Gottlob 1879 Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Nebert: Halle a. S.; repr. Frege 1977, 1–88; Engl. transl. Frege 1972, 101–203.Google Scholar
  3. Frege, Gottlob 1972 Conceptual Notation and Related Articles,transl. and ed. by Terrell Ward Bynum, Clarendon Press: Oxford.Google Scholar
  4. Frege, Gottlob 1977 Begriffsschrift und andere Aufsätze,3rd ed. by Ignacio Angelelli, Wissenschaftliche Buchgesellschaft: Darmstadt.Google Scholar
  5. Hilbert,David 1905a “Über die Grundlagen der Logik und der Arithmetik,” in: Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904,ed. Adolf Krazer, Teubner: Leipzig, 174–185.Google Scholar
  6. Hilbert 1905b “On the Foundations of Logic and Arithmetic,” The Monist 15 338–352.Google Scholar
  7. Hilbert 1905c Logische Principien des mathematischen Denkens,lecture course delivered in the summer term 1905, lecture notes by Max Born (Niedersächsische Staats-und Universitätsbibliothek Göttingen, Handschriftenabteilung, Cod. Ms. D. Hilbert 558a).Google Scholar
  8. Hilbert 1905d Logische Principien des mathematischen Denkens,lecture course delivered in the summer term 1905, lecture notes by Ernst Hellinger (Library of the Mathematics Institute of the University of Göttingen).Google Scholar
  9. Moore,Gregory H. 1982 Zermelo’s Axiom of Choice. Its Origins, Development and Influence,Springer: New York/Heidelberg/Berlin (= Studies in the History of Mathematics and Physical Science; 8).Google Scholar
  10. Moore, 1987 “A House Divided Against Itself: The Emergence of First-Order Logic as the Basis for Mathematics,” in: Studies in the History of Mathematics,ed. Ester R. Phillips, MAA: Washington D.C. (= MAA Studies in Mathematics; 26), 98–136.Google Scholar
  11. Peano, Giuseppe 1897 “Logique mathématique,” in: Peano, Formulaire de mathématiques, vol. 2, § 1, Bocca: Turin; reprinted in Peano, Opere Scelte, vol. 2: Logica matematica, interlingua ed algebra della grammatica, Cremonese: Roma 1958, 218–281.Google Scholar
  12. Peckhaus,Volker 1990a Hilbertprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie,Vandenhoeck & Ruprecht: Göttingen (= Studien zur Wissenschafts-, Sozial-und Bildungsgeschichte der Mathematik; 7). Google Scholar
  13. Peckhaus, 19906 ‘Ich habe mich wohl gehütet, alle Patronen auf einmal zu verschießen’. Ernst Zermelo in Göttingen,“ History and Philosophy of Logic 11 19–58.Google Scholar
  14. Peckhaus, 1991 “Ernst Schröder und die ‘pasigraphischen Systeme’ von Peano und Peirce,” Modern Logic 1 174–205.Google Scholar
  15. Peckhaus, 1992 “Hilbert, Zermelo und die Institutionalisierung der mathematischen Logik in Deutschland,” Berichte zur Wissenschaftsgeschichte 15 27–38.Google Scholar
  16. Post,Emil Leon 1921 “Introduction to a General Theory of Elementary Propositions,” American Journal of Mathematics 43 163–185.Google Scholar
  17. Schroder, Ernst 1873 Lehrbuch der Arithmetik und Algebra fir Lehrer und Studirende,vol. 1: Die sieben algebraischen Operationen,Teubner: Leipzig.Google Scholar
  18. Schroder, Ernst 1890 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, Teubner: Leipzig, repr. Chelsea: Bronx, New York 1966.Google Scholar
  19. Whitehead, Alfred North/Bertrand Russell 1910 Principia Mathematica,vol. 1, Cambridge University Press: Cambridge, England.Google Scholar
  20. Wittgenstein, Ludwig 1921 “Logisch-philosophische Abhandlung,” Annalen der Naturphilosophie 14 185–262.Google Scholar
  21. Zermelo, Ernst 1908a Mathematische Logik. Sommer-Semester 1908,lecture notes, Universitätsbibliothek Freiburg i.Br., Zermelo papers, box 2.Google Scholar
  22. Zermelo, Ernst 1908b Mathematische Logik. Vorlesungen gehalten von Prof. Dr. E. Zermelo zu Göttingen im S.S. 1908,lecture notes by Kurt Grelling, Universitätsbibliothek Freiburg i.Br., Zermelo papers, box 2.Google Scholar
  23. Zermelo, Ernst 1908c “Untersuchungen über die Grundlagen der Mengenlehre. I,” Mathematische Annalen 65, 261–281.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Volker Peckhaus
    • 1
  1. 1.Universität Erlangen-NürnbergGermany

Personalised recommendations