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 In this paper I intend to give a brief presentation of these calculi using hitherto unpublished material from the Nachltsse of Hilbert in Göttingen1 and Zermelo in Freiburg i.Br.2
I would like to thank Christian Thiel (Erlangen), Thony Christie (Erlangen), and Bernd Buldt (Bochum) for Valuable comments on earlier versions of this paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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.
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.
Frege, Gottlob 1972 Conceptual Notation and Related Articles,transl. and ed. by Terrell Ward Bynum, Clarendon Press: Oxford.
Frege, Gottlob 1977 Begriffsschrift und andere Aufsätze,3rd ed. by Ignacio Angelelli, Wissenschaftliche Buchgesellschaft: Darmstadt.
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.
Hilbert 1905b “On the Foundations of Logic and Arithmetic,” The Monist 15 338–352.
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).
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).
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).
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.
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.
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).
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.
Peckhaus, 1991 “Ernst Schröder und die ‘pasigraphischen Systeme’ von Peano und Peirce,” Modern Logic 1 174–205.
Peckhaus, 1992 “Hilbert, Zermelo und die Institutionalisierung der mathematischen Logik in Deutschland,” Berichte zur Wissenschaftsgeschichte 15 27–38.
Post,Emil Leon 1921 “Introduction to a General Theory of Elementary Propositions,” American Journal of Mathematics 43 163–185.
Schroder, Ernst 1873 Lehrbuch der Arithmetik und Algebra fir Lehrer und Studirende,vol. 1: Die sieben algebraischen Operationen,Teubner: Leipzig.
Schroder, Ernst 1890 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, Teubner: Leipzig, repr. Chelsea: Bronx, New York 1966.
Whitehead, Alfred North/Bertrand Russell 1910 Principia Mathematica,vol. 1, Cambridge University Press: Cambridge, England.
Wittgenstein, Ludwig 1921 “Logisch-philosophische Abhandlung,” Annalen der Naturphilosophie 14 185–262.
Zermelo, Ernst 1908a Mathematische Logik. Sommer-Semester 1908,lecture notes, Universitätsbibliothek Freiburg i.Br., Zermelo papers, box 2.
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.
Zermelo, Ernst 1908c “Untersuchungen über die Grundlagen der Mengenlehre. I,” Mathematische Annalen 65, 261–281.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Peckhaus, V. (1994). Logic in Transition: The Logical Calculi of Hilbert (1905) and Zermelo (1908). In: Prawitz, D., Westerståhl, D. (eds) Logic and Philosophy of Science in Uppsala. Synthese Library, vol 236. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8311-4_19
Download citation
DOI: https://doi.org/10.1007/978-94-015-8311-4_19
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4365-8
Online ISBN: 978-94-015-8311-4
eBook Packages: Springer Book Archive