Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T02:36:41.022Z Has data issue: false hasContentIssue false
  • English
  • Français

DEDEKIND AND HILBERT ON THE FOUNDATIONS OF THE DEDUCTIVE SCIENCES

Published online by Cambridge University Press:  07 November 2011

ANSTEN KLEV
ANSTEN KLEV*
Affiliation:
Institute for Philosophy, Leiden University
*
*INSTITUTE FOR PHILOSOPHY, LEIDEN UNIVERSITY, POSTBUS 9515, 2300 RA LEIDEN, THE NETHERLANDS. E-mail:: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas Hilbert dismisses elucidation and consequently treats the primitives as schematic.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 4 , Issue 4 , December 2011 , pp. 645 - 681
Copyright
Copyright © Association for Symbolic Logic 2011

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

BIBLIOGRAPHY

Avigad, J. (2006). Methodology and metaphysics in the development of Dedekind’s theory of ideals. In Ferreirós, J., and Gray, J. J., editors. The Architecture of Modern Mathematics. Oxford, UK: Oxford University Press, pp. 159–186.CrossRefGoogle Scholar
Barnes, J., editor. (1993). Aristotle. Posterior Analytics. Translated with a Commentary (second edition). Oxford, UK: Clarendon Press.Google Scholar
Barnes, J., Schofield, M., & Sorabji, R., editors. (1975). Articles on Aristotle: Science. London: Duckworth.Google Scholar
Bernays, P. (1942). Review of Steck (1940). Journal of Symbolic Logic, 7, 92–93.CrossRefGoogle Scholar
Betti, A., & de Jong, W. R. (2010). The Classical Model of Science: A millennia-old model of scientific rationality. Synthese, 174, 185–203.Google Scholar
Blumenthal, O. (1935). Lebensgeschichte. In Hilbert, D.Gesammelte Abhandlungen. Dritter Band. Berlin, Germany: Springer, pp. 388–429. Available from: http://gdz.sub.unigoettingen.de/Google Scholar
Bolzano, B. (1837). Wissenschaftslehre. Sulzbach, Germany: Seidel.Google Scholar
Bolzano, B. (1975). Einleitung zur Grössenlehre und Erste Begriffe der allgemeinen Grössenlehre, Vol. 7 of Bernard Bolzano – Gesamtausgabe. Reihe II: Nachlass. Stuttgart-Bad Cannstatt, Germany: Friedrich Frommann Verlag.Google Scholar
Boolos, G. (1971). The iterative conception of set. Journal of Philosophy, 68, 215–232.CrossRefGoogle Scholar
Breckenridge, W., & Magidor, O. (forthcoming). Arbitrary reference. Philosophical Studies.Google Scholar
Cantor, G. (1883). Über unendliche, lineare Punktmannichfaltigkeiten. Nummer 5. Mathematische Annalen, 21, 545–591.CrossRefGoogle Scholar
Carnap, R. (1927). Eigentliche und uneigentliche Begriffe. Symposion, 1, 355–374.Google Scholar
Carnap, R. (1947). Meaning and Necessity. A Study in Semantics and Modal Logic. Chicago, IL: University of Chicago Press.Google Scholar
Corry, L. (1996). Modern Algebra and the Rise of Mathematical Structures. Basel, Switzerland: Birkhäuser.Google Scholar
Corry, L. (1997). David Hilbert and the axiomatzation of physics (1894-1905). Archive for History of Exact Sciences, 51, 83–198.CrossRefGoogle Scholar
Corry, L. (2004). David Hilbert and the Axiomatzation of Physics (1898-1918). Archimedes. New Studies in the History and Philosophy of Science and Technology. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
Curry, H. B. (1941). Some aspects of the problem of mathematical rigor. Bulletin of the American Mathematical Society, 47, 221–241.CrossRefGoogle Scholar
Dedekind, R. (1872). Stetigkeit und irrationale Zahlen. Braunschweig, Germany: Vieweg und Sohn. Cited from Dedekind (1932b).Google Scholar
Dedekind, R. (1877). Sur la théorie des nombres entiers algébriques. Paris, France: Gauthier-Villars.Google Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen? Braunschweig, Germany: Vieweg und Sohn. Cited from Dedekind (1932b).Google Scholar
Dedekind, R. (1890a). Letter to Keferstein. Dated February 27, 1890. Cited from Sinaceur (1974).Google Scholar
Dedekind, R. (1890b). Über den Begriff des Unendlichen. Unpublished reply to Keferstein. Cited from Sinaceur (1974).Google Scholar
Dedekind, R. (1894). Über die Theorie der ganzen algebraischen Zahlen. Supplement XI in Dirichlet (1894). Cited from Dedekind (1932b).Google Scholar
Dedekind, R. (1897). Über Zerlegungen von Zahlen durch ihre größten gemeinsame Teiler. In Beckurts, H., editor. Festschrift der Herzoglichen Technischen Hochschule Carolo-Wilhelmina. Braunschweig, Germany: Vieweg und Sohn, pp. 1–40. Cited from Dedekind (1932a).Google Scholar
Dedekind, R. (1932a). Gesammelte mathematische Werke, Vol. 2. Braunschweig, Germany: Vieweg und Sohn. Available from: http://gdz.sub.uni-goettingen.de/.Google Scholar
Dedekind, R. (1932b). Gesammelte mathematische Werke, Vol. 3. Braunschweig, Germany: Vieweg und Sohn. Available from: http://gdz.sub.uni-goettingen.de/.Google Scholar
Dedekind, R. (1932c). Gesammelte mathematische Werke, Vol. 1. Braunschweig, Germany: Vieweg und Sohn. Available from: http://gdz.sub.uni-goettingen.de/.Google Scholar
Dedekind, R. (1996). Theory of Algebraic Integers. Cambridge, UK: Cambridge University Press. Translated by Stillwell, J.CrossRefGoogle Scholar
Demopoulos, W. (1994). Frege, Hilbert, and the conceptual structure of model theory. History and Philosophy of Logic, 15, 211–225.CrossRefGoogle Scholar
Dirichlet, P. G. L. (1894). Vorlesungen über Zahlentheorie (fourth edition). Braunschweig, Germany: Vieweg und Sohn. Edited and with supplements by Dedekind, R.Google Scholar
Dugac, P. (1976). Richard Dedekind et les Fondements des Mathémathiques (avec de nombreux textes inédits). Paris, France: Vrin.Google Scholar
Ferreirós, J. (1996). Traditional logic and the early history of sets, 1854-1908. Archive for History of Exact Sciences, 50, 5–71.CrossRefGoogle Scholar
Ferreirós, J. (1999). Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics. Science Networks – Historical Studies. Basel, Switzerland: Birkhäuser.CrossRefGoogle Scholar
Ferreirós, J. (2007). Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics (second edition). Basel, Switzerland: Birkhäuser. Reprint of Ferreirós (1999) with added postscript.Google Scholar
Ferreirós, J. (2009). Hilbert, logicism, and mathematical existence. Synthese, 170, 33–70.CrossRefGoogle Scholar
Ferreirós, J. (2012). On Dedekind’s logicism. In Arana, A., and Alvarez, C., editors. Analytic Philosophy and the Foundations of Mathematics, History of Analytic Philosophy. London: Palgrave/Macmillan.Google Scholar
Fine, K. (1985). Reasoning with Arbitrary Objects. Aristotelian Society Series. Oxford, UK: Blackwell.Google Scholar
Frege, G. (1879). Begriffsschrift. Halle, Germany: Louis Nebert.Google Scholar
Frege, G. (1893). Grundgesetze der Arithmetik I. Jena, Germany: Hermann Pohle.Google Scholar
Frege, G. (1903). Über die grundlagen der Geometrie. Jahresbericht der Deutschen Mathematiker-Vereinigung, 12, 319–324, 368–375.Google Scholar
Frege, G. (1906). Über die grundlagen der Geometrie. Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, 293–309, 377–403, 423–430.Google Scholar
Frege, G. (1976). Wissenschaftlicher Briefwechsel. Hamburg, Germany: Felix Meiner Verlag.CrossRefGoogle Scholar
Gabriel, G. (1978). Implizite Definitionen—eine Verwechslungsgeschichte. Annals of Science, 35, 419–423.CrossRefGoogle Scholar
Gauss, C. F. (1966). Disquisitiones Arithmeticae. New Haven, CT: Yale University Press. Translated by Clarke, A. A.Google Scholar
Hacker, P. (1986). Insight and Illusion (second revised edition). Oxford, UK: Clarendon Press.Google Scholar
Halbfass, W. (1971). Evidenz. In Ritter, J., et al. , editors. Historisches Wörterbuch der Philosophie. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft.Google Scholar
Hallett, M. (1994). Hilbert’s axiomatic method and the laws of thought. In George, A., editor. Mathematics and Mind. Oxford, UK: Oxford University Press, pp. 158–200.CrossRefGoogle Scholar
Hallett, M. (2008). Purity of method in Hilbert’s Grundlagen der Geometrie. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 198–255.CrossRefGoogle Scholar
Hallett, M. (2010). Frege and Hilbert. In Potter, M., and Ricketts, T., editors. Cambridge Companion to Frege. Cambridge, UK: Cambridge University Press, pp. 413–464.CrossRefGoogle Scholar
Hallett, M., & Majer, U., editors. (2004). David Hilbert’s Lectures on the Foundations of Geometry. Heidelberg, Germany: Springer.Google Scholar
Heck, R. G. (1995). Definition by induction in Frege’s Grundgesetze der Arithmetik. In Demopoulos, W., editor. Frege’s Philosophy of Mathematics. Cambridge, MA: Harvard University Press, pp. 295–333.Google Scholar
Hilbert, D. (1897). Die Theorie der algebraischen Zahlkörper. Jahresbericht der Deutschen Mathematiker-Vereinigung, 4, 175–546.Google Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig, Germany: Teubner. Reprinted with notes and introduction in Hallett & Majer (2004).Google Scholar
Hilbert, D. (1900a). Mathematische Probleme. Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, 253–297. Cited from Hilbert (1935).Google Scholar
Hilbert, D. (1900b). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, 180–184.Google Scholar
Hilbert, D. (1905). Über die Grundlagen der Logik und der Arithmetik. In Krazer, A., editor. Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904. Leipzig, Germany: Teubner.Google Scholar
Hilbert, D. (1914). Zur Begründung der elementaren Stralungstheorie. Dritte Mitteilung. Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, 275–298. Reprinted in Hilbert (1935).Google Scholar
Hilbert, D. (1918). Axiomatisches Denken. Mathematische Annalen, 78, 405–15.CrossRefGoogle Scholar
Hilbert, D. (1924). Die Grundlagen der Physik. Mathematische Annalen, 92, 1–32. Reprinted in Hilbert (1935).CrossRefGoogle Scholar
Hilbert, D. (1935). Gesammelte Abhandlungen. Dritter Band. Berlin: Springer. Available from: http://gdz.sub.uni-goettingen.de/.Google Scholar
Husserl, E. (1891). Philosophie der Arithmetik. Halle, Germany: C.E.M. Pfeffer. Cited from Husserl (1970).Google Scholar
Husserl, E. (1900). Logische Untersuchungen. Erster Theil: Prolegomena zur reinen Logik. Halle, Germany: Max Niemeyer.Google Scholar
Husserl, E. (1913). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch. Halle, Germany: Max Niemeyer.Google Scholar
Husserl, E. (1929). Formale und Transzendentale Logik. Halle, Germany: Max Niemeyer.Google Scholar
Husserl, E. (1970). Philosophie der Arithmetik. Mit ergänzenden Texten (1890-1901). Number XII in Husserliana. Den Haag, The Netherlands: Martinius Nijhoff.Google Scholar
Keferstein, H. (1890). Über den Begriff der Zahl. Mitteilungen der Mathematische Gesellschaft zu Hamburg, 2, 119–125.Google Scholar
Kelly, T. (2008). Evidence. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Fall 2008 Edition). Retreived fromhttp://plato.stanford.edu/entries/evidence/Google Scholar
Kleene, S. C. (1952). Introduction to Metamathematics. University Series in Higher Mathematics. New York, NY: Van Norstrand.Google Scholar
Lafuma, L., editor. (1963). Pascal. Æuvres Complètes. Paris, France: Éditions du Seuil.Google Scholar
Lambert, J. H. (1764). Neues Organon. Zweyter Band. Leipzig, Germany: Wendler.Google Scholar
Leśniewski, S.. (1931). Über Definition in der sogenannten Theorie der Deduktion. Compte Rendus des séances de la Société des Sciences et des Letters de Varsovie (cl. III), 24, 289–309. Translated in McCall (1967).Google Scholar
Locke, J. (1975). An Essay Concerning Human Understanding. Oxford, UK: Oxford University Press. Based on the fourth edition (1700).Google Scholar
Mancosu, P., editor. (2008). The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Martin-Löf, P. (1984). Intuitionistic Type Theory. Studies in proof theory. Naples, Italy: Bibliopolis.Google Scholar
Mehrtens, H. (1979). Die Entstehung der Verbandstheorie. Arbor Scientarum. Hildesheim, Germany: Gerstenberg.Google Scholar
Neurath, O. (1932). Protokollsätze. Erkenntnis, 3, 204–214.CrossRefGoogle Scholar
Oeing-Hanhoff, L. (1971). Axiom – Geschichte. In Ritter, J., et al. , editors. Historisches Wörterbuch der Philosophie. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft.Google Scholar
Pascal, B. (1657). De l’esprit géométrique. Cited from Lafuma (1963).Google Scholar
Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig, Germany: Teubner.Google Scholar
Petri, B., & Schappacher, N. (2007). On arithmetization. In Goldstein, C., Schappacher, N., and Schwermer, J., editors. The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae. Heidelberg, Germany: Springer.Google Scholar
Quine, W. V. O. (1982). Methods of Logic (fourth edition). Cambridge, MA: Harvard University Press.Google Scholar
Reck, E. H. (2003). Dedekind’s structuralism: An interpretation and partial defense. Synthese, 137, 369–419.CrossRefGoogle Scholar
Reck, E. H., & Price, M. P. (2000). Structures and structuralism in contemporary philosophy of mathematics. Synthese, 125, 341–383.CrossRefGoogle Scholar
Resnik, M. D. (1974). The Frege-Hilbert controversy. Philosophy and Phenomenological Research, 34, 386–403.CrossRefGoogle Scholar
Schlick, M. (1918). Allgemeine Erkenntnislehre. Berlin: Julius Springer.Google Scholar
Schlimm, D. (2000). Richard Dedekind: Axiomatic foundations of mathematics. Master’s thesis, Carnegie Mellon University, Pittsburgh, PA.Google Scholar
Schlimm, D. (2010). Pasch’s philosophy of mathematics. Review of Symbolic Logic, 3, 93–118.CrossRefGoogle Scholar
Schlimm, D. (2011). On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others. Synthese, 183, 47–68.CrossRefGoogle Scholar
Scholz, H. (1930). Die Axiomatik der Alten. Blättern für deutsche Philosophie, 4, 259–278. Cited from Scholz (1961). English translation in Barnes et al. (1975).Google Scholar
Scholz, H. (1942). David Hilbert, der Altmeister der mathematischen Grundlagen-forschung. In Scholz, H. editor. Mathesis Universalis. Abhandlungen zur Philosophie als strenger Wissenschaft. Basel, Switzerland: Benno Schwabe & Co Verlag, pp. 279–290. Not otherwise published.Google Scholar
Scholz, H. (1945). Pascals Forderungen an die mathematische Methode. In Festschrift zum 60. Geburtstag von A. Speiser. Zürich, Switzerland: Orell Füssli. Cited from Scholz (1961).Google Scholar
Scholz, H. (1961). Mathesis Universalis. Abhandlungen zur Philosophie als strenger Wissenschaft. Basel, Switzerland: Benno Schwabe & Co Verlag.Google Scholar
Schröder, E. (1890). Vorlesungen über die Algebra der Logik, Vol. 1. Leipzig, Germany: Teubner.Google Scholar
Shoenfield, J. (1977). The axioms of set theory. In Barwise, J., editor. Handbook of Mathematical Logic, Studies in logic and the foundations of mathematics. Amsterdam, The Netherlands: North-Holland, pp. 321–344.CrossRefGoogle Scholar
Sieg, W. (1990). Relative consistency and accessible domains. Synthese, 84, 259–297.CrossRefGoogle Scholar
Sieg, W., & Schlimm, D. (2005). Dedekind’s analysis of number: Systems and axioms. Synthese, 147, 121–170.CrossRefGoogle Scholar
Sinaceur, M. A. (1974). L’infini et les nombres. Revue d’histoire des sciences, 27(3), 251–278.CrossRefGoogle Scholar
Steck, M. (1940). Ein unbekannter Brief von Gottlob Frege über Hilberts erste Vorlesung über die Grundlagen der Geometrie. Sitzungberichte der Heidelberger Akademie der Wissenschaften. Math-Nat Klasse, (6).Google Scholar
Stein, H. (1988). Logos, logic, and logistiké. In Aspray, W. and Kitcher, P., editors. History and Philosophy of Modern Mathematics, Vol. XI of Minnesota Studies in the Philosophy of Science. Minneapolis, MN: University of Minnesota Press, pp. 238–259.Google Scholar
Sundholm, B. G. (2002). What is an expression? In Logica Yearbook 2001. Prague, Czech Republic: Filosofia Publishers, Czech Academy of Science, pp. 181–194.Google Scholar
Sundholm, B. G. (2009). A century of judgement and inference, 1837-1936: Some strands in the development of logic. In Haaparanta, L., editor. The Development of Modern Logic. Oxford, UK: Oxford University Press, pp. 263–317.CrossRefGoogle Scholar
Tait, W. W. (1997). Frege versus Cantor and Dedekind: On the concept of number. In Tait, W. W., editor. Early Analytic Philosophy: Frege, Russell, Wittgenstein. Chicago, IL: Open Court, pp. 213–248.Google Scholar
Tappenden, J. (2005). The Caesar Problem in its historical context: Mathematical background. Dialectica, 59, 237–264.CrossRefGoogle Scholar
Tappenden, J. (2008a). Mathematical concepts and definitions. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 256–275.CrossRefGoogle Scholar
Tappenden, J. (2008b). Mathematical concepts: Fruitfulness and naturalness. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 276–301.CrossRefGoogle Scholar
Tichý, P. (1988). The Foundations of Frege’s Logic. Foundations of Communication. Berlin: Walter de Gruyter.CrossRefGoogle Scholar
Tolley, C. (2011). Frege’s elucidatory holism. Inquiry, 54, 226–251.CrossRefGoogle Scholar
van der Waerden, B. L. (1930). Moderne Algebra. Berlin: Springer.CrossRefGoogle Scholar
Weber, H. (1893). Die allgemeinen Grundlagen der Galois’schen Gleichungstheorie. Mathematische Annalen, 43, 521–549.CrossRefGoogle Scholar
Wittgenstein, L. (1922). Tractatus Logicco-philosophicus. International Library of Psychology, Philosophy, and Scientific Method. London: Routledge & Kegan Paul.Google Scholar
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261–281.CrossRefGoogle Scholar