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Hilbert’s Axiomatic Method and His “Foundations of Physics”: Reconciling Causality with the Axiom of General Invariance

  • Katherine A. Brading
  • Thomas A. Ryckman
Chapter
Part of the Einstein Studies book series (EINSTEIN, volume 12)

Abstract

In November and December 1915, Hilbert gave two presentations to the Royal G?ottingen Academy of Sciences under the common title ‘The Foundations of Physics’. Distinguished as ‘First Communication’ (Hilbert, 1915b) and ‘Second Communication’ (Hilbert, 1917), the two ‘notes’, as they are widely known, eventually appeared in the Nachrichten of the Academy. The first quickly entered the canon of classical general relativity but has recently become the object of renewed scholarly scrutiny since the discovery (Corry, Renn and Stachel 1997) of a set of printer’s proofs dated December 6, 1915 ((Hilbert, 1915a), henceforth ‘Proofs’). Hilbert’s second presentation has not received the same detailed reconsideration, with the recent exception of an extended study offered by Renn and Stachel (1999/2007). While we agree with much of their detailed technical reconstruction, we profoundly disagree with the assessment of Renn and Stachel that the second note shows that Hilbert had abandoned his own project (set out in the first note), and is working on a variety of largely unrelated problems within Einstein’s. In our opinion, this assessment rests on misunderstandings concerning the aims, content, and significance of the second communication, as well as its links to the first. Our aim in this paper is to offer an alternate narrative, according to which Hilbert’s second note emerges as a natural continuation of the first, containing important and interesting further developments of that project, and above all shedding needed illumination on Hilbert’s assessment of the epistemological novelty posed by a generally covariant physics.

Keywords

General Covariance General Invariance Timelike Curve Covariant Theory Causality Principle 
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.

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References

  1. Born, Max. (1914). “Der Impuls-Energie-Satz in der Elektrodynamik von Gustav Mie”, Nachrichten von der Königliche Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalisch Klasse 12, 23–36.Google Scholar
  2. ——. (1922). “Hilbert und die Physik”, Die Naturwissenschaften 10, 88–93.CrossRefGoogle Scholar
  3. Brading, Katherine A. (2005). “A Note on General Relativity, Energy Conservation, and Noether’s Theorems”, in A. J. Kox and J. Eisenstaedt (eds.), The Universe of General Relativity (Einstein Studies v. 11). Boston: Birkh‥auser, 125–35.Google Scholar
  4. Brading, Katherine A. and Thomas A. Ryckman (2008). “Hilbert’s ‘Foundations of Physics’: Gravitation and Electromagnetism Within the Axiomatic Method”, Studies in the History and Philosophy of Modern Physics 39, 102–53.CrossRefMATHMathSciNetGoogle Scholar
  5. Corry, Leo (1999). “Hilbert and Physics (1900–1915)”, in Jeremy Gray (ed.), The Symbolic Universe: Geometry and Physics 1890–1930. Oxford: Oxford University Press, 145–88.Google Scholar
  6. ——. (2004). David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik. Dordrecht: Kluwer Academic Publishers.Google Scholar
  7. Corry, Leo, J‥urgen Renn, and John Stachel (1997). “Belated Decision in the Hilbert–Einstein Priority Dispute”, Science 278, 1270–3.CrossRefMathSciNetGoogle Scholar
  8. Earman, John and Clark Glymour (1978). “Einstein and Hilbert: Two Months in the History of General Relativity”, Archive for History of Exact Sciences 19, 291–308.CrossRefMathSciNetGoogle Scholar
  9. Einstein, Albert (1915). “Die Feldgleichungen der Gravitation”. Königlich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte, 844–847.Google Scholar
  10. Reprinted in A. J. Kox, Martin Klein, and Robert Schulmann (eds.), The Collected Papers of Albert Einstein, volume 6. Princeton: Princeton University Press, 1996, 244–9.Google Scholar
  11. ——. (1916). “Die Grundlage der allgemeinen Relativit¨atstheorie”, Annalen der Physik 49: 769–822. Reprinted in A.J. Kox, Martin Klein, and Robert Schulmann (eds.), The Collected Papers of Albert Einstein, volume 6. Princeton: Princeton University Press, 1996, 283–339.Google Scholar
  12. Ewald, William (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford: Clarendon Press.Google Scholar
  13. Gray, Jeremy (2000). The Hilbert Challenge. Oxford: Oxford University Press.Google Scholar
  14. Guth, E. (1970). “Contribution to the History of Einstein’s Geometry as a Branch of Physics”, in Moshe Carmeli et al. (eds.), Relativity. New York: Plenum Press, 161–207.Google Scholar
  15. Hallett, Michael (1994). “Hilbert’s Axiomatic Method and the Laws of Thought”, in Alexander George (ed.),Mathematics andMind. New York: Oxford University Press, 158–200.Google Scholar
  16. Hilbert, David (1899). Grundlagen der Geometrie. In Festschrift zur Feier der Enthullung des Gauss-Weber Denkmals in Göttingen. Leipzig, Teubner.Google Scholar
  17. ——. (1900). “Mathematische Probleme”. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris. Königlichen Gesellschaft der Wissenschaften zu Göttingen, Nachrichten, 253–97. Translation in Jeremy Gray (2000), 240–82.Google Scholar
  18. ——. (1915a). “Die Grundlagen der Physik (Erste Mitteilung)”. Annotated “Erste Korrektur meiner erste Note”, printer’s stamp date “6 Dez. 1915”. 13 pages with omissions. Göttingen, SUB Cod. Ms. 634. Translation in Renn and Schemmel (2007), 989–1001.Google Scholar
  19. ——. (1915b). “Die Grundlagen der Physik: Erste Mitteilung”, Königliche Gesellschaft der Wissenschaften zu Göttingen. Nachrichten. Mathematische-Physikalische Klasse, 395–407. Translation in Renn and Schemmel (2007), 1003–15.Google Scholar
  20. ——. (1916a). “Die Grundlagen der Physik”. Typescript of summer semester lecture notes. Bibliothek des Mathematisches Institut, Universit‥at Göttingen. 111 pages.Google Scholar
  21. ——. (1916b). “Das Kausalit¨atsprinzip in der Physik”. Typescript of lectures, dated 21 and 28 November 1916 by Sauer (2001). Bibliothek des Mathematisches Institut, Universit‥at Göttingen. 17 pages.Google Scholar
  22. ——. (1917). “Die Grundlagen der Physik: Zweite Mitteilung”, Königliche Gesellschaft der Wissenschaften zu Göttingen. Nachrichten. Mathematische-Physikalische Klasse, 53–76. Translation in Renn and Schemmel (2007), 1017–38.Google Scholar
  23. ——. (1918). “Axiomatisches Denken”, Mathematische Annalen, 78, 405–15.Google Scholar
  24. Translated by William Ewald as “Axiomatic Thought”, in Ewald (1996), 1105–15.Google Scholar
  25. ——. (1919–1920). Natur und mathematisches Erkennen. Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von P. Bernays. Published and edited by David Rowe. Basel: Birkh‥auser, 1992.Google Scholar
  26. ——. (1922). “Neubegr¨undung der Mathematik. Erste Mitteilung”, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universit¨at, 1, 157–77.Google Scholar
  27. Translated by William Ewald as “The New Grounding of Mathematics. First Report”, in Ewald (1996), 1115–34.Google Scholar
  28. ——. (1924). “Die Grundlagen der Physik”, Mathematische Annalen 92, 1–32.Google Scholar
  29. ——. (1925). “U¨ ber das Unendliche”, Mathematische Annalen 95, 161–90. Translated by Stefan Bauer-Mengelberg as “On the Infinite”, in J. van HeijenoortGoogle Scholar
  30. (ed.) , From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, 367–92.Google Scholar
  31. ——. (1930). “Naturerkennen und Logik”, Die Naturwissenschaften 18, 959–63.Google Scholar
  32. Translation by William Ewald as “Logic and the Knowledge of Nature”, in Ewald (1996), 1157–65.Google Scholar
  33. Hoefer, Carl (2000). “Energy Conservation in GTR”, Studies in History and Philosophy of Modern Physics 31, 187–99.CrossRefMATHMathSciNetGoogle Scholar
  34. Klein, Felix (1917). “Zu Hilberts erster Note ¨uber die Grundlagen der Physik”, Königliche Gesellschaft der Wissenschaften zu Göttingen. Nachrichten. Google Scholar
  35. Mathematisch-Physikalische Klasse. As reprinted in Felix Klein, Gesammelte Abhandlungen Bd I. Berlin: Julius Springer, 1921, 553–67.Google Scholar
  36. Majer, Ulrich (1995). “Hilbert’s Finitism and the Concept of Space”, in Ulrich Majer and Heinz-J‥urgen Schmidt (eds.), Semantical Aspects of Spacetime Theories. Mannheim:Wissenschaftsverlag, 145–57.Google Scholar
  37. ——. (2001). “The Axiomatic Method and the Foundations of Science: Historical Roots of Mathematical Physics in Göttingen”, in M. Redei and M. Stöltzner (eds.), John von Neumann and the Foundations of Quantum Physics. Dordrecht: Kluwer Academic Publishers, 11–33.Google Scholar
  38. Mehra, Jagdish (1974). Einstein, Hilbert and the Theory of Gravitation. Dordrecht: Reidel.Google Scholar
  39. Mie, Gustav (1917). “Die Einsteinsche Gravitationstheorie und das Probleme der Materie”, Physikalische Zeitschrift 18, 574–80; 596–602.Google Scholar
  40. Pais, Abraham (1983). ‘Subtle is the Lord. . . ’The Science and Life of Albert Einstein. New York: Oxford University Press.Google Scholar
  41. Pauli,Wolfgang, Jr. (1921). Relativit¨atstheorie. In Arnold Sommerfeld (ed.), Encyklop ¨adie der mathematischen Wissenschaften, mit Einschluss ihrer Anwedungen. Leipzig: B.G. Teubner, 539–775. Translated as The Theory of Relativity. Oxford: Pergamon Press, 1958.Google Scholar
  42. Peckhaus, Volker (1990). Hilbertprogramm und Kritische Philosophie. Göttingen: Vandenhoeck & Ruprecht.Google Scholar
  43. Renn, J‥urgen and Matthias Schemmel (eds.). (2007). The Genesis of General Relativity, vol. 4. Gravitation in the Twilight of Classical Physics: The Promise of Mathematics. Dordrecht: Springer.Google Scholar
  44. Renn, J‥urgen and John Stachel (1999). Hilbert’s Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity. Berlin: Max-Planck-Institut f‥ur Wissenschaftsgeschichte, Preprint 118. Reprinted in Renn and Schemmel (2007), 857–973.Google Scholar
  45. Rowe, David (2001). “Einstein Meets Hilbert: At the Crossroads of Mathematics and Physics”, Physics in Perspective 3, 379–424.CrossRefMATHMathSciNetGoogle Scholar
  46. Sauer, Tilman (1999). “The Relativity of Discovery”, Hilbert’s First Note on the Foundations of Physics”, Archive for History of Exact Sciences 53, 529–75.MATHMathSciNetGoogle Scholar
  47. ——. (2001). “The Relativity of Elaboration: Hilbert’s Second Note on the Foundations of Physics”, ms. dated September 10, 2001.Google Scholar
  48. ——. (2005). “Einstein Equations and Hilbert Action: What is Missing on p.8 of the Proofs for Hilbert’s First Communication on the Foundations of Physics”, Archive for History of Exact Sciences 59, 577–90.Google Scholar

Copyright information

© The Center for Einstein Studies 2012

Authors and Affiliations

  • Katherine A. Brading
    • 1
    • 2
  • Thomas A. Ryckman
    • 1
    • 2
  1. 1.University of Notre DameNotre DameUSA
  2. 2.Stanford UniversityStanfordUSA

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