Arithmetical Foundations for Physical Theories

  • Yvon Gauthier
Part of the Studies in Universal Logic book series (SUL)


The notion of analytical apparatus was introduced in the paper by Hilbert, Nordheim and von Neumann (1928) ≪ Über die Grundlagen der Quantenmechanik ≫ : The analytical apparatus <der analytische Apparat> is simply the mathematical formalism or the set of logico-mathematical structures of a physical theory. Von Neumann used the notion extensively in his 1932 seminal work on the mathematical foundations of Quantum Mechanics and he stressed particularly the auxiliary notion of conditions of reality <Realitätsbedingungen>. I want to show that this last notion corresponds grosso modo to our notion of model in contemporary philosophy of physics. Hermann Weyl, who has initiated group theory in Quantum Mechanics (1928), also exploited the idea in his conception of the parallelism between a mathematical formalism and its physical models.


Hilbert Space Physical Theory Hermitian Operator Local Observer Analytical Apparatus 
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.


  1. E. Bishop (1967): Foundations of Constructive Analysis. New York, McGraw-Hill.zbMATHGoogle Scholar
  2. L. E. J. Brouwer (1910): Beweis der Invarianz der Dimensionenzahl. Math. Annalen, 70:161–165.MathSciNetCrossRefGoogle Scholar
  3. L. E. J Brouwer (1912a): Beweis der Invarianz des n-dimensionalen Gebietes. Math. Annalen, 71:305–313.zbMATHCrossRefGoogle Scholar
  4. L. E. J. Brouwer (1912b): Über Abbildungen von Mannigfaltigkeiten. Math, 38(71):97–115.Google Scholar
  5. V. Busek and M. Hillery (1996): Quantum Copying: Beyond the No-Cloning Theorem. arXiv:quant-ph/19607018v1.Google Scholar
  6. G. Cantor (1966): Über einer Satz aus der Theorie der stetigen Mannigfaltigkeiten. Abhandlungen mathematisch und philosophischen Inhalts, Georg Olms, Hildesheim:134–138.Google Scholar
  7. J. Conway and S. Kochen (2006): The free will theorem. Found. Physics, 36:1441–1473.zbMATHMathSciNetCrossRefGoogle Scholar
  8. J. Conway and S. Kochen (2009): The strong free will theorem. Not. Amer. Math. Soc., 56:226–232.zbMATHMathSciNetGoogle Scholar
  9. D. Diecks (1982): Communication by EPR devices. Phys. Lett. A, 92:271–272.CrossRefGoogle Scholar
  10. J. Dowker and D. Kent (1996): On the Consistent Histories Approach to Quantum Mechanics. Journal of Statistical Mechanics, 82((5/6)): 1575–1646.Google Scholar
  11. Y. Gauthier (1983a): Le constructivisme de Herbrand. Journal of Symbolic Logic, 48(4):1230.MathSciNetGoogle Scholar
  12. Y. Gauthier (1983b): Quantum Mechanics and the Local Observer. Int. J. Theor, Physics, 72:1141–1152.Google Scholar
  13. Y. Gauthier (1985): A Theory of Local Negation. The Model and Some Applications. Archiv für mathematische Logik und Grundlagenforschung, 25:127–143.Google Scholar
  14. Y. Gauthier (2002): Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. Kluwer, Synthese Library, Dordrecht/Boston/London.zbMATHGoogle Scholar
  15. Y. Gauthier (2005): Hermann Weyl on Minkowskian Space-Time and Riemannian Geometry. International Studies in the Philosophy of Science, 19:262–269.MathSciNetCrossRefGoogle Scholar
  16. Y. Gauthier (2009a): The Construction of Chaos Theory. Found. Sci., 14:153–165.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Y. Gauthier (2009b): Classical Functional Theory and Applied Proof Theory. International Journal of Pure and Applied Mathematics, 56(2): 223–233.zbMATHMathSciNetGoogle Scholar
  18. Y. Gauthier (2013b): A General No-Cloning Theorem for an Infinite Multiverse. Reports in Mathematical Physics, 72:191–199.zbMATHMathSciNetCrossRefGoogle Scholar
  19. A.M. Gleason (1957): Measures on the Closed Subspaces of a Hilbert Space. Journal of Mathematics and Mechanics, 6:885–893.zbMATHMathSciNetGoogle Scholar
  20. S. Goldstein and D.N. Page (1995): Linearly Positive Histories Probabilities for a Robust Family of Sequences of Quantum Events. Physical Review Letters, 74:19.MathSciNetCrossRefGoogle Scholar
  21. R.B. Griffiths (1984): Consistent Histories and the Interpretation of Quantum Mechanics. Journal of Statistical Physics, 36:219.zbMATHMathSciNetCrossRefGoogle Scholar
  22. P. Halmos (1957): Introduction to Hilbert Space and the Theory of Spectral Multiplicity. 2nd ed., New York, Chelsea.zbMATHGoogle Scholar
  23. D. Hilbert (1932): Gesammelte Abhandlungen III. Chelsea, New York.Google Scholar
  24. E. Hrukovski and I. Pitkowsky (2004): Generalizations of Kochen and Specker’s Therorem and the effectiveness of Gleason’s Therorem. Studies in History and Philosophy of Science Part B, 35(2):177–184.MathSciNetCrossRefGoogle Scholar
  25. J.M. Jauch (1968): Foundations of Quantum Mechanics. Reading, Mass.zbMATHGoogle Scholar
  26. S. Kochen and E.P. Specker (1967): The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics, 17:59–87.zbMATHMathSciNetGoogle Scholar
  27. L. Kronecker (1889): Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Werke, II:47–208.Google Scholar
  28. G. Lindblad (1999): A General No-Cloning Theorem. Letters in Mathematical Physics, 47:189–196.zbMATHMathSciNetCrossRefGoogle Scholar
  29. H. Minkowski (1967): Gesammelte Abhandlungen,. hrsg. v. D. Hilbert (Chelsea, New York).Google Scholar
  30. E. Nelson (1987a): Predicative Arithmetic. Number 32 of Mathematical Notes. Princeton University Press, Princeton, N.J.Google Scholar
  31. E. Nelson (1987b): A Radically Elementary Probability Theory. Princeton University Press, Princeton, N. J.Google Scholar
  32. M.C.G. Redhead (1987): Incompleteness, Nonlocality and Realism. Oxford, Clarendon Press.zbMATHGoogle Scholar
  33. B. Riemann (1990): Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachtrage. Collected Papers. neu hrsg. v. R. Naramsihan, (B.G. Teubner, Springer-Verlag, Berlin, New York, Leipzig).CrossRefGoogle Scholar
  34. C. Rovelli (1996): Relational Quantum Mechanics. Int. J. Theor, Physics, 35:1637–1678.Google Scholar
  35. D. Schwarz-Perlow and A. Vilenkin (2010): Measures for a Transdimensional Universe. arXiv: 1004. 4567v2.Google Scholar
  36. B. van Fraassen (1991): Quantum Mechanics. An Empiricist View. Oxford, Clarendon Press.CrossRefGoogle Scholar
  37. A. Vilenkin (2006): A Measure of the Universe. arXiv: hep-th/0609193v3.Google Scholar
  38. J. von Neumann (1932): Mathematische Grundlagen der Quantenmechanik. Berlin, Springer.zbMATHGoogle Scholar
  39. J. von Neumann (1961): Collected Works, volume I, Eine Axiomatisierung der Mengenlehre, 24–33. Pergamon Press, Oxford.Google Scholar
  40. H. Weyl (1918): Das Kontinuum, Veit, Leipzig.Google Scholar
  41. H. Weyl (1920): Das Verhältnis der kausalen zur statistischen Betrachtungsweise in der Physik. Schweizerische Medizinische Wochenschrift, 50:537–541.Google Scholar
  42. H. Weyl (1950): Space, Time, Matter. trans. By H. L. Brose (Dover Publications, New York).Google Scholar
  43. H. Weyl (1960): Philosophy of Mathematics and Natural Science. Atheneum, New York.Google Scholar
  44. H. Weyl (1968): Gesammelte Abhandlungen. hrsg. v. K. Chandrasekharan (Springer-Verlag, Berlin, Heidelberg, New York).Google Scholar
  45. W. Wootters and W. Zurek (2009): The no-cloning theorem. Phys. Today, 62(2):76–66.CrossRefGoogle Scholar
  46. W.K. Wootters and W.H. Zurek (1982): A single quantum cannot be cloned. Nature, 299:802–803.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yvon Gauthier
    • 1
  1. 1.University of MontrealMontrealCanada

Personalised recommendations