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Epistemic Horizons and the Foundations of Quantum Mechanics


In-principle restrictions on the amount of information that can be gathered about a system have been proposed as a foundational principle in several recent reconstructions of the formalism of quantum mechanics. However, it seems unclear precisely why one should be thus restricted. We investigate the notion of paradoxical self-reference as a possible origin of such epistemic horizons by means of a fixed-point theorem in Cartesian closed categories due to Lawvere that illuminates and unifies the different perspectives on self-reference.

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  1. 1.

    Barrow, J.D., Davies, P.C.W., Harper Jr., C.L.: Science and Ultimate Reality: Quantum Theory, Cosmology, and Complexity. Cambridge University Press, Cambridge (2004)

    MATH  Book  Google Scholar 

  2. 2.

    Grinbaum, A.: Elements of information-theoretic derivation of the formalism of quantum theory. Int. J. Quantum Inf. 1(03), 289–300 (2003)

    MATH  Article  Google Scholar 

  3. 3.

    Rovelli, C.: Relational quantum mechanics. Int. J. Theor. Phys. 35(8), 1637–1678 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Brukner, Č., Zeilinger, A.: Information and fundamental elements of the structure of quantum theory. In: Castell, L., Ischebeck, O. (eds.) Time, Quantum and Information, pp. 323–354. Springer, Berlin (2003)

    MATH  Chapter  Google Scholar 

  5. 5.

    Fuchs, C. A.: Quantum mechanics as quantum information (and only a little more) (2002). arXiv:quant-ph/0205039

  6. 6.

    Masanes, L., Müller, M.P., Augusiak, R., Pérez-García, D.: Existence of an information unit as a postulate of quantum theory. Proc. Natl. Acad. Sci. 110(41), 16373–16377 (2013)

    ADS  Article  Google Scholar 

  7. 7.

    Höhn, P.A., Wever, C.S.P.: Quantum theory from questions. Phys. Rev. A 95(1), 012102 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    von Weizsäcker, C.F., Görnitz, T., Lyre, H.: The Structure of Physics. Springer, Berlin (2006)

    Google Scholar 

  9. 9.

    Spekkens, R.W.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75(3), 032110 (2007)

    ADS  Article  Google Scholar 

  10. 10.

    Curtright, T.L., Zachos, C.K.: Quantum mechanics in phase space. Asia Pac. Phys. Newsl. 1(01), 37–46 (2012)

    Article  Google Scholar 

  11. 11.

    Grinbaum, A.: Information-theoretic princple entails orthomodularity of a lattice. Found. Phys. Lett. 18(6), 563–572 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Chaitin, G.J.: Undecidability and randomness in pure mathematics. In: Cornwell, J. (ed.) Information, Randomness & Incompleteness, pp. 307–313. World Scientific, Singapore (1990)

    MATH  Chapter  Google Scholar 

  14. 14.

    Yurtsever, U.: Quantum mechanics and algorithmic randomness. Complexity 6(1), 27–34 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Bendersky, A., Senno, G., de la Torre, G., Figueira, S., Acin, A.: Nonsignaling deterministic models for nonlocal correlations have to be uncomputable. Phys. Rev. Lett. 118(13), 130401 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Calude, C.S., Svozil, K.: Quantum randomness and value indefiniteness. Adv. Sci. Lett. 1(2), 165–168 (2008)

    Article  Google Scholar 

  17. 17.

    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1969)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6–7), 467–488 (1982)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Edis, T.: How Gödel’s theorem supports the possibility of machine intelligence. Minds Mach. 8(2), 251–262 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Chaitin, G.J.: A theory of program size formally identical to information theory. J. ACM 22(3), 329–340 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. J. Math. 58(345–363), 5 (1936)

    MATH  Google Scholar 

  22. 22.

    Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin \(\Omega \) numbers. In: Annual Symposium on Theoretical Aspects of Computer Science, pp. 596–606. Springer, Berlin (1998)

  23. 23.

    Svozil, K.: Randomness and Undecidability in Physics. World Scientific, Singapore (1993)

    MATH  Book  Google Scholar 

  24. 24.

    Svozil, K.: Physical (A)Causality. Fundamental Theories of Physics, vol. 192. Springer, Cham (2018)

    MATH  Book  Google Scholar 

  25. 25.

    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38(1), 173–198 (1931)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Howard, W.A.: The formulae-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To HB Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 479–490. Academic Press, London (1980)

    Google Scholar 

  27. 27.

    Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Chaitin, G.J.: Gödel’s theorem and information. Int. J. Theor. Phys. 21(12), 941–954 (1982)

    MATH  Article  Google Scholar 

  29. 29.

    Cubitt, T.S., Perez-Garcia, D., Wolf, M.M.: Undecidability of the spectral gap. Nature 528(7581), 207–211 (2015)

    ADS  Article  Google Scholar 

  30. 30.

    Berger, R.: The Undecidability of the Domino Problem, vol. 66. American Mathematical Society, Providence (1966)

    MATH  Google Scholar 

  31. 31.

    Lloyd, S.: Quantum-mechanical computers and uncomputability. Phys. Rev. Lett. 71(6), 943 (1993)

    ADS  Article  Google Scholar 

  32. 32.

    Lloyd, S.: Necessary and sufficient conditions for quantum computation. J. Mod. Opt. 41(12), 2503–2520 (1994)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Eisert, J., Müller, M.P., Gogolin, C.: Quantum measurement occurrence is undecidable. Phys. Rev. Lett. 108(26), 260501 (2012)

    ADS  Article  Google Scholar 

  34. 34.

    Popper, K.R.: Indeterminism in quantum physics and in classical physics. Part I. Br. J. Philos. Sci. 1(2), 117–133 (1950)

    Article  Google Scholar 

  35. 35.

    Rothstein, J.: Thermodynamics and some undecidable physical questions. Philos. Sci. 31(1), 40–48 (1964)

    Article  Google Scholar 

  36. 36.

    Fuchs, C.A.: On participatory realism. In: Durham, I.T., Rickles, D. (eds.) Information and Interaction, pp. 113–134. Springer, Berlin (2017)

    Chapter  Google Scholar 

  37. 37.

    Wheeler, J.: Add “Participant” to “Undecidable Propositions” to arrive at Physics (1974).

  38. 38.

    Bernstein, J.: Quantum Profiles. Princeton University Press, Princeton (1991)

    Book  Google Scholar 

  39. 39.

    Chiara, M.L.D.: Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics. J. Philos. Logic 6(1), 331–347 (1977)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Breuer, T.: The impossibility of accurate state self-measurements. Philos. Sci. 62, 197–214 (1995)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Breuer, T., von Neumann met Kurt Gödel, J.: Undecidable statements in quantum mechanics. In: Chiara, M.L.D., Giuntini, R., Laudisa, F. (eds.) Language. Quantum, Music: Selected Contributed Papers of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995, pp. 159–170. Springer, Dordrecht (1999)

  42. 42.

    Aerts, S.: Undecidable classical properties of observers. Int. J. Theor. Phys. 44(12), 2113–2125 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Zwick, M.: Quantum measurement and Gödel’s proof. Specul. Sci. Technol. 1(2), I978 (1978)

    Google Scholar 

  44. 44.

    Peres, A., Zurek, W.H.: Is quantum theory universally valid? Am. J. Phys. 50(9), 807–810 (1982)

    ADS  Article  Google Scholar 

  45. 45.

    Brukner, Č.: Quantum complementarity and logical indeterminacy. Nat. Comput. 8(3), 449–453 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Paterek, T., Kofler, J., Prevedel, R., Klimek, P., Aspelmeyer, M., Zeilinger, A., Brukner, Č.: Logical independence and quantum randomness. New J. Phys. 12(1), 013019 (2010)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Calude, C.S., Jürgensen, H.: Is complexity a source of incompleteness? Adv. Appl. Math. 1(35), 1–15 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Calude, C.S., Stay, M.A.: From Heisenberg to Gödel via Chaitin. Int. J. Theor. Phys. 46(8), 2013–2025 (2007)

    MATH  Article  Google Scholar 

  49. 49.

    Lawvere, F.W.: Diagonal arguments and Cartesian closed categories. In: Hilton, P.J. (ed.) Category Theory, Homology Theory and Their Applications II, pp. 134–145. Springer, Berlin (1969)

    Chapter  Google Scholar 

  50. 50.

    Yanofsky, N.S.: A universal approach to self-referential paradoxes, incompleteness and fixed points. Bull. Symb. Log. 9(03), 362–386 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Cantor, G.: Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung 1, 75–78 (1892)

    MATH  Google Scholar 

  52. 52.

    Russell, B.: Letter to Frege. In: van Heijenoort, J. (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge (1967)

    Google Scholar 

  53. 53.

    Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philos. 1, 261–405 (1936)

    MATH  Google Scholar 

  54. 54.

    Russell, B.: Mathematical logic as based on the theory of types. Am. J. Math. 30(3), 222–262 (1908)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Ord, T., Kieu, T.D.: The diagonal method and hypercomputation. Br. J. Philos. Sci. 56(1), 147–156 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    De Broglie, L.: La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J. Phys. Radium 8(5), 225–241 (1927)

    MATH  Article  Google Scholar 

  57. 57.

    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables, i and ii. Phys. Rev. 85(2), 166 (1952)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Valentini, A.: Signal-locality in hidden-variables theories. Phys. Lett. A 297(5–6), 273–278 (2002)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Karrass, Abraham: Some remarks on the infinite symmetric groups. Math. Z. 66(1), 64–69 (1956)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Richard, J.: Les Principes des Mathématiques et le Problème des Ensembles. In: van Heijenoort, J. (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge (1967)

    Google Scholar 

  61. 61.

    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299(5886), 802–803 (1982)

    ADS  MATH  Article  Google Scholar 

  62. 62.

    Svozil, K.: A constructivist manifesto for the physical sciences-constructive re-interpretation of physical undecidability. In: Kohler, E., Stadler, F. (eds.) The Foundational Debate, pp. 65–88. Springer, Dordrecht (1995)

    MATH  Chapter  Google Scholar 

  63. 63.

    Kolmogorov, A.N.: On tables of random numbers. Sankhyā 25, 369–376 (1963)

    MathSciNet  MATH  Google Scholar 

  64. 64.

    Levin, L.: On the notion of a random sequence. Sov. Math. Dokl. 14, 1413–1416 (1973)

    MATH  Google Scholar 

  65. 65.

    Schnorr, C.-P.: Process complexity and effective random tests. J. Comput. Syst. Sci. 7(4), 376–388 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  66. 66.

    Chaitin, G.J.: Information-theoretic incompleteness. Appl. Math. Comput. 52(1), 83–101 (1992)

    MathSciNet  MATH  Google Scholar 

  67. 67.

    Shimony, A.: Metaphysical problems in the foundations of quantum mechanics. Int. Philos. Q. 18(1), 3–17 (1978)

    Article  Google Scholar 

  68. 68.

    Bohr, N.: The causality problem in atomic physics. New Theor. Phys. 147, 11–30 (1939)

    Google Scholar 

  69. 69.

    Svozil, K.: Undecidability everywhere? In: Casti, J.L., Karlqvist, A. (eds.) Boundaries and Barriers: On the Limits to Scientific Knowledge, pp. 215–237. Basic Books, New York (1996)

    Google Scholar 

  70. 70.

    von Neumann, J.: Mathematical Foundations of Quantum Mechanics, vol. 2. Princeton University Press, Princeton (1955)

    MATH  Google Scholar 

  71. 71.

    Van den Nest, M., Briegel, H.J.: Measurement-based quantum computation and undecidable logic. Found. Phys. 38(5), 448–457 (2008)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    Baez, J.C.: Quantum quandaries: a category-theoretic perspective. In: Rickles, R.C., French, S.R., Saatsi, J.T. (eds.) Structural Foundations of Quantum Gravity, pp. 240–267. Oxford University Press, Oxford (2006)

    MATH  Chapter  Google Scholar 

  73. 73.

    Baez, J., Stay, M.: Physics, topology, logic and computation: a Rosetta Stone. In: Coecke, R. (ed.) New Structures for Physics, pp. 95–172. Springer, Berlin (2010)

    MATH  Chapter  Google Scholar 

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My first and foremost thanks is due to Dagmar Bruß and Hermann Kampermann, whose guidance and tutelage I had the great privilege to receive, and who have been instrumental in the sharpening of the ideas presented here. Furthermore, I wish to thank Karl Svozil and Noson Yanofsky for invaluable discussion of the material compiled in this article.

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Correspondence to Jochen Szangolies.

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Szangolies, J. Epistemic Horizons and the Foundations of Quantum Mechanics. Found Phys 48, 1669–1697 (2018).

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  • Quantum foundations
  • Diagonal arguments
  • Self-reference