Skip to main content

Epistemic Horizons and the Foundations of Quantum Mechanics

Foundations of Physics volume 48, pages 1669–1697 (2018)Cite this article

Abstract

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  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)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

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

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Chapter  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  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. Wheeler, J.: Add “Participant” to “Undecidable Propositions” to arrive at Physics (1974). https://jawarchive.files.wordpress.com/2012/03/twa-1974.pdf

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

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

  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)

    Chapter  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Im Weidenbruch 113, 51061, Cologne, Germany

    Jochen Szangolies

Authors
  1. Jochen Szangolies
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jochen Szangolies.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Szangolies, J. Epistemic Horizons and the Foundations of Quantum Mechanics. Found Phys 48, 1669–1697 (2018). https://doi.org/10.1007/s10701-018-0221-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-018-0221-9

Keywords

  • Quantum foundations
  • Diagonal arguments
  • Self-reference