Foundations of Physics

, Volume 32, Issue 7, pp 989–1029 | Cite as

Towards a Coherent Theory of Physics and Mathematics

  • Paul Benioff


As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently strong, and it must maximally describe its own validity and sufficient strength. The mathematical logical definition of validity is used, and sufficient strength is seen to be a necessary and useful concept. The requirement of maximal description of its own validity and sufficient strength may be useful to reject candidate coherent theories for which the description is less than maximal. Other aspects of a coherent theory discussed include universal applicability, the relation to the anthropic principle, and possible uniqueness. It is suggested that the basic properties of the physical and mathematical universes are entwined with and emerge with a coherent theory. Support for this includes the indirect reality status of properties of very small or very large far away systems compared to moderate sized nearby systems. Discussion of the necessary physical nature of language includes physical models of language and a proof that the meaning content of expressions of any axiomatizable theory seems to be independent of the algorithmic complexity of the theory. Gödel maps seem to be less useful for a coherent theory than for purely mathematical theories because all symbols and words of any language must have representations as states of physical systems already in the domain of a coherent theory.

coherent theory physics mathematics 


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  1. 1.
    S. Weinberg, Dreams of a Final Theory (Vintage Books, New York, 1994).Google Scholar
  2. 2.
    B. Greene, The Elegant Universe (Vintage Books, New York, 2000).Google Scholar
  3. 3.
    J. Casti and A. Karlqvist, eds., Boundaries and Barriers, On the Limits to Scientific Knowledge (Perseus Books, Reading, MA, 1996).Google Scholar
  4. 4.
    M. Tegmark, Ann. Phys. 270, 1-51 (1998).Google Scholar
  5. 5.
    J. R. Shoenfield, Mathematical Logic (Addison-Wesley, Reading, MA, 1967).Google Scholar
  6. 6.
    R. Smullyan, Gödel's Incompleteness Theorems (Oxford University Press, New York, 1992).Google Scholar
  7. 7.
    K. Gödel, “Ñber formal unentscheidbare Sätze der Principia Mathematica und Vervandter Systeme I,” Monatschefte für Mathematik und Physik 38, 173-198 (1931).Google Scholar
  8. 8.
    A. A. Frankel, Y. Bar-hillel, A. Levy, and D. van Dalen, Foundations of Set Theory, 2nd revised edn. (Studies in Logic and the Foundations of Mathematics, Vol 67) (North-Holland, Amsterdam, 1973).Google Scholar
  9. 9.
    P. Benioff, J. Math. Phys. 17, 618, 629 (1976).Google Scholar
  10. 10.
    J. Väänänen, Bull. Symbolic Logic 7, 504-519, (2001).Google Scholar
  11. 11.
    P. Benioff, Phys. Rev. A 59, 4223 (1999).Google Scholar
  12. 12.
    A. Heyting, Intuitionism, An Introduction, 3rd revised edn. (North-Holland, New York, 1971).Google Scholar
  13. 13.
    E. Bishop, Foundations of Constructive Analysis (McGraw-Hill, New York, 1967).Google Scholar
  14. 14.
    M. J. Beeson, Foundations of Constructive Mathematics Metamathematical Studies (Springer, New York, 1985).Google Scholar
  15. 15.
    R. Penrose, The Emperor's New Mind (Penguin Books, New York, 1991).Google Scholar
  16. 16.
    P. Davies and R. Hersh, The Mathematical Experience (Birkhäuser, Boston, 1981).Google Scholar
  17. 17.
    M. Kline, Mathematics, The Loss of Certainty (Oxford University Press, New York, 1980).Google Scholar
  18. 18.
    W. W Marek and J. Mycielski, “The foundations of mathematics in the twentieth century,” Amer. Math. Monthly 108, 449-468 (2001).Google Scholar
  19. 19.
    C. H. Papadimitriou, Computational Complexity (Addison-Wesley, Reading, MA, 1994).Google Scholar
  20. 20.
    D. Deutsch, A. Ekert, and R. Lupacchini, Bulletin Symbolic Logic 6, 265-283 (2000).Google Scholar
  21. 21.
    P. Shor, in Proceedings, 35th Annual Symposium on the Foundations of Computer Science, S. Goldwasser, ed. (IEEE Computer Society Press, Los Alamitos, CA, 1994), pp. 124-134.Google Scholar
  22. 22.
    L. K. Grover, Phys. Rev. Lett., 79, 325 (1997). G. Brassard, Science 275, 627 (1997). L. K.Google Scholar
  23. 23.
    Grover, Phys. Rev. Lett. 80, 4329 (1998).Google Scholar
  24. 24.
    R. P. Feynman, Int. J. Theoret. Phys. 21, 467 (1982).Google Scholar
  25. 25.
    C. Zalka, “Efficient simulation of quantum systems by quantum computers,” Los Alamos Archives preprint quant-ph/9603026.Google Scholar
  26. 26.
    S. Wiesner, “Solutions of many body quantum systems by a quantum computer,” Los Alamos Archives preprint quant-ph/9603028.Google Scholar
  27. 27.
    D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 79, 2586-2589 (1997).Google Scholar
  28. 28.
    S. Lloyd, Science 273, 1073-1078 (1996).Google Scholar
  29. 29.
    R. Landauer, Physics Today 44(5), 23 (1991); Phys. Lett. A 217, 188 (1996); in Feynman and Computation, Exploring the Limits of Computers, A. J. G. Hey, ed. (Perseus Books, Reading, MA, 1998).Google Scholar
  30. 30.
    P. Bridgman, Nature of Physical Theory (Dover, New York, 1936).Google Scholar
  31. 31.
    K. Svozil, Found. Phys. 25, 1541 (1995).Google Scholar
  32. 32.
    D. Finkelstein, Int. J. Theoret. Phys. 31, 1627 (1992); “Quantum sets, assemblies and plexi,” in Current Issues in Quantum Logic, E. Beltrametti and B. van Frassen, eds. (Plenum, New York, 1981), pp. 323-333; Quantum Relativity (Springer, New York, 1994).Google Scholar
  33. 33.
    G. Takeuti, “Quantum Set theory,” in Current Issues in Quantum Logic, E. G. Beltrametti and B. C. van Fraasen, eds. (Plenum, New York, 1981), pp. 303-322.Google Scholar
  34. 34.
    H. Nishimura, Int. J. Theoret. Phys. 32, 1293 (1993); 43, 229 (1995).Google Scholar
  35. 35.
    A. Odlyzko, “Primes, quantum chaos and computers,” in Number Theory, Proceedings of a Symposium, May 4 1989, Washington, D.C., Board on Mathematical Sciences, National Research Council, 1990, pp. 35-46.Google Scholar
  36. 36.
    R. E. Crandall, J. Phys. A: Math. Gen. 29, 6795 (1996).Google Scholar
  37. 37.
    S. Okubo, “Lorentz-invariant Hamiltonian and Riemann hypothesis,” Los Alamos Archives, quant-ph/9707036.Google Scholar
  38. 38.
    Y. Ozhigov, “Fast quantum verification for the formulas of predicate calculus,” Los Alamos Archives Rept. No. quant-ph/9809015.Google Scholar
  39. 39.
    H. Buhrman, R. Cleve, and A. Wigderson, “Quantum vs. Classical Communication and Computation,” Los Alamos Archives Rept. No. quant-ph/9802040.Google Scholar
  40. 40.
    C. Schmidhuber, Los Alamos Archives preprint hep-th/0011065.Google Scholar
  41. 41.
    S. Blaha, “Cosmos and consciousness,” 1stbooks Library, Bloomington, IN, 2000; “A quantum computer foundation for the standard model and superstring theory,” Los Alamos Archives Rept. No. quant-ph/0201092.Google Scholar
  42. 42.
    D. Spector, J. Math. Phys. 39, 1919 (1998).Google Scholar
  43. 43.
    L. Foschini, “On the logic of quantum physics and the concept of the time,” Los Alamos Archives Preprint, quant-ph/9804040.Google Scholar
  44. 44.
    G. W. Mackey, Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963).Google Scholar
  45. 45.
    R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, New York, 1992).Google Scholar
  46. 46.
    G. Birkhoff and J. von Neumann, Ann. Math. 37, 743 (1936).Google Scholar
  47. 47.
    J. M. Jauch and C. Piron, Helv. Phys. Acta 36, 837 (1963); C. Piron, ibid. 37, 439 (1964).Google Scholar
  48. 48.
    L. Hardy, Los Alamos Archives Preprint quant-ph/0101012.Google Scholar
  49. 49.
    E. Wigner, Comm. Pure and Applied Math. 13, 001 (1960); reprinted in E. Wigner, Symmetries and Reflections (Indiana University Press, Bloomington, IN, 1966), pp. 222-237.Google Scholar
  50. 50.
    P. C. W. Davies, “Why is the physical world so comprehensible?,” in Complexity, Entropy, and Physical Information (Proceedings of the 1988 Work-shop on Complexity, Entropy, and the Physics of Information, May-June 1989, Santa Fe New Mexico), W. H. Zurek, ed. (Addison-Wesley, Redwood City, CA, 1990), pp. 61-70.Google Scholar
  51. 51.
    C. C. Chang and H. J. Keisler, Model Theory (Studies in Logic and the Foundations of Mathematics, Vol 73) (American Elsevier, New York, NY, 1973), pp. 36-45.Google Scholar
  52. 52.
    G. Chaitin, Information Theoretic Incompleteness (World Scientific Series in Computer Science, Vol. 35) (World Scientific, Singapore, 1992); Information Randomness &;;; Incompleteness (Series in Computer Science, Vol 8), 2nd edn. (World Scientific, Singapore, 1990); Scientific American 232, 47-52 (1975); American Scientist 90, 164-171 (2002).Google Scholar
  53. 53.
    P. J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin, New York, NY, 1966).Google Scholar
  54. 54.
    H. Woodin, Notices, Amer. Math. Soc. 48, 567-576 (2001).Google Scholar
  55. 55.
    J. Stillwell, Am. Math. Monthly 109, 286-298 (2002).Google Scholar
  56. 56.
    C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999). C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev A, 59, 1070 (1999).Google Scholar
  57. 57.
    P. Benioff, Phys. Rev. A 64, #052310 (2001); Los Alamos archives quant-ph/ 0104061.Google Scholar
  58. 58.
    A. Turing, Proc. London Math. Soc. 42, 230-265 (1936).Google Scholar
  59. 59.
    D. Deutsch, Proc. Roy. Soc. London, Series A 400, 997 (1985).Google Scholar
  60. 60.
    E. Bernstein and U. Vazirani, SIAM J. Comput. 26, 1541-1557 (1997).Google Scholar
  61. 61.
    M. Tegmark, Los Alamos Archives preprint quant-ph/9709032.Google Scholar
  62. 62.
    H. Everett, Rev. Mod. Phys. 29, 454-462, (1957).Google Scholar
  63. 63.
    J. A. Wheeler, Rev. Mod. Phys. 29, 463-465, (1957).Google Scholar
  64. 64.
    H. P. Stapp, Mind, Matter, and Quantum Mechanics (Springer, Berlin, 1993).Google Scholar
  65. 65.
    E. Squires, Conscious Mind in the Physical World (IOP Publishing, Bristol, England, 1990).Google Scholar
  66. 66.
    D. Page, Los Alamos Archives preprint quant-ph/0108039.Google Scholar
  67. 67.
    E. Wigner, “Remarks on the mind-body question,” in The Scientist Speculates, I. Good and W. Heinemann, eds. (Putnam, London, 1962).Google Scholar
  68. 68.
    D. Albert, Phys. Lett. A 98, 249 (1983); Philos. Sci. 54, 577 (1987); “The quantum mechanics of self-measurement” in Complexity, Entropy and the Physics of Information, (Proceedings of the 1988 workshop in Santa Fe, New Mexico, 1989), W. Zurek, ed. (Addison-Wesley, Redwood City, CA, 1990).Google Scholar
  69. 69.
    P. Benioff, Phys. Rev. A 63, #032305 (2001).Google Scholar
  70. 70.
    P. Benioff, Los Alamos Archives preprint, quant-ph/0103078, Accepted for publication in special issue of Algorithmica.Google Scholar
  71. 71.
    J. D. Barrow and F. J. Tipler, The Anthropic Cosmologic Principle (Oxford University Press, 1989).Google Scholar
  72. 72.
    C. Hogan, Rev. Mod. Phys. 72, 1149 (2000).Google Scholar
  73. 73.
    G. Greenstein and A. Kropf, Am. J. Phys. 58, 746 (1989).Google Scholar
  74. 74.
    P. Benioff, Los Alamos Archives preprint quant-ph/0106153.Google Scholar
  75. 75.
    P. Benioff, Phys. Rev. A 58, 893-904 (1998).Google Scholar
  76. 76.
    A. Berthiaume, W. van Dam, and S. Laplante, J. Comp. Syst. Sciences 63, 201-221 (2001).Google Scholar
  77. 77.
    P. Vitanyi, “Three approaches to the quantitative definition of information in an individual pure quantum state,” Proceedings 15th IEEE Conference on Computational Complexity (Piscatawy, NJ, 2000), pp. 263-270; Los Alamos Archives preprint quant-ph/9907035.Google Scholar
  78. 78.
    P. Gacs, J. Phys. A, Math Gen. 34, 6859-6880 (2001).Google Scholar
  79. 79.
    W. V.O. Quine, Mathematical Logic (Norton, 1940).Google Scholar
  80. 80.
    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys. Rev. Lett. 77, 198 (1996); D. P. DiVincenzo and P. W. Shor, Phys. Rev. Lett. 77, 3260 (1996). E. M. Raines, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 79, 954 (1997).Google Scholar
  81. 81.
    W. H. Zurek, Phys. Rev. D 24, 1516 (1981); 26, 1862 (1982).Google Scholar
  82. 82.
    E. Joos and H. D. Zeh, Zeit. Phys. B 59, 23 (1985). H. D. Zeh, quant-ph/ 9905004; E. Joos, quant-ph/ 9808008.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Paul Benioff
    • 1
  1. 1.Physics DivisionArgonne National LaboratoryArgonne

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