Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics


In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above.

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  1. 1.

    Angeles, P.: Dictionary of Philosophy. Barnes & Noble, New York (1981)

    Google Scholar 

  2. 2.

    Antoniou, I., Misra, B.: The relativistic internal time operator. Int. J. Theor. Phys. 31(1) (1992)

  3. 3.

    Audretsch, J.: Entangled Systems. Wiley-VCH, Bonn (2006)

    Google Scholar 

  4. 4.

    Bateson, G.: Steps to an Ecology of Mind. Paladin Books (1973)

  5. 5.

    Bateson, G.: Mind and Nature: A Necessary Unity. Bantam, New York (1988)

    Google Scholar 

  6. 6.

    Battersby, S.: Einstein on acid. New Sci. 180(2426), 40 (2003)

    Google Scholar 

  7. 7.

    Beloussov, L.V., Voeikov, V.L., Martynyuk, V.S. (eds.): Biophotonic and Coherent Systems in Biology. Springer, Berlin (2006)

    Google Scholar 

  8. 8.

    Berlin, B., Kay, P.: Basic Color Terms: Their Universality and Evolution. University of California Press, Berkeley (1969)

    Google Scholar 

  9. 9.

    Bohm, D., Hiley, B.: The Undivided Universe. Routlegde, London (1993)

    Google Scholar 

  10. 10.

    Buccheri, R., di Gesù, V., Saniga, M.: Studies on the Structure of Time: From Physics to Psycho(patho)logy. Springer, Berlin (2000)

    Google Scholar 

  11. 11.

    Buccheri, R., Elitzur, A., Saniga, M.: Endophysics, time, quantum and the subjective. In: Proceedings of the ZIF Interdisciplinary Research Workshop, 17–22 January 2005, Bielefeld. Springer, Berlin (2005)

    Google Scholar 

  12. 12.

    Buccheri, R., Saniga, M., Stuckey, W.M.: The Nature of Time: Geometry, Physics and Perception. NATO Science Series II: Mathematics, Physics and Chemistry. Springer, Berlin (2003)

    Google Scholar 

  13. 13.

    Cartan, E.: The Theory of Spinors. Dover, New York (1952)

    Google Scholar 

  14. 14.

    Conte, E., Khrennikov, A., Tadarello, O., Federici, A., Zbilut, J.: J. Neuroquantology 7(2), 204–212 (2009)

    Google Scholar 

  15. 15.

    Daddesio, T.: On Minds and Symbols. de Gruyter, Amsterdam (1995)

    Google Scholar 

  16. 16.

    Damasio, A.: Looking for Spinoza; Joy, Sorrow, and the Feeling Brain. Harvest, New York (2003)

    Google Scholar 

  17. 17.

    Damasio, A.: Descartes’ Error: Emotion, Reason, and the Human Brain. Penguin, London (2005)

    Google Scholar 

  18. 18.

    Damasio, A.: The Feeling of What Happens: Body and Emotion in the Making of Consciousness. Harvest, New York (2005)

    Google Scholar 

  19. 19.

    Definitions of Cybernetics. American Society of Cybernetics.

  20. 20.

    Fanchi, J.: Parametrized Relativistic Quantum Theory. Kluwer, Boston (1993)

    Google Scholar 

  21. 21.

    Finkelstein, D.R.: Cosmic computation. In: Weibel, P., Ord, G., Rossler, O. (eds.) Space Time Physics and Fractality, pp. 65–82. Springer, Berlin (2005)

    Google Scholar 

  22. 22.

    Fock, V.: The Theory of Space, Time and Gravitation. Pergamon, London (1958)

    Google Scholar 

  23. 23.

    Gisin, N., Percival, I.: arXiv:quant-ph/9701024v1

  24. 24.

    Grynberg-Zylberbaum, J.: Bases Psicofisiologicas de la Percepcion Visual. Editorial Trillas, Mexico (1981)

    Google Scholar 

  25. 25.

    Gunther, G.: Cybernetic ontology and transjunctional operations. In: Yovits, M.C., Jacobs, G.T., Goldstein, G.D. (eds.) Self-organizing Systems, pp. 313–392. Spartan Books, Washington (1962)

    Google Scholar 

  26. 26.

    Gunther, G.: Time, timeless logic, self-referential systems. Ann. N.Y. Acad. Sci. 138, 317–346 (1967)

    Google Scholar 

  27. 27.

    Harms, J.: Time-lapsed reality visual metabolic rate and quantum time and space. Kybernetes 32(7/8), 113–1128 (2003)

    Article  Google Scholar 

  28. 28.

    Heelan, P.: Space Perception and the Philosophy of Science. University of California Press, Berkeley (1989)

    Google Scholar 

  29. 29.

    Hegel, G.F.: Wissenschaft der Logik II. Suhrkampf, Berlin (1969)

    Google Scholar 

  30. 30.

    Hegel, G.F.: Logic. Evergreen Review (2007)

  31. 31.

    Hehl, F., von der Heyde, P., Kerlick, G.D., Nester, J.M.: Rev. Mod. Phys. 48, 3 (1976)

    Article  Google Scholar 

  32. 32.

    Hehl, F., Dermott McCrea, J., Mielke, E., Ne’eman, Y.: Phys. Rep. 258, 1–157 (1995)

    MathSciNet  ADS  Article  Google Scholar 

  33. 33.

    Heidegger, M.: Time and Being. Harper & Row, New York (1962), pp. 1–24

    Google Scholar 

  34. 34.

    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  35. 35.

    Honig, W.M.: Nonstandard Logics and Nonstandard Metrics in Physics. Series on Knots and Everything, vol. 10. World Scientific, Singapore (1995)

    Google Scholar 

  36. 36.

    Horwitz, L.P., Piron, C.: Helv. Phys. Acta 46, 316 (1973)

    Google Scholar 

  37. 37.

    Horwitz, L.P., Piron, C.: Helv. Phys. Acta 66, 694 (1993)

    MathSciNet  Google Scholar 

  38. 38.

    Horwitz, L.: Hypercomplex quantum mechanics. Found. Phys. 26(6), 851 (1996)

    MathSciNet  ADS  Article  Google Scholar 

  39. 39.

    Horwitz, L.P., Shnerb, N.: Found. Phys. 28, 1509 (1998)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Horwitz, L.: Quantum interference in time. Found. Phys. 37(3/4), 734 (2007)

    MATH  MathSciNet  ADS  Article  Google Scholar 

  41. 41.

    Horwitz, L., Ben Zion, Y., Lewkowicz, M., Schiffer, M., Levitan, J.: Geometry of Hamiltonian chaos. Phys. Rev. Lett. 98, 234–301 (2007)

    Article  Google Scholar 

  42. 42.

  43. 43.

    Hu, H., Wu, M.: Spin as primordial self-referential process driving quantum mechanics, spacetime dynamics and consciousness. J. Neuroquantology 2, 41 (2004)

    Google Scholar 

  44. 44.

    Hubel, D.H.: Eye, Brain and Vision. Scientific American Library. Freeman, New York (1988)

    Google Scholar 

  45. 45.

    Illert, C.R.: Nuovo Cimento D 9(7), 791–784 (1987)

    MathSciNet  ADS  Article  Google Scholar 

  46. 46.

    Illert, C.R.: Nuovo Cimento D 11(5), 761–780 (1989)

    MathSciNet  ADS  Article  Google Scholar 

  47. 47.

    Illert, C.R.: Nuovo Cimento D 12(10), 1405–1421 (1990)

    MathSciNet  ADS  Article  Google Scholar 

  48. 48.

    Illert, C.R.: Nuovo Cimento D 12(2), 1611–1632 (1990)

    MathSciNet  ADS  Article  Google Scholar 

  49. 49.

    Illert, C.R.: Foundations of Theoretical Conchology, 1st edn. Hadronic Press, Palm Harbor (1992)

    Google Scholar 

  50. 50.

    Illert, C.R., Santilli, R.M.: Foundations of Theoretical Conchology, 2nd edn. Hadronic Press, Palm Harbor (1995)

    Google Scholar 

  51. 51.

    Indow, T.: The Global Structure of Visual Space. Advanced Series on Mathematical Psychology, vol. 1. World Scientific, Singapore (2004)

    Google Scholar 

  52. 52.

    Iran-Nejad, A.: A non-connectionist schema theory of understanding surprise-ending stories. Discourse Process. 12, 127–148 (1989)

    Article  Google Scholar 

  53. 53.

    Johansen, S.: Outline of a Differential Epistemology. English translation of the original Norwegian published by University of Trondheim (1991). To appear

  54. 54.

    Kappraff, J.: Beyond Measure. World Scientific Series on Knots and Everything. World Scientific, Singapore (2002)

    Google Scholar 

  55. 55.

    Kauffman, L.: De Morgan algebras, completeness and recursion. In: Proceedings of the Eighth International Symposium in Multiple Valued Logics.

  56. 56.

    Kauffman, L.: Laws of Form-An Exploration in Mathematics and Foundations. Book in progress,

  57. 57.

    Kauffman, L.: Space and time in computation and discrete physics. Int. J. Gen. Syst. 27(1), 249–273 (1998)

    MATH  MathSciNet  Article  Google Scholar 

  58. 58.

    Kauffman, L.: Formal systems, eigenform. Kybernetes 34(1/2), 129–150 (2004)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Khrennikov, A.: Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena. Fundamental Theories of Physics. Springer, Berlin (2004)

    Google Scholar 

  60. 60.

    Kleinert, H.: Multivalued Fields: In Condensed Matter, Electromagnetism, and Gravitation. World Scientific, Singapore (2008)

    Google Scholar 

  61. 61.

    Lacan, J.: Ecrits. Norton, New York (2007)

    Google Scholar 

  62. 62.

    Land, M.C., Shnerb, N., Horwitz, L.P.: J. Math. Phys. 36, 3263 (1995)

    MATH  MathSciNet  ADS  Article  Google Scholar 

  63. 63.

    Libet, B., Gleason, C.A., Wright, E.W., Pearl, D.K.: Time of conscious intention to act in relation to onset of cerebral activity (readiness potential). The unconscious inhibition of a freely voluntary act. Brain 106(3), 623–642 (1983)

    Article  Google Scholar 

  64. 64.

    Liebling, M., Blu, T., Unser, M.: Fresnelets: new multiresolution wavelet bases for digital holography. IEEE Trans. Image Process. 12(1), 29–43 (2003)

    MathSciNet  ADS  Article  Google Scholar 

  65. 65.

    Lin, Y.: Systemic Yoyos: Some Impacts of the Second Dimension. Auerbach Publications, Boca Raton (2008)

    Google Scholar 

  66. 66.

    Luneburg, R.: Mathematical Analysis of Binocular Vision. Edwards, Ann Arbor (1948)

    Google Scholar 

  67. 67.

    Luhmann, N.: Theories of Distinction: Redescribing the Descriptions of Modernity. Cultural Memory in the Present. Stanford University Press, Stanford (2002)

    Google Scholar 

  68. 68.

    Marcelja, S.: Mathematical description of the responses of simple cortical cells. J. Opt. Soc. Am. 70, 1297–1300 (1980)

    MathSciNet  ADS  Article  Google Scholar 

  69. 69.

    Merleau Ponty, M.: Phenomenology of Perception. Routledge & Kegan Paul, London (1962)

    Google Scholar 

  70. 70.

    Merleau Ponty, M.: The Visible and the Invisible. Northwestern University Press, Evanston (1968)

    Google Scholar 

  71. 71.

    Musès, C.: Applied hypernumbers: computational concepts. Appl. Math. Comput. 3, 211–226 (1976)

    Article  Google Scholar 

  72. 72.

    Musès, C.: Hypernumbers-II 4, 45–66 (1978)

    MATH  Google Scholar 

  73. 73.

    Musès, C.: Destiny and Control in Human Systems. Frontiers in System Research. Kluwer-Nijhoff, Boston (1984)

    Google Scholar 

  74. 74.

    Nagarjuna: The Middle Way. SUNY, New York (1986). David Karlupahana, translator.

    Google Scholar 

  75. 75.

    Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985)

    Google Scholar 

  76. 76.

    Raju, C.K.: The Eleven Pictures of Time. Sage, New Delhi (2003)

    Google Scholar 

  77. 77.

    Rapoport, D.L.: On the unification of geometric and random structures through torsion fields: Brownian motions, viscous and magneto-fluid-dynamics. Found. Phys. 35(7), 1205–1244 (2005)

    MATH  MathSciNet  ADS  Article  Google Scholar 

  78. 78.

    Rapoport, D.L.: Cartan-Weyl Dirac and Laplacian operators, Brownian motions: the quantum potential and scalar curvature, Maxwell’s and Dirac-Hestenes equations, and supersymmetric systems. Found. Phys. 7, 1383–1431 (2005)

    MathSciNet  ADS  Article  Google Scholar 

  79. 79.

    Rapoport, D.L.: Torsion fields, Cartan-Weyl space-time and state-space quantum geometries, their Brownian motions, and the time variables. Found. Phys. 37(4–5), 813–854 (2007)

    MATH  MathSciNet  ADS  Article  Google Scholar 

  80. 80.

    Rapoport, D.L.: In: Adenier, G., Khrennikov, A.Yu., Lahti, P., Man’ko, V.I. (eds.) Quantum Theory: Reconsideration of Foundations—4. AIP Conference Proceedings. Springer, Berlin (2007)

    Google Scholar 

  81. 81.

    Rapoport, D.L.: Torsion fields, propagating singularities, nilpotence, quantum jumps and the eikonal equations. In: Dubois, D., et al. (eds.), Proceedings of Casys09. AIP Proceedings Series. Springer, Berlin (December 2009, to appear).

  82. 82.

    Rescher, N.: Many-Valued Logic. McGraw Hill, New York (1969)

    Google Scholar 

  83. 83.

    Rosch, E., Varela, F., Thompson, E.: The Embodied Mind. MIT Press, Cambridge (1991)

    Google Scholar 

  84. 84.

    Rosen, S.M.: Dimensions of Apeiron: A Topological Phenomenology of Space, Time and Individuation. Rodopi, Amsterdam (2004)

    Google Scholar 

  85. 85.

    Rosen, S.M.: Topologies of the Flesh: A Multidimensional Exploration of the Lifeworld. Series in Continental Thought. Ohio University Press, Athens (2006)

    Google Scholar 

  86. 86.

    Rosen, S.M.: Quantum gravity and phenomenological philosophy. Found. Phys. 38, 556–582 (2008)

    MATH  MathSciNet  ADS  Article  Google Scholar 

  87. 87.

    Rosen, S.M.: The Self-Evolving Cosmos: A Phenomenological Approach to Nature’s Unity-in-Diversity. Series on Knots and Everything. World Scientific, Singapore (2008)

    Google Scholar 

  88. 88.

    Rossler, O.: Endophysics: The World As an Interface. World Scientific, Singapore (1998)

    Google Scholar 

  89. 89.

    Rowlands, P.: From Zero to Infinity. World Scientific, Singapore (2008)

    Google Scholar 

  90. 90.

    Ruhnau, E.: In: Metzinger, T. (ed.) Conscious Experience. Imprint Academic, Exeter (1996)

    Google Scholar 

  91. 91.

    Russell, B.: Introduction to Mathematical Philosophy. MacMillan, New York (1918)

    Google Scholar 

  92. 92.

    Pattee, H.H.: The Physics of Symbols: Bridging the Epistemic Gap. BioSystems 60, 5–21 (2001)

    Article  Google Scholar 

  93. 93.

    Penrose, R.: Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press, Oxford (1996), and references therein

    Google Scholar 

  94. 94.

    Post, E.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921)

    MATH  MathSciNet  Article  Google Scholar 

  95. 95.

    Purcell, M.C.: Towards a new era (epistemological resolution analysis by through and from the Klein bottle wholeness). Ph.D. Thesis, Department of Philosophy, University of Newcastle, Australia (2006)

  96. 96.

    Purcell, M.C.: In: CASYS09, August 03–08, 2009, Univ. of Liège (2009, forthcoming).

  97. 97.

    Pylkkanen, P.T.I.: Mind, Matter and the Implicate Order. The Frontiers Collection. Springer, Berlin (2006)

    Google Scholar 

  98. 98.

    Schouten, J.: Ricci Calculus. Springer, Berlin (1954)

    Google Scholar 

  99. 99.

    Sidharth, B.G.: Chaotic Universe: From the Planck Scale to the Hubble Scale. Nova Science, New York (2002), and references therein

    Google Scholar 

  100. 100.

    Sidharth, B.G.: The Universe of Fluctuations. Springer Series on the Fundamental Theories of Physics. Springer, Berlin (2004)

    Google Scholar 

  101. 101.

    Snyder, H.S.: Phys. Rev. 72(2), 68–71 (1947)

    MATH  ADS  Article  Google Scholar 

  102. 102.

    Spencer-Brown, G.: Laws of Form. Allen & Unwin, London (1969)

    Google Scholar 

  103. 103.

    Spinoza, B.: Ethics. Dover, New York (1955)

    Google Scholar 

  104. 104.

    Stern, A.: Matrix Logic and Mind. Elsevier, Amsterdam (1992)

    Google Scholar 

  105. 105.

    Stern, A.: Quantum Theoretic Machines. Elsevier, Amsterdam (2000)

    Google Scholar 

  106. 106.

    Stueckelberg, E.C.: Helv. Phys. Acta 14, 322–588 (1941)

    MathSciNet  Google Scholar 

  107. 107.

    Swindale, N.V.: Visual cortex: Looking into a Klein bottle. Curr. Biol. 6(7), 776–779 (1996)

    Article  Google Scholar 

  108. 108.

    Taborsky, E.: Architectonics of Semiotics. Palmgrave MacMillan, New York (1998)

    Google Scholar 

  109. 109.

    Tanaka, S.: Topological analysis of point singularities in stimulus preference maps of the primary visual cortex. Proc. R. Soc. Lond., B Biol. Sci. 261, 81–88 (1995)

    ADS  Article  Google Scholar 

  110. 110.

    Tenen, S.:

  111. 111.

    The Klein bottle.

  112. 112.

    The Klein bottle.

  113. 113.

    The Klein bottle.

  114. 114.

    The Necker Cube.

  115. 115.

    Varela, F.: Principles of Biological Autonomy. North-Holland, New York (1979)

    Google Scholar 

  116. 116.

    Vidyabhusan, S.C.: A History of Indian Logic. Calcutta (1921). Reprinted: Munshiram Manoharlal, New Delhi (1979)

  117. 117.

    von Foerster, H.: Objects: tokens for eigenbehaviours. In: Observing Systems. The Systems Inquiry Series, pp. 274–285. Intersystems Publications, Salinas (1981).

    Google Scholar 

  118. 118.

    Vrobel, S., Rossler, O.: Simultaneity: Temporal Structures and Observer Perspectives. World Scientific, Singapore (2008)

    Google Scholar 

  119. 119.

    Wu, Y., Lin, Y.: Beyond Nonstructural Quantitative Analysis: Blown-ups, Spinning Currents and Modern Science. World Scientific, Singapore (2002)

    Google Scholar 

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Rapoport, D.L. Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics. Found Phys 41, 33–76 (2011).

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  • Time operator
  • Time waves
  • Self-reference
  • Torsion geometry
  • Quantum mechanics
  • Quantum computation
  • Spin
  • Radical recursion
  • Muses hypernumbers
  • Nilpotents
  • Neurology
  • Klein bottle
  • Cartesian cut
  • Calculus of distinctions
  • Multivalued logics
  • Matrix logics
  • Philosophical phenomenology
  • Cognition
  • Perception
  • Eikonal equation
  • Photon
  • Cybernetics
  • Fibonacci sequence
  • Systems theory
  • Semiotics
  • Endophysics
  • Implicate and explicate orders
  • Holomovent
  • Mind-matter problem
  • Meta-algorithmic level
  • Moebius surface