Abstract
Let us assume that our world is discretely organized, and that it is governed by constructive, i.e., effectively computable, laws [1]. By that assumption, there exists a “blueprint”, a complete set of rids or laws governing the universe. This seems unlike mathematics for which Gödel, Tarski, Turing and others proved that no reasonable (i.e., strong enough and consistent) formal system will ever be able to prove all true well-formed statements. Indeed, Chaitin showed that certain mathematical entities are as random as a sequence produced by the tossing of a fair coin [2, 3]. Hence, let us contemplate the assumption that, when it comes to an enumeration of laws and initial values, nature is finitely “shallow” while mathematics is infinitely “deep” [4].
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References
In contradistinction, contemporary physical theories are expressed in terms of continua: time, position, momentum, wave amplitudes,… The very notion of continuum embodies indeterminism insofar as “almost all” (i.e., with probability 1), elements of continua are (Martin-Löf/Solovay/Chaitin) random. Physical chaos, if it exists, is the necessary consequence of this fact. In these models, indeterminism is “put-in” from the very beginning. There is no reasonable machine representation and no conceivable “explanation” corresponding to such models. They (together with classical, non-constructive, mathematics) are irrational at heart, and, while interesting in many respects, we shall not dwell deeper on this subject.
C. Calude, Information and Randomness - An Algorithmic Perspective ( Springer, Berlin, 1994 ).
In this context, the terms “shallow” and “deep” refer to algorithmic information [2] rather than to Bennett’s notion of “logical depth”, cf. Ch. H. Bennett, “Logical Depth and Physical Complexity”, in The Universal Turing Machine. A Half-Century Survey, ed. by R. Herken (Kammerer & Unverzagt, Hamburg, 1988 ). The apparent “paradox” that a complex phenotype originates from low-complex initial values and evolution is not paradoxical at all. Indeed, that the world appears complex by all means does not necessarily mean that its laws have a high algorithmic information content.
T. Toffoli, “The role of the observer in uniform systems”, in Applied General Systems Research, ed. by G. Klir (Plenum Press, New York, London, 1978 ).
O.E. Rössler, “Endophysics”, in Real Brains, Artificial Minds, ed. by J. L. Casti and A. Karlquist (North-Holland, New York, 1987), p. 25; Endophysics, Die Welt des inneren Beobachters, ed. by P. Weibel ( Merwe Verlag, Berlin, 1992 ).
K. Svozil, “Il Nuovo Cimento” 96B, 127 (1986); Randomness and Undecidability in Physics (World Scientific, Singapore, 1993 ); Quantum Logic, ( Springer, Singapore, 1998 ).
Archimedes (= 287–212 b.c.) encountered the mechanical problem “to move a given weight by a given force”. According to Plutarch “Marcellus, `that he declared… that any given weight could be moved by any given force (however small)’ and boasted that, `if he were given a place to stand on, he could move the earth’ [cited from T. Heath, A History of Greek Mathematics, Volume II’ (Clarendon Press, Oxford, 1921), p. 18].
J. von Neumann, Theory of Self-Reproducing Automata, ed. by A. W. Burks ( University of Illinois Press, Urbana, 1966 ).
H. Rogers, Theory of Recursive Functions and Effective Computability ( MacGraw-Hill, New York 1967 ).
P. Odifreddi, Classical Recursion Theory ( North-Holland, Amsterdam, 1989 ).
K. Gödel, ‘Monatshefte für Mathematik und Physik 38’, 173 (1931); English translation in [13] and in Davis, ref. [14].
K. Gödel, Publications 1929–1936, ed. by S. Feferman, J. W. Dawson, Jr., St. C. Kleene, G.H. Moore, R.M. Solovay, J. van Heijenoort ( Oxord University Press, Oxford, 1986 ).
M. Davis, The Undecidable ( Raven Press, New York, 1965 ).
Here “H(p)” is defined [2, 3, 16] as the length of the smallest program p* (in prefix code) which runs on a universal (Chaitin) computer and outputs p.
M. Li and P.M.B. Vitânyi, “Kolmogorov Complexity and its Applicatinos’, in Handbook of Theoretical Computer Sciences, Algorithms and Complexity, Volume A (Elsevier, Amsterdam and MIT Press, Cambridge, MA., 1990 ).
E.M. Gold, “Information and Control” 10, 447 (1967).
D. Angluin and C.H. Smith, “Computing Surveys’ 15, 237 (1983).
M. Li and P.M.B. Vitânyi, “Journal of Computer and System Science” 44, 343 (1992).
L.M. Adleman and M. Blum, “Journal of Symbolic Logic” 56, 891 (1991).
J.E. Hoperoft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation ( Addison-Wesley, Reading, MA, 1979 ).
W. Brauer, Automatentheorie ( Teubner, Stuttgart, 1984 ).
E.F. Moore, “Gedanken-Experiments on Sequential Machines’, in ”Automata Studies“, ed. by C.E. Shannon & J. McCarthy ( Princeton University Press, Princeton, 1956 ).
A.M. Turing, “Proc. London Math. Soc.” (2), 42, 230 (1936–7), reprinted in [14].
H.D.P. Lee, Zeno of Elea (Cambridge University Press, Cambridge, 1936; reprinted by Adolf M. Hakkert, Amsterdam, 1967 ).
K.R. Popper, “The British Journal for the Philosophy of Science” 1, 117, 173 (1950).
E. Schrödinger, “Naturwissenschaften” 23, 807; 823; 844 (1935) [English translation in J.A. Wheeler and W.H. Zurek, eds., Quantum Theory and Measurement (Princeton University Press, Princeton, 1983), p. 152–167].
A. Einstein, B. Podolsky and N. Rosen, “Phys. Rev.” 47, 777 (1935).
J.S. Bell, Speakable and Unspeakable in Quantum Mechanics ( Cambridge University Press, Cambridge, 1987 ).
D.M. Greenberger, M. Horne and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, ed. by M. Kafatos (Kluwer, Dordrecht, 1989); D.M. Greenberger, M.A. Horne, A. Shimony and A. Zeilinger, “Am J. Phys.” 58, 1131 (1990).
A. Peres, “Am. J. Phys”. 46, 745 (1978).
A. Peres, Quantum Theory: Concepts & Methods ( Kluwer Academic Publishers, Dordrecht, 1993 ).
G. Krenn and K. Svozil, “Stronger-than-quantum correlations” (TUVienna preprint, 1995 ).
J.A. Wheeler, “Law Without law”, in J.A. Wheeler and W.H. Zurek, eds., Quantum Theory and Measurement ( Princeton University Press, Princeton, 1983 ), p. 182–213.
A. Shimony, “Controllable and uncontrollable non-locality”, in “Proc. Int. Symp. Foundations of Quantum Mechanics”, ed. by S. Kamefuchi et al. (Physical Society of Japan, Tokyo, 1984), reprinted in [36], p. 130; A. Shimony, “Events and Processes in the Quantum World”, in Quantum Concepts in Space and Time, ed. by R. Penrose and C.I. Isham (Clarendon Press, Oxford, 1986), reprinted in [36], p. 140.
A. Shimony, Search for a Naturalistic World View, Volume II ( Cambridge University Press, Cambridge, 1993 ).
N. Herbert, “Foundation of Physics” 12, 1171 (1982).
W.K. Wooters and W.H. Zurek, “Nature” 299, 802 (1982); L. Mandel, “Nature” 304 188 (1983); P.W. Milonni and M.L. Hardies, “Phys. Lett.” 92A, 321 (1982); R.J. Glauber, “Amplifiers, Attenuators and the Quantum Theory of Measurement”, in Frontiers in Quantum Optics’, ed. by E.R. Pikes and S. Sarkar (Adam Hilger, Bristol 1986); C.M. Caves, Phys. Rev. D 26, 1817 (1982).
] R. Landauer, “Irreversibility and Heat Generation in the Computing Process”, “IBM J. Res. Dev.” 5, 183191 (1961); reprinted in: Maxwell’s Demon, ed. by H.S. Leff and A.F. Rex (Princeton University Press, 1990), pp. 188–196; R. Landauer, “Wanted: a physically possible theory of physics, in IEEE Spectrum” 4, 105–109 (1967); R. Landauer, “Fundamental Physical Limitations of the Computational Process; an Informal Commentary”, in “Cybernetics Machine Group Newsheet” 1/1/87; R. Landauer, `Computation, Measurement, Communication and Energy Dissipation“, in Selected Topics in Signal Processing, ed. by S. Haykin (prentice Hall, Englewood Cliffs, NJ, 1989 ), p. 18.
C.H. Bennett, “Logical Reversibility of Computation”, “IBM J. Res. Dev.” 17, 525–532 (1973); reprinted in: Maxwell’s Demon, ed. by H.S. Leff and A.F. Rex (Princeton University Press, 1990 ), pp. 197–204.
One may think of a virtual reality installation corresponding to the physical universe, and the player who is connected to it via a passive interface corresponding to the “super-observer”.
I. Lakatos, The Methodology of Scientific Research Programs ( Cambridge University Press, Cambridge, 1978 ).
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Svozil, K. (1999). On Self-Reference and Self-Description. In: Carsetti, A. (eds) Functional Models of Cognition. Theory and Decision Library, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9620-6_12
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