Mathematical jujitsu: Some informal thoughts about Gödel and physics

Complexity

Volumn 5, Issue 5, 2000, Pages 28-34

  Barrow, John D ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

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EID: 0012323096     PISSN: 10762787     EISSN: 10990526     Source Type: Journal    
DOI: 10.1002/1099-0526(200005/06)5:5<28::AID-CPLX5>3.0.CO;2-S     Document Type: Article

References (26)

1. Jaki, S. Cosmos and Creator; Scottish Academy Press: Edinburgh, 1980, p 49.
2. Jaki, S. The Relevance of Physics; University of Chicago Press: Chicago, 1966, p 129.
3. Chaitin, G. Information, Randomness and Incompleteness; World Scientific: Singapore, 1987.
4. Lloyd, S. The Calculus of Intricacy, The Sciences, Sept/Oct 1990, p 38
5. Barrow, J.D. Pi in the Sky; Oxford University Press: Oxford, 1992, p 139.
6. Gödel, K. What is Cantor's Continuum Problem? Philosophy of Mathematics; Benacerraf, P.; Putnam H., eds.; p 483.
7. Recorded by Chaitin; see Bernstein, J. Quantum Profiles; p 140
8. Svozil, K. Randomness and Undecidability in Physics; World Scientific: Singapore, 1993, p112.
9. Wolfram, S. Cellular Automata and Complexity, Collected Papers; Addison Wesley: Reading, MA, 1994.
10. Calude, C. Information and Randomness—An Algorithmic Perspective; Springer: Berlin, 1994
11. Calude, C.; Jürgensen, H.; Zimand, M. Is Independence an Exception? Appl Math Comput 1994, 66, 63
12. Svozil, K. Boundaries and Barriers: On the Limits of Scientific Knowledge; Casti, J.; Karlqvist, A., eds.; Addison Wesley: Reading, MA, 1996, p 215.
13. N. Presburger arithmetic allows us to talk about positive integers, and variables whose values are positive integers. If we enlarge it by permitting the concept of sets of integers to be used, then the situation becomes almost unimaginably intractable. It has been shown that this system does not admit even a K-fold exponential algorithm, for any finite K. The decision problem is said to be nonelementary in such situations. The intractability is unlimited.
14. That the terms in the sum get progressively smaller is a necessary but not a sufficient condition for an infinite sum to be finite. For example, the sum 1 + 1/2 + 1/3 + 1/4 + 1/5 +.. is infinite.
15. Rosen, R. Boundaries and Barriers: On the Limits of Scientific Knowledge; Casti, J.; Karlqvist, A., eds.; Addison Wesley: Reading, MA, 1996, p 199.
16. John A. Wheeler has speculated about the ultimate structure of spacetime being a form of “pregeometry” obeying a calculus of propositions restricted by Gödel incompleteness. We are proposing that this pregeometry might be simple enough to be complete; see Misner, C.; Thorne, K.; Wheeler, J.A. Gravitation; W.H. Freeman: San Francisco, 1973, p 1211.
17. The situation in superstring theory is still very fluid. There appear to exist many different, logically self-consistent superstring theories, but there are strong indications that they may be different representations of a much smaller number (maybe even just one) theory.
18. da Costa, N.C.; Doria, F. Int J Theor Phys 1991, 30, 1041
19. da Costa, N.C.; Doria, F. Found Phys Letts 1991, 4, 363
20. Actually, there are other more complicated possibilities clustered around the dividing line between these two simple possibilities, and it is these that provide the indeterminacy of the problem in general.
21. Geroch, R.; Hartle, J. Found Physics 1986, 16, 533. The problem is that the calculation of a wave function for a cosmological quantity involves the sum of quantities evaluated on every four-dimensional compact manifold in turn. The listing of this collections of manifolds is uncomputable.
22. Pour-El, M.B.; Richards, I. Ann Math Logic 1979, 17, 61. The wave equation with computable initial data such that its unique solution is not computable. See also Adv Math 1981, 39, 215; Int J Theor Phys 1982, 21, 553.
23. Traub, J.F.; Wozniakowski, H. Mathematical Intelligencer 1991, 13, 34.
24. Wolfram, S. Phys Rev Lett 1985, 54, 735; Phys Rev Lett 1985, 55, 449; Int J Theoret Phys 1982, 21, 165.
25. If the metric functions are polynomials, then the problem is decidable, but is computationally double-exponential. If the metric functions are allowed to be sufficiently smooth, then the problem becomes undecidable; see the article on Algebraic Simplification by Buchberger and Loos in Buchberger, Loos and Collins, Computer Algebra: Symbolic and Algebraic Computation, 2nd ed.; Springer: Wien, 1983. I am grateful to Malcolm MacCallum for supplying these details.
26. For a fuller discussion of the whole range of limits that may exist on our ability to understand the universe see the author's discussion in Barrow, J.D. Impossibility; Oxford University Press, 1998, on which the discussion here is based.