{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T03:56:40Z","timestamp":1778558200292,"version":"3.51.4"},"reference-count":26,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2015,9,1]],"date-time":"2015-09-01T00:00:00Z","timestamp":1441065600000},"content-version":"tdm","delay-in-days":5722,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2000,5]]},"DOI":"10.1002\/1099-0526(200005\/06)5:5<28::aid-cplx5>3.0.co;2-s","type":"journal-article","created":{"date-parts":[[2002,8,25]],"date-time":"2002-08-25T19:00:59Z","timestamp":1030302059000},"page":"28-34","source":"Crossref","is-referenced-by-count":8,"title":["Mathematical jujitsu: Some informal thoughts about G\ufffddel and physics"],"prefix":"10.1002","volume":"5","author":[{"given":"John D.","family":"Barrow","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2000]]},"reference":[{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB1","first-page":"49","volume-title":"Cosmos and Creator","author":"Jaki","year":"1980","unstructured":"Cosmos and Creator; Scottish Academy Press: Edinburgh, 1980, p 49."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB2","first-page":"129","volume-title":"The Relevance of Physics","author":"Jaki","year":"1966","unstructured":"The Relevance of Physics; University of Chicago Press: Chicago, 1966, p 129."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB3","doi-asserted-by":"crossref","DOI":"10.1142\/0531","volume-title":"Information, Randomness and Incompleteness","author":"Chaitin","year":"1987","unstructured":"Information, Randomness and Incompleteness; World Scientific: Singapore, 1987."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB4A","author":"Lloyd","year":"1990","unstructured":"The Calculus of Intricacy, The Sciences, Sept\/Oct 1990, p 38"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB4B","first-page":"139","volume-title":"Pi in the Sky","author":"Barrow","year":"1992","unstructured":"Pi in the Sky; Oxford University Press: Oxford, 1992, p 139."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB5","first-page":"483","volume-title":"What is Cantor's Continuum Problem? Philosophy of Mathematics","author":"G\u00f6del","unstructured":"What is Cantor's Continuum Problem? Philosophy of Mathematics; ; eds.; p 483."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB6A","unstructured":"Recorded by Chaitin; see Bernstein, J. Quantum Profiles; p 140"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB6B","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1142\/1524","volume-title":"Randomness and Undecidability in Physics","author":"Svozil","year":"1993","unstructured":"Randomness and Undecidability in Physics; World Scientific: Singapore, 1993, p112."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB7","volume-title":"Cellular Automata and Complexity, Collected Papers","author":"Wolfram","year":"1994","unstructured":"Cellular Automata and Complexity, Collected Papers; Addison Wesley: Reading, MA, 1994."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB8A","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03049-3","volume-title":"Information and Randomness?An Algorithmic Perspective","author":"Calude","year":"1994","unstructured":"Information and Randomness?An Algorithmic Perspective; Springer: Berlin, 1994"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB8B","first-page":"63","volume":"66","author":"Calude","year":"1994","journal-title":"Appl Math Comput"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB8C","first-page":"215","volume-title":"Boundaries and Barriers: On the Limits of Scientific Knowledge","author":"Svozil","year":"1996","unstructured":"Boundaries and Barriers: On the Limits of Scientific Knowledge; ; eds.; Addison Wesley: Reading, MA, 1996, p 215."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB9","unstructured":"The decision procedure is in general double-exponentially long, though. That is, the computational time required to carry out N operations grows as (2N)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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB10","unstructured":"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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB11","first-page":"199","volume-title":"Boundaries and Barriers: On the Limits of Scientific Knowledge","author":"Rosen","year":"1996","unstructured":"Boundaries and Barriers: On the Limits of Scientific Knowledge; ; eds.; Addison Wesley: Reading, MA, 1996, p 199."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB12","author":"John","year":"1973","unstructured":"Wheeler has speculated about the ultimate structure of spacetime being a form of ?pregeometry? obeying a calculus of propositions restricted by G\u00f6del 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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB13","unstructured":"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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB14A","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1007\/BF00671484","volume":"30","author":"da Costa","year":"1991","journal-title":"Int J Theor Phys"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB14B","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF00665895","volume":"4","author":"da Costa","year":"1991","journal-title":"Found Phys Letts"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB15","unstructured":"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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB16","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/BF01886519","volume":"16","author":"Geroch","year":"1986","journal-title":"Found Physics"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB17","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0003-4843(79)90021-4","volume":"17","author":"Pour-El","year":"1979","journal-title":"Ann Math Logic"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB18","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/BF03024085","volume":"13","author":"Traub","year":"1991","journal-title":"Mathematical Intelligencer"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB19","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1103\/PhysRevLett.54.735","volume":"54","author":"Wolfram","year":"1985","journal-title":"Phys Rev Lett"},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB20","year":"1983","unstructured":"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."},{"key":"10.1002\/1099-0526(200005\/06)5:5<28::AID-CPLX5>3.0.CO;2-S-BIB21","year":"1998","unstructured":"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. 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