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Computable Chaos

Published online by Cambridge University Press:  01 April 2022

John A. Winnie
John A. Winnie*
Affiliation:
Department of History and Philosophy of Science, Indiana University
*
‡Send reprint requests to the author, Department of History and Philosophy of Science, Goodbody Hall 114, Indiana University, Bloomington, IN 47405, USA.
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Abstract

Some irrational numbers are “random” in a sense which implies that no algorithm can compute their decimal expansions to an arbitrarily high degree of accuracy. This feature of (most) irrational numbers has been claimed to be at the heart of the deterministic, but chaotic, behavior exhibited by many nonlinear dynamical systems. In this paper, a number of now classical chaotic systems are shown to remain chaotic when their domains are restricted to the computable real numbers, providing counterexamples to the above claim. More fundamentally, the randomness view of chaos is shown to be based upon a confusion between a chaotic function on a phase space and its numerical representation in R”.

Type
Research Article
Information
Philosophy of Science , Volume 59 , Issue 2 , June 1992 , pp. 263 - 275
Copyright
Copyright © Philosophy of Science Association 1992

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Footnotes

†

I am very grateful to Stephen Kellert, Marco Giunti, John Ewing, Zeno Swijtink, Lawrence Husch, and, especially, Linda Wessels for a number of helpful comments and conversations.

References

Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; and Stacey, P. (1992), “On Devaney's Definition of Chaos”, American Mathematical Monthly 99(4): 332–334.CrossRefGoogle Scholar
Barnsley, M. (1988), Fractals Everywhere. Boston: Academic Press.Google Scholar
Beltrami, E. (1987), Mathematics for Dynamic Modeling. Boston: Academic Press.Google Scholar
Chaitin, G. (1987), Algorithmic Information Theory. Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
Devaney, R. L. (1987), An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley.Google Scholar
Falconer, K. (1990), Fractal Geometry: Mathematical Foundations and Applications. New York: Wiley.Google Scholar
Ford, J. (1986), “Chaos: Solving the Unsolvable, Predicting the Unpredictable!”, in Barnsley, M. F. and Demko, S. G. (eds.), Chaotic Dynamics and Fractals. New York: Academic Press, pp. 1–52.Google Scholar
Ford, J. (1989), “What is Chaos that We Should be Mindful of It?”, in Davis, P. (ed.), The New Physics. Cambridge, England: Cambridge University Press, pp. 348–372.Google Scholar
Hardy, G. H. and Wright, E. M. (1938), An Introduction to the Theory of Numbers. Oxford: Clarendon Press.Google Scholar
Hirsch, M. W. and Smale, S. (1974), Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press.Google Scholar
Kamke, K. (1950), Theory of Sets. New York: Dover Publications.Google Scholar
Martin-Löf, P. (1966), “The Definition of Random Sequences”, Information and Control 9: 602–619.CrossRefGoogle Scholar
Minsky, M. (1967), Computation: Finite and Infinite Machines. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Niven, I. and Zuckerman, H. S. (1972), An Introduction to the Theory of Numbers. 3rd ed. New York: Wiley & Sons.Google Scholar
Robert, F. (1986), Discrete Iterations: A Metric Study. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Schuster, H. G. (1988), Deterministic Chaos: An Introduction. 2d ed., revised. New York: VCH Publishers.Google Scholar
Turing, A. M. (1937), “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society 42: 230–265.CrossRefGoogle Scholar