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Complex Systems and Binary Networks pp 2–21Cite as

Randomness & complexity in pure mathematics

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Part of the book series: Lecture Notes in Physics ((LNP,volume 461-461))

Abstract

One normally thinks that everything that is true is true for a reason. I've found mathematical truths that are true for no reason at all. These mathematical truths are beyond the power of mathematical reasoning because they are accidental and random.

Using software written in Mathematica that runs on an IBM RS/6000 workstation, I constructed a perverse 200-page algebraic equation with a parameter N and 17,000 unknowns:

Left-Hand-Side(N)=Right-Hand-Side(N).

For each whole-number value of the parameter N, ask whether this equation has a finite or an infinite number of whole number solutions. The answers escape the power of mathematical reason because they are completely random and accidental.

This work is an extension of famous results of Gödel and Turing using ideas from a new field called algorithmic information theory.

Originally appeared in International Journal of Bifurcation and Chaos 4 (1994), pp. 3–15

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Bibliography

  1. G. J. Chaitin, Algorithmic Information Theory, revised third printing, Cambridge University Press, 1990.

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  2. G. J. Chaitin, Information, Randomness & Incompleteness, second edition, World Scientific, 1990.

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  3. G. J. Chaitin, Information-Theoretic Incompleteness, World Scientific, 1992.

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  4. G. J. Chaitin, “Exhibiting randomness in arithmetic using Mathematica and C”, IBM Research Report RC-18946, 94 pp., June 1993. (To obtain this report in machine readable form, send e-mail to chaitin@watson.ibm.com.)

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Authors and Affiliations

  1. IBM Research Division, P. O. Box 704, 10598, Yorktown Heights, NY, USA

    G. J. Chaitin

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  1. G. J. Chaitin
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Ramón López-Peña Henri Waelbroeck Riccardo Capovilla Ricardo García-Pelayo Federico Zertuche

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© 1995 Springer-Verlag Berlin Heidelberg

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Chaitin, G.J. (1995). Randomness & complexity in pure mathematics. In: López-Peña, R., Waelbroeck, H., Capovilla, R., García-Pelayo, R., Zertuche, F. (eds) Complex Systems and Binary Networks. Lecture Notes in Physics, vol 461-461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103568

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  • DOI: https://doi.org/10.1007/BFb0103568

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60339-9

  • Online ISBN: 978-3-540-44937-9

  • eBook Packages: Springer Book Archive

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