Randomness & complexity in pure mathematics
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:
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.
KeywordsDecision Procedure Pure Mathematic Diophantine Equation Mathematical Reasoning Riemann Hypothesis
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- G. J. Chaitin, Algorithmic Information Theory, revised third printing, Cambridge University Press, 1990.Google Scholar
- G. J. Chaitin, Information, Randomness & Incompleteness, second edition, World Scientific, 1990.Google Scholar
- G. J. Chaitin, Information-Theoretic Incompleteness, World Scientific, 1992.Google Scholar
- 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 email@example.com.)Google Scholar