Thinking about Godel and Turing: Essays on Complexity, 1970-2007

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Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable [Omega] number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as Godel and Turing.

This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of Godel and Turing on the limits of mathematical methods, both in logic and in computation. Chaitin argues here that his information-theoretic approach to metamathematics suggests a quasi-empirical view of mathematics that emphasizes the similarities rather than the differences between mathematics and physics. He also develops his own brand of digital philosophy, which views the entire universe as a giant computation, and speculates that perhaps everything is discrete software, everything is 0's and 1's. Chaitin's fundamental mathematical work will be of interest to philosophers concerned with the limits of knowledge and to physicists interested in the nature of complexity.

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Product Details

  • ISBN-13: 9789812708960
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 8/28/2007
  • Pages: 368
  • Product dimensions: 7.00 (w) x 10.00 (h) x 0.80 (d)

Table of Contents

Introductory note     1
On the difficulty of computations: IEEE Transactions on Information Theory, 1970     3
Information-theoretic computational complexity: IEEE Transactions on Information Theory, 1974     17
Randomness and mathematical proof: Scientific American, 1975     31
Godel's theorem and information: International Journal of Theoretical Physics, 1982     47
Randomness in arithmetic: Scientific American, 1988     65
Randomness in arithmetic and the decline & fall of reductionism in pure mathematics: Bulletin of the European Association for Theoretical Computer Science, 1993     75
A century of controversy over the foundations of mathematics: Calude & Paun, Finite versus Infinite, 2000     99
A century of controversy over the foundations of mathematics: Complexity, 2000     129
Metamathematics and the foundations of mathematics: Bulletin of the European Association for Theoretical Computer Science, 2002     153
Paradoxes of randomness: Complexity, 2002     169
Two philosophical applications of algorithmic information theory: Lecture Notes in Computer Science, 2003     189
On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility: Hogrebe & Bromand, Grenzen und Grenzuberschreitungen, 2004     201
Leibniz, information, math & physics: Loffler & Weingartner, Wissen und Glauben,2004     227
Leibniz, randomness & the halting probability: Mathematics Today, 2004     241
Complexity & Leibniz: Academie Internationale de Philosophie des Sciences, Tenerife, 2005     247
The limits of reason: Scientific American, 2006     251
How real are real numbers?: International Journal of Bifurcation and Chaos, 2006     267
Epistemology as information theory: From Leibniz to [Omega]: Collapse: Journal of Philosophical Research and Development, 2006     281
Is incompleteness a serious problem?: Lolli & Pagallo, La complessita di Godel, 2007     299
Speculations on biology, information & complexity: Bulletin of the European Association for Theoretical Computer Science, 2007     303
How much information can there be in a real number?: International Journal of Bifurcation and Chaos, 2007     313
The halting probability [Omega]: Irreducible complexity in pure mathematics: Milan Journal of Mathematics, 2007     319
The halting probability [Omega]: Concentrated creativity: Obrist, Formulas for the Twenty-First Century, 2007     333
List of publications     335
Acknowledgements     343
About the author     347
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