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Abstract

G. J. Chaitin is at the IBM Thomas J. Watson Research Center in New York. He has shown that God plays dice not only in quantum mechanics, but even in the foundations of mathematics, where Chaitin discovered mathematical facts that are true for no reason, that are true by accident. This book collects his most wide-ranging and non-technical lectures and interviews, and it will be of interest to anyone concerned with the philosophy of mathematics, with the similarities and differences between physics and mathematics, or with the creative process and mathematics as an art. "Chaitin has put a scratch on the rock of eternity." Jacob T. Schwartz, Courant Institute, New York University, USA "(Chaitin is) one of the great ideas men of mathematics and computer science." Marcus Chown, author of The Magic Furnace, in NEW SCIENTIST "Finding the right formalization is a large component of the art of doing great mathematics." John Casti, author of Mathematical Mountaintops, on Godel, Turing and Chaitin in NATURE "What mathematicians over the centuries - from the ancients, through Pascal, Fermat, Bernoulli, and de Moivre, to Kolmogorov and Chaitin - have discovered, is that it ÄrandomnessÜ is a profoundly rich concept." Jerrold W. Grossman in the MATHEMATICAL INTELLIGENCER

Content

A Century of Controversy over the foundations of mathematics.- How to be a mathematician.- The creative life: science vs art.- Algorithmic information theory and the foundations of mathematics.- Randomness in arithmetic.- The reason for my life.- Undecidability and randomness in pure mathematics.- Math, science and fantasy.- Sensual mathematics.- Final thoughts.

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Abstract

This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Gödel views of the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the autor web site

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Abstract

Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of Gödel's incompleteness theorem, using an information theoretic approach based on the size of computer programs. One half of the book is concerned with studying the halting probability of a universal computer if its program is chosen by tossing a coin. The other half is concerned with encoding the halting probability as an algebraic equation in integers, a so-called exponential diophantine equation.