Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable O 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 Gödel 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 Gödel 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.
On the Difficulty of Computations
Information-Theoretic Computational Complexity
Randomness and Mathematical Proof
Gödel's Theorem and Information
Randomness in Arithmetic
Paradoxes of Randomness
Complexity and Leibniz
The Limits of Reason
How Real Are Real Numbers?
Is Incompleteness a Serious Problem?
How Much Information Can There Be in a Real Number?
and other papers