- Paperback: 368 pages
- Publisher: World Scientific Publishing (6 Oct 2007)
- Language English
- ISBN-10: 9812708960
- ISBN-13: 978-9812708960
- Product Dimensions: 25.9 x 18 x 1.7 cm
- Amazon Bestsellers Rank: 1,005,647 in Books (See Top 100 in Books)
- See Complete Table of Contents
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Product Description
Product Description
Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable 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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2 of 2 people found the following review helpful
Chaitin Does Not Stand Up to Formal Scrutiny
23 April 2012
Format:Paperback
For many years Gregory Chaitin has been talking about Godel's Incompleteness Theorem and how he has provided a better proof. However, a few years after Godel's 1931 paper, Barkley Rosser published his extension which is universally recognized as being a more powerful theorem since it reduces the requirement of w-consistency to simple consistency. Chaitin never mentions Rosser's improvement. It is meaningless to talk about improving upon a theorem that has already been improved upon.
Chaitin argues that mathematics is random and incompleteness theorems are unprovable because they are just randomly true. But it is not that Godel's sentence G cannot be proven. It simply cannot be proven within the formal system provided by Godel. Godel in fact proves that G is true, and notes, "The theorem which is undecidable within the system PM has hence been decided by metamathematical considerations."
Chaitin's number Omega is not a number but rather a function of the universal Turing Machine used, which can be in any system of computing and between any two given numbers between 0 and 1. There in fact is no probability that a Turing Machine will halt because the incidence of halting can fluctuate wildly as the size of its input increases and does not converge over infinity.
Chaitin attempts to analyze formal mathematical concepts with informal statements about randomness that do not stand up to scrutiny.
Chaitin argues that mathematics is random and incompleteness theorems are unprovable because they are just randomly true. But it is not that Godel's sentence G cannot be proven. It simply cannot be proven within the formal system provided by Godel. Godel in fact proves that G is true, and notes, "The theorem which is undecidable within the system PM has hence been decided by metamathematical considerations."
Chaitin's number Omega is not a number but rather a function of the universal Turing Machine used, which can be in any system of computing and between any two given numbers between 0 and 1. There in fact is no probability that a Turing Machine will halt because the incidence of halting can fluctuate wildly as the size of its input increases and does not converge over infinity.
Chaitin attempts to analyze formal mathematical concepts with informal statements about randomness that do not stand up to scrutiny.
10 of 14 people found the following review helpful
Very interesting
7 Jan 2008
Format:Paperback|Amazon Verified Purchase
I bought this book, from a New Scientist review, for my husband who is extremely gifted in math, and not so much verbally. Reading is a chore. So when, out of all the Christmas books past, he pronounced this volume a really, really good book and read some every day - I knew it was extraordinary. Now I hope he finishes soon so I can start it!
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