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47 of 48 people found the following review helpful:
3.0 out of 5 stars
Thought-provoking but sloppy and self-important, December 27, 2006
This review is from: Meta Math!: The Quest for Omega (Peter N. Nevraumont Books) (Hardcover)
Chaitin is a good mathematician, not a great one as he seems to think. His invention of algorithmic complexity (independent of the parallel work of the truly great mathematician Kolmogorov) is a permanent feature of the mathematical landscape, and will ensure the immortality of his name, but his other mathematical work, while sound and original, is of technical interest only.
However, he is a better philosopher than he is usually given credit for. His views on the foundations and meaning of mathematics are very original. By avoiding, on the one hand, the formalistic view that mathematical statements are meaningless, and, on the other hand, the conventional view that the current mathematical foundations (for the specialist, I am referring to the ZFC axioms for set theory augmented by large cardinal axioms) are adequate, he is able to show that mathematics is ultimately an empirical science.
The overwhelming inexhaustibility of mathematics is clearer in Chaitin's formulation than in Godel's -- the sense that everything we know about math is an infinitesimal fraction of what there is to know about it. The other major theme which Chaitin clarifies is that mathematics is not logically prior to physics, which Godel also knew, but which is now much more sharply established. And his approach provides a very intuitive way, for those familiar with computer programming, to understand the work of Godel and Turing that avoids the usual self-referential fussing.
That doesn't mean this is a good book. It is badly written, unnecessarily self-congratulatory, and at an uneven technical level. It would have been better for Chaitin to simply state his main results clearly, discuss their implications in the main part of the book, and give the proofs in an Appendix which would be at a higher technical level (but still accessible to those with mathematical ability). Instead, he goes on and on with vague descriptions of his arguments that satisfy neither the casual reader nor the careful one, and unnecessary remarks about how brilliant it all is.
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166 of 186 people found the following review helpful:
2.0 out of 5 stars
Disappointing, March 9, 2006
This review is from: Meta Math!: The Quest for Omega (Peter N. Nevraumont Books) (Hardcover)
As someone who studied meta-mathematics at Caltech and UCLA, I found this book disappointing-stylistically, mathematically and philosophically. To paraphrase the physicist Pauli, this isn't right; this isn't even wrong. This well-meaning man's editors should do a little bit of legwork before reprinting a man's inflated self-appraisal. I am so disappointed in this book that I am seriously considering returning it for a refund.
I guess I should blame myself. My first response to the editorial comment naming the author as the intellectual heir to Gödel and Turing was, "Gregory who?" Shelah, Solovay, Martin: these are names I know, but who is Gregory Chaitin? I should have gone with my gut. In retrospect, it is telling that all the jacket quotes are from freewheeling authors of popularizations, not from respected philosophers, logicians, or scientists.
The entire book is written in an embarrassingly gushing, adolescent style full of boldface and exclamation points. I know that the author was trying to write an enthusiastic, accessible book of philosophical and methodological advocacy, but this doesn't excuse shoddy editorial craftsmanship.
Don't take my word for it. Let the author speak for himself. From page 7, "Gödel's 1931 work on incompleteness, Turing's 1936 work on uncomputability, and my own work on the role of information, randomness and complexity have shown increasingly emphatically that the role that Hilbert envisioned for formalism in mathematics is best served by computer programming languages[.]"
Imagine if a working composer wrote, "Bach's preludes and fugues, Beethoven's symphonies, and my own string quartets have shown increasingly emphatically..." This man's reputation in his declared field is nowhere near his apparent stature in his own mind. The ideas discussed in this book are worthy of late-night musings over a nice brandy, or maybe a Scientific American article, but only after extensive revision. They are not ready for publication in a monograph.
From pages 148-149, "This book is full of amazing case studies of new, unexpected math ideas that reduced the complicated to the obvious. And I've come up with a few of these ideas myself. How does it feel to do that? [...] You have to be seized by a demon, and our society doesn't want too many people to be like that! [...] In fact, I only really feel alive when I'm working on a new idea, when I'm making love to a woman (which is also working on a new idea, the child we might conceive) or when I'm going up a mountain! It's intense, very intense. [...] I push everything else away. [...] I don't pay the bills. [...] And you can't force yourself to do it, any more than a man can force himself to make love to a woman he doesn't want. [...] People may think that something's wrong with me, but I'm okay, I'm more than okay."
And there you have it. I was hoping for a book to catch me up on some of the recent advances in meta-mathematics and how these ideas bear on science and philosophy. For a far better viewpoint on how information science influences modern physics, check out Charles Seife, Decoding the Universe.
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38 of 40 people found the following review helpful:
3.0 out of 5 stars
Interesting, but excess ego, February 22, 2006
This review is from: Meta Math!: The Quest for Omega (Peter N. Nevraumont Books) (Hardcover)
Modesty is a most desirable quality in a scientist, even in the most brilliant. I have no doubt that Gregory Chaitin is a fine and respected mathematician. However, his book "Meta Math!: The Quest for Omega" is written in a style that transmits the impression that he lacks the humility that usually characterizes a true great master scientist. I cannot imagine, for instance, Professor Stephen Hawking comparing his own mind to that of Isaac Newton and referring to himself as the successor of Einstein. Even if it were true, and many people think it is, it would certainly be arrogant if stated by Hawking himself. In fact, this is exactly what Chaitin appears to be doing in his book. He is not embarrassed at all when he claims to be at the same intellectual level (even as a teenager) as Leibniz. He describes his proof for diophantine equations as being side by side with those of Euclid and Euler. Moreover, he refers to Kurt G?del as his "predecessor", something that I think is absolutely unfair considering G?del as the original author of the incompleteness theorem. And, amazingly enough, there is a section included in the book that deals with egotism in science and how it should be avoided- very subtly written. In this section, Chaitin claims that no scientific idea should have only one name associated with it and as an example, he describes the chain of thoughts that inspired mathematicians since Zeno up to himself. I am in total agreement with this concept. However, in the next paragraph, Chaitin explains how, "...the best minds in the human race...." join together to create these theories. Hence, he has almost subliminally included himself in this group of distinguished intellectuals. I have read many books on scientific dissemination and this is the first book of this type that I have encountered where the author's picture appears on the cover. Is this another example of Chaitin "tooting his own horn"? This is really an insignificant detail although I think it fits into the whole context of the author's self-centeredness.
With respect to the content of the book, in my opinion Chaitin partially succeeds in transmitting the essence of his ideas. There is, for instance, an excess of exclamation points within the text for my taste. There are actually some single paragraphs that contain exclamation points in every sentence. Personally, this overabundance of emphasis makes the reading stressing and difficult. For the sake of clarity, I decided to ignore them throughout, and the intelligibility and ease of reading actually improved. I am assuming that the objective of such an overuse of exclamation points is to transmit Chaitin's own personal enthusiasm for the subject at hand. However, I think that the final result somehow underestimates the reader's capacity to comprehend and appreciate the ideas. It is, in my opinion, a childish approach.
Despite these inconveniences, I feel that the book finally reaches its goal of explaining what Ω is. However, according to my impartial reading of the book, I deduce that Chaitin's proof on randomness is just a different approach to understanding G?del's incompleteness and an extension of Turing's results on the limits of proof and computation. Chaitin, in contrast, gives the impression that his quest for Ω is the summit in the line of mathematical thought since the Pythagorean school. I am not a mathematician but, however, I am cautious so I suspect it is just another of the author's self-magnanimous claims. Chaitin pompously defines his results on incompleteness as "the jewel in the crown" of Algorithmic Information Theory. According to my modest understanding, contribution of Kolmogorov to AIT is at least as important as that of Chaitin. However, Kolmogorov is cited only once throughout the book, and his name is suspiciously absent from the index. I consider this a clear contradiction to Chaitin's assertion of being against egotism in science.
In summary, I have certainly learned new and interesting concepts after reading this book. I particularly enjoyed the timeline established by the author to exemplify the continuum, complexity and incompleteness problems from a historical perspective. Unfortunately, I did not appreciate the hyper-emphasizing text style and the self-centered attitude of the author.
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