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At a glance
- Condition
- New
- Dimensions
- 5.19 x 0.6 x 8 in
- Weight
- 7.2 oz
- Publisher
- Knopf Doubleday Publishing Group
- Publication date
- November 14, 2006
- Edition
- Reprint
- Language
- English
Description
Gregory Chaitin, one of the world’s foremost mathematicians, leads us on a spellbinding journey, illuminating the process by which he arrived at his groundbreaking theory. Chaitin’s revolutionary discovery, the Omega number, is an exquisitely complex representation of unknowability in mathematics. His investigations shed light on what we can ultimately know about the universe and the very nature of life. In an infectious and enthusiastic narrative, Chaitin delineates the specific intellectual and intuitive steps he took toward the discovery. He takes us to the very frontiers of scientific thinking, and helps us to appreciate the art—and the sheer beauty—in the science of math. Read more
Specs 9
| Publisher | Knopf Doubleday Publishing Group |
|---|---|
| Publication date | November 14, 2006 |
| Edition | Reprint |
| Language | English |
| Print length | 240 pages |
| ISBN-10 | 1400077974 |
| ISBN-13 | 77 |
| Item Weight | 7.2 ounces |
| Dimensions | 5.19 x 0.6 x 8 inches |
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Top Amazon Reviews
Got it for a penny, it's worth that for sure...
Format: Paperback
Guy is full of ego but NOT full of crap as some reviews have said. At the prices this has dropped to, it's well worth a scan, even if just for the breathtaking feeling of how little we really know about math, and the possible solutions for tensions between math and computing. This is really an odd little book, and I can see why you either love it or hate it. Although the author has a very high self opinion (or, more kindly, not a big problem with self esteem), he's also a mathematician and not afraid to make statements like "I used to think God was a mathematician, but now I realize He's a programmer..." which takes courage for a science type. I know a group of transplant surgeons who claim that without big egos there would BE no transplants, so who knows? The author (later) even makes FUN of ego... quite the cornucopia. Maybe some of his personal syllogisms are tongue in cheek and we're just not getting them... I mean for a bright guy to equate: "And as Galileo and I like to say..." (doesn't that HAVE to be spoofing???). Ok, another apologist theory. They guy really has a great sense of humor and really IS as brilliant as his "fans" think. So, those of us who get the stench of ego might just be walking into a carefully laid trap where he is demonstrating that his use of Berry's paradox in proving Chaitin incompleteness necessarily includes self referential constructs, just as mathematical induction requires, and in fact is a form of, recursion. Ok, I tried. The book is kindof a rambling, semi (psychological) autobiography of wacky and amazing ideas. It will deviate off into details about LISP you never really wanted to know, but then give profound insights into math you won't find anywhere else. At a penny, take a chance!!!
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Reviewed in the United States on May 27, 2013 by Professor dot biz
The best book on math I have ever read. Bar none.
Format: Paperback
This book is great. Other reviewers might bash it because they are coming from the perspective of a mathematician or because Chaitin writes about his OWN accomplishments (rather than pretend that he is a disinterested third party) which I guess comes across to some people as vain self-congratulation, but I think they are missing the point!: This book is to get you excited about the mystery of math. It is a book written for intelligent people who think math is about crunching numbers or doing tried and true algorithms to solve specific problems. To me, as a non-mathematician with a philosophy background who has read a LOT of math and science non-fiction, it was easily the best book about math that I have ever read. He was enthusiastic and filled with joy about the topic and it showed in a way that no other author on the subject that I have read has been able to convey. The book bounced around gleefully but never left any single topic half-explained and it all intertwined together to create what I thought was a great reading experience. But even more important, it left me with a much higher opinion of the subject of math as human CREATIVE endeavor - that there is so much more to create and do, and that it isn't a dusty topic relegated to the semi-autistic, but instead a playground waiting for rebels and the truly inspired to break new ground. And that alone is worth the price of admission. Plus the book is super cheap now. So why not!
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Reviewed in the United States on November 30, 2015 by Chuck J
A real boundary between what one can know of, but not know?
Format: Hardcover
The omega number arises in the context of what is called algorithmic information theory, which the author of this book has been instrumental in developing. It is not a difficult concept to understand at least from the standpoint of where it stands in the deductive ladder of modern mathematics. The goal of algorithmic information theory was to formulate a notion of randomness that was not only rigorous from a mathematical standpoint but was also embedded in the notion of an algorithm. Recognizing that it is impossible to construct and infinite random sequence using a (deterministic) algorithm, the author's main contribution was to define a random infinite (binary) sequence using what he has called the `halting probability.' The omega number, or `halting probability' is defined with respect to a (prefix) machine that halts for some input program. Since the machine can halt, the omega number is positive, but is less than 1 since the machine does not always halt. The omega number is a probabilistic notion since it is the probability that the machine halts if its program is given by a sequence of fair coin tosses. The author shows that the omega number is a random real number, is not computable, and therefore transcendental. All of these notions he discusses in fair detail in the book. The omega number, as he defines and discusses it, is quite an astounding quantity, for he concludes that it measures what can be known by human reason. If the entire edifice of mathematics were compressed to a particular number of bits, then the omega number (for this number of bits) can be used to decide whether every result in this edifice is true or false (or independent). The book is interesting reading, the reader obtaining insight into how the omega number was discovered, its role in the philosophy of mathematics, and its ramifications in the automatic discovery of mathematics. In addition, there are many places in the book where the author gives sound advice on how best to pursue research in science and mathematics, and even philosophy. For example, he encourages the sharing of ideas in order for them to become successful. He also complains of the excessive egotism that permeates the scientific community, and describes this as "poisoning" science. In this regard, he correctly notes that scientific results are the product of many minds, and that their prioritization to one individual is wrong and instead is diffused over the population of researchers. The author is correct in saying that mathematics is not very different from physics, and that the creation of mathematics involves intuition and guesswork. But he is not convincing in showing that this intuition cannot be emulated in a (calculating) machine, but merely takes it to be an activity that must take place due to the limitations arising from the omega number. It is one thing to prove that the omega number places limits on mathematics. It is quite another to characterize "intuition" explicitly, freeing it from its current mystical connotations, and showing how it operates to bring about creative ideas in mathematics. And yes, mathematics must be beautiful, as the author states in the book. Aesthetic quality in pure mathematics has driven many of the research programs in mathematics. But mathematical beauty is in the eye of the beholder, and varies radically from one mathematician to another. Each mathematician deploys their own hedonic function when judging the beauty of particular mathematical constructions. The author also mentions AM (Automated Mathematician) in the book, calling it, interestingly, a "program." At the time it first appeared, AM was thought of as "intelligent" and had the capability of creating original concepts in mathematics. After this initial confidence, it was then subjected to intense criticism, and due to this criticism AM was abandoned both by its creator and everyone else in the automated reasoning community of researchers. It has now become merely a "program", i.e. a collection of statements that cannot possibly be thought of as intelligent or creative. The author though exaggerates the ramifications of the incompleteness theorems of Godel and in the formalist program of Hilbert. The everyday practice and discovery of mathematics is not done formally, and if it were it is doubtful that mathematicians could be as productive as they are. Mathematical results are reported using a mixture of natural language and mathematical syntax, and in no way are like the formal languages that Godel and Hilbert insist on. So the results of Godel do not cause any trouble for mathematics, since all mathematical research is expressed in informal language. And there are no examples of statements of the Godelian type being derived in the everyday practice of mathematics. Godelian statements must be artificially contrived and even then using a non-constructive diagonalization argument. Therefore it is of no surprise at all to find out that all of mathematics has not been derived from a small collection of mathematics. It is certainly not discovered that way, but rather arises as vague, random mental associations in the mind of the mathematician. Once discovered though, the results are published, using as much rigor as practical, with this rigor being constrained by the use of natural language. It remains to be seen whether the use of natural language can be eliminated ala the strategy of Bourbaki. If it can, maybe using a variant of discourse representation theory or some other strategy, then it is not unreasonable to believe that all mathematical results can be derived from a few axioms. If they cannot, this is no reason for worry, as they are intrinsically beautiful in themselves, and their applications will still go on and on.
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Reviewed in the United States on October 9, 2005 by Dr. Lee D. Carlson
Best Book About Math I've Read
Format: Hardcover
MetaMath is the best book about math that I have read. I say that because of the author's uniquely refreshing attitude about how real math is really done. I wish that I had had this book to read when I was pursuing a degree in Math some years back. It would have given me permission to be creative, use my imagination, be more bold and adventurous in playing with matematical concepts and getting down to their vital roots. My math classes gave me none of that, and left me with the feeling that the study of mathematics was nothing more than the study of polished and perfected theorems stripped out of their historical context. The specific topic of MetaMath is the Incompleteness Theorem. I have also read several books on that (Goedel's contribution specifically.) But MetaMath helped me really really understand what the Incompleteness Theorem is all about, and why it is important. Now it seems so clear. A set of closed axioms can never suffice to explain something that is intrinsically open ended and infinite. New axioms must always be added as our knowledge of the mathematical frontiers expand. I thoroughly enjoyed this book. Kudos to the author for his very healthy and encouraging attitude to all explorers of scientific and mathematical truth. I cannot imagine that anyone but the most stodgy stuffed shirt would not find benefit and enjoyment from reading this mind expanding work.
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Reviewed in the United States on October 15, 2005 by Amazon Customer
Thought-provoking but sloppy and self-important
Format: 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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Reviewed in the United States on December 28, 2006 by Joe Shipman
Fundamental
Format: Paperback
Besides Chaitan's Omega there is only one other fundamental physical known constant that has as great a weight: alpha, the fine structure constant in physics. Funny since they are near the same in value: alpha =0.007297353079644819 ( 10 places known in government standards) omega=0.0078749969978123844 ( 17 places known in tables: OEIS) From the definition of Omega given as a sum of inverse powers of two, there is obviously another constant: Not_Omega= 2-omega=Sum[2^(-n), {n, 0, Infinity}]-Sum[2^(-p), {p_halting, 0, Infinity}] Not_Omega=Sum[2^(-q), {q_not_halting, 0, Infinity}] Another equation that the book suggests to me is the elliptic: w^2=(x+2)*(x-omega)*(x+omega)=x^3+2*x^2-omega^2*x-2*omega^2 I'm not really ready to write this review: I somehow doubt I will ever be. This guy Chaitin has discovered a fundamental property or law of nature. I doubt it will be much better understood in an hundred years, but more places will have been calculated. Maybe even a formula for the digits will be derived like that for Pi. He is a modern Pythagoras or Euclid.
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Reviewed in the United States on October 12, 2007 by Roger Bagula
Agree with negative reviews
Format: Hardcover
Rather than writing a full review as I intended to do, I see that other readers have stated very clearly my problem with this book. I wish I had read those reviews prior to ordering... It is very sloppily written, and should have been much better edited. (Or rather, much better edited!! as he would have written it.) In fact it is one of the poorest written popular science books I've read among the dozens I've accumulated over the years. Although his thesis is very convincing, and the concept of omega is very fascinating with regards to incompleteness and noncomputable numbers, etc., the actual informational content is larded with numerous references to how brilliant the writer is, what a genius he was as a child, how important his work is, how well he compares with Turing and Godel (how reasonable is that?), and multiple comments that are just downright weird, as when he talks about the information content of having sex with a woman (in terms of genetic transfer of information, that is). So the earlier reviewer who mentioned that he seems to be hiding low self-esteem with self-aggrandizement seems to me to be right on the mark. Either that, or the author is living on a planet of his own where he doesn't understand how embarassing and unsuitable this sort of heavy-handed arrogance is for this kind of book. Eventually I gave up reading it because of the odd writing style. Another minor drawback is that the content is very repetitive, particularly in comparison with other popular science books which appear dense with information.
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Reviewed in the United States on May 16, 2007 by J. Jenkins
Elegant
Format: Paperback
Paraphrasing elegant program definition: No smaller book in the same language has the same output, I'd to say that this book is almost Elegant!. It's a piece of art, is filled not only with interesting ideas, it also has references and histories and feelings, this book is like a ride at high speed within deep ideas, but a trip where you feel safe, because you trust the driver. He suggest demostrations, avoid unuseful details or spoon feeding (that would fill the books with redundance, and I repeat it's elegant)
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Reviewed in the United States on February 13, 2012 by hernan_e