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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
Publisher : Knopf Doubleday Publishing Group
Publication date : November 14, 2006
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
Best Sellers Rank: #1,075,294 in Books (See Top 100 in Books) #385 in Mathematical Analysis (Books) #425 in Mathematics History #1,365 in Philosophy Metaphysics
#385 in Mathematical Analysis (Books):
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Top Amazon Reviews
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
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
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