- Paperback: 240 pages
- Publisher: Vintage (November 14, 2006)
- Language: English
- ISBN-10: 1400077974
- ISBN-13: 978-1400077977
- Product Dimensions: 5.2 x 0.7 x 8 inches
- Shipping Weight: 12.6 ounces (View shipping rates and policies)
- Average Customer Review: 33 customer reviews
- Amazon Best Sellers Rank: #1,555,627 in Books (See Top 100 in Books)
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Meta Math!: The Quest for Omega Paperback – November 14, 2006
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“A startling vision of the future of mathematics. . . . The Chaitinesque intellectual future will be eternally youthful and anarchic.”–American Scientist
“Math’s dark secret is out. . . . Chaitin explains why omega, a number he discovered thirty years ago, has him convinced that math is based on randomness.”
“Captivating. . . . With extraordinary skill and a gentle humor, Chaitin shares his profound insights.”
–Paul Davies, author of How to Build a Time Machine
“A clearly written and witty look at a difficult subject. . . . Chaitin explains with infectious enthusiasm how mathematics doesn't equal certainty.” –Science News
About the Author
Gregory Chaitin works at the IBM Thomas J. Watson Research Center in Westchester County, New York, and is a visiting professor in the Computer Science Department of the University of Auckland, New Zealand. The author of eight previous books on mathematics, he lives in New York.
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For example, it took Kurt Gödel many many pages of dense, exotic symbolism to set out his famous result, the incompleteness (if not inconsistency) of ordinary arithmetic. Viz., there are "theorems" which cannot be derived from the axioms of arithmetic, but which we can nonetheless see to be true because they state, in effect, their own non-derivability. That proof is profoundly complex in part because, as Chaitin points out, Gödel has to concoct a sort of programming language to get his result: this (1931) five years before there was any good idea of what a computer might be (Turing, 1936).
Using that notion of a computer, and more particularly the "elegance" of a computer program -- its being the simplest to generate a particular output -- Chaitin can convince you of the same result by a much simpler route.
Axiomatic theories can be represented as computer programs capable of generating, one by one, all the possible valid theorems; and of course a program can be represented as a string of bits, "1"s and "0"s. Bits also measure information, à la Claude Shannon, so we immediately have a measure of the amount of information contained in that axiomatic theory. So, what if we construct a statement that contains more information than that (which, pretty much, is what Gödel did)? You cannot gain information via transmission, you can only lose it (the information-theoretic incarnation of the Second Law of Thermodynamics) -- you cannot get more information from less. So obviously, that statement cannot be derived in that axiomatic system, whatever it says!
Chaitin has his faults. His prose can be off-puttingly gung-ho: he is inordinately fond of exclamation points. And he can be very full of himself and his discoveries, somewhat mitigated by the free acknowledgement of others' contributions and influences (he is also inordinately fond of Leibniz). But he does not go quite so far as to proclaim himself a great lover, and when he supposes one has a greater understanding of a mathematical proof when one has derived it oneself, he is only echoing his hero Leibniz: "the sources of invention ... are, in my opinion, more interesting than the inventions themselves" (p. 56).
I could hardly praise this book more than to say it would be a great read for any numerately-inclined high school or college student, or indeed any inquisitive person with a taste for philosophically-infused mathematics. You can find scholarly criticisms of Chaitin and his approach, e.g. in his Wikipedia entry. But if you get that far, he has already accomplished a major goal of his little book: to get you engaged in these intriguing metamathematical issues of Infinity And Beyond.
(Thanks to Steve Schwartz for correcting some oversights.)
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!