Gregory Chaitin

Meta Math!:The Quest for Omega

Gregory Chaitin has done significant work on how computability is limited by complexity. In Meta Math!:The Quest for Omega he gives an account of some of his work aimed at the non-specialist reader. The book starts with his early fascination with Gödel's incompleteness theorem as well as his interest in computing, and in particular the LISP programming language. He explains how several of the philosophical ideas which seem to have arisen with the advent of the computer age, such as the universe being built out of information, were in fact thought of by Leibniz several centuries before.

Chaitin then moves on to consider the real numbers, and in particular the idea of a random real - since the real numbers form an uncountable set, most of them will not have any compact description. In the final number he gets on to the particular random real Ω. and shows what powerful results can be obtained from the idea that a given system has a certain complexity, and this limits the complexity of what can be generated from that system.

The book might seem hard to follow sometimes, but I think it's worth sticking at it, as any problems aren't so much due to technicalities as to the number of new concepts which Chaitin introduces. Indeed his enthusiasm for his work is obvious, and if you read this books then, who knows, maybe you'll pick up some of that enthusiasm too.

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Paperback 240 pages
ISBN: 1400077974
Salesrank: 840294
Weight:0.05 lbs

Published: 2006 Vintage

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Paperback 240 pages
ISBN: 1400077974
Salesrank: 624777
Weight:0.05 lbs

Published: 2006 Vintage Books USA

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Paperback 240 pages
ISBN: 1400077974
Salesrank: 252586
Weight:0.05 lbs

Published: 2006 Vintage

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Product 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.

Product 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.

Inconsistently Interesting ***
I was originally interested in this book because of D. Zeilberger's positive appraisal of G. Chaitin's mathematical work. Zeilberger has a nose for interesting mathematics and he was spot on as usual. However, it's mostly just the math that is interesting here. Chaitin as a spokesman for mathematical ideas and philosophy is amateurish and inconsistent. As a spokesman for himself - well, you can read the other reviewers to get an idea. Let me give you a telling example of Chaitin's loose philosophizing. On page 7 he says "formalism in mathematics is best served by computer programming languages, which are in fact formalisms that can be mechanically interpreted--but they are formalisms for computing and calculating, NOT for reasoning, NOT for proving theorems, and most emphatically NOT for inventing new mathematical concepts NOR for making new mathematical discoveries" (p. 7, caps are mine). But then when he is praising an apparently useful method in number theory, maximally divisible integers, he states "the brilliantly intuitive mathematician Ramanujan...and Doug Lenat's artificial intelligence program AM both came up with just such a concept" (p. 16). Ok. So he contradicts his earlier, emphatic, remark that computers cannot invent new mathematical concepts. But this is not the only false claim he makes. Where is says that computers cannot prove theorems, this also is patently false. Not only do computers verify existing proofs, they also find alternative proofs and new proofs. W. McCune's proof of the Robbins Conjecture is a well known example of a new proof generated by a computer program (a modified version of OTTER). His proof was known before the publication of Chaitin's book as well so Chaitin is apparently spouting off without checking the facts. For more info on automated theorem proving see work by L. Wos. So one of Chaitin's statements he shows is false, while I showed that another is false. This gives just a brief illustration of the clumsy philosophical dialogue throughout. In sum, the math here is well worth the read (and the price). The verdict might still be out on the depth of chaitin's results, but there are some fascinating layers to algorithmic information theory that Chaitin touches on. Just beware that there is a lot (a lot!) of peripheral trash to wade through as well.

Inconsistently Interesting ***

I was originally interested in this book because of D. Zeilberger's positive appraisal of G. Chaitin's mathematical work. Zeilberger has a nose for interesting mathematics and he was spot on as usual. However, it's mostly just the math that is interesting here. Chaitin as a spokesman for mathematical ideas and philosophy is amateurish and inconsistent. As a spokesman for himself - well, you can read the other reviewers to get an idea.

Let me give you a telling example of Chaitin's loose philosophizing. On page 7 he says "formalism in mathematics is best served by computer programming languages, which are in fact formalisms that can be mechanically interpreted--but they are formalisms for computing and calculating, NOT for reasoning, NOT for proving theorems, and most emphatically NOT for inventing new mathematical concepts NOR for making new mathematical discoveries" (p. 7, caps are mine). But then when he is praising an apparently useful method in number theory, maximally divisible integers, he states "the brilliantly intuitive mathematician Ramanujan...and Doug Lenat's artificial intelligence program AM both came up with just such a concept" (p. 16). Ok. So he contradicts his earlier, emphatic, remark that computers cannot invent new mathematical concepts.

But this is not the only false claim he makes. Where is says that computers cannot prove theorems, this also is patently false. Not only do computers verify existing proofs, they also find alternative proofs and new proofs. W. McCune's proof of the Robbins Conjecture is a well known example of a new proof generated by a computer program (a modified version of OTTER). His proof was known before the publication of Chaitin's book as well so Chaitin is apparently spouting off without checking the facts. For more info on automated theorem proving see work by L. Wos.

So one of Chaitin's statements he shows is false, while I showed that another is false. This gives just a brief illustration of the clumsy philosophical dialogue throughout.

In sum, the math here is well worth the read (and the price). The verdict might still be out on the depth of chaitin's results, but there are some fascinating layers to algorithmic information theory that Chaitin touches on. Just beware that there is a lot (a lot!) of peripheral trash to wade through as well.

too full of himself *
sorry, i just cannot read this book - try it and you will realize soon why! the author believe that he is a genius. if your answer now is - who cares - please note that 50% of the book is about math while 50% is just about how great the author is... maybe this book works for Americans, but if you are from Europe do not even touch it.....

too full of himself *

sorry, i just cannot read this book - try it and you will realize soon why! the author believe that he is a genius. if your answer now is - who cares - please note that 50% of the book is about math while 50% is just about how great the author is... maybe this book works for Americans, but if you are from Europe do not even touch it.....

Very very interesting ideas, and yet... ****
not altogether intelligibly presented. Towards the beginng of MM, the author discusses a certain book that exerted a tremendous impact on his early intellectual development: Nagel and Newman's Godel's Proof. The same book was also instrumental in shaping my own deepening interest in questions having to do with the foundations of mathematics. I wish Chaitin had taken N+N's model more to heart. Whereas their own book is a model of clarity, systematically laying out the proof of Godel's Theorem, more or less as Godel had proved it himself, Chaitin's treatment of his own very important contributions to the field is full of explanatory holes. Not errors (so far as I can tell), but simply gaps. For the non-expert reader, these can be very frustrating, particularly as (s)he pursues Chaitin's argument to its increasingly opaque conclusion. I think Chaitin could have done a better job of laying out his Theorem from A to Zed. For that reason, and because the text is larded with endless exclamation points and other juvenalia, 4 stars. But the 'story' MM tells is 5-star stuff without doubt. A fascinating chapter in the history of information and computation theory, and metamathematics.

Very very interesting ideas, and yet... ****

not altogether intelligibly presented. Towards the beginng of MM, the author discusses a certain book that exerted a tremendous impact on his early intellectual development: Nagel and Newman's Godel's Proof. The same book was also instrumental in shaping my own deepening interest in questions having to do with the foundations of mathematics. I wish Chaitin had taken N+N's model more to heart. Whereas their own book is a model of clarity, systematically laying out the proof of Godel's Theorem, more or less as Godel had proved it himself, Chaitin's treatment of his own very important contributions to the field is full of explanatory holes. Not errors (so far as I can tell), but simply gaps. For the non-expert reader, these can be very frustrating, particularly as (s)he pursues Chaitin's argument to its increasingly opaque conclusion. I think Chaitin could have done a better job of laying out his Theorem from A to Zed. For that reason, and because the text is larded with endless exclamation points and other juvenalia, 4 stars. But the 'story' MM tells is 5-star stuff without doubt. A fascinating chapter in the history of information and computation theory, and metamathematics.

A great pleasure; a great book *****
This is one of my favorite math/science books, ever. It's right up in the company of George Gamow's One, Two, Three... Infinity; Erwin Schrodinger's What is Life; and Richard Feynman's Lectures on Physics. Chaitin's work conveys that great rarity, a powerful new idea, presented in a form that rewards reading over and over until it gradually sinks in. Thank you, Greg.

A great pleasure; a great book *****

This is one of my favorite math/science books, ever. It's right up in the company of George Gamow's One, Two, Three... Infinity; Erwin Schrodinger's What is Life; and Richard Feynman's Lectures on Physics. Chaitin's work conveys that great rarity, a powerful new idea, presented in a form that rewards reading over and over until it gradually sinks in. Thank you, Greg.

An outstanding book *****
An outstanding book in meta mathematics, taking us from beginnings of meta mathematics with Hilbert and Godel through to the randomness embodied in omega, the halting probability. The trail moves through Leibniz, Godel's incompleteness, Turing's halting and incompleteness, to Algorithmic Information Theory which proves the key to finding Omega, the number whose bits show irreducible randomness in mathematics. The writer assumes no major existing knowledge, but leads you quickly into high ground. Some discussions also touch on philosophy and the history of ideas. His enthusiasm is infectious; a great read.

An outstanding book *****

An outstanding book in meta mathematics, taking us from beginnings of meta mathematics with Hilbert and Godel through to the randomness embodied in omega, the halting probability.
The trail moves through Leibniz, Godel's incompleteness, Turing's halting and incompleteness, to Algorithmic Information Theory which proves the key to finding Omega, the number whose bits show irreducible randomness in mathematics.
The writer assumes no major existing knowledge, but leads you quickly into high ground.
Some discussions also touch on philosophy and the history of ideas.
His enthusiasm is infectious; a great read.

Amazing book *****
This book is amazing, and the Central primary idea of the book is just mind bogglingly interesting and important. The incompleteness of Formal systems and a real number that is only just uncomputable. Maths is now can be less over-precious than it used to be. The author's style is to empasize the secondary central theme of the book that maths is a creative process, and backs this up with developments in maths which lead him to this conclusion, and of course his own creation of Omega. Thank you Gregory Chaitin (et al.).

Amazing book *****

This book is amazing, and the Central primary idea of the book is just mind bogglingly interesting and important. The incompleteness of Formal systems and a real number that is only just uncomputable. Maths is now can be less over-precious than it used to be.

The author's style is to empasize the secondary central theme of the book that maths is a creative process, and backs this up with developments in maths which lead him to this conclusion, and of course his own creation of Omega.

Thank you Gregory Chaitin (et al.).

Poorly written book with little new to say *
What an ANNOYING (?!?) Book. The style of WRITING with many (!) at the end of sentences and randomly BOLD words makes for a very distracting read. And it seems (to me) that Mr. Chaitin has merely found a different way of expressing a well known mathematical proposition - better expressed by the likes of Godel and Turing (who were true mathematical geniuses). He seems to be trying to edge himself into the pantheon of genius on their coat tails. Not so much "standing on the shoulders of giants", as "trying to sneak into the party through the back door".

Poorly written book with little new to say *

What an ANNOYING (?!?) Book. The style of WRITING with many (!) at the end of sentences and randomly BOLD words makes for a very distracting read. And it seems (to me) that Mr. Chaitin has merely found a different way of expressing a well known mathematical proposition - better expressed by the likes of Godel and Turing (who were true mathematical geniuses). He seems to be trying to edge himself into the pantheon of genius on their coat tails. Not so much "standing on the shoulders of giants", as "trying to sneak into the party through the back door".

spoiled by bias, egomania and screeching emphasis *
I bought this book blind, and it is the only math book I have ever simply thrown away. If the book had all the paragraphs that contained bold text or insistent exclamation marks cut out, and the names of other people working in the same field (Levin, Kolmogorov) added to the index and acknowledged in the text, then it would be worth reading for the math (but there again, it would only be a couple of dozen pages long).

spoiled by bias, egomania and screeching emphasis *

I bought this book blind, and it is the only math book I have ever simply thrown away.

If the book had all the paragraphs that contained bold text or insistent exclamation marks cut out, and the names of other people working in the same field (Levin, Kolmogorov) added to the index and acknowledged in the text, then it would be worth reading for the math (but there again, it would only be a couple of dozen pages long).

Wade through the bravado and find a jewel? ****
Readers tend to have a problem with Chaitin's style of writing. While reading, I feel the need to bite my tongue when dealing with the author's familiar, cheery, exclamation-point-based writing, and his self-assured self-importance - this does seem to me to be a man who wishes to impress the reader through friendly bravado. The difficulty is not to let that approach detract from the notions presented within. These are interesting, thought-provoking, and accessible explanations of an area with which I am not, was not, familiar. It's always a joy to find disparate areas that fuse into something logically solid and coherent. The author has succeeded in perform this magic trick. He offers his ideas as a silver bullet, a cure-all viewpoint, a Theory of Everything that is based on Maths. Unfortunately, the author's writing style can make his ideas appear just that - a magic trick, performed by a charlatan. Me, I need to see this man's work criticised for its content. I need to read other books dealing this area. For me, it holds water - and this very fact makes me think that it's exciting and dangerous. But I am not an expert - I'm barely a novice (altho I am a computer programmer for some years now).

Wade through the bravado and find a jewel? ****

Readers tend to have a problem with Chaitin's style of writing.

While reading, I feel the need to bite my tongue when dealing with the author's familiar, cheery, exclamation-point-based writing, and his self-assured self-importance - this does seem to me to be a man who wishes to impress the reader through friendly bravado. The difficulty is not to let that approach detract from the notions presented within. These are interesting, thought-provoking, and accessible explanations of an area with which I am not, was not, familiar.

It's always a joy to find disparate areas that fuse into something logically solid and coherent. The author has succeeded in perform this magic trick. He offers his ideas as a silver bullet, a cure-all viewpoint, a Theory of Everything that is based on Maths.
Unfortunately, the author's writing style can make his ideas appear just that - a magic trick, performed by a charlatan.

Me, I need to see this man's work criticised for its content. I need to read other books dealing this area. For me, it holds water - and this very fact makes me think that it's exciting and dangerous. But I am not an expert - I'm barely a novice (altho I am a computer programmer for some years now).