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ByDr. Lee D. Carlsonon October 9, 2005

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.

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.

68 people found this helpful

ByJoe Shipmanon December 27, 2006

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.

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.

ByJoe Shipmanon December 27, 2006

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.

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.

Byan informed readeron March 9, 2006

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.

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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ByJordi Bozzo Muleton February 22, 2006

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.

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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ByD. Rubinoon August 16, 2006

The quest for the limits of deduction is a well-precedented and exciting topic, and is worth of deep study. If this is your opinion, then this book is worth glancing at. Certainly Chaitin's unique approach to tackling the notion of incompleteness cuts to the core of the matter, and is free from the beautiful self-referential chains that bind the reader from a clear understanding of Godel's first and second incompleteness theorems.

Even so, Chaitin's opinions of science, his iconic worship of Leibniz (and consequent condemnation of Newton), his accusative implication that Godel's contributions to deductive logic are overblown and confusing, make this book difficult to read. On the one hand, Chaitin appears to be a haughty mathematician reaching to mask his low self esteem. He constantly references his 'teenage' work, and the technical reading that he was able to accomplish at the age of 12. These comments are ostentatiously interweaved into a forest of exclamation points, and the thesis that he alone understood and extrapollated on Leibniz's understanding of algorithmic complexity. I believe that this book is very much about an author desperately trying to establish his legacy as a great mathematician, and not just a simple read about the limits of knowledge. If you can stomach pages of writing from a self-aggrandizing braggart (establishing himself as Leibniz's natural successor in the understanding of information), and have a deep need to understand the notion of incompleteness in a new and innovative light, then I recommend this book only to you.

Even so, Chaitin's opinions of science, his iconic worship of Leibniz (and consequent condemnation of Newton), his accusative implication that Godel's contributions to deductive logic are overblown and confusing, make this book difficult to read. On the one hand, Chaitin appears to be a haughty mathematician reaching to mask his low self esteem. He constantly references his 'teenage' work, and the technical reading that he was able to accomplish at the age of 12. These comments are ostentatiously interweaved into a forest of exclamation points, and the thesis that he alone understood and extrapollated on Leibniz's understanding of algorithmic complexity. I believe that this book is very much about an author desperately trying to establish his legacy as a great mathematician, and not just a simple read about the limits of knowledge. If you can stomach pages of writing from a self-aggrandizing braggart (establishing himself as Leibniz's natural successor in the understanding of information), and have a deep need to understand the notion of incompleteness in a new and innovative light, then I recommend this book only to you.

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ByJ. Jenkinson May 16, 2007

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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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.

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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Byadishon January 28, 2007

I was looking forward to reading this book. The ideas presented, Chaitin's life's work, are fascinating and incredibly interesting. Although most of the material can be found online, e.g. on Chaitin's website, they are presented here in a more organized and more ordered form. There aren't many mathematical proofs, mostly just hand-waving and intuitions, but the ideas are usually clearly conveyed.

However, it seems that Chaitin couldn't really decide what kind of book to write, math, philosophy or romantic novel. The result is a sometimes incoherent blend of 2-bit philosophy and romantic musings about math, sex and life. I would rather have liked to read each in a separate book, and not all jumbled up, sometimes in the same paragraph. Seems like a good editor is the thing most missing from this book.

However, it seems that Chaitin couldn't really decide what kind of book to write, math, philosophy or romantic novel. The result is a sometimes incoherent blend of 2-bit philosophy and romantic musings about math, sex and life. I would rather have liked to read each in a separate book, and not all jumbled up, sometimes in the same paragraph. Seems like a good editor is the thing most missing from this book.

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ByRegis P. Digiacomoon April 10, 2006

This work is difficult to read. There are enough exclamation points to exhaust most readers. A lack of humility and the self aggrandizement soon makes the reader quite weary. The subject matter is well woth exploring but the author gets in the way. I cannot recommend this book.

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ByPatrickon August 7, 2006

This book has some very interesting mathematical and philosophical tidbits, and is well worth a read if you are interested in epistemology, the foundations of mathematics, or information theory. I find the author to have some rather annoying tendencies, such as his over-use of exclamation points (!) and his tendency to over-state what conclusions his arguments actually support. For example, the author argues that mathematics should not be viewed as primarily the work of constructing formal axiomatic systems. But while he argues convincingly that there are inherent limits to what can be proved by any formal axiomatic system, this does not imply that mathematics is not still primarily the work of constructing such systems (and it's not clear what mathematics would be other than the construction of such systems). In fact, the author's fascination with the algorithmic perspective suggests that he mainly practices mathematics from the perspective of algorithms and formal systems. But aside from these annoying tendencies, this book still presents a fascinating discussion of our mathematical understanding of what can and cannot be achieved by the axiomatic method.

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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.

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byGregory J. Chaitin

$138.09

byGregory J. Chaitin

$159.00

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