Meta Math!: The Quest for Omega (Peter N. Nevraumont Books) | 
 enlarge | Author: Gregory Chaitin
Publisher: Pantheon
Category: Book
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Rating:  20 reviews
Sales Rank: 579930
Media: Hardcover
Pages: 240
Number Of Items: 1
Shipping Weight (lbs): 1.1
Dimensions (in): 9 x 6.3 x 1
ISBN: 0375423133
Dewey Decimal Number: 510
EAN: 9780375423130
Publication Date: October 4, 2005
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Product Description
In Meta Math!, Gregory Chaitin, one of the world’s foremost mathematicians, leads us on a spellbinding journey of scientific discovery and illuminates the process by which he arrived at his groundbreaking theories.
All of science is based on mathematics, but mathematicians have become painfully aware that math itself has serious limitations. This notion was first revealed in the work of two giants of twentieth-century mathematics: Kurt Goedel and Alan Turing. Now their successor, Gregory Chaitin, digs even deeper into the foundations of mathematics, demonstrating that mathematics is riddled with randomness, enigmas, and paradoxes.
Chaitin’s revolutionary discovery, the Omega number, is an exquisitely complex representation of unknowability in mathematics. His investigations shed light on what, ultimately, we can know about the universe and the very nature of life. But if unknowability is at the core of Chaitin’s theories, the great gift of his book is its completely engaging knowability. In an infectious and enthusiastic narrative, Chaitin introduces us to his passion for mathematics at its deepest and most philosophical level, and delineates the specific intellectual and intuitive steps he took toward the discovery of Omega. In the final analysis, he shows us that mathematics is as much art as logic, as much experimental science as pure reasoning. And by the end, he has helped us to see and appreciate the art––and the sheer beauty––in the science of math.
In Meta Math!, Gregory Chaitin takes us to the very frontiers of scientific thinking. It is a thrilling ride.
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| Customer Reviews: Read 15 more reviews...
 Very thought-provoking and exciting!!! October 30, 2005
A. Lupsasca (Cambridge, MA)
47 out of 52 found this review helpful
Hello, I am a student in high school (second year) and I read this book over the last vacation. I am writing this review because this book has become one of my favorites, and I want to share my appreciation with other (potential) readers.
First of all, this is a book about math, and very deep math at that! (It also gets into Meta-Math -hence the title- which is an exciting branch of mathematics that examines the limits of what can and cannot be accomplished determined with calculations). I have read few other books about math (except my textbooks :-) and those that I read, "Fermat's Last Theorem" and "The Man Who Loved Only Numbers" (which, by the way, I also recommend) were actually more concerned with the history of mathematicians and of their ideas, than with math itself.
Here, the author doesn't hesitate to give you, for each theorem he mentions, a proof of that theorem and also an analysis of the consequences it entails. This means this book is rather technical and contains many calculations! Now don't get too scared: the author explains everything step by step and if you put in some time and effort, you'll certainly understand even the most difficult parts of the book (after all, I haven't finished high school and I managed to read it).
Also, I was pleasantly surprised by the wealth of ideas that it contains. Since it's about 200 pages, I expected to finish it in a few hours. However, it took me much more that (almost a week): indeed, some passages are so deep and provide so much insight into the world of math, that I had to spend half an hour for each in order to figure out their meaning.
Although it was quite a difficult read, I found it quite enjoyable.
As a matter of fact, because of its difficulty, this book is the most challenging, but also the most rewarding one I have ever come across. As the author also says in his book, instant gratification is not that gratifying, so even if I didn't understand all of his ideas at first, I clung to them and thought about the concepts until I understood them. And the book is well worth the effort: at the end, I felt I had really done something worthwhile.
Indeed, Mr. Chaitin, who is very funny and spirited, guides you step by step through a series of magnificent proofs that ultimately lead you to his wonderful and astonishing result: the Omega number. And the journey is not at all boring, like reading a school textbook on math: Mr. Chaitin is funny and obviously very passionate about his work, and it inevitably rubs off on you. By the end of the book, I felt almost as enthusiastic as him about his discovery (and mathematics in general).
Also, at the end of the book, in the appendix, there are two essays (about 20 pages long) that explain the gist of the book: they give a pretty good idea about what Mr. Chaitin's discovery is all about, but with more emphasis on the history of the mathematical ideas and less on those ideas themselves. So when you finish the book, if you want to clarify the information you've assimilated and recapitulate the main ideas without getting into all the details again, you can read the appendix and your memory will be refreshed, which I found really useful!
Of course, the book isn't only about math: it's also quite philosophical. Mr. Chaitin doesn't just write a bunch of calculations and expose his results. He also discusses them, what they mean, and what they represent. So the last chapter is more of a discussion on his life work's findings, and their significance for the world of math. The great thing is that he always remains modest: some authors, when writing about their discoveries, keep boasting and bragging, which really grates on my nerves. Mr. Chaitin thankfully avoids this trap.
In conclusion, if you're interested in math and also in philosophy, then Meta Math! is for you! It provides a lot of food for thought and will keep you entertained even a long while after you've read it.
You can also read it if you're just plain curious and want to spend a thought-stimulating time!
However, beware: in order to understand this book, you must be quite knowledgeable about computers and more specifically, have a basic idea of how the binary system works. Nonetheless, a few quick searches on Wikipedia will soon provide you with the necessary knowledge to get the most out of the author's explanations.
Thank you for writing this book, Mr. Chaitin!
PS: if you want to make comments about this book, e-mail me at alupsasca[AT]yahoo[DOT]fr. I'll be quite delighted to discuss it with you! Also, if anyone has a thorough explanation of the proof on page 42, please send it to me, because it's the only passage in the book I haven't quite fully understood yet.
Best Book About Math I've Read October 15, 2005
Kat Bakhu (Albuquerque, NM United States)
46 out of 62 found this review helpful
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.
"Without mathematics we cannot penetrate deeply into philosophy." - Leibniz. January 16, 2007
Prometheus (EVROPA.)
9 out of 11 found this review helpful
_Meta Math!: The Quest for Omega_ by mathematician Gregory Chaitin (co-discoverer of Algorithmic Information Theory (AIT), among other things) is one of the best and most fascinating accounts of some recent developments in mathematics (and "meta-mathematics") and what these developments mean and have to say in terms of philosophy that I have encountered. Chaitin is obviously brilliant, but he does recognize the fact that ideas do not originate in any one person. Thus, he notes that important precursors of his ideas may be found in the writings of Leibniz (who argued that God had chosen the best and simplest of all possible worlds) and Borel (who argued that the existence of a so-called know-it-all real number was evidence against the reality of the reals) as well as in biology (where DNA plays the role of a computer program). Thus, Chaitin provides an argument against egotism in science and related fields. Nevertheless, some have claimed that he gets carried away on certain points and believe this is a criticism of the book. I do think he may get carried away when trying to make his points and he obviously is an eccentric and hyper-enthusiastic individual, but I don't think this should detract one from reading the book. For example, he does make the claim that Leibniz is more important than Newton (whom he denigrates). While in terms of philosophy and metaphysics Leibniz may indeed be more important, it obviously is wrong to denigrate the genius of Newton (particularly as concerns his important discoveries in physics) or to take the side of Leibniz in the controversy over the calculus merely because of this. Both were obviously geniuses in their own right and both deserve an important place in the history of science, mathematics, and philosophy. Some have argued that Chaitin attempts to put himself on a par with Godel and Turing. While this would be arrogant on his part (only history can really tell where he will stand), it is important to note that his own work is in line with the developments of Godel and Turing, and that in many respects it simplifies and clarifies certain aspects of their works. In addition to being a work about mathematical discovery, this is also a work about philosophy (and the philosophical implications of mathematical discovery). In particular, Chaitin will argue that certain results obtained in meta-mathematics (such as Godel's Incompleteness Theorems, or the solution of the "Halting Problem" by Turing, or even his own results concerning randomness and the discovery of the halting probability (Omega)) have important philosophical consequences. For one thing, Chaitin argues that the incompleteness results demolish the formalist program of David Hilbert. But, also Chaitin will argue that the randomness of his halting probability Omega, shows that certain mathematical theorems are "true merely by accident", thus casting doubt on both mathematical formalism and Platonism. Chaitin will also argue that in view of such results as these, it is best to conceive mathematics as an empirical science (and he offers a program for doing "math" without recourse to proof, but instead making use of "experiment"). This is the post-modern perspective. Obviously computers will play an important role in numerical "experiment". (Nevertheless, while it is true that the results obtained through "experiment" using computers are important, I believe that his conclusions regarding the incompleteness results are overblown. For instance, in the book _Godel's Theorem: An Incomplete Guide to Its Use and Abuse_, Torkel Franzen finds some of Chaitin's claims to be problematic. For example, given the incompleteness theorem, it is not immediately obvious that any question occurring in normal mathematics will be subject to it. This calls into doubt the widely repeated claim that Godel's result demolished Hilbert's formalist program. But, further, Franzen directly addresses the claim made by Chaitin that certain mathematical theorems are "true for no reason", and he suggests that to make this claim it must be shown what is meant by reasoning. Franzen also points out several other difficulties with this approach and I recommend his book as providing a useful understanding of exactly what the incompleteness results entail.) Chaitin is obviously indebted to Leibniz, and he frequently seems to argue that mathematical theorems are "true in the mind of God". This leads Chaitin to refer to himself as possessing a "monotheistic temperament". Finally, it should be noted that while the "proofs" in this book are pretty basic and easy to follow, this book is probably not for those without some background in mathematics or philosophy.
Chaitin begins his books by commenting upon the beauty of and the nature of mathematics. He demonstrates this by showing three proofs for the fact that there are an infinite number of primes: Euclid's (the ancient proof and also the most succinct), Euler's (the modern proof, which has consequences for the Riemann hypothesis), and his own proof (a post-modern proof, relying on the notion of complexity). He then comments some on Godel's theorems, showing how the results originally appeared non-obvious to him, but that after he reformulated the theorems in terms of complexity they became more obvious. Chaitin comments on Hilbert's formalist program and explains what is meant by a Formal Axiomatic System (FAS) and relates this to Hilbert's 10th Problem and Diophantine Equations. (Indeed it will turn out that a Turing machine can be written as a Diophantine equation; thus, showing that Hilbert's 10th Problem is unsolvable because the "Halting Problem" is unsolvable.) Chaitin also makes some remarks about the programming language LISP (emphasizing the beauty of this language). In the next chapter, Chaitin discusses Leibniz and Newton and shows how Leibniz anticipated binary arithmetic as well as complexity (in his claim that "God has chosen that which is the most perfect, that is to say, in which at the same time the hypotheses are as simple as possible, and the phenomena are as rich as possible."). In this sense, Leibniz may be seen to offer a stronger argument than that used by Occam in his famous "razor". Chaitin also discusses biological complexity and DNA as the "software of life". (Although some of his biological claims may be problematic.) Chaitin then discusses the idea of self-delimiting information and AIT. In the next brief chapter, Chaitin provides an "Intermezzo". Here, among other things, he discusses the idea of "digital philosophy" in physics (this is the idea that the continuum is just an illusion and that space is actually digital, i.e. a real world argument against real numbers, which was anticipated by Zeno). In the next chapter, Chaitin discusses the continuum, distinguishes between algebraic and transcendental numbers, and proves many startling facts about real numbers (e.g. real numbers are uncomputable and un-nameable with probability 1). He uses these facts to offer a philosophical argument against the continuum (i.e. against the existence of real numbers). The next chapter is devoted to complexity and randomness. Here, he defines complexity and randomness (showing Borel's argument that the definition of randomness leads to an infinite regress). He also considers other notions of "randomness" such as k-normality. Chaitin uses these results about randomness to argue against the rationalist paradigm of Leibniz and Wolfram, for example. Chaitin then considers the "Halting Problem" for Turing machines, showing also that it is impossible to prove that a program is "elegant". He defines the halting probability Omega as an "oracle for the halting problem", and then he proves that Omega is random and k-normal. He does show a procedure for attaining bits of Omega (by successive approximations), and he relates this procedure to Diophantine equations. He also shows that the number Omega is an interesting one. (In particular, the number Omega may be related back to Borel's know-it-all number which provides an answer to every question in French, and which Borel used to argue against the existence of the reals.) Finally, he ends with some remarks on discovery and "egotism". In the conclusion, Chaitin relates his results to philosophy and physics (and to the Theory of Everything (TOE)) and then makes some rather cheesey comments on creativity. The book ends with two appendices: some comments on computers, Godel's theorems, and Hilbert's program and the paper "On the Intelligibility of the Universe and the Notions of Simplicity, Complexity and Irreducibility".
This book is fascinating and has much to say of mathematical and philosophical importance. If you want to read about what is occurring at the very forefront of mathematical research into complexity then this is an interesting and important book to read.
Math IS sexy...(ladies take note) August 5, 2006
W. Stevens
5 out of 7 found this review helpful
I found a book of this style to be refreshing. What struck me the most, and what I liked the most about the book, was the enthusiasm in Chaitin's tone and delivery--it was contagious. As someone who's slogged through a math graduate program, I have all too often been subject to dreadfully dry math texts where the author seems to delight in wringing every last drop of intuition (and emotion) out of an explaination, to leave a long string of theorems with very short proofs (Chaitin addresses this phenomenon early in his book). His writing produces a math related book that can serve to motivate(!)--what an idea. He also rolls back the curtain a little bit on the intuitive process and openly discusses how mathematical ideas are born and evolve into common acceptance. Chaitin is not afraid to try something new here and, for my part, succeeds.
To keep things simple, Chaitin keeps explainations quite tame. This didn't bother me until a hundred or so pages into the book, he unveils the book's punchline--the number omega--without much of a hint as to why it must be between 0 and 1 (that is, a probability in the strict sense). A little footnote there about Kraft's inequality would have been nice.
Yes, there's an ego behind these ideas, and yes, the author is self-promoting (as he mentions early in the book, it's likely not enough to just HAVE an idea, it has to be PUSHED into the awareness of ones peers and, perhaps, the general public)--I just didn't find those aspects that distracting (history has a way of giving merit where it's due). In all, the book might be nice for someone who likes (or wants to return to) a sense of PLAY in math. It is NOT a book for the mathematical ascetic.
Fundamental October 12, 2007
R. Bagula (Lakeside, Ca United States)
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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