Meta Math!: The Quest for Omega (Peter N. Nevraumont Books) [Paperback]

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.

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.

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.