Sans les mathématiques on ne pénètre au fond de la philosophie
Sans les philosophie on ne pénètre au fond de la mathématiques
Sans les deux on ne pénètre au fond de rien.
-- Gottfried Wilhelm von Leibniz
Without mathematics one cannot penetrate deeply into philosophy. Without philosophy one cannot penetrate deeply into mathematics. Without both one cannot penetrate deeply into anything. This is a central theme of the new book by the famous mathematician Gregory Chaitin, Meta Math! The Quest for Omega. This is Chaitin's attempt to present the ideas of metamathematics, algorithmic information theory, and the philosophy it implies in a form accessible to the layman, in short to become for the foundations of mathematics what A Brief History of Time was for physics. In my own opinion, it seems that Chaitin may be slightly more successful than Stephen Hawking was in his efforts to bring a glimpse of the frontiers of physics to the non-technical reader.
I'm actually a professional computer scientist, and as such I'm almost one of those people who can actually take a shot at understanding Chaitin's technical oeuvre on the subject (as some of my other nodes attest), so perhaps my judgment on what may and may not be comprehensible to the general reader might be a bit skewed. Nonetheless, it seems to me that the book is supremely accessible, only going beyond the standard high school mathematics in a few places (he even goes into a brief diversion into LISP programming at one point!), but even where Chaitin dives into some complicated proofs and explanations of the most cutting edge mathematics he somehow keeps it from degenerating into a pedantic text on abstract theory. In this respect he seems to have far exceeded Hawking. What he gives the reader is the real thing, not dumbed-down and bowdlerized for mass consumption, not simplified beyond recognition.
One of the things I liked best about the book was its presentation of a colorful living history of mathematics. Unlike most other books on mathematics, which present dull ideas divorced from historical context, away from what was going on in the minds of those who first imagined them, Chaitin manages to sprinkle the text with delightful little anecdotes on some of the greatest names in mathematical history, from Euclid to Georg Cantor to Alan Turing, leaving the reader with a sense of why these ideas were developed, and what brought the ideas to life. This approach goes a long way to making the mathematics that much more accessible: instead of presenting math as abstract ideas with all the clarity and bluntness of a "divine revelation" he presents mathematics as a uniquely human endeavor that evolves and changes with the people who made it happen.
Chaitin's style is infectiously enthusiastic and highly entertaining, and in several places he goes into amusing diversions that nevertheless don't distract from the subject matter. In one place, he actually presents some applications of algorithmic information theory to biology, and in particular human sexual attraction:
When a couple falls madly in love, what is happening in information-theoretic terms is that they are saying, what nice subroutines the other person has, let's try combining some of her subroutines with some of his subroutines, let's do that right away! That's what it's really all about!
That really made me laugh, and the fact that it's also very much in keeping with Charles Darwin's theories on sexual selection makes it all the more engaging.
The main thrust of the book is, of course, the epistemological and ontological implications of his work on incompleteness and algorithmic information theory. This he covers very well, humbly acknowledging that even he stood upon the shoulders of giants; Leibniz had anticipated his own theories almost four centuries before:
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.
This statement by Leibniz, stronger yet than Occam's razor (because it actually gives an ontological argument why Occam's razor should work), reaches its apotheosis in algorithmic information theory, which gives a way to determine whether a set of natural laws (computer program) of the universe (computer) is actually more coincise than the phenomena it generates (data output by the program). It's a powerful idea, and bears with it the implication that the whole universe may be thought of as a cosmic computer, and God as the supreme programmer who wrote the program that is the laws of nature! Another implication is that in order to understand something, it must be compressed, the laws of nature cannot be as complicated as the phenomena they describe, or else there is no reason for the laws of nature to be true. These are the intriguing and fascinating ideas that Chaitin explores.
While some people might object to Chaitin's apparent hubris in presenting his own theories and discoveries on par with Kurt Godel and Alan Turing's results at the beginning of the last century, I would say that this pride he has in his discoveries is not totally unwarranted. His own incompleteness result, the number Omega, shows that even in the heart of the purest of the pure mathematics incompleteness is all-pervasive. If that's not a result significant enough to bring changes to all of mathematical endeavor almost as much as Gödel's and Turing's discoveries have, I don't know what is.
I cannot recommend this book too highly. In only 157 pages Chaitin manages to explain in plain English the greatest ideas at the frontiers of mathematics. Even better, while the book has not yet gone to press (it's scheduled for release in 2005), the entire book is available for free on Chaitin's own website:
If you'd prefer it in PDF, it's available here:
Try reading it, you won't be disappointed!
It also makes a fine companion to Stephen Wolfram's A New Kind of Science.