I read this new book by IBM researcher Gregory Chaitin in one big gulp. While Meta Math! has some actual math included, it’s not particularly necessary to follow his story. The subtitle is ‘The Search for Omega,’ and Omega is a infinitely precise real number which cannot be compressed (alternatively it is ‘irreduceable’ or ‘random’). The reason for this search is linked to Godel’s Incompleteness Theorem, the concept that any formal axiomatic system will contain true statements that can’t be proved by the axioms.
Chaitin is working in the same area, but with computers and programs, and so some of the key questions are about whether one can prove that a particular program is the ‘most elegant’ or shortest expression possible. He proves that one cannot prove such a thing. But he’s more interested in the philosophy behind these findings.
If we think of scientific theories as ‘compressed knowledge’ (ie. a short statement that implies a lot about the behavior of the world), then numbers like Omega that can’t be compressed represent the notion that some things that are true in the world cannot be linked back to simple theories. If randomness is an important quality of the universe, then apparently God is throwing dice!
Chaitin notes that Wolfram has a different point of view; that the universe contains seemingly random, pseudo-random, complexity that is the result of fairly simple rules (cellular automata). And there are interesting discussions of the strangeness of real numbers, pointing to the idea that perhaps these numbers of infinite precision are not good representations of things in the universe (that in fact the universe is in some sense digital and not continuous, that you reach a level where you can’t split things any further).
In the end, my understanding is that Chaitin recommends that Mathematics take a more experimental approach, since there are limits to what the formal approaches can discover. This idea is quite similar to what physicist Robert Laughlin recommends in his recent book A Different Universe (see my earlier post).
The book’s written in a breathless style with plenty of exclamation points, and it’s main body is just about 150 pages, so it pushes along quite nicely (except when Chaiten decides to talk a bit about the pleasures of making love). There’s good history of math included, in particular material on Leibniz, Cantor, Turing and others.