Gregory Chaitin has written an article “The Limits of Reason” in the latest Scientific American (March 2006) (the obligatory link to scientific american is www.sciam.com). As an aside, Australian newsagencies sell editions of Scientific American about 2 months after their publication date, but I receive it promptly in the mail as I subscribe to it – very nice.
Chaitin writes that there cannot be any “theory of everything” in mathematics as there exist pieces of mathematics (eg his number Omega) that cannot be effectively compressed into a theory. While I don’t fully understand his article (I’ve only read through it quickly), I recall his previous article introducing Omega in Scientific American a few years ago made sense.
In the “Overview box”, the article is summed up (in the first dot point) by:
Kurt Godel demonstrated that mathematics is necessarily incomplete, containing true statement that cannot be formally proved. A remarkable number known as omega reveals even greater incompleteness by providing an infinite number of theorems that cannot be proved by any finite system of axioms. A “theory of everything” for mathematics is therefore impossible.
Interesting! Previously I’ve felt that mathematics could be seen as a beautiful gem, that different aspects of it were like different facets of the gem, and that there was an overall goal of unifying all the different aspects of mathematics, but I havn’t felt like this for a few years. Now I’m more interesting in constructing bits of mathematics.