This interesting article in the New Yorker about ultra-geniuses Albert Einstein and Kurt Gödel is well worth reading (it’s in part a review of Rebecca Goldstein’s new book called "Incompleteness: The Proof and Paradox of Kurt Gödel," also discussed here in the NYT). Einstein, as everyone knows, revolutionized physics by, among other things, showing that space and time do not actually behave in the way that ordinary experience suggests that they do. And although Gödel is a much less well-known figure, what Gödel did for mathematics is possibly even more profound than what Einstein did for physics. (According to the article, Einstein often went to his office at Princeton "just to have the privilege of walking home with Kurt Gödel.") In grotesquely oversimplified terms which I hope are roughly accurate, Gödel showed that no mathematical system can be both consistent and complete – that is, all mathematical systems contain propositions which, while true, can be proven true only from outside the system itself. Talk about thinking outside the box. The article notes that, in the view of some, "Gödel’s incompleteness theorems have profound implications for the nature of the human mind. Our mental powers, it is argued, must outstrip those of any computer, since a computer is just a logical system running on hardware, and our minds can arrive at truths that are beyond the reach of a logical system."
I studied Gödel’s incompleteness theorems many years ago, and while the details have long since slipped out of my consciousness, I clearly recall being struck by their beauty. It was a revelation to discover that beauty exists not only in art, or music, or literature, or nature, but also in the highly abstract realm of pure mathematics. Great mathematics, like great art, tells us something profound about ourselves, and about the universe. And that’s beautiful.
