The following quote from science-fiction writer Greg Egan's new novel Incandescence beautifully expresses the unifying power of mathematics. I mention it also, because I used it to start a long posting to the n-Category Café blog about what category theory can do for Artificial Intelligence and cognitive science. Because this posting is in a blog for category theorists, it uses terminology that I'd avoid when posting to Dobbs; but if you're prepared to skim that, take a look. Topics covered include neural nets, holographic reduced representations, and a bit of logic programming. And analogical reasoning. I've always been excited by what Douglas Hofstadter and his colleagues had to say about this, and I've given references to his work, which I find very inspiring. And do look at Egan's quote below:
"Interesting Truths" referred to a kind of theorem which captured subtle unifying insights between broad classes of mathematical structures. In between strict isomorphism — where the same structure recurred exactly in different guises — and the loosest of poetic analogies, Interesting Truths gathered together a panoply of apparently disparate systems by showing them all to be reflections of each other, albeit in a suitably warped mirror. Knowing, for example, that multiplying two positive integers was really the same as adding their logarithms revealed an exact correspondence between two algebraic systems that was useful, but not very deep. Seeing how a more sophisticated version of the same could be established for a vast array of more complex systems — from rotations in space to the symmetries of subatomic particles — unified great tracts of physics and mathematics, without collapsing them all into mere copies of a single example.