Why is our scientific description of the universe based on mathematics? If you've taken physics or chemistry classes, it might seem obvious that the laws of nature are mathematical, but in fact it's a very deep mystery. I should admit right from the start that I am not a particular expert on this subject, but since this is a blog, I am entitled to spout off about all sorts of things that I know nothing about. Caveat emptor. By the way, I also don't speak Latin. I just use it to make myself appear smarter than I really am.
Eugene Wigner, who was one of the great figures in quantum mechanics, was one of the first people to think about this problem. He wrote a famous article on it: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," which you can read here. But why is this even a mystery? The way we learn mathematics in school obscures the true nature of math. In grade school and high school, math is firmly embedded in physical reality -- it's a way of solving real-world problems. We learn arithmetic in order to balance our check books (does anyone besides me do that anymore?) and we learn algebra in order to determine the age of our friends, like Suzy, who is twice as old as Jim was when Jim was as old as Suzy is now. But "real" mathematics, as practiced by professional mathematicians, is nothing like that. Mathematics involves the construction of increasingly complex mathematical structures, which seem to have no basis in physical reality. If you doubt my view, take a look at a random entry at the Mathworld website. And yet abstract mathematical structures frequently turn up in physical theories, often decades after they were first invented. Differential geometry, which examines curved spaces that seem to have no relation to the physical universe, turns out to be the basis of general relativity (whose centenary we are celebrating this month). Abstract algebra (not the algebra you learned in high school, but things like group theory and linear algebra) lies at the foundation of quantum mechanics. So why do these inventions of mathematicians turn up so reliably in physical theories?
To approach this question, I think we first have to take a step back and ask what, exactly, is mathematics? Is it just a game that mathematicians play, putting together logical structures like kids assembling tinker toys? Or do mathematicians "discover" mathematics, in the same way the physicists discover the laws of nature? In that case, there has to be an abstract mathematical reality out there that's independent of the human mind. (This would make Plato happy!) Every mathematician I've talked to seems to believe that the second of these is indeed the case. That makes sense -- if mathematics is just an arbitrary game, why not work on real games and make more money in the computer industry? But most physicists probably tend toward the first option -- if mathematics is simply a creation of the human mind, then it's a short step to a simple explanation of why it works so well in science -- mathematics becomes simply another way of making sense of the universe. From this point of view, science becomes fundamental, and mathematics is just a tool that we've developed to do science.
But if mathematics is a sort of preexisting reality then its effectiveness in science becomes even more mysterious. The extreme Platonist view (which is espoused by my colleague Max Tegmark at MIT) is that the underlying reality of the universe is mathematical. In Max's view, the structure of mathematics is fundamental, and the universe has to be the way it is in order to conform to the laws of mathematics. That should make all you mathematicians out there happy! You can read a long exposition of Max's ideas in this paper. Would most physicists agree with Max? Probably not. But in reality, most of us don't spend much time thinking about these issues at all.