*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.

## 6 comments:

Does math describe, and therefore discover, abstract relationships between concrete objects?

Certainly some mathematics is derived from our experience of the real world, e.g., using numbers to count. But much of mathematics, at least what mathematicians would label "pure" math (as opposed to "applied" math) is completely abstract. It's an interesting topic, and one that I haven't really done full justice to here.

The universe involves finitness at every level of its construction and mathematics is finiteness personified, and that's the reason that it works to the extent that it does. However, it's far from prefect in representing the Universe. Indeed, the unquestioned commitment to mathematics and measurements by the physics establishment means that it has failed to grasp the fundamental nature of the Universe. See the essay "Waves and The Cave and the true nature of the Universe" for a strictly rationalist and materialist perspective of the Universe: home.spin.net.au/paradigm/waves.pdf

As far as I can tell, the people who remark on the "unreasonable effectiveness of mathematics" are physicists. But physicists are good at math (or at least they were in high school). Not surprisingly, they are interested in those parts of the observable universe which lend themselves to quantititive observation and mathematical descriptions. Stuff they can't measure and write equations for are simply NOT physics (OK, there's some chemistry in there too - the kind that physicists like). Show me a biologist or a psychologist who is as impressed by mathematics as Wigner was. In other words, maybe there is no mystery; just a selection effect.

"just a selection effect". Excellent point.

Since at least the 1930s, those biologists interested in evolutionary theory have been impressed enough with mathematics to accept Fisher's mathematical characterization of core portions of evolutionary processes (via rather simple probabilistic processes). In general, though, biological and ecological systems present a much greater challenge than the systems physicists generally choose to focus on due to the importance of temporal dependencies ('historical contingencies' due to evolutionary processes) and nonlinear scaling. These challenges seem to be less matters of needing new mathematical structures than matters requiring radically different computational approaches.

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