Things like the Koide formula make me kinda happy. It’s exciting that we have already developed frameworks to understand so many things, but that so much still remains. We clearly know very little about the universe.

A show of brute computation doesn’t interests me nearly as much as an elegant result. If I had to define elegance in this context, I think I would say that a subject with a high results to complexity ratio is very elegant, especially for math. I think first-year calculus (I and II) is very elegant, as it makes many very difficult problems easy, and makes insoluble problems possible.

To me, the most elegant result obtained by first-year calculus is Euler’s formula, or in its most famous special case, that e^(i*pi) = -1. The fact that this can be conclusively proved to or even discovered by high-schoolers with knowledge of Taylor series is, I think, a testament of how elegant calculus is. Of course, there are other ways to prove it, but I think Taylor series are the most elegant way–the clearest, most straightforward, least contrived; no clever logic is required at all.

Of course, Euler’s many formulae remain the gold standard of elegant results in mathematics. While he didn’t always convincingly prove his results, I think it was more important that he simply generated them. Sometimes simpler logic goes further than more bulletproof logic. I certainly don’t believe I should be mentioned in the same paragraph as Euler, but as to myself, I was quite happy with my intuitive proof using Gram-Schmidt that, for vector space V and subspace W, that dim(W) + dim(W perp) = dim(V). The idea is too conceptual to forge into any sort of formal proof, but it makes perfect sense; I would be interested in seeing a more rigorous proof, but I haven’t gone out of my way to look for one. I am, for the most part, satisfied with rationality.

Andrew Wiles’s proof of Fermat’s last theorem was certainly a tour de force in the mathematical world, and certainly, as I understand, contains many interesting ideas (which no, I am not myself qualified to appreciate). It’s certainly nice that the theorem–and subsequently, the full Shimura-Taniyama theorem–have been proven, but the proofs are long, complex, involved, and *gasp* even, in some places, significantly case-oriented. It’s a true achievement, but to me isn’t as awesome (in the pre-slang sense of the word) as much shorter results. The epsilon conjecture hinted at great possibilities, but to me, V – E + F = 2 has for centuries been quite profound enough.

I don’t mean to say that I don’t consider the above to be true accomplishments; they certainly are. I suppose it’s simply that I am more interested in the concept than the execution itself. This is why I doubt I will be entering math itself as a field, though I fully expect to be using math in whatever I end up doing.

And it’s also, I suppose, why I have always been so happy to help somebody with math who seems actually interested in the ideas themselves, not just how to do well on the next test. It’s easy to tell the students who truly want to hear the concepts and understand them for themselves apart from the students who simply want an algorithm to memorize. And it’s perfectly understandable why math could be so hated by students who view it as simply a set of algorithms. Because then, as John once sagely pointed out, math is no more than a “demented mind-game.”