Every day I leave physics with a new amazement at the coolness of this universe. Actually, with a lot of frustration, but also with the amazement thing. The whole idea that you should be able to put together a set of artificial mathematical rules to describe the world, and that it should really work and be consistent and predict new things and all fit together- pretty crazy, really.
Like take electromagnetism- here we have all of these different forms of energy, and everyone is just having a lovely time analyzing them, and then boom! All of a sudden it turns out that they're really the same thing, sort of, and can convert back and forth in nice mathematical ways. And then Maxwell wanders along and Boom! We discover that this whole craziness will simply yield the speed of light. I mean, who saw that one coming? And why should it have been coming? What reason on earth is there that the world should all fit together with such simple beauty?
And then I went to math. Today we learned about the coolness of relating e^x to sin and cos. Everybody was just fiddling around with these things, which just happened to have all these crazy properties, which are pretty cool in and of themselves, and then along comes De Moivre and discovers that e^(i*pi)+1=0. How crazy is that? We have e, this mysterious number that for some reason has crazy properties and does stuff like always being its own derivative, and also happens to be irrational and transcendental and nobody knows how it got there, it just is. Then we have pi, which likewise is just one of these numbers that just hangs about, inexplicably being the number that circles always come in. Then we have i, which is totally and entirely made-up. In that mathematicians were like, "wouldn't it be cool if there was a square root of -1? Let's pretend that there is." And they did, and it turned out to work! And so all of these totally crazy numbers, end up yielding the basic normal numbers of 0 and 1, off of which everything is built. Why? Who knows? (I mean, I vaguely get the math behind it...It has to do with Taylor Series and stuff, but that's how, not why)
Our math prof said that De Moivre's formula was so overwhelmingly cool that mathematicians used it to prove that there is some kind of Platonic ideal of math, so that math isn't really constructing these truths but discovering them. I find myself agreeing entirely- the sheer compact, logical beauty of the maths seem to be an irrefutable proof of some higher logical order- in fact, one of the most convincing proofs of G-d's existance that I have yet encountered.