Scott complains that “art snobs” want to have it both ways: to assert that the beauty of art is an aesthetic experience, a gut feeling, and, at the same time, that certain tastes are more valid than others. What is this, do they think that their guts are better than ours? That’s certainly not how we do things in complexity theory. Or is it?
When we talk about the “beauty” of a theorem or of a proof, we rarely refer to the literal statement of the theorem, much less to the fact that the proof is correct.
The beauty of a theorem is typically found in the way it fits into the bigger fabric of a theory, how it is explained by, and how it explains, other results. The statement of a theorem can feel comforting, surprising, or even unsettling, or worrisome. In a proof, we appreciate economy, a way of getting to the point in what feels like the “right” way, an unexpected use of techniques, a quick turn and a surprising ending.
If we think of, say, Reingold’s proof that L=SL, we (meaning, I and some other people) appreciate the statement for being a major milestone in the program of derandomization, and we appreciate the proof for the simplicity of its structure and for the cleverness of the way its technical tools are used.
Although, in the end, it is a matter of taste to see that the statement is important and that the proof is beautiful, I don’t think that the taste of the expert in the area is equally valid as the taste of the non-expert who says, oh this is an algorithm for connectivity that runs in n100 time and it does not even work on directed graphs?
(I am so delighted to have a discussion where one can take the notion of mathematical beauty for granted, and then use it to argue about artistic beauty.)
This wouldn’t be a non-technical In Theory post without an unnecessary personal story, so let me conclude with my experience with modern dance.
At one point, a few years ago, I was taken to see several modern dance performances. My first experience was not unlike Woody Allen’s in Small Time Crooks. While the sound system was playing annoying and discordant sounds, people on stage were moving around in what looked like a random way. I was convinced that the nearly sold-out audience at Zellerbach (it’s a big theater) was there simply to feel good about themeselves and nobody could really like that stuff. There was something that puzzled me, however: at one point, everybody laughed, presumably because something (intentionally) funny happened in the choreography. Everybody got it (except me, of course), so perhaps there was something going on in those random movements, after all. After a few more shows, the experience started to feel less agonizing, and I started to notice that I would like some segments better than others, and that this would agree with what others thought. Finally, one night, I laughed out when a dancer did something really funny on stage, and so did the rest of the audience. Aha, the brainwashing had succeeded! I left the theater feeling good about myself… No, wait, this is so not the point I wanted to make…