This idea has been tossing around in my head recently, since I’ve been reading the book The Last Avant-Garde: The Making of the New York School of Poets. I know I’ve done the art/math comparison before. I think I’ve even done the math education/art education comparison before. But I want to re-cap it and go one step further this time.
I don’t really want to make the arguments that math is an art form. It is a rather easy argument to make (the hard parts are creating an aesthetic theory, etc). The comparison this time is that of education. I find it extremely disconcerting that math and science are taught so differently from art. Art education tries to bring out the original ideas in the student so that new and exciting progress can be made, whereas math and science education tries to squash creativity and teach that there are only right and wrong answers (at least in the American primary/secondary/undergraduate system).
Now I hear the complaints already, “But there are right and wrong answers in math. There is no such thing as right and wrong in art. Anything goes.” And so on. But hold on a minute. The person that makes these claims has certainly never taken music lessons. There are just as many black and white correct and incorrect ways of doing things. The rhythm is either right or wrong. You’re either in tune or you aren’t. You are either playing the right note or you aren’t. You must rigorously train all of these mechanical techniques just as mathematicians must train the mechanical techniques of symbol manipulation and logical argument.
The problem is that in art education, the student is constantly reminded that these mechanics must be learned, but they are not the art itself. Once you’ve internalized the mechanics you must transcend them in order to create new original art, but it is only after the necessary evil of these basics that one can do this.
In math education, we face the problem that not only is the second step of the art not emphasized, but it is often never even mentioned. You must learn basic arithmetic and how to solve quadratic equations, but this is not math by any mathematician’s standards. In order to create new and original math, you have to transcend these basics just as the artist has to. This is much harder for mathematicians (in my opinion) solely due to this educational reason. We get indoctrinated with old methods and techniques, and never are explicitly reminded that new techniques need to be invented. It is the creation aspect of math.
I don’t want to dwell on that since I believe I’ve posted about it before (I could go on forever about it), but I did want to bring up those points again so that the main idea here has context. All other arts have had “avant-garde movements” in some sense. These avant-garde movements have essentially rendered it nearly impossible for truly original things to come out of the arts now. The rules have been followed, the rules have been completely broken, and now if you sit somewhere in between it can be said to be a conglomeration of the two. The importance of avant-garde movements should not be underestimated, though. The avant-garde has opened up the freedom to create precisely what you want, and it will have a context. Creativity for these arts is also now a two-way street and not just a one-way as before the movement. Creativity comes not only from pushing the limits of the rules, but also from toning down the lack of rules of the avant-garde.
This is what I propose. There has been no avant-garde mathematical movement (well, maybe…). One reason that it is hard to produce original math is that mathematicians constantly have to push out in creativity. An avant-garde movement always will open up the door for originality by toning down. It cracks open an avenue for incredibly new techniques that can produce lots of results.
So why did I write “well, maybe…”? I think there have been some potential quasi-avant-garde movements already, and they have been wonderfully fruitful as I said should happen. I think the idea of categorification, or category theory as a foundation for mathematics as opposed to set theory was a pretty radical shift in perspective. I think anytime a major shift in notation useage happens, this could boarder on the avant-garde as with string diagrams.
The one true avant-garde that first got me thinking of this, though, is Grothendieck. I truly am not qualified to talk about any of his work (I’ve only recently started learning about schemes even), but it completely revolutionized algebraic geometry, not because of the standard mathematical method of proving/improving theorems. Not because of the standard mathematical method of inventing new techniques and tools. But because he actually completely changed the way people view and think about the subject. In the way of the avant-garde he decided to throw out not just some, but all of the old rules and invent his own. Now this can be applied all over mathematics.
So when I call for an avant-garde movement in math, I don’t mean throw out all of the rules in the sense of logic and sound reasoning, I mean dare to think radically differently than your predecessors. After all, the avant-gardes of music still used sound and instruments, the poets and authors still wrote things using words and English, painters still used paint. Avant-garde doesn’t mean you stop using what foundationally makes your art identifiable, but it might render some branches as unrecognizable (I don’t think a pre-Grothendieck algebraic geometer would recognize current algebraic geometry as such).