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).

Anybody in?

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.

I respectfully suggest that you check out the composer John Cage. His purpose is purposelessness and you will find anything goes.

My suggestion to you and others is to quit trying to make the math is art connection and focus either on a fusion of the two or focus on the aesthetics of mathematics as something worthy of studying. There classes on the aesthetics of art why not have classes on the aesthetics of math. Why not have museums to collect items that demonstrate the aesthetics of math. However you are correct in that aesthetic theories for mathart need to be written and debated.

http://mathematicalpoetry.blogspot.com/2007/02/delineations-between-aesthetics-of-math.html

Good luck on your endevors!

Kaz

I know John Cage well. I think his music fits perfectly what I’m saying. He was an avant-garde (as was Penderecki, George Crumb, etc), but you can’t compose music that doesn’t follow the rules if you don’t know the rules that you aren’t following. (We could debate that, but the main purpose was to call for an avant-garde movement in math, since it is an art form that has not had this happen yet.)

On your second point, hurrah!! I completely agree! My undergraduate thesis was a huge attempt at creating an aesthetic theory for mathematics. (Note I didn’t attempt to make the math is art connection and rather said it was an easy argument). The problem is that many people don’t think so, and as a result we don’t teach math and art in the same way and don’t have people working seriously on aesthetic theories or making museums. Until the world at large is convinced, it will be very hard to accomplish these things.

Just since the “essay” above wanders a bit, I’ll concisely restate its points:

1. Since math is an art, an avant-garde movement would be beneficial.

subpoint of 1: an avant-garde movement is possible in math despite an “anything goes” approach.

2. The main reason math hasn’t had one, but most other arts have, is that the way math is taught causes a major block to creativity.

John Cage has one rule. 1.) There are no rules. – his work was nothing like any other avant-garde composer … he had more in common with the sex pistols than he did George Crumb

Even Escher had trouble calling himself an artist for he knew his aesthetic had more to do with math than it did art — that said you are not going to get far in the art world calling math an art and for good reason … You need to come up with a different word. If Budweiser made bicycles no one would buy them for they are known for beer. Aesthetics and art are two different things … art resides in the subset of aesthetics as well as math. Furthermore, the aesthetic of art and math is quite different. If you want to metaphorically say math is ‘an’ art the same as plumbing is ‘an’ art or accounting is ‘an’ art then so be it. However math ‘is’ not art and never will be.

On another note:

Mathematics has already had examples of Avant-garde. Cantor and Godel serve that distinction quite well although Cantor more so than Godel.

I agree that math has had avant-gardes, as I wrote about Grothendieck (and Cantor and Godel work as well), but single individuals are not a “movement”.

As for John Cage, of course he wasn’t like any other avant-garde composer. No avant-garde is like any other. It would defy the meaning of the term. He was, however, a trained composer (by at least Weiss and Schoenberg), and I think this is important. An avant-garde artist without training isn’t really a true avant-garde, for they don’t have any boundaries they are pushing. And there are implicit rules that John Cage was following. His compositions involved sound. Thus it was “music.”

This is precisely my point. You can’t play a clarinet and call it a painting. Avant-gardes may violate all rules, but what they do stays within the definition of their art. When you think of an art form as exterior to its definition (a foundation that cannot be shaken), then math is art.

This brings me to the last point. I don’t understand why math ‘is not’ art. I’ve never heard a defining quality of art that math does not also satisfy. Maybe I should go through the points of your distinction that you linked to that I don’t agree with?

The reason art has avant-garde movements is that in art it’s easy to throw down some crap that defies all conventions and claim it’s a masterpiece, and if you have the right combination of friends/connections/image/circumstances then lots of people will join in and proclaim how beautiful the Emperor’s new clothes are.

As Dale Carnegie pointed out, some people would rather go insane or behave insanely than recognize that they are unimportant.

In math, it’s just not easy to fake it.

I ran into your page serendipitously and thought you might like to know that there is a fair amount of work in curriculum studies that has generalized a lot of this for education of/in most any school subject area … even so, I am sure it would not surprise you that it is very challenging to share these ideas with people who don’t already “get it.” One of my more recent attempts within mathematics education was a plenary lecture last August at the Children’s Mathematical Education Conference, “Sense & Representation in Elementary Mathematics.” The abstract that appeared in the program was:

Two concepts are central to contemporary mathematics education theory and practice: the support of sense-making by pupils, and support for developing facility with representations. This presentation problematizes and recasts both of these concepts by framing learners as artists – creators and producers – within a curriculum that usually wants them to “consume and use” instead. A common assumption is that mathematics curriculum is content that represents and interprets. Applying work of Sontag that argued against representational art, we can generate new forms of learning activities where artists evoke parody, abstraction, decoration, and non-art in ways that make mathematics vibrant and relevant to several of our conference themes.

The presentation was included in a conference book that is available free at this address:

http://www.cme.rzeszow.pl/img/part_1.pdf

I’d be curious to get your reaction.

Thanks,

–Peter

See Paul Lockhart’s essay: http://www.maa.org/devlin/devlin_03_08.html. It makes the same basic points as you do about the way math is taught at much greater length.

Well, I did feel a resonance with that one, so I think we are sharing some themes here.

I guess this is sort of a dead blog post, but I found it interesting and it looks to me like hilbertthm’s point was never actually addressed. Here’s a rambling reply that will probably be rather painful to read:

The idea that somehow creating a mathematical avant-garde has a certain appeal to it, and in a kind of stilted manner, perhaps, I find myself asking, “what is there to rebel against?”

First: what does an avant-garde do. I like your basic idea that it throws down old rules while leaving the core of the subject untouched. But probably it would be more accurate to say that it redefines what the untouchable core is. John Cage retained the core fact that music must involve sound. The point is that before him, most people would have insisted that the core of music included more than just sound, but also a certain structure, perhaps certain instruments,… or whatever it is.

There is a certain destructive element to being the avant-garde: you’re generally rejecting some form of structure, retaining only a fragment of what was previously considered important before. But you also have to put in some new rules / structure to replace it– even if just for experimentation’s sake.

So the question is: what is the core of mathematics? Or rather, what might we want it to be — what is the next underlying assumption that we should let go of to see what happens? First, let’s take a ridiculously incomplete and inaccurate look at some past math avant-gardes:

One thing Cantor did was to let go of the assumption mathematics deals with things that we can visualize in some sense (perhaps?): today, we are more comfortable talking about things formally, although, we frequently limit ourselves to the more visualizable things (compact manifolds, …) simply because they are more interesting, more manageable– more can be said about them.

Godel showed that we can’t hold on to the assumption that a given true statement can be proved. We usually ignore this, though. Perhaps this is justified because problems that might be formally independent may still generate a lot of interesting mathematics in pursuit of their proof. Gregory Chaitin seems to consider something of an avant-garde — or at least a heretic — in calling for a more experimental approach to math.

The category theory revolution let go of the assumption that what we study in math is certain sets with extra structure. I like to think of category theory as expressing the fact that what we study in math is really certain conceptual packages. Or at least that the way the human mind works is to think of things in conceptual packages– objects with morphisms predefined, and the freedom to define new morphisms only sort of opens up on a higher level of abstraction or something….

But what now? Maybe the problem is that most people are agreed that logic at least provides a pretty solid core for math. But logicians are already busy at work looking at what we can do dropping certain logical rules and so forth… Perhaps there should be more applications of intuitionistic and other logics to math– more topos theory?

I like to think of math as “formalizing” certain types of intuitions. One possible theory about the way the mind works it that it has lots of more-or-less independantly – functioning ways of processing information: Maybe the human brain has a geometric module, a logical module, an interpersonal module, a probability module, an emotional module,… (to give a cartoonish division which would have to be totally inaccurate). One thing math might do is to “formalize” the way each module of the brain works. So maybe we should looks for new sorts of human intuition to formalize and see if they give us interesting mathematics. Of course, this would be a module with motivations in some sense external to mathematics (although perhaps it would be just more honest to math to admit that we don’t really have new modes of thinking, but just formalize ones that are already there…)

Or if math’s common core is not logic, is it certain objects of study? Should we study mathematical objects in non-logical ways? Experimental mathematics? There have been calls for such directions from people like Chaitin…

Maybe math’s common core is a certain “mode” of understanding… This is obviously false as stated, since there are so many different modes of understanding used in math. Is it a particular sort of elegance and interestingness, perhaps as defined here: http://arxiv.org/abs/0812.4360v2.

All right. I think that’s well past the level of kookiness where I should stop!

Thanks!

I think the specifics also depend on certain branches of math. Grothendieck was an avant-garde in the sense that he completely threw out what was being studied and replaced it with something that on the surface looked completely different. Now from what I gather, it seems as if Jacob Lurie is doing the same thing again.

This fits with your changing what the “core” is approach. What is it that algebraic geometry studies? What is the core of this branch of math? Before each of these people the answer was entirely different (well, probably not Lurie…yet).

I ran into an art student dojng a math project on negative numbers and who incorporated Long Cat. Long Cat is crazy! Long Cat’s a cult!