An Avant-Garde Movement for Math

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?

Artistic Influences

I personally subscribe to the idea that art is meaningless without its context. The question I’ve recently been asking is: How important are specific influences to the context? I’ve been on a kick of reading books and listening to music that my favorite artists have said have been major influences. This is an attempt to better understand where they are coming from.

A weird thing has happened, though. I’ve started to change my mind about how important this is. It seems as if the non-major influences are more important. The stuff that has randomly seeped in from all over the place is sort of what makes it original.

Some examples. In The Broom of the System, not only does one of the main characters carry a copy of Wittgenstein’s Philosophical Investigations around, but the title also refers to an important aphorism in that work. Clearly you will have a hard time reading this without at least some peripheral knowledge of that work. This is a case where reading the influence is very important.

Panic at the Disco’s new album is supposedly a reworking of Sgt. Pepper’s Lonely Hearts Club Band. To me this reference is almost not worth noting. The band itself does note this influence, but truly the similarities are that the songs are upbeat and there is a brass section. If every band that used a brass section with their upbeat songs had to note the Beatles as an influence, then we would be missing the point of influences. The interesting parts are where they differ from this traditional pop sound. There are definitely folk and other non-pop influences like skipping beats on the lead in to the chorus. This is a case where major influences can safely be ignored.

Then there are in between cases where it is probably good to know what the tradition that the work is coming from is about, but the specifics aren’t necessary (like the Beat generation writers).

What do you guys think? How important are specific influences to interpreting a work of art?

Film Analysis

Today I saw a great film, yesterday I saw a horrible film. The great film was The Hawk is Dying (THID) and the horrible film was Ellie Parker (EP). It is rare that I see such a contrast so close together, so I’ve decided to analyze it. I haven’t analyzed any films yet on here (in my memory). Here is an interesting tidbit about why I’ve chosen these; both are independent. This means that to the average box office movie-goer, both of these films would probably be written off as “unbearable,” “plotless,” or “arty.” To the average independent film-goer, both of these would be considered huge successes. This gives me great opportunity to explore at a deeper level than just genre basing my opinions.

My typical analysis of whether a film is bad, OK, good, or great goes through many layers. So let’s lay out a bunch of parts of a film first. There is: acting, script, editing, directing, and cinematography as major categories (of course there are others, but these catch my attention first). To pass the first level, a film has to show proficiency in each of these areas. THID definitely passes round one in each of these categories. EP had some slight issues with acting. It was superficial at times. I’ll get to directing later, but this was definitely very immature directing, as well. Overall, both pass round one, but EP shows some weaknesses that could pose problems in later rounds.

Round two: originality vs predictability. In my mind, you either follow the formula or you break out of it. Now these are both independent films, so by their very nature they don’t follow the formula or else they would have funding from a major studio. Here is the interesting thing, though. THID decided to take a rather mundane idea (taming a hawk) and pull out a very original and intense film. EP decided to take the originality route from the ground up. This is sort cliche in my opinion. It is much more impressive to take something that is common place and make it original than to just toss together all sorts of randomness to make it original. Whatever the case may be, both pass round two.

Round three: balancing the creative and technical. This is the eternal struggle in creating art. You must know and be able to follow the rules in order to break them. A musician must have incredible technical proficiency that comes from training scales, arpeggios, and etudes with the relentless click of a metronome before they can play expressively, bending the tempo and tone to fit the mood. A film can easily fall prey to being too technical and dry or too artistic and without any technique. The two go hand-in-hand. THID blends this beautifully. The technique is often awe inspiring. Angles, lighting, and acting are pitch perfect to fit the scene. Every shot is thought about. EP goes over-the-top. It is what I often call “young director’s syndrome.” A director that is overflowing with ideas often lets too many of them spill over into the film. EP is filled with rapid editing showing a tour de force in both camera work and editing. It serves no purpose, though. It is a mindless show of flash. It lacks the artistic side. This can be explained perfectly, though. This was Scott Coffey’s first directing experience. He had been an actor in several Lynch films. For an aspiring director, it is clear that he was trying to imitate Lynch. The thing he didn’t realize is that Lynch does things with a great vision and masterful artistry. The technical display by Lynch is vacuous like what Coffey did. THID is clearly superior in this round.

Finally, I fall to overall purpose. Now I am a big supporter of art for art’s sake (meaning a work of art doesn’t need a purpose), but that is part of the process. I determine if there was a purpose or not. If not, did it accomplish that? If so, how well and accessible was it? THID also masterfully pulled this off. Nothing was handed to the watcher. This is important to me. If the purpose is handed to me, then I feel like the director and writer didn’t trust me (a common problem with major pictures). I like that I had to work and figure things out. Overall, nothing was beyond the watcher’s capabilities, though. EP on the other hand had another young director’s faux pas. It confused its purpose. It posed as a truly pure art film without a purpose, while at the same time it had one. This is a common mistake. Young directors want arty directors to respect them, so they think if their film has purpose then they will be written off as not serious or commercial or something. What they don’t realize is that trying to come across as something your not ruins the film.

Overall, THID was excellent. It had great technical proficiency balanced with the perfect amount of artistry to keep the watcher interested. It took the common place and made it original. EP tried to be too much. It was like freshmen writing class. You have an overambitious student take a grand topic and write a short essay on it. They try to cover everything. The true art of writing comes when you can take a very small idea and develop it in a long work. The artist that tries too much ends up digging their own grave. Yes, I was taught this by my freshmen writing professor when she took me aside and harshly told me that I was trying to write a lifetime’s work into a 10 page essay. Narrow the focus. Use the tools that suit the purpose. Be ambitious, but don’t let every idea that pops into your head appear in your work. This is the difference between an experienced director and a new one.

Hope you now have a better idea of how I critique films

The House of Mirth

I lied to you again. It turns out that Gravity’s Rainbow has been delayed by a week. I started looking up influences and references in it, and it turns out I have a lot of reading to do before starting the novel. Second, the book club I joined meets this Sat, and I haven’t even started the book. The book is Edith Wharton’s The House of Mirth.

OK. Since I’m interested in post-modernism, I would never dream of reading Wharton on my own. She writes social realism, which seems to be a wasted genre. Realism definitely has its pluses, since it points out faults with society, but to be considered literary enough for a book club is beyond me. When your main concern is to get things right, how can you include the millions of other aspects of a work of literature that actually requires interpretation and hence would make good discussion.

I started it, and it is quite enjoyable. At the same time it is highly frustrating. Wharton’s intent is to point out many dreadful aspects of society (of her time), but unintentionally she is pointing out something much bigger in my mind. The clearest theme (and I’m quite sure it is unintentional) that comes across is how unreflective western society is. These people are going through their entire lives and not once asking themselves any of what I consider the “big questions.” They don’t even realize that there could be something to think and talk about outside of their shallow thoughts.

I could be wrong about whether Wharton wanted this to come across or not, but I don’t think so. Her themes (as realism dictates) are quite obvious. Since she is portraying realistically, though, the theme I just discussed inevitably comes across as well to someone concerned with that aspect of society.

This just reminds me how far art has come in such a short period of time. This was only about 20 years before Faulkner, but Faulkner is at such a higher level. Other than theme, I can’t really think of good things to talk about in Wharton. If we were to discuss Faulkner, we would have to start at such a more fundamental level. What is the plot of the novel? There would be disagreement. What are all these crazy devices being used? Why? Look at all these different levels and patterns emerging, etc.

When we move even further along the literary timeline we get even crazier things like: Character A is thinking about Character B. Suddenly the narrative shifts to B, and travels via analepsis back in time to her life with Character C. Character C takes up the narrative and analeptically shifts the focus back in time farther to an event that shaped his life; then the focus returns to C’s “present.” The prose is then recycled back to its starting point with Character A, who is currently inhabiting the reader’s “real-time” present.

I just don’t understand why a book club would pick a novel that is so one-dimensional with so many other novels out there that have extreme interpretations worth discussing.

When Art is Meaningless

I recently read Kafka on the Shore by Haruki Murakami. Murakami brings up many fantastic themes in a very complicated plot, but simple writing style. I would like to focus on a very often overlooked function of art.

Page 393:

“Writing things was important, wasn’t it?” Nakata asked.
“Yes, it was. The process of writing was important. Even though the finished product is completely meaningless.”

This quote reminds me that most of the time artists create to express themselves; to explore something that is bothering them, maybe. When interpreting a work, we often forget about this. This brings to mind two things. First, maybe we don’t have any right to judge a work of art. The work could be completely meaningless to everyone except the artist, but it did its job. It helped the artist through something. Who are we to judge whether it is good or not? Second, if we are to make judgments, write papers, critique, interpret, etc, then we really should take into consideration that the work could be meaningless to us.

This seems to have some interesting ramifications in analyzing math as an art form. When the lay person looks at a truly beautiful proof, all they get is something that is meaningless. The mathematician (eh erm, Andrew Wiles), may have been tormented by the concept for years. The end product that gets spewed out is something that the mathematician “needed” to do. It is a personal thing that doesn’t need to make sense to others.

This post is sort of interesting, since now that I finished this novel I am moving on to what I am calling the “Gravity’s Rainbow Challenge.” I will attempt to read (and probably write here about what I read) all of Gravity’s Rainbow in a two month period. This work is often considered one of the most challenging novels of all time. I need to bookmark this post to remind myself that despite thousands of Ph.D. thesis and many full length books being written on how to interpret this novel, the novel could be something that the author needed to get out and is not supposed to be something that outsiders get.

In other news. I gave up on that proof from the last post. It is too difficult, and was sort of a spur of the moment thing that I don’t have enough interest in to finish.

On Plot

I feel like I’ve posted this before… I rewatched Margot at the Wedding. It really is just fantastic. It is definitely the culmination of all of Noah Baumbach’s past efforts. I often hear the same complaint over and over about the movie (from the people that I make watch it). There was no plot. People loafed around and nothing happened. My argument is that plot is a sufficient but not necessary condition for great art.

Let’s look at film in particular. Off the top of my head there is script, acting, costume design, sound, music, all the aspects of cinematography, editing, directing, symbolism, etc. It is traditional that plot should take precedence, but should we condemn a movie for lacking one (out of hundreds) aspect? Say a movie had a fantastic plot, but was lacking in cinematography. Would the lay audience even notice? I’ll take a film with great cinematography over plot any day.

So we have sort of strayed from the point a little. Art seems to me to be about expressing ideas in an original way (“what is art?” could be a 1000 word post in itself). If you don’t need plot to express your point, then you will be doing it not only in an original way, but in a much purer form. Why use devices that you don’t need, when that could potentially interfere with what you are trying to do? Margot definitely gets its point across.

Let’s tie this back to math (since I have been philosophizing a lot and ignoring math and physics which are supposedly part of this blog). Most arguments against an aesthetic theory for math come from precisely this. Math lacks in some of the traditional parts that art has. Margot is a good example of a great work of art that lacks on purpose. I think that traditionally math cannot be considered art, but under modern considerations (i.e. my definition) math is just a pure form of art in which ideas are expressed in an original way.

There probably isn’t anything new in this, but seeing that again reminded me of all the arguments about the necessity of plot that I’ve had.

Table some issues

Now that I’m done with undergrad, I’ve decided to throw out there some of the things I’ve attempted to have answered, but people always seemed to beat around the bush. Maybe my readers (which I’ve calculated to be somewhere between three and five readers) can help.

1. Setting: Sophomore year: Eastern religions. My prof is amazing and a devout Buddhist. He claims that the religion is one of the few perfectly internally consistent religions. So I ask, how can it be consistent that the principle teaching is that the world is an illusion, and another primary teaching is that you shouldn’t harm another living being? If you truly believe the world is an illusion, then harming someone is an illusion, thus cannot be bad. I was given some mumbo-jumbo about karma, but I replied that this assumes you are putting negativity out while you harm this living thing. This went on, but I’m convinced that it is possible to harm something without feeling hatred etc, he claims that this is still against the teachings. Is this a contradiction? (I think yes).

2. Setting: Recurrent throughout all four years. Would independent art still be considered good if a large mainstream audience started taking interest? So I should elaborate, I guess. Often times it seems like small independent artists that create extremely challenging art are considered “true artists” while mainstream people are considered “pop artists” or “money makers.” It seems to me that even if the art doesn’t change, a sudden burst into fame can cause the art-lovers to despise this artist. As if it is the struggling artist that is the cause of good art and not the art itself that is under consideration. I guess a good comparison would be the filmmakers David Lynch vs Spielberg (although Lynch has had some commercial success).

3. Setting: Senior year: 20th Century philosophy. Is empathy or language epistemically primary? (I won’t even go there). Okay. Yes I will. Suppose we find a “savage” that has been in the woods on their own since birth (i.e. they have no language and have never encountered another human being). I claim that if they saw someone crying, they would remember when they cried and understand what that meant (i.e. the empathy transcends language and is thus epistemically primary). My prof claims that without language the “savage” wouldn’t have memories, or at least wouldn’t be able to access memories of when they cried (i.e. language is epistemically primary). This seems to be a question more directed at cognitive scientists than philosophers though, since it seems as if this could be empirically tested quite easily.

4. Setting: Probably junior year conversation with brother. Why is science fiction as a genre not considered serious as an artistic genre when there are so many examples that show that it is? E.g. Dhalgren, Slaughter-House Five, The Fountain, The Ender Quartet, etc.

5. Setting: Trying to get into grad school (also came up in an ethics class discussion). Is there any fair way to measure ability other than a standardized test? The GRE is a horrid measure of ability, but I just can’t think of any other way to do it that would be more fair (does the concept of “more fair” exist?).

I actually had more when I decided to write this (and some of these weren’t on my list), but once I got through the first one, I couldn’t remember any of the ones that I originally thought of. Maybe I’ll add more later.