Mathematics and Art -- Expression

So my friend Paul -- a fellow math grad student at Penn who does combinatorics -- sent me the following email:

I was talking to someone last night, and they said they loved math because it was "beautiful". This maybe always struck me as a rather odd, if widespread, idea; it at least seemed odd then. My idea of "beautiful", aesthetically or artistically speaking, involves expressiveness, or at least something to do with human emotions, some human connection. Maybe that's *just* my idea, but in any case mathematics has nothing whatsoever *expressive* about it, so it doesn't fit at least my idea of beautiful in any artistic sense (yeah, a sunset is pretty, but, I wouldn't go see a movie that was just a sunset). This got me thinking about what I do find, if not exactly beautiful, at least appealing, about some mathematical work. I posted it because I think it's not the same thing a lot of my peers find appealing, and, I was curious as to their reaction.

Basically, I think of mathematics as a bag of tautologies--statements that are trivially true, and thus in a sense devoid of content, not really saying anything at all, like "If we're going to the movies, either we'll see 'Star Trek' or something else". (Or "If you take a bunch of apples and put them in 10 boxes, at most one to a box, you have 10 boxes you can put the first one in, and then only 9 boxes you can put the second one in. Because it's one to a box. Can't put the second one in the same box as the first one. Ten minus one is nine. Etc."). That's why I don't find mathematical theorems, as such, very interesting. Theorems in the physical sciences have content, they describe how the world works or might work. Mathematical theorems have no content, they're theorizing about nothing, a bag of tautologies; they're trivialities in disguise. Who cares?

That's (certainly) not to say they're trivially true to me, or to anyone else necessarily, but, to the extent I, or everyone else, *don't* think of a mathematical result as tautological, I believe it reflects on a failure of my personal, or our collective, understanding. Overall, mathematical language doesn't seem like something we're terribly well-suited to, as a species. We have quirky and limited human brains that are good at other things, like figuring out how to bite things or passing immediate judgment on the hotness of people we just met. So in one sense, mathematical results don't say anything; in another sense, they say something trivial that we can't recognize as trivial because we're idiots about that kind of thing. The interesting thing about the triviality in disguise is the disguise, not the triviality, and when I work on math, I'm much more motivated by frustration over not being able to see through the disguise than any particular interest in the triviality. This is a "problem-solver"'s attitude, I suppose, and a "theorist" would be more interested in the triviality (or, more likely, characterize the whole thing differently).

What I like in a mathematical argument, and maybe this is what people think of as "beautiful", is when it manages to make the essentially trivial or tautological nature of the result apparent to our quirky and limited human brains. That's why I don't like long or messy arguments and why I like visual proofs of combinatorial identities and why I feel compelled to redo certain proofs--if they "work", but don't make the essentially tautological nature of the result clear, then a "new" result may be established, but the basic failure of understanding--the recognition that the result *isn't saying anything*--remains, and it's that basic failure of understanding that interests me. But if the essentially tautological nature of the result *is* made clear, you can think of it as making our quirky and limited human brains a little more universal and a little less limited, which provides some human connection to the enterprise. It doesn't feed a single starving child, but, it has an appeal, as activities go. Is it something to spend 80 years doing until you die of old age? YMMV.

This conjured up lots of possible responses and this wasn't something meant to stir debate, but I felt like I had to be a voice for "beauty" in mathematics. This is a subject I have thought about for quite some time, but I eventually settled on the following response:
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Well articulated Paul, but I disagree on two primary issues:

1. Mathematics is not expressive -- I will argue that mathematics is the pinnacle of human expressiveness.
2. Mathematics does not connect with human emotion. -- Mathematics is both something to be appreciated passively and carried out actively. The first expresses satisfaction in certainty, elegance and other archetypical emotions. The second is the consequence of curiosity and mystery.

First I will address the second issue because it refers directly to your use of the word "expressive". Then I will move to broaden the definition.

There are several emotions that mathematics appeals to. The ones that come most readily to mind are: elegance, permanence, purity and certainty. Beauty is a multi-faceted sense but I would argue that several of the aforementioned components do account for a classic sense of beauty. The entire pursuit of art is to express feelings of forlorning, hope, despair, love, abdonment, achievement, failure and the whole palette of human experience.

What emotions are conjured when appreciating the marble statues of Greco-Roman past? Although their world bears little resemblance to our own they were familiar with much the same range of emotion that we are. Their art expresses archetypes and reflects universal stories or ideals that humans spread across millenia can appreciate. Art has the same purpose of philosophy -- it is meant to make explicit these emotions so that one may stand back and reflect. It is an attempt to purify a given aspect of human experience and abstract away irrelevant details. Art accomplishes this through physical instantiation, whereas philosophy and thus mathematics (since the two were always part of the same enterprise and only recently have grown separate) sets aside the object for study inside of one's own mind so that introspection may be applied. One cannot say that one method is better than another just because the philosopher and mathematician frames his or her art in words and symbols. If one can appreciate the marble rapture of two bodies intertwined, but not Aristotle's words "Love is composed of a single soul inhabiting two bodies" then I concede the issue at hand.

On a meta-level the age of statues and words alike -- their permanence -- evokes a sense of solidity and foundation -- a rock that even the most tortured souls can cling to. Mathematics is without a doubt the most permanent human enterprise (meaning: as long as there are humans to support its meme pool). When a theorem is proved it is acknowledged as true for the rest of time. It is changeless. It echoes Zeno and other members of the Parmenidean school's belief that reality is one, change is impossible, and existence is timeless, uniform, and unchanging.

This way of thinking is spawned out of a desire for certainty. This is perhaps the most primordial human emotion. It is the impetus for all mythology, all religions and all science. Religion and Science are manifestations of a desire to narrate human existence. They both advocate methodologies for how to construct this narrative, but ultimately they are just developments of the more basic role of mythology -- coherent story-telling. Mathematics is the most coherent form of story-telling yet devised by human beings. One might argue that this is like comparing apples to modules, but the distinction is superficial. In one, the scene and characters are put in place and the governing dynamics are given by the personalities and tempers. In the other, a background universe and cast of definitions are put in place with logic and inspiration for what might be true determining the dynamics at hand. The success of physics since the time of Archimedes has been the application of abstraction to human perception to create a model with initial conditions and letting mathematics provide a story that unfolds in logical, clock-work fashion. The simple models our ancestors had pale in comparison with our current (and probably still over-simplified) model of the universe as a fiber bundle with local sections equipped with SU(3)xSU(2)xU(1) (or E8) symmetry group. Perhaps faith provides the input but reason creates the output. The individual steps may be tautologies, but the unfolding pattern is the most beautiful, spectacular, psychadelic creation humankind has ever conceived.

What is it that makes this patterned way of thinking beautiful? It is the sucess of expression unparalled by any other human endeavor. The example of putting apples in boxes may lack expressiveness because the initial conditions and structure of the problem are transparent. The action of solving the problem is just a process of deduction (or of building a general algorithm and then deducing the answer). What really makes mathematics expressive is the process of identifying features of system or processes and then naming them. This resonates with the old Christian/Greek notion that naming something gives one power over that thing. The tautological statement that "plugging in numbers into this polynomial in different ways leaves the result unchanged" seems like a silly thing to say, but when the correct notion of a Galois group is harnessed suddenly problems of the ancients come tumbling down (trisecting the angle, doubling the cube, squaring the circle). The ability to transplant one problem from a completely unrelated domain into another is at the core of human intelligence. More fundamentally, meaning comes from the isomorphism discovered between two things. Reasoning by analogy -- a method perfected in mathematics -- is all that expression is and can ever be. It is all literature, psychology and art has ever done. One must take some personal internal state that is largely unspeakable, unnameable, undrawable, irrational and point to some isomorphic system sitting outside of ourselves and say "That is me!" We are ourselves manifolds, only locally similar to anything that is understood. Although antithetical in nature, I believe that Riemann would have agreed with Emerson, a contemporary on a different continent using different methods to grasp at transcendental truths, when he said

"Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth."

A Brighter Future

Hello Everyone!

Quick Update:

1. I placed out of the core Complex Analysis course at Penn, freeing me up to do a reading course on Symplectic Topology with Prof. Block.

2. I have been neglecting my reading course because I have been occupied with the details of the Thom-Pontryagin Construction which relates homotopy classes of maps $f: M^n\to S^p$ (the p-sphere) and framed cobordism classes of closed $n-p$-dimensional manifolds. It was fun but I wasn't planning on the extra labor.

3. I just gave the pizza seminar talk on "Soliton Solutions of Integrable Systems and Hirota's Method" Friday that went really well.

4. I just looked at Penn Math Courses for Fall 2009 and there is an awesome selection!

Notable entries include:

- P. Freyd's "Calculus of Variations" Course.

- Jonathan Block will continue the 3 course sequence on Algebraic Topology with 618.

- M. Ballard's "Complex Algebraic Geometry" Course: It should be interesting to see fresh blood teach this staple of Penn.

- Joachim Krieger is teaching a course on PDEs: A subject I hope to take as a minor oral topic. Calabi Fellow Joachim Kriger was recently cited five times Terry Tao's AMS Bulletin (Jan. 09) article on "Why Are Solitons Stable?"

- Ron Donagi is teaching a course on "Mathematical Foundations of Theoretical Physics" a course that enticed me into coming to Penn, but hasn't been taught in several years.

- The Eugenio Calabi is teaching a topics course on Differential Geometry backed with W. Wylie's Ricci Flow madness.

Next year is going to be awesome!

My Catharsis of a Year Past

It's been many months since I've decided to update this blog. These have been months of success and failure, brief moments of ecstasy framed by doubt, insecurity and regret: I have survived my first semester of graduate school.

This shouldn't sound so dramatic and I realize that in the grand scheme of things, my life is great. I graduated from MIT in June 2008 with a perfect GPA in mathematics and a minor in Philosophy (not perfect in all subjects though). I was accepted by more than one pure mathematics graduate school with full funding. I am currently attending an Ivy League university with 2 years and 3 summers free of any teaching duties and the remaining 3 years supported by teaching. I have a beautiful girlfriend and we have been together over 3 years and she is also attending an Ivy League university less than an hour away and pursuing a PhD in physics. I can imagine that many many people would be envious of my circumstances. So what's the rub?

The rub is that despite my rational decision-making there are parts of me that perpetually fret that I made a bad-decision for my graduate institution. These thoughts are mostly irrational fears based on my own insecurities and adjustment to graduate school. I realize this and part of the symbolic quality of writing this on New Years Day is to fulfill a resolution: To move on with my life at Penn and to actualize my passion and enthusiasm by playing the cards I've been dealt rather than wishing my circumstances were different. Before I can move on I must throw out the old. This is my Catharsis.

Over the summer I was engaged in incredibly rewarding work with Victor Guillemin at MIT. I got to combine my passion for geometry and writing by completely rewriting the core chapter on manifolds in Victor Guillemin's new work-in-progress, a reworked take on the classic "Differential Topology", but with a major focus on multilinear algebra, differential forms and DeRham Cohomology. Those of you who have read the preface of the original classic will know these topics were intentionally left out and all manifold theory occurs in R^n. This book places a focus on doing "Grown-Up" Manifold theory with an emphasis on the previously neglected topics. The praise that Prof. Guillemin handed down to me was both warming and hurtful. The knowledge that had I delayed my graduate applications, taken the math GRE during a less chaotic semester and had the fortune of getting letters of recommendation from both my research advisor and VWG, that I might have fared much better in the admissions game, incenses me from time to time.

Having enjoyed my summer work greatly, it was sad to stop and head off to Philadelphia to start my August lease, away from girlfriend and friends, in short, my support system, so that I could prepare for 6 hours of preliminary exams at the end of August. I knew that my finite group theory was weak and that I had spent my previous year focusing entirely on the beauty of algebraic topology, smooth manifold theory, Riemann surfaces and Integrable Systems (the last two being a brief introduction). I began my review with the excellent "Berkeley Problems in Mathematics" and got my butt kicked, but whenever I turned to past Penn prelims they were usually predictable and elementary. Nevertheless I set to work on these and did many years worth, getting stuck occasionally, but eventually solving several years worth of problems. Nearing exam time there was an exam or two where I could do an entire 3 hour section in less than 45 minutes. The odd thing would be that a question or two would pop up that was much more difficult than the others and would take well over an hour to solve. The pattern, which I only realized later is that these sorts of questions became slightly more frequent in the last year or two. On an exam day I was oddly relaxed and was actually cocky that I might be able to tear through a section in less than hour. What actually happened was that I would recognize a hard question as being related to an older question and spend half an hour trying to remember how the old question went instead of trying to solve the problem in front of me. I became nervous and made several small errors and then later huge conceptual mistakes. After the day was done and discussion ensued, I realized that I had made the same mistakes as many of my classmates, yet instead of simply having common intersection I had the union of say two people's mistakes. I began to worry immensely.

The next three days were filled with incredible nausea and sleeplessness. I had dreams where I professors would yell at me and say "You want to go to Caltech!? Here let me call them and see if they'll take you now!" Immediately my mind fractured into two lives, one real, one imagined. In my imagined life I was at Caltech in Sunny Pasadena, surrounded by an elite community of scientists and engineers, buttressed by an undergraduate community reminiscent of my fond memories as an undergraduate at MIT. Here I had no preliminary exams, no quals, and was never confronted with a feeling of intense inadequacy or failure. My real life was painted with black, an immediate branding as sub-par in my department, and a city filled with poverty, crime, racial and socio-economic tension. My university well known, but not immediately recognized by the layman as elite mathematically as Caltech. After three days I was told that I had failed at the PhD level but had the highest pass at the Master's level. Basically, they had set me as the cut-off, the next person was 6 points ahead of me and another 2 points would have done the trick.

So needless to say I had a rough start to the semester. I was forced to take a remedial course (proseminar) meant to patch-up faults from my undergraduate education (this meant for me supplementing my 8 week course on Groups, Rings, Modules at Cambridge as my only algebra education). This scared me for two reasons. One was that traditionally attrition from people taking Prosem is quite high, meaning I was more likely to drop out of the program. Second was that my intentions on getting ahead on advanced course material and getting into research would be delayed. I initially had planned to place out of the required Geometric Analysis course and take Lie Algebras in its stead, but after having all the fight kicked out of me and a rather difficult placement exam I decided against it. I was also promised new and exciting topics on the Hodge Decomposition Theorem, Gauss Linking Integrals, Calibrated Geometry and all sorts of extra topics I had not been exposed to. Taking the normal first-year courses did not sound so bad, but then I was informed there was a scheduling conflict between this course and the proseminar and I might have to push back my geometry education, which was the whole reason I came to Penn in the first place! Fortunately after some last minute gifts given by the faculty, I was able to meet at another time for prosem and take the normal first year curriculum.

So my courses totaled: Real Analysis (Lebesgue Theory, Frechet/Banach/Hilbert Space theory, etc.) Algebra (Sylow Theorems, Group (Co)Homology, Category Theory, Rings, Modules), Geometric Analysis (Smooth Manifold Theory, Vector Calc, DeRham Cohomology, Frobenius Integrability), Riemannian Geometry (Do Carmo), and proseminar. As I began to recover from my initial setback I was energized to succeed in my courses, yet I found myself perpetually switching back and forth between my real life at Penn and my imagined life at Caltech. Most of my courses were good, but I felt that my Geometric Analysis course moved too slowly on the elementary material and too quickly on the important material. The lectures themselves consisted of mostly intuitive arguments and hardly was a rigorous proof demonstrated in class. To repair this, the instructor would assign problem sets with anywhere from 10 to 20 problems that were either heavily computational and unenlightening or were major results that were routinely in most textbooks and better copied. There would be pretty routinely one problem that was novel and interesting, but would require tremendous effort given the tools at hand or easily done using material not yet studied. There was also an absurd amount of focus on vector calculus that would be interesting for physics undergraduates, but only alienated most of my classmates from the course material. This also resulted in tons of tedious problems that filled my week with what could have been spent on pretty much anything else.

In my parallel life I was receiving stellar lectures filled with exacting detail and rigor. Topics that were "fun" were left to the side and room was made instead for Fundamental groups and covering spaces, homology, cohomology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups and exact sequences of fibrations, structure of differentiable manifolds, degree theory, de Rham cohomology, elements of Morse theory. Geometry of Riemannian manifolds, covariant derivatives, geodesics, curvature, relations between curvature and topology. (This is the list of topics for Caltech's Ma151 abc) All things which I consider to be simultaneously fun and important...

ENOUGH COMPLAINING! This where I stop my critique of things past.

What I came to realize is that a lot of graduate school is less about how well other people can educate you than it is about how well you can educate yourself.

Most of my internal struggle can be pinpointed to faulty thinking. Part of my insecurity comes from the fact that when you are at a place like MIT or Caltech you have some external verification of your own self worth. Granted it is true that well taught classes and a community of bright, competitive colleagues and professors can help significantly in the training of a mathematician, a lot of work needs to be invested by one's self into one's own study of mathematics (Not to suggest that Penn lacks bright, achieved faculty). Many of my classmates have learned most of their undergraduate mathematics, not because they had top-notch professors that told them what to study when and which problems to solve, but because they put in their own effort and pushed their professors to teach them more.

The bottom line is that a professional mathematician must learn a large chunk of requisite mathematics and this must be done by any means necessary. Beyond the first year or two of material suddenly everything that one needs to learn is self-taught. Success in research ultimately depends on this self-initiative and ability to acquire material independent of the environment around you.

So what about the future? I believe that over all many of my courses at Penn will be quite good. Ironically, based on a cursory glance of some of the problem sets at Caltech and the ones assigned at Penn, I would say that ours were of equivalent if not greater sophistication (Geom/Top not counted since they learn the subjects in opposing order to us). Tony Pantev is a superb instructor and mathematician. I've also heard very good things about Jonathan Block, my start-up advisor, who will most likely be responsible for most of my education here. Also, Peter Storm and Natasa Sesum will be he here next Fall and I expect that I will have the chance to learn much from both of them. Finally, the thing which excites me most about my circumstances is that Helmut Hofer has received a life-long appointment at the IAS starting this year. Given my plans to move to New Jersey and commute next year, I will have some of the best living practicing Symplectic Geometers in my backyard.

I am also pleased to know that I have the option of pursuing symplectic geometry as a major topic for my orals next year. I have had several interesting conversations with other Penn graduate students, and I do believe that I am not the only one with some of my interests. There seems to be several postdocs working on Mirror Symmetry -- a very cool and modern pairing of algebraic and symplectic geometry -- and at least one other graduate student intends to have his dissertation on Homological Mirror Symmetry. Aside from interests, my classmates are overall a fun group and I actually enjoy a social life outside of my studies. All the reasons outlined in my decision post below seem to be borne out, but the true fruits of the dissertation phase are still waiting for the picking.

The past is immutable and the only rational action is to seize the present. I share this story not so much to dwell on the specifics of my life, but so that you might find some common thread in your own life story.