On Hating Geometry: A Correspondence

I received yesterday the following astonishing email from a student in my sophomore mathematics tutorial, to which I append my response. I feel that my response is inadequate and doesn’t fully address the force of the challenge.
Mr. Venkatesh — In light of ending St. Augustine’s Confessions, I feel I myself have some “confessions” to share with you in regards to math class. There are a number of things I’d like to talk with you about next semester, i.e. creating a game plan for approaching spring semester math, but I wanted to send you an email ahead of time.
I believe it seems clear that since we started Apollonius, I have been more distant and removed from the material and the class in general. We did have that small chat about approaching Apollonius with the mindset of “beauty for the sake of beauty”, but it seems the more I’ve begun to reflect on my qualms, there is actually more lurking beneath the surface that I need to address.
What I am about to say may seem tangential, but it relates to the end point I wish to make.
I have begun to realize something within myself; I cannot stand Aristotle, as well as any other author who carries the same strict, logical rhetoric. I imagine Aristotle as taking a human, lying him/her on a chopping block, and dicing them into an infinitesimal amount of pieces. In the attempt to learn about what it means to be human, it seems his end result is anything but human. Might this interpretation be due to my youth and maybe even loose reading of Aristotle on my behalf? I believe that is a possibility. As I grow and mature, I may be able to approach Aristotle and the like in a more compassionate manner. However, why am I bringing up Aristotle, you might be wondering?

Apollonius, Euclid, etc. seem to me as the Aristotles of math. Ptolemy, Copernicus, and mostly Kepler, are as the Platos of mathematics. I simply feel estranged to the material of Apollonius, and admittedly even feel his mathematics to be somewhat dark, rather than beautiful. Kepler, on the other hand, has been one of the most amazing men I’ve studied on the program thus far; he is mystical, magical, humble, adventurous, brave, and Human. I can feel his genuine struggle ooze through the pages centuries later when I read him; he moves me. I feel he has become my friend. Apollonius, on the other hand, seems to be unapproachable and lacking a pulse.

However, I can’t allow myself to be satisfied with this apparent disposition I have acquired. Although it seems evident that, as humans, we all find ourselves individually migrating towards particular fascinations, I don’t want to allow myself to become blinded my own fancies. In fact, Kepler himself is a perfect example of someone who went into the realm of the uncomfortable because he knew the reward was worth the sweat. I want to do the same, but how?

The reason this “confession” is so important to me is because it is not just about math; it’s about life. When I graduate from St. John’s, if I am faced with something that is seemingly foreign or threatening to me, I cant simply bury my head in the sand. What would be demanded of me is the ability to approach the apparent impasse with confidence and workmanship. It is funny to think that I had figured myself to be “mature” upon my return to St.John’s this fall, due to the independence and jobs I immersed myself in over the past two years. But now I realize I am anything but ready to be called a Man. This Great Books education forces unexplored crevices within my soul to be perennially cracked open, leaving me to tend the wounds; but when those wounds are tended, the end result is infinitely stronger than what was previously there.  

So it seems Apollonius is actually a part of my psyche that I am not comfortable facing. He is opening a crevice within me I have not been willing to explore with integrity. It is thus a psychological experiment, in dealing with the Conics. What part of my ψυχή are the Conics forcing me to wrestle? I must wrestle with it; if not, I will not grow.

I’m writing this to give you an account for what may have been perceived as negligence on my behalf. Although I do admit I could have handled my aloofness in a better manner.

On a side note, I have a question in regards to my paper. Is there any possibility I can get it to you Sunday, as opposed to Saturday? If I attempt to have it completed by tomorrow morning, I feel I simply wont be in the least bit proud of it. Between juggling work and two other papers, I’ve allowed myself to get sloppy in regards to prioritizing. This paper is also important to me, because its my last stab at Kepler, whom I adore.

Dear Mr.K — What is even more beautiful than your sincere desire both to learn and to become the fullest possible human being is that you don’t see these as separate endeavors. It is also moving to me that you can be moved by Kepler and see a complex human being in mere ink and paper.

   Your view of Euclid, Apollonius and Aristotle makes sense to me. Your resistance comes partly — and only partly — from a feisty young man’s distrust of the seemingly authoritative. I think that this distrust should be respected because it is often right, but the distrust itself should be distrusted because it is sometimes wrong. I understand how Aristotle might make this impression on you, but I wonder if this might actually come from more from the over-stiff, formal-sounding translations we use, with their opaque Latin terminology derived from the medieval scholastic tradition that did turn Aristotle into a system. I have grown to like Aristotle through struggling with his Greek, where he comes across as an audacious, experimental thinker who succeeds in finding new ways to think about our lives, our minds, and the world we live in. Like Kepler, he is a discoverer; and like Kepler, he is open to anything and will consider anything.
   The Greek geometers, however, are a different kettle of fish. There is a secretive, hieratic quality to their work, as if they are communicating truths that are mysterious to most of us but that have been being revealed to them as a whole. There is a Pythagorean “feel” to Euclid’s obsession with things like perfect numbers and regular solids, and their contempt for the useful and mundane. Why define a straight line, for instance, as “a line which lies evenly with the points on itself,” rather than “the shortest distance between two points”? And the crucial definitions of “point” and “proportion” can generate hours of discussion because of pregnant but needless obscurity, which appeals to the kind of mind fascinated by mystery. Indeed, the whole of Euclid and Apollonius is characterized by a sense of mystery, and Apollonius especially by a feeling of the uncanny: how did they see such subtleties, how does anyone think of such theorems? Apollonius heightens this effect by concealing his tracks, so that we are often presented with the eerily beautiful flower of his intellectual process but almost never the roots. It is as if the natural, the human, the fallible have been eradicated from the intellectual vision, and what we have is a flower floating in ethereal geometrical space, which bears little resemblance to the space we live in. You are suspicious of this; it seems not just inhuman but dishonest.  In contrast, Kepler is all-too-human. He lays bare his own mistakes, the twists and turns of his mind as he grapples with the muddy and imperceptible, his limitations when he realizes that there are things he is not able to mathematize. In his efforts to understand he is willing to throw anything at the mystery: magnets, gravity, but also pretzels, ferries, oars, the “fat-bellied sausage.” Nothing is in itself extraneous to the quest for truth.
   This moral objection to the priestly obscurations of Euclid and Apollonius might also be an aesthetic objection: you don’t find those texts beautiful, they conceal too much, they pretend too much, they project an authority that is not grounded in what you feel to be real. Their aesthetic is one of “pure” beauty, a beauty purged of the material, the changing, the faulty, the mismatchings of existence — whereas Kepler’s beauty requires messiness, error, and approximations. These are wholly opposed ideas of beauty.
   To answer this profound qualm, we can note that a large part of the wonder of these texts is their demonstration of the power of reason — that all of it comes, astoundingly, from just two pages of axioms, from which intellect manages to build a granite skyscraper of “truths” by inference alone. Even in the Chinese and Indian traditions, reason had no “success story” as compelling as this. Watching how  something like this arises and develops may be in itself a fascinating and satisfying study, but in the case of Apollonius there is almost a “poetic” or “musical” level to the geometry. He is a trickster, a magician, a riddler, always full of surprises and conundrums; this may make him a villain to you, a deceiver, a wily snake — but he is honest in his playfulness, and seems to signal it even as early as his initial definition of the conic surface, where he almost makes a point of freeing himself from the rigid and solemn incarceration in the right-angled triangle, which is the mark of Euclid. He won’t spell out everything, but provokes you to “see” them, to intuit connections and “hear” resonances. This is what makes Apollonius so interesting to me: his project is not to build structures and bound forms, but rather to spread wings into the boundless and indefinite, to give articulation to forms that are open and go on forever, and that have no inside or outside. I think this is why Kepler likes him too, as an exemplar of the free but disciplined flight of intelligence.
   I hope this goes some way to addressing your concerns.

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