Friday, November 30, 2007

Mathematical masturbation

Professor Alexander Ol'Shanski'i worked wonders with his magical constants.

He'd begin a proof by laying out several numbers that were related to one another through precisely premeditated proportions: "we let epsilon [always pronounced "ep-SIGH-lon"] be given, and we choose lambda less than 53 epsilon, and we let c be chosen so that c squared is greater than 7 lambda minus 3 epsilon over 2..." These numbers were always given from memory, as though he were performing a familiar liturgical paean in some gnostic ritual.

As the proof progressed, the pieces of the proof, including the magical constants, fell into place like tumblers in a lock. If you waited patiently and bothered to put the pieces together you'd see why it was each particular choice was made, you'd see why every constant had been chosen to have exactly the value that it had, no more and no less. The tapestry was woven so finely as to deflect the sharpest logical blade. The resulting proof was a thing of unrivalled beauty.

Professor Ol'Shanski'i was the architect of several other, relatively constant-free, of my favorite proofs from my graduate school days. I recall a day on which he led our Representation Theory class through the verification of some theorem or another whose content now escapes me. At the time the proof left me with a warm feeling of empowerment, of being in on a deep secret. That's how such a proof made me feel: with its careful construction, its meticulous and methodical progression, its ultimate culmination in an elegant and often surprising result from the most arcane abysses of mathematical thought, it could hardly fail to leave an avid acolyte like me spellbound, basking in the warm glow of its ethereal nimbus.

Thus, it seems like sacrilege to me to say that I worry about the message the non-mathematically inclined might get from all of this mathematical masturbation.

Now, when I exhort my mostly mid-level math students in 280 to make their proofs clear, readable, and intuitive, I'm asking them to construct the antithesis of a "magical constant" proof. A good proof should let its reader in, not shut her out. It should beckon to the reader and invite understanding through a gentle mental climb. It should provide toeholds and grips, enough for the reader to reach its peak. Don't say "let epsilon be at least 7 delta...," say instead "notice that our goal is to find a value of epsilon so that if this condition is met...because of this, we must choose an epsilon that is at least 7 delta..."

Mathematicians on the whole do a whole lot of mutual ego-stroking and chest-thumping. "We're so damned smart that no one really understands how smart we are."

Mathematicians on the whole are lousy ambassadors for their field.

What are we doing to let non-math-type people in on the tricks of our trade? It's clear that such in-letting needs to be done: have you ever known another subject in which people are so proud to boast loathsome incompetence?

"What do you do?"

"I'm a mathematician."

"Oh, I hate math. I'm so bad at it."

I'd kill to hear someone say in earnest, "You're a sex therapist? Oh, I hate sex. I'm so bad at it."

Many view math as an inaccessible sanctuary, a place where only geniuses may tread, and there with only the lightest and carefullest steps. Instilling confidence in my students, convincing them that yes they too can do mathematics, encouraging them to embrace their inner mathematicians (yes, we all have one)...these are the hardest parts of my job. Anyone can add, subtract, multiply, divide, differentiate, integrate...given the will to do so and the dedication to give it an honest effort.

I really believe that.

Maybe I'm just naive. Maybe I'm just an idealist, seeing the world through rose-tinted glasses.

Can you blame me? I'm having a really good week. Every teacher will recognize those "breakthrough" moments, at which a crack appears and a shaft of sunlight blazes through. I've had a dozen or so in the past three days.

Today's most momentous breakthrough came when one of my students dropped by my office with a rough draft of her poem, titled simply "Frustration."

"This is really cool," I said after finishing the first read. "This is really cool," I said again. And then again. "I realize I just said the same thing three times," I said, "but...this is really cool!" Then we had a somewhat more lucid and fruitful conversation about the poem.

It spoke sincerely of her feelings on mathematics, how sometimes its "cumbersome intricacies" frustrate her to no end. "Cumbersome intricacies": I love that choice of words! I wonder how conscious she was of that choice. As I noted to her, the words "cumbersome intricacies," cumbersomely intricate themselves, phonetically capture the sense of building tension and frustration she's attempting to convey; in those words the form and the content of the poem combine and give her writing the energy it needs to reach the climax at the piece's middle.

The only suggestion I could offer her was to think about anchoring it more firmly to mathematics by providing a metaphorical object for her frustration: is there one thing she could point to, specifically, that frustrates her?

We both hit upon the same concept, simultaneously: "those functions!" we declared, in unison. We'd worked on those together.

"Don't be afraid to say something like 'why can't those fucking functions behave!' " I told her, knowing from a prior conversation that she'd not object to the salty language. (One of my colleagues was walking by my office at the time and made a comment about how violent math had become.) "Like I said in the prompt for this assignment, you should feel free to use any words you want, obscenity, profanity, whatever you want to say, as long as it means something to you." She loved the idea, and she set about revising her work right away.

I was positively tickled by her work. As I see it, she's gotten out of this project exactly what I'd hoped my students will get. I hope they'll find out something about themselves, as students, as students of math, as human beings. I hope they'll explore the way they think about math, and how it makes them feel: is it frustrating? Is it warming? Is it exciting, sexy, fun? I hope they'll learn how it is that they learn, I hope they'll learn that writing about math can be useful, it can help them better access and organize their own ideas, it can help them make sense of their own thoughts, and to fit them together with others' thoughts as they work together to construct new knowledge.

She's done that.

This is really cool.

The first two of several student presentations in 280 came today. Although Dewey could stand to work on eye contact a little bit, he showed great mastery of the content he'd chosen to work on (a couple of countability proofs involving unions of countable sets), and his organization and boardwork were solid. With only a minor slip here and there, I thought his presentation was splendid.

Twyla and Calliope, my pair of graduating seniors, followed this with a flashy (methinks at times distractingly flashy?) PowerPoint presentation on fuzzy logic. I appreciated the central example they chose: how might two people of different ages construct different fuzzy membership functions for the same concept, namely, age itself.

"Twyla, who is 21," began one slide, "views anyone under 25 as definitely young." After that, your youth slinks quickly away.

"Calliope, who is 31," began the next slide, "believes you're definitely young if you're under 40." For her, youth slid downward, but not nearly so precipitously as in Twyla's measure.

By Twyla's measure I had partial membership in geezerhood, but I was still a spring chicken in Calliope's eyes, which are only a year younger than my own.

After their presentation, they told me how glad they were that I'd suggested the topic to them, that they'd both learned a lot and had had a lot of fun in preparing their presentation. I told them that their presentation had made me want to get back into fuzzy logic, with the aim of teaching a special topics course on the subject at some point soon.

I'm going to miss them. I've had the pleasure of having Calliope in three of my classes now, and I'm sure she'll do well in grad school.

It is now after 11:00 p.m.

I've had no more than five hours of sleep on any given night this week.

I am going to bed.

Fare thee well, fair reader, fare thee well.

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