October 3, 2008

The beauty of the fundamental theorem of calculus

Michael Bérubé is back with Arbitrary but Fun Friday, and other bloggers have Friday entries on fish, movies, fun, ... so what the heck am I supposed to do? I got the staid bearded portrait up there in the corner of the blog. It's staring at me right now, daring me to be frivolous. Or giving me guilt trips for spending a few minutes early on Friday evening on frivolity. In honor of Tampa Bay Rays rookie sensation Evan Longoria, who hit two home runs yesterday afternoon into and through and out of left field, I will name this feature Out of Left Field Friday. (Complete tangent: A friend and his son were sitting in the outfield seats right under the first home run and a section away from where the second one landed.) So onto the first Out of Left Field Friday...

I wasn't a math major, and I probably wasn't going to become one, but that decision was solidified my first semester in college when I was taking a linear-algebra course with a visiting faculty member, and the moment one morning when he stopped in the middle of a proof, looked down at his notes, looked up at us, looked down again, smiled nervously, and said, "I seem to have left my notes for the rest of this proof at home." He waved at the board with the half-scrawled lines and said, "You can ... see how it goes." He was so engaging as a teacher that one of our classmates decided to ask questions frequently to keep herself awake. (At the time, several of us thought she was highly annoying, but I became friends with her over the next few years, and she finally explained her strategy.)

But at the end of my junior year, I finally acceded to all of my friends who were math majors and told me I had to take a specific math professor before I graduated. So I signed up for real analysis with Kyewon Park. I am surely one of the few history Ph.D.'s who have taken real analysis, but the point is not me but her and the class. First, my friends were right: Kyewon was a wonderful teacher, especially for a non-major taking what should have been an impossible class for me. More than 20 years later, I even recall what a compact metric space is. I think. That's a testament to her.

The second point is that the class reminded me how beautiful the fundamental theorem of calculus is. In basic calculus (either high school as an AP class or college calculus), you typically speed through the gist of differentiation proofs and the rationalization for integration, and then if you get a day or two free (as my high school teacher made sure of), you get exposed to the proof connecting the two. For me, the key link was the intermediate value theorem, which makes everything else pretty trivial. (My high school calculus teacher was also very good.) Beautiful proof structure, oohs, ahs, Louis Kahn, it's triangular (go there and look at the fireplaces). (I warned you this was Out of Left Field Friday.)

But that day or so is nothing compared with a good real analysis class, which builds up things from the ground up (or from the assumptions up). You start with all of this weird arcane stuff at the beginning that can only come from 19th century central-European mathematicians (hi, Leonhard!). If presented well, it feels like weird arcane stuff that you just trust will add up but seems pretty interesting in an "I'd rather learn this than a new language group" way. At the end of the first semester, I felt as if I had a very firm grasp on the nebulous mist of weird arcane stuff about measures invented out of whole cloth by 19th century central-European mathematicians; if you will, imagine driving a car backwards up a steep mountain road because the person sitting in the passenger seat assures you that there's a fabulous view at the end of the road. Fortunately, very few students die in the middle of a real analysis class.

Around February, I began to get glimpses of where this was all heading. "Oh, this is the firmer analog of X!" Well, "firmer" is true less in an epistemological sense than in the sense that some things were looking more familiar than they had in September and I had been exposed to a good number of theorems about the more abstract versions. March, and we were starting to see the contours of the bigger version of the intermediate value theorem. Okay, that gave us a second wind, and if you just keep the car on the road going backwards, you might find your way to a place where you can finally turn the car around and head to the summit forward for once.

I'm not going to tell you what it felt like to get to the point in the class where I thought I had a real grasp on the fundamental theorem of calculus. That was over two decades ago, and if I took pictures at the summit, they've been lost somewhere. I remember enough about the scenery to describe it in a vague way, and I do vividly remember the terror I felt along the way, the thrill on getting there, and the appreciation I still have for my guide.

Every once in a while, everyone should take a course in real analysis, or whatever your equivalent is. Find something that you know is absolutely gorgeous but currently incomprehensible and work to understand it better. Find a guide. Follow the guide and know that you're still going to be doing most of the work. Keep at it. Know that the thrill of understanding may outlast the understanding itself, and that's okay.

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Posted in Out of Left Field Friday on October 3, 2008 6:52 PM |