Tuesday, May 28, 2013

Catching change

Sometime about the second or third week of every university calculus course, the student is introduced to the concept of a limit. In my textbook, the definition was as follows. I quote it in full, perversely, knowing that I risk losing my readers:
DEFINITION: Let ƒ(x,∆x) be defined for some fixed value of x and for all values of ∆x (different from zero) in some interval -h < ∆x < +h, that is, for -h < ∆x < 0 and for 0 < ∆x <+h. Let there be a number L(x) (which may depend upon x), such that to any positive number ε, there corresponds a positive number δ, 0 < δ < h, having the property that ƒ(x, ∆x) differs from L(x) by less than ε when |∆x| is different from zero and is less than δ. That is, if 0 < |∆x| < δ, then |ƒ(x,∆x) - L(x)| < ε.
This passage was the most incomprehensible thing I had hitherto encountered in my life. It was appended to the statement "These preliminary remarks should now enable us to understand. . ."

Of course, I did not understand. I doubt if any first-year calculus student reaches this point in the course with understanding (although maybe these days textbooks take a more intuitive approach).

Still, I was smart enough to know that if I sidestepped this initial hurdle I would never grasp what followed. So I beat my head against it for a week until the light bulb finally went on. I figured out the definition of a limit.

The rest, as they say, was a piece of cake. The study of calculus became pure bliss. No kidding. Maybe I was weird or something, but I remember calculus as being the neatest thing I encountered in college (excluding my life partner). And looking back on it today, I'd say the idea embodied in that definition above is one of the greatest innovations of our species: A way of applying the logical rigor of mathematics to continuous change. That is to say, to the world we live in.

Flow. Flux. One cannot step in the same river twice. The hawk soars on an infinitely variable wind. The buttercups bend in the breeze. I breathe in, I breathe out. The blood ebbs and flows in my veins. The universe's clock doesn't stop for geometry; it only yields to that mysterious thing called a limit, a way of confidently treating the infinite and the infinitesimal.