The Abiding ‘Mystery’ of Calculus

I first encountered calculus in the eleventh grade. A mysterious symbol had made an appearance in my physics text–in the section on dynamics–as we studied displacement, velocity and acceleration. What was this ds/dt thing anyway? I had, at that point in time, never studied calculus of any variety; to suddenly encounter a derivative was to be confronted with mystery of the highest kind. I asked for explanation and clarification; I received less than satisfactory obfuscation in response. Something about ‘instantaneous rate of change’, whatever that was.

A few months later, having encountered differential calculus in the mathematics syllabus, I was considerably, if not totally, edified. Functions, curves, graphs, tangents; somehow, I was able to partially relate the material we had studied in the physics class to this mathematical paraphernalia. And then, a little later, in the twelfth grade, having encountered integral calculus and then differential equations, other pieces of the puzzle fell into place as the relationship between mathematical apparatus, the models they comprised, and the physical world became a little clearer.

But as the story of my introduction to calculus–an abrupt exposure to its application and formalism in dynamical analysis–shows, calculus had an initial air of mystery that took some shaking. It had been suddenly introduced as a mathematical tool to enable grappling with a problem of physical mechanics, but the formal insights that lay at its core–especially the concept of a limit–were decidedly unfamiliar. More to the point, its use seemed utterly gratuitous; I could not see how my understanding of the physical details of velocity and acceleration had been improved in any way. And even when I did study differential calculus, I felt as if I became an expert manipulator of its many recipes and techniques well before I understood what my activity entailed. Syntactical manipulation, the transformation of one set of mathematical symbols into another according to a well-specified algorithmic procedure, was easy enough; understanding what those meant, and how they underwrote our understanding of the world of becoming and change, was a different matter.

We were science students in high school, ostensibly preparing ourselves for careers in engineering, medicine, and perhaps even basic research in the physical sciences; calculus was one of our most important tools. But we remained befuddled by its place in the conceptual apparatus of our studies for a very long time. This should be, and was then, a matter of some perplexity, especially when I consider how enlightened I felt when I better understood its place in making a changing world comprehensible.

Years on, when I became embroiled in debates over curricula in computer science undergraduate education, it occurred to me little had changed; many students remained perplexed by calculus’ importance in their education, by its most foundational presumptions and applications.Nothing quite exercises pedagogues like mathematics education, and in their catalog of perplexities, the failure to properly contextualize calculus should rank especially high. I’m almost tempted to describe it as a civilizational failure, so convinced am I of the judgment of any extraterrestrial visitors when confronted with this peculiar combination of indispensability and incomprehensibility in our epistemic scheme of things.

Manil Suri on the Beauty and Beguilement of Mathematics

Manil Suri has an interesting Op-Ed on math–How To Fall In Love with Math–in The New York Times today. As befitting someone who is both a mathematician and a novelist, there are passages of writing in it that are both elegant and mathematically sound. The examples he provides of mathematical beauty–the natural numbers, n-sided regular polygons that become circles as n approaches infinity, fractals–are commonly used, but for all that they have not lost any of their power to beguile and fascinate.

I fear though that Suri’s message–that math is beautiful, creative, elegant, not to be confused with routine number crunching, and worthy of wonder and exaltation and careful study–will not be heard through the haze of the very math anxiety he seeks to cure. Suri needn’t feel too bad about this though; many others have tried and failed in this very endeavor in the past.  If, as Suri suggests, we are wired for math, we also sometimes give the appearance of being chronically, congenitally, incurably anxious about it.

A personal note: I came to a realization of mathematics’ beauty late myself. Like many of the math-phobic that Suri refers to in his article, through my school years my attitude toward the study of math was fraught with fear and befuddlement. I was acceptably competent in the very junior grades but a harsh teacher in the seventh grade ensured that I would earn my worst grades then. My concerned father took it upon himself to drill me in algebra and I regained a little confidence. Not enough though, to want to study it at the higher levels that were made available to us in the ninth and tenth grades. But the pressure to study engineering at the university level meant I had to return to the study of more advanced mathematics for the eleventh and twelfth grades.

In those two years, I was exposed to calculus for the first time and came to love it; its connection with motion, the slopes of curves, the use of differential equations to model complex, dynamic systems; these all spoke to me of a logical system deeply implicated in the physical world around me. Once I had learned calculus, I saw it everywhere around me: in a stone’s dropping, a car’s acceleration, a rocket’s launch, an athlete’s push off the starting block.

I majored in mathematics and statistics at university, but if I saw any beauty in mathematics in those years, it came when I saw some good friends of mine working out complex problems with some style. I had lost motivation and interest and barely survived my college years. My graduate studies in computer science didn’t help; I was able to–unfortunately–skip the theory of computation and graduate.

Years later, when I encountered mathematical logic as part of my studies in philosophy, some of my appreciation for the world of abstract symbol manipulation came back. It helped that my dissertation advisor was an accomplished mathematician whose theorems and proofs sparkled with style and substance alike. From him, I regained an appreciation for the beauty of the symbolic world.

I was never a mathematician, and don’t work in logic–mathematical or philosophical–any more. But my brief forays into that world were enough to convince me of the truths that Suri refers to in his piece:

[Mathematics] is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.