the paper I’d assign to a calculus class if everyone shared my slightly skewed sense of intellectual fun and my excessive fondness for mathematical metaphors
Forget the history of calculus. Write me a paper on the calculus of history.
You won’t be the first. In War and Peace, Tolstoy compared civilization to a vast integral. Only by summing all “the individual tendencies of men,” Tolstoy wrote, “can we hope to arrive at the laws of history.” His was a true people’s history. Each peasant and prince gets the same weight in Tolstoy’s great Riemann sum. To give the monarchs disproportionate weight (thereby silencing the masses) would be a perversity, a paradox. No delta functions in Tolstoy’s mathematics.
History as an integral. That’s one way to see it.
Or imagine history as an infinite series. Each day adds a new term to the massive sum that precedes it. The question arises: Does history converge? Are we inching, year by year, towards some fixed destination? Will history roll slowly to a stop? Or will it diverge—oscillating between two extremes, or perhaps cascading slowly out of control, millennium after millennium? Will the decades ultimately add up to something unrecognizable?
Or perhaps the sum is finite, and the human story will end abruptly.
Another approach would take history as a solution to a vast set of partial differential equations. First, distill civilization to a set of variables—aesthetic trends, political wills, technological breakthroughs. Second, chart the ways the variables change, their dependences on one another. Third, summarize these interactions with a complex system of relations. The history of the world must be a solution to this system.
But is this solution unique? Or could it be merely a particular solution, one of many?
In other words, was our timeline inevitable, or could some other arrangement have satisfied the forces of history? Are we missing out on an entirely different version of human civilization, with alternative institutions, powers, and lifestyles?
Or tell me about limits. Are there discontinuities in the human experience? Does life advance from one moment to the next in smooth and fluid motion, offering no true surprises, every aspect of the future buried somewhere in the derivatives of the present? Or does it occasionally jump, like a historical step function, the next moment completely unlike the last?
The history of calculus? Heck, anyone can tell me about how humans discovered the mathematics of continual change. It’s right there on Wikipedia.
I want you to make something new. Tell me about the calculus of history.