The Calculus of History

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.

22 thoughts on “The Calculus of History

  1. History is discontinuous, because scientific breakthroughs (on which technological changes depend) are discontinuous. Nobody can say on a timeline longer than a few years what they expect to happen in science; all is uncertainty. The same is true of artistic developments, which may be equally important to history: it was the Iliad that nailed down once and for all, at least in the West, that the death of an enemy is tragic, not comic.

    1. History is continuous, because scientific breakthroughs build on each previous discovery. That is, small changes in output are a result of small changes in input, but how fast these changes occur is not relevant. It is possible to measure the slope of the tangent lines for each point in history, which eventually leads to the discovery that new science is dependent on more precise instrumentation and new research (new research occurs as a result of old research becoming widespread knowledge—or, small input gives small output). The same could be said of art, since once an artistic breakthrough is made it endures through the repeated tests and experiments of later artists, who eventually take it to new and unexplored heights based on those proofs. The uncertainties of the future do not mean history is discontinuous, because as soon as the future becomes history there will be measurable influence from the past’s past (e.g., it is not piecewise, which dictates different expression for different intervals). Fast forwarding to the end of time (assumed to be infinite) where every event is historical, will result in a differentiable function, which is, by definition, continuous.

    2. Ooh, I like this debate.

      The Iliad strikes me as a bad example of discontinuity, given that it evolved gradually through centuries of oral tradition before Homer codified it in writing. Scientific breakthroughs may be more “discontinuous”–perhaps you can model them with a function that has extremely large second derivatives (as the Childlike Author suggests), or perhaps the best model for them really is a discontinuous (or at least nondifferentiable) one, as John suggests.

      In fact, I wonder whether history isn’t best described with a function that’s continuous everywhere (because each moment flows into the next) but differentiable nowhere (because the rates of change can never be precisely pinned down).

      1. Indeed how about the Brownian motion (continuous everywhere but differentiable nowhere) as a model?

        As an aside, finance guys quite ubiquitously use Brownian motion for modelling stock prices and many such variables!

      2. You’re right about that, but it was the written form (set down, according to tradition, around the year -550) that influenced all future ages. There is not a scrap of purely oral Greek poetry surviving since Homer’s time.

  2. I’m very disappointed, Ben.

    One of your “partial differential equations” in an inequality. Strike one. And your smooth history’s future is not embedded in the derivatives of the present unless history is analytic. Strike two. You’re on thin ice.

    I will retract this complaint as soon as I am presented with evidence that you have completed a thorough analysis sequence

    1. *shakes a fist to the sky*


      BTW, I’m totally working my way (slow as a snail) through Stein’s first book on analysis. Lately I’ve been computing Fourier series for super easy functions, until I find out I screwed up integrating by parts somewhere, and go eat jellybeans instead. But hey – progress!

  3. I feel like we would need to apply L’Hopital’s rule a few times to figure it all out. Possibly start considering partial differential equations with all the variables involved

    1. I’m sure there’s some poor 24-year-old political science PhD student somewhere out there, furiously applying L’Hopital’s to political history, over and over, hoping that eventually he gets a definite form. Good luck to him.

    1. Well, if pi is indeed a “normal” number, then somewhere in those digits you’ll find a well-written (though coded) version of the entire history of humanity, with every person’s biography spelled out in immaculate detail.

      Still, I’d have liked to see more bad pictures with your paper. A minus.

      1. Haha pi is extremely AB normal. It rambles on forever, never really going anywhere like my mother…oh wait, perhaps it’s the MOST normal number ever, and it reflects our future…
        Sorry no bad drawings; every time I pick up a pencil I create a masterpiece. It’s a curse, especially when I’m just trying to make a grocery list.

    1. +200 for Frozen references

      +400 for Hilbert’s paradox

      +infinity for doing both at once, rendering your other two bonuses functionally meaningless

  4. This is absolutely awesome. The idea of reversing the typical analysis of “history of _____” to the “___ of history”, especially when applied to calculus, is brilliant. What an awesome way to both measure understanding of content and allow students to express personal beliefs and what they find interesting. A friend brought me to this site just twenty minutes ago, and already you have a bookmark. Well done.

    1. Thanks! I’d be curious to see what “the rock-climbing of history” would look like… there’d have to be some pretty crazy features on that wall.

  5. Yay, a small comment that’s months late. Days are getting longer, not shorter, due to Earth’s rotation slowing down, although I don’t understand the mechanism.

    Also, this is my first comment here. Great blog!

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