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. Continue reading

Anxiety, Mathematics, and Words of Kindness

Last April 13th, I emailed a few friends to let them know I was starting a blog. “I’m a little afraid it will land with a dull thud against the hard pavement of the internet,” I wrote.

Two weeks later, I posted an essay called What It Feels Like to Be Bad at Math, about my struggles with topology. It was stubbornly hard to write. I spat out 500 words of excuses and hedges (which I later deleted) before I could bring my fingers to type the truth.


Then the post started getting passed around. Continue reading

A Teaching Philosophy I’m Not Ashamed Of

I’ve always dreaded being asked for my “teaching philosophy.”


For years, I gave nonsense or scattershot answers. “Logic and critical thinking are paramount.” “I care more about conceptual understanding than computational skill.” “A balanced, student-centered approach is always best.” “We buzzword to buzzword, not for the buzzword, but for the buzzword.” At best, each of my disjointed half-theories captured only a piece of the puzzle.


Worse still, none of my replies explained why I devote so much class time to plain old practice. Continue reading

Sorting the Tools from the Toys

I don’t usually struggle to distinguish toys from tools. Gas stove? That’s a tool. Easy-Bake oven? That’s a toy. Bricks? Tools. LEGO bricks? Toys.

Mathematical tools are similarly distinctive. They harness industrial-strength power—think of Taylor series, or completing the square. Mathematical tools shine floodlights into dark corners. They unlock doors, solve problems, and make attentive students utter, “Whoa, deep.” They often come with complex instruction manuals, requiring weeks (or months (or years!)) of technical training to master.

Mathematical toys… not so much. They’re simple to grasp, fun to handle, and not much substantive good to anyone. Think of Sudoku puzzles, or differentiating cos(cos(cos(cos(x)))). We might get a kick out of poking and prodding such problems, but solving them won’t teach us anything fundamental about the workings of the universe or the necessities of logic. Toy problems aren’t floodlights; they’re more like flashlights dangling off of a keychain.

But just as the Incas mistook the wheel for a mere toy, sometimes mathematicians get it wrong. Sometimes what seems to be a toy is, in fact, a powerful tool.

Sometimes a toy is just a tool in waiting. Continue reading

The Real Bracketology

“Hooray, it’s tax season!” said nobody ever, except for the clinically ill and the clinically sarcastic. But I’m here, in this season of paperwork and low spirits, to offer a hymn of praise to the poor, misunderstood public servants that make income taxes work. No, not IRS agents, although goodness knows those sorry devils could use a defender or two.

I’m talking about tax brackets.

Continue reading