The Emergency Black-Box Checklist

I’ve been having a little argument with five-years-ago me. The question is this:

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Five-years-ago me? Throw his drink in your face. He’d tell you that rote thinking is the bane of his working days, that deep understanding is the whole point of learning mathematics. He’d tell you: No black boxes, ever.

Today-me is less convinced. Don’t working mathematicians, from the ground floor all the way up to Andrew Wiles, sometimes use black boxes? Isn’t it common sense that sometimes you need to use tools that you can’t build for yourself?

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I’m still wary of equipping students with black boxes, but these days I’m willing to do it, so long as three conditions are met. I hesitate to share this crude checklist, knowing my colleagues out there in the profession will have wiser ways to frame the tradeoffs. (After all, aren’t checklists too binary, too black-and-white, for an idea as elusive and shaded as “understanding”?)

Nevertheless, my checklist goes something like this:

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Life, in Coordinate Planes

with gratitude to Dan Meyer and apologies to Randall Munroe

NOTE: These are 100% subjective and 110% definitive.

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Historians will look back at this period and ask, “What mass lunacy gripped these people, that so many of them sought pleasure in running long distances?” Their books will have titles like “The 21st-Century Illness: How Marathons Brought Civilization on the Brink” and “26-Mile Masochism: Had They Not Heard of Cars and Bicycles?” and “Running in Giant Meaningless Circles: You Were Right All Along, Ben.” Then they will go play dodgeball, because the future is a better place.
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Is Algebra Just a Series of Footnotes to the Distributive Property?

Last month, I was helping some 6th-graders prepare for their final exam when it became clear that their teacher had utterly failed them. “You poor souls!” I said. “Orphaned by intellectual negligence!”

They blinked hopefully, as children do.

“Who was it?” I demanded. “Who taught you – or should I say, failed to teach you?”

“Um…” they hesitated. “You?”

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“Well, whoever it was,” I said, “he denied you the chance to experience the beauty and centrality of the distributive property. For this, he shall have my undying scorn.”

The philosopher Alfred North Whitehead once described European philosophy as “a series of footnotes to Plato.” Twist his arm enough, and perhaps he’d have consented to describe algebra as “a series of footnotes to the distributive property.”

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The Three Phases of the Mathematical Life

This autumn, I got the chance to ask a few questions of Ngô Bảu Châu.

If your jaw is not on the floor, it’s because (A) you’ve spent shockingly little time browsing the list of Fields Medal winners, and (B) you’re not Vietnamese.

A helpful Vietnamese journalist I met explained to me that Châu is “the biggest celebrity in Vietnam.” Châu won his Fields Medal in 2010 for proving—hands inside the vehicle, please, because this is a wild ride—a key relationship between “orbital integrals on a reductive group over a local field” and “stable orbital integrals on its endoscopic groups.”

In Vietnam, that relationship is apparently the one sizzling on tabloid covers.

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Châu is not your prototypical superstar. Even in Vietnam, apparently, he is a cryptic figure; not a chatty TV celebrity, but a silent legend. At the press conference where I met him, at the Heidelberg Laureate Forum, he gave some journalists terse one-sentence answers. Not because he was being standoffish, but because a mathematician like Châu never proves in ten lines what he can prove in just one.

I didn’t know what to ask him. I’m not a research algebraist and have never been mistaken for one. So I asked about his education, his youth in Vietnam, his mathematical coming of age.

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The Joy of Slightly Fishy Proofs

I’ve got a proof stuck in my head.

It comes from a man as brilliant as he is grumpy. Last week in the staff room, one of my all-time favorite colleagues busted out a dozen-line proof of a world-famous theorem. Though I doubt it’s original to him, I shall dub it Deeley’s Ditty in his honor:

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This argument is sneaky like a thief. I admire but don’t wholly trust it, because it solves a differential equation by separation of variables, a technique I still regard as black magic.

What I can’t deny is that it’s about 17 million times shorter than the proof I usually show to students, which works via Taylor series:

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You might call this one Taylor’s Opus. It builds like a symphony, with distinct movements, powerful motifs, and a grand finale. It takes familiar objects (trig and exponential functions) and leaves them transformed. It’s rich, challenging, and complete.

It’s also as slow as an aircraft carrier making a three-point turn.

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Here we have two proofs. The first is a nifty shortcut, persuasive but not explanatory. Watching it unfold, I’m unsure whether I’m witnessing a scientific demo or a magic trick.

The second is an elaborate work of architecture. It explains but perhaps overwhelms. If an argument is a connector between one idea and another, then for some students, this will feel like building the George Washington Bridge to span a creek.

The question it prompts, to me, is: What do we want from a proof?

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