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 be 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.

How does Ngô Bảu Châu get to be Ngô Bảu Châu? Continue reading

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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Lemony Snicket’s “A Series of Unfortunate Series”

 

Let me be very clear, as clear as the vials of tears that I keep on my desk: This story is a long and sad one. It converges to no happy ending, and perhaps does not converge at all, although as you read, you will find your own joy and sanity both converging swiftly to zero.

If you were to abandon this text and go read about something pleasant, like butterscotch pudding or statistical sampling, I would applaud your good judgment, and humbly beseech you to statistically sample a pudding on my behalf.

As for me, I am compelled to tell this tale to its sour end, because I am an analyst—a word which here means “someone who fusses over agonizing details, bringing grief to many and enjoyment to none.”

But if you insist on reading further, then you ought to meet the three poor students at the heart of this tale:

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Lines Beyond y = mx + b

What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then ohhhhhhhh.

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Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for ohhhhhhh.

But there’s a common problem: too often, kids can beat the puzzles without feeling any of those things.

I find this, for example, with lines in the coordinate plane.

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