Clippings

There is a persistent story at Warwick University that I ended my first ever undergraduate lecture by walking into the broom cupboard. It is time to set the record straight. Yes, I admit that it was a broom cupboard, but it was also the emergency exit from the lecture hall. I had assumed, without finding out ahead of time, that when the students left the hall by the main doors, I would be able to leave by what looked like a side door. But when I tried it, I found myself surrounded by buckets and mops. Worse, I discovered that the only way to leave the building by that route was to push open an emergency exit, which would set off an alarm. I had noticed the EXIT sign over the door but had failed to spot the word “emergency” above it. So I was forced, rather sheepishly, to emerge from the so-called broom cupboard and join the students as they walked up the stairs to the back of the hall and out the main doors.

If a proof is a story, then a memorable proof must tell a ripping yarn. What does that tell us about how to construct proofs? Not that we need a formal language in which every tiny detail can be checked algorithmically, but that the story line should come out clearly and strongly. It isn’t the syntax of the proof that needs improvement: it’s the semantics.

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Psychologists now tell us that without emotional underpinnings the rational part of our mind doesn’t work. It seems that we can only be rational about things if we have an emotional commitment to such a recently evolved technique as rationality … I don’t think I could get very emotional about a structured proof, however elegant. But when I can really feel the power of a mathematical story line, something happens in my mind that I can never forget … I’d rather we improved the storytelling of proofs, instead of dissecting them into bits that can be placed in stacks of file cars and sorted into order.

…Nicolas Bourbaki is the pseudonym of a group of mathematicians — mostly French, mostly young — who tidied up the mathematics of the mid-20th century in a lengthy series of books. Their guiding principle was never to prove a theorem if it could be deduced as a special case of a more general theorem. To study planar geometry, work in n dimensions and then let n = 2.

Fashions change, and nowadays the presentation of mathematics has veered back toward specific examples and a preference for ideas that are more concrete, more down-to-earth. Though what counts as concrete today would have astonished the mathematicians of the 19th century to whom the general polynomial over the complex numbers was the height of abstraction, to us it is a single concrete example.

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There is nothing wrong with abstraction and generality — they are still cornerstones of the mathematical enterprise. But “abstract” is a verb as well as an adjective: general ideas should be abstracted from something, not conjured from thin air. Abstraction in this sense is highly non-Bourbakiste, best summed up by the counter-slogan “let 2 = n.” To do that we have to start with case 2, and fight our way through it using anything that comes to hand, however clumsy, before refining our methods into an elegant but ethereal technique which — without such preparation — lets us prove case n without having any idea of what the proof does, how it works, or where it came from.