Clippings

…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.