May 28, 2008

On A Generalization Of The Second Theorem Of Bourbaki

In Moe's I pick up a small paperback, "The Artist And The Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed" by Amir D. Aczel (Thunder's Mouth Press, NY). The world's crying out for a good Bourbaki biography, but this ain't it, unfortunately. It's a confused, repetitive, portentous, and rather plodding attempt to … well, what, exactly? And that's the problem, I think: it's trying to be a bunch of things, and doesn't really do any of them well.

It rather half-heartedly tries to play on the suspense of Bourbaki's identity, but the Bourbaki in-joke won't be any sort of mystery to maths insiders, or anyone who's read the jacket blurb, so that vein can't be mined for much. It's also a weird Grothendieck booster — but that falls flat, too, if only because most non-maths types won't understand why Grothendieck might deserve the adulation (especially since this will almost certainly be the first time they've ever heard of him), but more importantly because Aczel just lets that part of the story trail off, without actually explaining G's importance (he was important, to be sure, but he's the sort of guy — like Tesla, in a different field — who attracts True Believers). He seems to think it's self-evident; but without a good maths or maths history background, it's not clear at all.

In fact, the one thing it might have done to pull the whole thing together would have been to help explain the maths and the maths background, but the book seems to assume either (or both) that the reader can't or won't understand the maths, or that they already know it. It's a strange omission, for sure: a history of a mathematical identity (in several different usages of that term) that doesn't explain the maths at all.

The book's also a claim that Bourbaki was either a spark of Structuralism or sparked Structuralism, something that I hadn't heard claimed before and that struck me as potentially interesting. But as with so much of this book, that trail just sort of petered out after a lot of suggestive but inconclusive tidbits. I'd guess Bourbaki was very weakly both a spark of Structuralism and sparked Structuralism (there's a lot of vague metaphorical stuff in common if you don't spend too much time looking at the details), but it seems a real stretch to make him one of the great Structuralist prime movers.

And the book claims that Bourbaki almost single-handedly founded modern maths, which strikes me as ludicrous: Bourbaki was an interesting sidetrack or sideline at best, and, like the book's many claims, really went nowhere in a sea of words. I don't know any mathematicians who spend much time reading Bourbaki (I personally find him more unreadable than most maths writers, and given the field, that's really saying something), and few think of Bourbaki's rigid and scholastic attempts to reground mathematics as having led anywhere much at all.

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