Ars Mathematica

Investigating John Baez’ shocking (but true) claim that I misspelled Bourbaki’s first name, I discovered that the MacTutor History of Mathematics site has a detailed biography of Bourbaki, despite the obvious objections to a biography of that illustrious personage.

Surprisingly, this thread at Not Even Wrong (attached to a post about Harvard’s alumni magazine) has drifted into a discussion of the merits or demerits of Bourbaki.

I would argue that whatever the merits of Bourbaki’s purely mathematical contribution, the influence on expository style was negative. (Though it’s possible that Bourbaki merely typified the style, but did not cause it.) The austere theorem-proof style of mathematical writing was dominant for much of the last century, only beginning to fade in the 90s. (Compare Bourbaki’s Commutative Algebra, or Matsumura’s text of the same name, to Eisenbud’s Commutative algebra with a view towards algebraic geometry. The earlier books aim for an effect akin to Moses descending from Sinai. Eisenbud’s book is much more idiosyncratic, full of motivations, hand-wavy gestures towards geometric intuition, and asides.)

Some subjects are so compelling that they require no external motivation — they sell themselves. For me, group theory would be an example. For other subjects, you need some idea of how human beings ever arrived at a topic so outre. The first time I saw the definition of Lie algebra, my reaction was “Huh?” I needed to see the geometric motivation, plus a few unsophisticated derivatives of matrix equations, to see the point.

I take a few days vacation, and I see (via Scott Aaronson) and things promptly spin out of control. Now people are publicly dividing by zero. I shudder to think what would have happened if I’d been gone for a full week.

Andrew Ranicki has an excellent webpage devoted to the the Hauptvermutung; the conjecture, now known to be false, that for a triangulable space, all triangulations are equivalent. Even more surprising, it’s false even if you restrict yourself to only manifolds. The discovery spelled the end of the original combinatorial approach to algebraic topology (though I think the approach was largely superceded by the time the falsity of the conjecture was discovered.) Ranicki includes a link to a PDF of The Hauptvermutung Book, an introductory collection of papers on the subject that he edited.

I also came across these lecture notes that describe Milnor’s counterexample in detail.