Ars Mathematica

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.