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.
“The first time I saw the definition of Lie algebra, my reaction was “Huh?†”
It is one of the principal blind spots of undergraduate mathematics – the cross product of vectors in 3-dimensional Euclidean space is occasionally mentioned, but I had not seen an undergraduate textbook which would discuss a fundamental fact: the cross product is the Lie multiplication in the Lie algebra of the group SO(3) of rotations of 3-dimensional Euclidean space. We live inside of a Lie algebra.
Since when is his name spelled ‘Nicholas’?
Am I to take this as a recommendation for Eisenbud?
I think the example of so(3) may be the first time Lie algebras seemed interesting. When I studied vector calculus, I hated the cross product, just because it so obviously did not generalize to higher dimensions. I liked the so(3) explanation, since it obviously did generalize.
Bourbaki earned the superfluous ‘h’ two days ago. (In all these years, I never noticed that it was spelled without the ‘h’.)
Eisenbud is a terrific book, if you are interested in commutative algebra. (If you’re not intrinsically interested in the subject, then I’m not sure the book is so great that it will change your mind.)