[kzz102]

It's interesting that while Bourbaki had a large influence on modern mathematics, very few people read their books (at least among the people I know). In a sense, their project of producing a definitive exposition for a large part of mathematics has failed. I wonder whether it's because different branches of mathematics have their unique personalities, and therefore the attempt to provide a unified point of view are bound to fail.

[wrp]

I read once that the general attitude of the group was that their publications were not meant to be widely read, but just to provide the foundation for better expository work.

I also heard that part of the bad reputation that Bourbaki got was due to their being used in graduate education, despite warnings that they weren't suitable. In the 1950s/60s, there was a lack of good graduate texts. Of course, then Serge Lang came along...

[seanhunter]

Serge Lang, who happened to be a member of the Bourbaki group for some time coincidentally.

[madcaptenor]

Also mathematicians tend to not read "the classics" of the field. Do the people you know read other math books from the same time period?

[throwaway81523]

Yes, Whitaker & Watson (analysis), Hardy and Wright (number theory), Dieudonne (analysis and he was literally a Bourbaki member), heck, Euclid's Elements; Gauss Disquisitiones, etc. Bourbaki is more of a monument. Writing it was necessary, but for readers it suffices to know that it is there ;).

[Davidzheng]

while it's certainly not read by most mathematicians, Bourbaki (especially set theory & general topology) are still quite often read by mathematicians in training I believe.

[throwaway81523]

The set theory book is, at best, very outdated. No idea about topology.

[euiq]

General Topology is valuable, especially for the filter perspective; so are some of the Algebra volumes.

[kzz102]

I was applying a unfair standard to them of course. Every field has a few classics that last a long time, but most old books are not read. But I think Bourbaki maybe had grand ambitions that were eventually unrealized. My theory is that the presentation of mathematics is not based on unifying principles, but rather on the collective taste of mathematicians. So what end up being the most popular books is based on how the collective taste evolve.

[ysofunny]

they provided a unified point of view by explaining it all in terms of sets

ultimately they failed because they wrote such that it didn't matter if other people understood. it's a style that is only intelligible if you already know (from some other experience) what they are describing.

[bikenaga]

On the positive side:

Bourbaki is known for their "definition-theorem-proof" style, which for a while influenced a lot of mathematical writing. It makes the logic of the presentation easy to follow. The proofs are complete and fairly clear. The logical order within books and in the series of books as a whole is also pretty good - if you read pages 1 through n in the books, you have the prerequisites to read a proof on page n + 1. There is a good index, a table of notation, exercises (at the back, not by section), and a table of contents (at the back, since the books are in French).

They probably originated the "dangerous bend" symbol (a Z-shaped curve in the margin) to indicate a tricky or subtle point.

They're pretty good as references (to look up the proof of a result, or read about single topic).

On the negative side:

There is little exposition in the sense of motivation for what is presented, or applications.

I'm looking at "Algèbre - Chapitre 10 - Algèbre homologique" (the only Bourbaki I own). In the introduction, they say:

"Le mode d'exposition suivi est axiomatique et procède le plus souvent du général au particulier."

"L'utilité de certaines considérations n'apparaitra donc au lecteur qu'à la lecture de chapitres ultérieurs, à moins qu'il ne possède déjà des connaissances assez èntendues."

Thus, you won't find applications, or many examples - just definition-theorem-proof.

It's assumed you know why you're reading the material, and so don't need to be told.

This particular volume is a little unusual for the series in that it has lots of pictures, but that's only because this is homological algebra, so there are many commutative diagrams. Most of the volumes are just walls of text (though the formatting and the production tend to be very clear).

(I believe they actually wrote some historical remarks in some of the books which were collected in a separate volume - I don't see any historical material in the volume I'm looking at, however. The members were not unmindful of things like history: Dieudonne wrote an excellent history of algebraic and differential topology, and Andre Weil wrote a book on the history of numbers.)

The fact that it took a while for many of the volumes to be translated from French to English may have deterred some English readers (though mathematical French is not too hard to understand even if you don't know French [like me]).

On the whole, (in my opinion) the presentation is too relentlessly formal for most people to try learning a subject (as opposed to a small topic) by reading Bourbaki. They did produce a "definitive exposition" of the subjects they covered, in the sense that the results and proofs are there. It's just that most people would have a hard time learning any of the subjects by reading through the books.