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Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the opportunity to access them though) and was a bit amazed by the Wikipedia article suggesting they did mathematics very rigorously.

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Bourbaki may not be the best place to be introduced to a subject, because they are very abstract; for most people, it is hard to come to a new subject from the point of pure abstraction. They can be an invaluable resource and a good way to advance in a subject once you have the necessary "mathematical maturity", but I would not recommend it for high school students. I do not think either high school or undergraduate students are the "target audience". –  Arturo Magidin May 14 '12 at 20:03
    
I would not worry that the more typical texts for undergraduates (at least, the ones intended for prospective mathematicians; if you are learning analysis and topology, for example, then everything in Rudin and Munkres is nailed down) being insufficiently rigorous. This is usually the least of your problems. –  Dylan Moreland May 14 '12 at 21:01
    
Dear Sabyasachi, yes Bourbaki is very, very rigorous. Some critics claim that they sacrificed motivation and pedagogy in favor of rigor. –  Georges Elencwajg May 14 '12 at 21:01
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I should add that you can often find certain facts written down in these volumes that don't seem to be written down anywhere else—as Georges says, the Algebra, Commutative Algebra, and Lie Groups volumes are really quite good for this. I've also found good stuff in the Topological Vector Spaces book. But this was long after learning analysis and by the time you actually need something in Bourbaki then you're much more prepared. The hardest thing is to avoid letting your enthusiasm take over and cause you to skip the basics that are really important in the long run. –  Dylan Moreland May 14 '12 at 21:13
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The best way for an undergraduate to use Bourbaki is to have the series prominently displayed on his/her bookshelf when inviting math professors over for dinner. They will be very impressed. Of course, if the professor asks you something about the content of the series, you must be ready to change the subject quickly ("Oh! I think the roast is burning!"). –  Gerry Myerson May 14 '12 at 23:40
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1) No, you are not the target audience.

2) No undergraduate is.

3) For every subject an undergraduate might want to learn there are books better suited than Bourbaki: ask about them here.

4) In a strictly historic sense the target audience was Henri Cartan and the aim of Bourbaki was to give him a text from which he could teach Stokes' theorem rigorously, so that he would stop badgering André Weil. (Bourbaki had estimated in 1934 that it would take six months to write the book)

5) That aim has never been achieved.

6) The thousands of pages written by Bourbaki started as prerequisites for the proof of Stokes' theorems. Then they decided to forget about Stokes and write a treatise containing all of basic mathematics. They failed to achieve that goal too. (There is for example an allusion to a forthcoming book on numerical mathematics...)

7) At the graduate level certain books by Bourbaki are still arguably the best available references: some volumes in Algebra, Commutative Algebra and Lie Theory come to mind.

8) Bourbaki is not dead: they have published in 2012 a completely modified new version of Algèbre, Chapitre 8, about 3 times the size of the old version, published in 1958. (Recall that in 1934 they had estimated that the whole treatise would have been written within 6 months...)

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On 2): Harry Gindi is (or at least was) an undergraduate who likes to learn from Bourbaki. –  Michael Greinecker May 14 '12 at 20:48
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What about topology? My math teacher when I was in high school advised me both topology and algebra from Bourbaki. But I took only the one on analysis, because I didn't have enough money at that time and I thought that one the most important. Haha. :D –  Raskolnikov May 14 '12 at 20:49
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I would like to provide a slightly different answer to the question "Can the Bourbaki series be used profitably by undergraduates and high school students?" from personal experience.

I personally have read most of the Bourbaki volumes. I just finished my undergraduate education this past semester. However, I was fairly advanced as an undergraduate, and I took the graduate sequences at my school in a number of subjects.

I think that I learned a lot from Bourbaki, but it was almost never the first source that I went to. The only exception was the first volume on Set Theory. Because it is listed as a prerequisite for the later volumes, I read through it despite having only the basic undergraduate knowledge of set theory. This was fairly difficult, and in retrospect I would have been better off reading a more elementary book first on the subject. It is also not as much of a prerequisites as the writers would like you to believe; you really only need elementary undergraduate set theory to understand most of the later books.

As for the other volumes, I read all of them except the volume on spectral theory and about half of the volume on commutative algebra. None of them were easy reads. I did not read the algebra and topology ones until after I had taken graduate classes on the subjects, and the other ones after I had at least read another book on the same subject. I doubt I would have gotten anything out of them before that point. The easiest for me personally was topology, but this probably varies a lot from person to person. In most cases, the Bourbaki books were the last, rather than the first, introductory books I read (introductory here should be taken to mean that the exposition was relatively complete with few formal prerequisites).

So my answer is that the Bourbaki books can be used profitably by undergraduates, but only those undergraduates who are fairly advanced. If you're at the level of learning the things for the first time (where most undergraduates are), or you're expecting an easy read, then there are much better books. And the target audience is most definitely not undergraduates.

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I would submit that for this sort of question the most reasonable definition of an undergraduate is "someone who is taking undergraduate level math courses and reading undergraduate level math texts". If you've successfully completed multiple graduate courses, I think you are for this intent and this purpose a graduate student. (Congratulations.) In this sense: no, the Bourbaki books are not for undergraduates. –  Pete L. Clark May 14 '12 at 22:10
    
I have to disagree. Who is an undergraduate, vs. a graduate student/professional, is well-established, and answers like "No undergraduate should ever read Bourbaki" may mislead the (admittedly small) fraction of undergraduates who are perfectly capable of reading it. I know a reasonable number of undergraduates who have taken graduate courses, and I think some people like this might ask the same question in the future (as I did at one point). So I submit that, while your definition of undergraduate simplifies the question considerably, it may not be a desired simplification. –  Logan Maingi May 14 '12 at 22:48
    
I have been teaching undergraduates and graduates students -- both separately and together -- for several years now. Especially, I have seen undergraduates become graduate students, and what I'm saying is that -- insofar as they are students in my math classes -- there is no key qualitative difference between them. In fact my department offers courses cross-listed at both the undergraduate and graduate levels, and it is not uncommon for undergraduates to take the courses at the graduate level. When they do, they are functionally graduate students. –  Pete L. Clark May 14 '12 at 23:04
    
Also, when you wrote "No undergraduate should ever read Bourbaki," who are you quoting? I don't see that anyone said that. Anyway, the opinion that I expressed seems to be exactly the same as yours: it will be hard to appreciate a Bourbaki text without some graduate level coursework in the subject at hand. So I'm honestly a bit puzzled as to what, if anything, we're disagreeing about, other than academic terminology. –  Pete L. Clark May 14 '12 at 23:08
    
I agree entirely with what you have said, but this fact is not necessarily apparent to undergraduates who are in this situation. This answer is directed more towards those undergraduates than towards someone at a more advanced level such as yourself, who has long since realized that there are many undergraduates who are functionally graduate students. For these people, qualifying exactly what level one should be at before one is functionally a graduate student is useful, and that is the primary point of my answer. –  Logan Maingi May 14 '12 at 23:14
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Just to make things clear, I love Bourbaki.

However... These books are not made to teach, they aim rather at generality and exhaustivity. And they succeed very well I think. Some volumes are getting old fashioned, like Théorie des ensembles, or Intégration. Algèbre and Algèbre commutative are just great, I often need a generality I can only find there.

Concerning the rigor, they are rigorous in the narrow meaning of the term : they will always prefer generality and precision rather than readability and ease of use. But in the other hand, the logical frame is clear, references are precise, you know where are the definition, you know what they mean, always. They are rigorist.

With Bourbaki, all starts from the beginning. You might think that you can learn all math simply reading Bourbaki, but that just wrong. And they know that very well.

In the end, if your question is really “are you the target audience ?”, the answer is no. If the question is “is it interesting for me ?”, then you should read it, and you will see. Bourbaki is one of the most influential mathematician (fictional, but still) of the 20th century, and also one of the most controversial (is formalism a dead end ?), you should know what is it like, what is his philosophy, and have your opinion on it.

My advice is to read first the introduction of Théorie des ensembles which explains the general philosophy, and then read the volume about the field you know the best.

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Thanks for this clear and informative answer. The word "lisibility" is very rare in English. The closest common analog is "intelligibility". –  MJD May 14 '12 at 22:16
    
Thanks for pointing out ! I guess readability is even closer ? –  Lierre May 14 '12 at 22:38
    
"Readability" is also good. I'm glad I didn't offend you with my suggestion. –  MJD May 14 '12 at 23:11
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