I am an economics undergrad who wants to develop a firm grasp over mathematics, so will I be able to understand Bourbaki's books on my own without needing additional books? What about the books titled analysis released by Roger Godement?
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6I do not recommend any of Bourbaki's books. They are the opposite of pedagogical. – GReyes Sep 10 '21 at 04:01
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What kind of mathematics are you interested in? – GReyes Sep 10 '21 at 04:02
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From Bourbaki theoretically is possible to read "Theory of sets" without background, but practically you need solid mathematical thinking culture. Also note, that today a lot of people counted them outdated, not me. Godement's book are quite different world and you can start with Analysis I. But, as I wrote, these two authors are very different. – zkutch Sep 10 '21 at 04:09
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To be honest, I want to get a very general idea of the entire field of mathematics. – Slothrop Sep 10 '21 at 04:10
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As you are economics undergrad, then may by more appropriate is to look in mathematical economics direction. Most good will be to find somebody supervisor. – zkutch Sep 10 '21 at 04:20
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3If you want to get a good idea of Math in general I recommend "Mathematics, its contents, methods and meaning" by Kolmogorov, Aleksandrov and Lavrent'ev. You can use Bourbaki's books to stick under the leg of a wobbly table, if you have one. – GReyes Sep 10 '21 at 04:34
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https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163 – GReyes Sep 10 '21 at 04:36
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2For a general overview of math, one book to be aware of is The Princeton Companion to Mathematics. Mathematics and Its History by Stillwell is also a good book to be aware of. I'm also a fan of The Calculus Gallery by Dunham. – littleO Sep 10 '21 at 05:54
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2Oh lord, Bourbaki is so tough to read. In all probability, you have way easier alternatives to a given subject. – AlvinL Sep 10 '21 at 08:29
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1@Slothrop : who committed the sin to even tell you about Bourbaki's books? Decades ago there was certainly a time when terse books like this needed to be written. However, thank god that we moved on. The recommendations in some of the earlier comments are terrific. Bourbaki is for math PhD students who need to get down into their rabbit holes. – Kurt G. Sep 10 '21 at 08:52
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@GReyes. Your comment with "books to stick under the leg of a wobbly table" is impolite and rude. If someone told you that your thoughts should be stick under the leg of a wobbly table, would you call it polite? – zkutch Sep 11 '21 at 10:09
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I want to preface what I'm about to say by first expressing my admiration for the Bourbaki textbooks, or at least the ones I've read from (namely, Algebra and Commutative Algebra). The Bourbaki series is the distillation of countless hours of work from some of the greatest mathematicians of the 20th century on what they felt was the correct pedagogical approach to certain core areas of mathematics. Some of the books are a little dated - for instance, Bourbaki's book on algebra is conspicuously devoid of categorical language - but many are nevertheless deep, beautiful works which contain treasures that I have been unable to find in any other books. I am an algebraist by training, and found the two books mentioned above indispensible for conducting my own research. Frankly, to go a step further, I find a lot of the criticism directed at the Bourbaki books misplaced and undeserved.
All that being said, my advice would be to reconsider whether reading the works of Bourbaki will really get you closer to what you want. You write in the comments, for instance, that you want to get a general picture of the whole of mathematics. This is an ambitious goal, and I think in practice, the way that most people do this is by getting briefly acquainted with the basics in many fields, and then going deeper based on what they're interested in and what their needs are. It's also important to look at different resources and approaches to material, because different presentations of the same material might resonate with you in entirely different ways. It's hard to know a priori what will produce the most understanding within yourself, and for many people, the only remedy for this problem is to see the same material over and over in different guises.
The Bourbaki tomes are encyclopedic, extremely dry, and ponderous at times. They are not, as far as I am aware, intended to be read from start to finish by a neophyte who wants to learn the subject. If you had extraordinary focus and maybe 15 or 20 years to spare, you might get somewhere with your goal by just reading through Bourbaki. The problem is, the Bourbaki series isn't well suited to jumping around, flitting from subject to subject, if you're not already acquainted with the material being discussed. One of the aims of the Bourbaki series was to build modern mathematics from the foundations up, addressing every last rigorous detail in a self-contained way. The result is that the proof of a given statement in Bourbaki's "Algebra" might reference several other statements in "Algebra", or even in (say) their book on set theory. This makes it very difficult to peruse the material non-linearly if you're seeing it for the first time, since you will have to either look up every reference you come across, or already understand the statements referenced.
Just trying to bulldoze through the books from start to finish isn't something I'd recommend either. One could spend a very long time on (say) Bourbaki's book on set theory without learning anything substantial about mathematics as a whole. I should add that this is especially true without the guidance of great teachers, which was indispensable for me on my own journey through these texts.
Anyway, the above represents my personal feelings, and shouldn't be taken as gospel - a person could surely learn a great deal of deep mathematics just by reading Bourbaki. If you're determined to do this, don't let this post discourage you. But I think that there are better ways to get what you want, and in time, you'll be in a better place to appreciate these classic works for what they are. Good luck!
Alex Wertheim
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"..which contain treasures that I have been unable to find in any other books." I agree about this. – hm2020 Sep 13 '21 at 08:44
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Question: "What mathematical education should I have in order to study the books released under the pseudonym N. Bourbaki?"
Answer: For the algebra and commutative algebra books I recommend the algebra book of Lang and the commutative algebra books of Atiyah-Macdonald and Matsumura. When you have studied these books you will easily understand the Bourbaki books. The Bourbaki books in algebra and commutative algebra are used by researchers in the field. They contain much material and are self contained.
hm2020
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