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Analysis How are Bourbaki's book and Dieudonne's book?

  1. Mar 29, 2016 #1
    Dear Physics Forum friends,

    While vigorously studying Dugundji's Topology and Rudin's PMA, I found that the reference mentions the series of books written by N. Bourbaki, known as "Elements of Mathematics", and Dieudonne's Foundations of Modern Analysis. How are those books, specifically their treatment of the real analysis and topology compared to the books like Dugundji and Rudin. I always have been enjoying reading other books and articles, and I am very curious about Bourbaki and Dieudonne. Is there anything special about their exposition and problem sets compared to books I mentioned?
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  3. Mar 30, 2016 #2


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    Bourbaki is the pseudonym used by a collective of (mainly French) mathematicians, one of them Dieudonné.

    The Bourbaki books are quite terse, very abstract, but have been very influential (see https://en.wikipedia.org/wiki/Nicolas_Bourbaki#Influence_on_mathematics_in_general ).
    Some people love the Bourbaki books, others hate them.

    If you have the time, I would suggest to give one of the Bourbaki books a look. They have a very specific style, that you may, as I said, love or hate.
  4. Mar 30, 2016 #3
    Of Dieudonné's dictionary of mathematics (years 1700 to 1900), I own a copy of the original French version. Most recommendable as reference for the college level honors courses; its latest chapters are usefull for a university certificate in math, because it gives nice abstracts of the key notions. ___________ For modern analysis, I will only read the required part of infinitesimal analysis contained in ''Advanced Calculus'' by Foland; and a book on Fourier's analysis by the same author; and "Differential Topology: An Introduction" by David B. Gauld. Since my trade is industrial draftsman, I only need to learn the math supporting: statics, and strength of materials, and mechanics of hydraulics, and advanced strength of materials, and mechanics of vibrations. ____________ For any student in major in math, the authors of analysis whom you mentionned, are often recommended. Foland's latest book on analysis, lacks of illustrations; his errata are shown on Internet.
    Last edited: Mar 30, 2016
  5. Mar 31, 2016 #4
    I just taken a look at some of Bourbaki's books, mainly the "Functions of a Real Variable" and "Topology I". I actually like the exposition, but I am not sure if they are up-to-date for the subjects like analysis and topology, judging from the fact that the books were written at 1960. Are their books on analysis and topology enough to carry me to the graduate level?
  6. Mar 31, 2016 #5
    By "dictionary of mathematics", do you mean Dieudonne's series of books on the analysis (Treatise on Analysis)? I read some positions of Folland, but I never like his style.
  7. Mar 31, 2016 #6


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    It probably isn't a good idea to learn topology and analysis with the Bourbaki books. There are better textbooks available (there are some Insights about that, as well as quite a few threads in this subforum). But if you like the exposition, they are a nice addition.
  8. Apr 1, 2016 #7
    Folland's style: rather compacted, rare drawings, but examples on physics and engineering. He almost admits in his introduction that he grows mathematicians with readers. The difficulties with the reading of Dieudonné's (and moreover the N. Bourbaki's ) books and articles, are bigger, because they assume that the reader audience has, beyond the post high-school college math, a sophomore educational background of classic & modern algebra, linear algebra, infinitesimal calculus, analysis and a minimum of knowledge in thery of sets, structures of groups/ rings/ etc.. The Bourbaki team were in a mathematical religion which states that a more sustainable, the sure foundation of math, shall strictly avoid drawings/ pistures and shall be constructed absolutly of axioms. The student has all latitude for creating the best illustrations in his/her mind; isn'it wonderful !
    ''Abrégé d'histoire des mathématiques'' (1700- 1900), by J. Dieudonné, is 40% a dictionary of math accessible for whoever knows the math & physics of the French baccalaurate _in the USA, the courses Algebra I, the Linear Algebra I & a half of II, the Calculus I, II, III (and IV for the latter chapters)_. Together with ''Mathematical Universe'' by another author, translated in French, it is the ''royal way'' to learn the trade of technician in math. Bourbaki team joined in, or started the mathematical religion which commands the absence of drawing/ illustration in the books/ articles; and obliges to build knowledge upon heavy axiomatic footings; and suffers unclear definition(s) of mathematical objects/ structures. (This is imperfect view of the discipline, in my opinion.) This religion still prevails among the scholars and most reputated discoverers: they don't dare to criticize, or decide what is important from what is ephemere/ fanzy. A three hundred page prove is expected where a single page would have sufficed for celebrated amateur Fermat.) ______________________I agree that Folland's books are hard to understand. The new didactic approach is visual, geometric, reader-friendly.
    Last edited: Apr 1, 2016
  9. Apr 2, 2016 #8


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    Well, in some sense you can say that the Bourbaki style is the end of mathematical civilization (from the point of view of a theoretical physicist). I think that the attitude encouraged by this book is even dangerous for the development of math, because it hinders mathematicians to develop an intuition for their subject which they need to guess new theorems to prove. Of course math must be rigorous at the end, but you need intuition to find something to make rigorous. Books in the Bourbaki style are good to have the rigorous final version of math stored but it's not good as a textbook to learn math to become a creative mathematician.
  10. Apr 3, 2016 #9
    errata: 1_ "Mathematical Universe" is replaced by "The Mathematical Experience". 2_ last but one page, "prove" is replaced by "proof". _______________ The traditional point of view of theorical physicist(s) is that the notion of group as mathematical tool," is used" in the researches in physics, percisely at the galactic & intergalactic scale, and (excuse my ignorance) perhaps at the sub-atomic scale. Since a.k.a. mathematician Nicolas Bourbaki, the notion of group is used worldwide by the departments/ agencies of research in physics. Every adult knows that the theories succeed one after another; as a result a reference textbook for sophomore in electronics, waves physics or mostly in modern physics, is truely updated after seven years (it means completely improper for self teaching). ____Bourbaki's great contribution is the creation, upon the theory of sets (wich was created upon the early (and alias non-standard infinitesiamal) analysis, of the modern and completely abstract algebra. You already know around 1830, was born in the mind of the scientists, a shared desire to decide of a serious classification of all the existing/recorded mathematicS, and to merge & order them into The mathematic. In the 1830s, they were beleiving that the queen of sciences could help resolve all sorts of problems, thus saving manking or at least allowing victory in industrial competition and warfares and conquest of the world & its natural ressources. An imminent and religious quest for a unique & sure foundation for "la mathématique" was launched, while the French mathematicians were the foremost in Occident. The fiirst chapters of the first tome by Bourbaki, worths to be read and understood by a math student with sufficient preparation. Because it is part of international heritage; and because the "groups" are still in usage in modern physics. _____ I am in favor of working with all kind of mathematics, at the free choice of the researcher (whose intuition will not be hindered). If you solve a question in math or in physics, with the use of higher arithmetic and hyperbolic trigonometry, very good !; then you will have transform the sequence /process of your discovery into the Bourbaki's language, to satisfy the examiners and the magazines and the donators of schoolarship/ grant. I learned at a state university that in extremely rare cases of discovery, the "scientific method" has been used and with success at the firs blow. In reality, they discover in the way(s) that suit themselves, then they compose/ invent a marvelous/fanciful narration of their discovery/rediscovery, e.g. the apple fallen on asleep Isaac Newton, to suit/satisfy the educational system, the society in general and the public libraries, also to flatter the author's nation, and maintain a high level of nobility & honour in the sight of posterity. Often the educate/ learned has stolen a discovery/ work from the amateur or from the common, along the centuries & millenia. Authorship can stolen or purshased, e.g. the "rule of l'Hôpital" has been bought by the marquis from one of the two brothers Bernouilli. As you concludes, the Bourbaki's style is the rigorous final version for editing/ publishing. It is an undiscutable, yet very costy tradition: no wonder why our national Goverments are so indepted. It creates jobs and slow down the production of discoveries in math and, possibly in non-mechanics physics, who knows? In the past centuries, discoverers were obliged to hire in-Latin translators, to properly present/publish any serious thesis.
  11. Apr 9, 2016 #10


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    Bourbaki books are more abstract and rigorous, it is a mathematical style hard to read at the beginning but It gives a lot of satisfaction in a second time ... depends also by your level of experience.
  12. Apr 13, 2016 #11


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    i think dieudonne's analyis book you mention is one of the best books in existence. the amount of information and the depth to which it goes is virtually unmatched in other works. note however he says in the introduction that it is clear one should not approach his book without first having a firm grasp of classical analysis, which means perhaps a thorough grounding in rudin's work or a more example oriented advanced calculus book such as courant.

    i add however that i think dieudonne's approach to integration theory is very unhelpful. he assails the value of riemann's approach in that book and then later in his higher volumes gives a very abstract, complicated, and to me unintuitive approach to lebesgue integration. to be sure the elementary version in the book you cite is fine and interesting, but it unnecessarily disparages riemann integration in favor of lebesgue's, whereas the approach he gives later to lebesgue's theory is hardly one i would recommend. i.e. while i agree that lebesgue's integration theory has multiple virtues compared to riemann's, nonetheless dieudonne's choice of presentation of lebesgue's theory is not recommended (by me, admittedly not an expert) as an introduction.

    to cite one example of an admirable feature of dieudonne's book, how many books you know contain a self contained proof of the jordan curve theorem, (an appendix to the chapter on complex analysis). however, my recommendation is not toe fall for his snooty prference for doing analysis over a banach space of scalars, when the complex numbers will serve just fine. to be sure he says one may assume all scalar domains are reals or complexes but he says it in a sneering way that makes one embarrassed to do so. try to ignore these rude and snotty lapses and benefit from the magnificent content of his book. it was typical in the 1960's for some reason that some outstanding mathematicians felt the neccesity to insult their readers while teaching them. to be fair, they did not mean to insult the reader but rather the outmoded members of the previous generation of textbook writers, but the reader feels it just the same.

    as to bourbaki, these are books of excellent quality, written by outstanding mathematicians, but not attempting to motivate or really teach, merely to present material in a version thought to be the best available. so the material is excellent but the presentation is not at all user friendly, except for the great clarity. so what they say is very clear and correct, but why they say it that way is not mentioned. there are however potentially useful historical appendices seldom found elsewhere.
    Last edited: Apr 13, 2016
  13. Apr 15, 2016 #12
    I have "A Primer of Lebesgue Integration" by H.S. Bear. The riemanian integration has much evolved since Riemann. Lebesque's contribution is everlasting and every student interested in a math-based carreer should read an introduction of it, although not in the cussiculum (because the Ministry of Education is dishonest or chauvinist).
  14. Apr 17, 2016 #13


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    Let me be clear: I agree that one should learn Lebesgue integration. I do not however agree with Dieudonne' that one should NOT learn Riemann integration. I also do not agree with Dieudonne that when one begins to learn Lebesgue integration, the appropriate way to introduce it is via the rather esoteric and complicated approximation technique he uses in his later volumes of his treatise. I do agree that the approach in your chosen book by H.S.Bear, is reasonable, namely Bear first defines Riemann (and the even more simple Darboux) integration, then explains Caratheodory's inner and outer measure theory which enables him to define Lebesgue integration in an intuitive manner. I also think one might benefit from actually reading Riemann's own treatment of integration. If one does so, one will learn that Riemann already had the crirterion for Riemann integrability so often falsely attributed to lebesgue, i.e. that a function is Riemann integrable if and only if its set of discontinuities have measure zero. Yes Riemann already had a notion equivalent to that of "Lebesgue" measure zero (a countable union of sets of content zero). This result occurs in Riemann on the next page after he defines the integral.
  15. Apr 18, 2016 #14
    Thank you very much. I agree with everything written, and with your participation on the subject "Lebesgue vs Riemann integral". Bear's book contains 1.3-credit of knowledge. I will read it during my thorough self study of "Advanced Calculus" by Gerald B. Folland who uses Riemann measure & integration all across the book, except for a few pages on Lebesgue measure & integration.
  16. Apr 20, 2016 #15
    Thank you very much for all of your detailed advice. I have been actively reading both Bourbaki and Dieudonne. Bourbaki's General Topology I book is very, very well written with clear proofs, but I see that it has some missing contents such as co-finite topology and nets. Although I think that the spaces such as L-space and D-space are not important as the concepts forming them, but I am not sure why Bourbaki did not discuss the nets as they are useful techniques for proving the compactness and metrication properties. Regarding to General Topology I, perhaps I found it very easy as I had been reading Dugundji and Kelley before that book.

    Regarding to Dieudonne, I decided to read Sterling's book on the real analysis first.
  17. Apr 20, 2016 #16
    Because the members of Bourbaki are French and the French prefer filters over nets. Filters are superior to nets too.
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