A reply to some remarks by André Joyal

Previous post: The first post.


I am a fairly surprised by your criticism of Bourbaki work (I would be not surprised by a critique of Bourbaki from many other mathematicians). I hardly can say anything new in the defense of Bourbaki. The best defense was, probably, provided by J. Dieudonne. And one should not forget that, as some other former member of Bourbaki pointed out, that Bourbaki books succeeded in changing the style in which books in mathematics are written. There is no more such an urgent need in rigorous and systematic expositions as was in France before WWII, because many books by other authors are both rigorous and systematic.

Of course, applications are absent from Bourbaki books, as they are absent from almost all books in pure mathematics. I doubt that I ever read or used any book in pure mathematics discussing applications. And since I think that pure mathematics is not needed for any applications, I believe that it is better to separate them.

The issue of the presumed absence of motivation in Bourbaki books is more subtle. There are several sorts of motivations. A reference to applications is a common but very poor motivation to study proofs, because only statements and formulas are needed for applications. It seems that the most widespread sort of motivation is a pseudo-historical one: one invents some way how mathematicians could in principle arrive at a result or definition; in the best cases it resembles the actual history. Very often such a motivation uses tools developed only later and, moreover, under the influence of the original discovery. I consider such motivations as very misleading. The only true motivation in mathematics is the real history of the problem or of the theory at hand. I learned this point of view from my Ph.D. thesis adviser in the context of writing introductions to research papers, and later I saw that it applies to expository writings, textbooks, everywhere. Using it in the teaching or learning is possible, but by purely practical reasons only occasionally. This approach requires such amount of time that it will defer own research by decades. Another effective form of motivation is apparently already forgotten in the western countries. It is simply the trust to a more experienced person than you. If you are told that you will need to know the derived categories or it just worth to know about them, this should be a sufficient motivation and you will study the theory of derived categories. Finally, the organization of the material could be self-motivating. I think that Bourbaki books are superb in this respect. They are, in fact, much more readable than most of mathematicians think. Many other books are also self-motivating without any special efforts.


Next post: The times of André Weil and the times of Timothy Gowers. 1.