Founder of the Secret Society of Mathematicians

Anti-Globalism suggests an article at Science News on the passing of Henri Cartan, one of the founding members of a strange and influential group of French mathematicians in the twentieth century. "In the 1930s, a group of young French mathematicians led an uprising that revolutionized mathematics. France had lost most of a generation in the First World War, so the emerging hotshots in mathematics had few elders to look up to. And when these radicals did look up, they didn't like what they saw. The practice of mathematics at the time was dry, scattered and muddled, they believed, in need of reinvention and invigoration... Using the nom de plume Nicolas Bourbaki (after a dead Napoleonic general), they wrote a series of textbooks laying out mathematics the right way. Though the young mathematicians started out only intending to write a good textbook for analysis..., they ended up creating dozens of volumes which formed a manifesto for a new philosophy of mathematics. The last of the founders of Bourbaki, Henri Cartan, died August 13 at age 104... Two of his students won the Fields medal..., one won the Nobel Prize in physics and another won the economics Nobel."

Comments

And it all came down to this

[El Yanqui]

Hell is other mathematicians.

You meant the wrong way

[Anonymous Coward]

Bourbaki books are the most boring books you can buy. Avoid them at all costs. If you want to study mathematics, there are much better books.

Their motto is to never explain anything. These makes these books completely unreadable. Mathematics the right way ccroding to them is:
--no example: examples are evil being that stray us from the true path of pure abstraction.
--never mention any applications: are you nuts ? mathematics must remain pure. Applied mathematics are the spawn of the devil. If it serves some purpose, it's not mathematics anymore.
--Don't draw anything. Drawings are tools of the devil. 2D domains and geometrical figures should only exists as pure abstraction.
--The less explanations, the better: only idiots needs explanations.
--Never rewrite a theorem for the sake of clarity: having 20 references to other theorems
(usually in another volume) in a 5 lines proof
is better for clarity (don't even write the name of the Theorem you refer to, a true mathematician knows them by volume and page number).

And better they add insult to the injury in the preface: "no prerequisite knowledge of math is needed to read this book". Yeah whatever.

Re:You meant the wrong way

[msuarezalvarez]

If you really think there is no value in Bourbaki's texts, you know nothing of the history of mathematics, and probably you have not had to study that many textbooks from before their time...

Re:You meant the wrong way

[Alomex]

There were no gross errors found by the Bourbaki group. This is why their horrible formal writing style died out: it increased the pain of writing without the gain of the previous waves of math formalization.

Re:You meant the wrong way

[lysergic.acid]

yea, but how many of those authors published books in the 1930's? unless Walter Rudin started publishing his writings in his early-teens, and Hormander before he turned 10, i don't think you can make a comparison between their works and Bourbaki's.

Re:You meant the wrong way

[porpnorber]

I think perhaps you weren't born to be a mathematician. I seriously wish that some of my profs would have STFU'd about the applications and focused on the proofs and proof techniques. If you are studying biology, do you really want half your class hours devoted to what plants look nice in a garden? (Maybe you do, but if so, study gardening instead.) If you are studying software architecture, should the textbook assume that what you really care about is, I don't know, writing keyboard scanners? (Again, you might, but then why not buy a different book?) And do you want your general psych class turned into a course on methods of military indoctrination? It doesn't matter the field, I think we'd benefit from a lot less focus on applications and a lot more on mastery of content. Mathematics most of all, because the cultural content of mathematics is the collection of tools for thinking about pure problems, abstracted from any problem domain. Indeed, the best advances in mathematics come, it seems to me, from abstracting internal mathematical tools away from their original mathematical focus, and thereby making them available to the whole subject, and not just one small field.

As to issues with how theorems are referred to, I think this brings us to the root of the Bourbaki phenomenon. The cult of personality is not so productive in a field whose content is supposedly objective, and naming results after people is a barrier to objectivity in understanding and a barrier to communication. My girlfriend is Chinese. Do you suppose she knows, or cares, about Green's theorem or Taylor series, under those names? But five seconds with a pencil and paper and we are in sync.

Mathematics is not automotive mechanics and it is not pop music! And—I don't mean to be rude here, in making a cultural observation—that was a particularly hard lesson for French academia in particular—though for France we needed to write "it is not the civil service and it is not religion."

Re:You meant the wrong way

[Scott Carnahan]

I think perhaps you weren't born to be a mathematician.

The idea that career choices are predetermined at birth is a popular romantic view (cf. the human literary corpus of epics and fairy tales about Chosen Ones), but there is essentially no hard evidence for its validity, and I think it devalues the richness and variability of life experiences. Also, I don't think we should exclude people from mathematics just because they don't like the sort of dry abstraction you find in Bourbaki texts. There are plenty of reasonably successful mathematicians who are more comfortable learning things when they have an example or application in mind. For example, Timothy Gowers wrote two posts on his blog, suggesting that exposition is improved by starting with examples to motivate an idea.

It doesn't matter the field, I think we'd benefit from a lot less focus on applications and a lot more on mastery of content. Mathematics most of all, because the cultural content of mathematics is the collection of tools for thinking about pure problems, abstracted from any problem domain.

From a practical standpoint, I don't think we should try to change teaching methodology too much at a time, because there are almost always weaknesses in revolutionary plans that don't show up at the thought experiment stage. More abstractly, I think people tend to learn content better when it is motivated with a useful context. Exactly where this balance should be struck is still a contentious issue (see math wars), but I don't think Bourbaki is the answer. Even among pure academics, we value theoretical work by some notion of applicability. We say that etale cohomology is a good theory, not because it lets you think abstractly about pure stuff (although it does), but because you can use it to prove hard quantitative statements like the Weil conjectures, the Adams conjecture, and many theorems in representation theory.

Re:You meant the wrong way

[NaveNosnave]

The principal drawback of the Bourbaki school is not that its output is "boring", but that it presents the final results of a theorem in a pristine format while removing all references to the messy drudgework and dead ends it took to get there. As such, it's pleasingly elegant for practicing mathematicians, but terrible for students. Students need to see examples of math as a process, not just as a finished product.

Why are we celebrating these books?

[dlenmn]

From TFA:

The result was austere books with almost no examples, guide for intuition or pictures. Philip Davis of Brown University described them in an article in SIAM News as "mathematics with all its juices extracted; bare bones, skeletonic, anorexic stuff; Twiggy dressed in the tunic of Euclid." Michael Atiyah of the University of Edinburgh says: "They're not designed to be read. They're designed to set out a the is for how mathematics ought to be done."

Any they thought other books were dry? I guess books like this may have some use for hard core math types, but they sound like horrible books for almost anyone else. Examples, pictures, and the likes are very important for learning. Designing books not to be read seems like silly exercise.

Re:New Math was Horrible!

[linzeal]

No, the point of teaching math is to give the skill sets needed for analyzing other subjects mathematically. No one should be teaching math in a purely abstract way to anyone but grad students.