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

Remove the stone of shame...

[Anonymous Coward]

Attach the stone of triumph!

secret?

[bfields]

"Secret society" is a bit over the top! I always had the impression it had the feeling more of a running joke; from the article:

In the 1950s, Ralph Boas of Northwestern University wrote an article for the Encyclopaedia Britannica on Bourbaki, explaining that it was the pseudonym for a consortium of French mathematicians. The editors of the encyclopedia soon received a scalding letter signed by Nicolas Bourbaki himself, declaring that he would not allow anyone to question his right to exist. In revenge, Bourbaki began spreading the rumor that Ralph Boas himself didn't exist, and that B.O.A.S was an acronym of a group of American mathematicians.

Though it wasn't *just* a joke--they wrote a lot of very serious mathematics!

Re:secret?

[DeadDecoy]

It's like having a secret society D&Ders with regular meetings in the basement, erm ... Painkeep. It's so secret nobody cares.

Mwahahaha this plan is brilliant. Now if only I could get with a girl and spread my brood upon the world.

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

[PiNtoS]

Bourbaki books are the most boring books you can buy. Avoid them at all costs.

This is a bit extreme. While they're certainly not the best books to begin learning a subject, they're great reference books. They're well written, (generally) correct, and what's more, they've got some seriously elegant proofs. And from what I recall, they do have some diagrams (e.g., their commutative algebra book).

Re:You meant the wrong way

[killmofasta]

Umm. I have a few texts from the late 1800s, and they are absolutly completely worse. Bourbaki's is the abstraction. ( I have their volume on abstract algebra, that I referred to while I was taking that graduate level course. Theirs is a terse work, much more accurate, and well though out. ) A few Springer-Verlag texts are worse also.

Re:You meant the wrong way

[msuarezalvarez]

Indeed. And the much more literary style that was deemed acceptable before resulted not only in inaccuracy but in gross errors.

Bourbaki's work is an amazing feat, which nowadays can be appreciated maybe only with a considerable amount of historical perspective---mostly because it was extremely successful: it set (maybe by using an elaborate, laborious, hyperbole that is, among many other things, a display of love for the subjects treated) standards against which mathematical writing was (and is!) compared, if not jugded, and the student of today has the false impression that the textbooks he reads today are of the same kind as those that were read at all times, simply because he does not know history.

The effort spent in coming up with clear, precise definitions, detailed proofs, even with usable notation, is easy to disparage once one can enjoy its benefits.

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.

Metascore

[Anonymous Coward]

From what I can tell, Metascore is an attempt by mathematicians to take over the government. In fact, every government.
http://www.metascore.org/
Difference is they do not seem to be very secretive.

They really are radicals!

[CaptainPatent]

I don't want to get off on a tangent, but the whole group seem kind of irrational.
How does someone become a member of this finite group? Do they have to stand in the middle while the rest of them form a perfect square around them? I wonder if they have to hide their identity?

oh well, at least at the end there's pi!

Re:They really are radicals!

[exp(pi*sqrt(163))]

> How does someone become a member of this finite group?

You make sure you have an inverse and that you associate nicely with the other elements.

Not so secret...

[mbone]

This was common knowledge when I was taking advanced Math classes in mid 1970's.

Re:when things don't add up & everything's 'se

[Anonymous Coward]

Oh, Anonymous Coward, how I love your posts. The complete disregard to logic, spelling and grammar appearing in every topic has become my warm security blanket. You are the nougat center of my candybar and I will henceforth use random caps keys while starting all sentences with a lowercase letter - or better an ampersand - as a show of the esteem I hold you in.

& sINCE yOU post this in every topic perhaps you might want to edit the oRIGINAL.

pS, I work for the gOVERNMENT and we are tracking you. yOU should quit doing that thing with anime porn or yOU"LL go blind.

Re:Their biggest achievement

[4D6963]

They made Steve Guttenberg a star.

Who??

Re:Imagine their meetings!

[Anonymous Coward]

The He-Man-Godel-Hating Club!

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:Why are we celebrating these books?

[Anonymous Coward]

Bourbaki was writing for a graduate student or professional level mathematician, not a public audience. The writing is dry by today's standards - indeed, many texts today use more prose and include diagrams. However, Bourbaki was very good at getting the mathematics itself clearly defined.

Re:Why are we celebrating these books?

[L'homme de Fromage]

Many people blame Bourbaki for the horrendous "new math" which infected mathematics teaching in the 1960's. And there is some validity to that accusation. A scathing indictment of Bourbaki was given by the renowned mathematician V.I. Arnold, author of famous books on classical mechanics and differential equations. Arnold tears apart the dry, lifeless and phony "rigor" and "purity" of Bourbaki and others who divorce mathematics from reality, which he describes as "sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people." Here's a link to his full comments:
http://pauli.uni-muenster.de/~munsteg/arnold.html

As a mathematician, I have to agree with the critics of Bourbaki. It put mathematics on the wrong path, in my opinion, and much of that influence continues on today. Mathematicians would do well to heed Arnold's advice on the direction mathematics needs to take. "Pure mathematicians" and other people who still think that Bourbaki was "doing mathematics the right way" are simply misguided.

Re:Why are we celebrating these books?

[ObsessiveMathsFreak]

However, Bourbaki was very good at getting the mathematics itself clearly defined.

It's just a pity they were never able to clearly get it across.

Bourbaki, and the Bourbaki style, makes great reference material. But that's all it makes. There is more to mathematics, and pictures and example are part of that "more". A big part. Bourbaki did not just forget these topics. They actively excluded them. Jean Dieudonne stood up in the middle of a conference and shouted "Down with Euclid! Death to Triangles!". It was an irrational zealotry, but mathematicians are people too, and they followed the trend setters.

You cannot learn mathematics from a reference book, or from books and people that try to be like those reference books. This applies to graduate students and professionals studying mathematics just as much as it applies to preschoolers learning about shapes. Here's a link to another view of mathematics, and how it should be taught by V.I. Arnold, a famous Russian mathematician.

New Math of the 1960's - they invented it

[Cliff Stoll]

Aargh! From these mathematicians grew the "New Math" of the early 1960's.

During the 1950's, high school math was mainly geometry, algebra, trig, and calculus.

Then came the New Math. Imported from France, it emphasized set theory, number bases, and abstract number theory. Students learned cardinality, commutative laws, associative laws, and "pure" math, with less applied math and problem-solving.

  Many educators (and even more parents) saw the New Math as being too abstract for daily use and undercutting concrete skills such as computation. Physicists, especially objected, when college freshmen could calculate in multiple number bases, but couldn't solve algebraic equations.

    Mathematician/singer Tom Lehrer wrote the song,"New Math", with the line, "It's so very simple that only a child can do it!"

    One book, Why Johnny Can't Add - the Failure of the New Math, pointed out that in the mid 1970's teachers applauded the death of the New Math. By the late 70's, algebra was back in style, and even trigonometry was being taught. So ended the French invasion of high school math classes.

    The latest, of course, is the new-new-math, also called rain-forest math. Don't get me started...

Re:gna:a

[Anonymous Coward]

"From steadily fucking," eh? Well, sir -- I wish I had YOUR problems.

Re:Imagine their meetings!

[CRCulver]

Imagine them finding an equation to discover the solution for all the problems of the world, and then the awkard silence as it turns out to be 42.

Secret Society

[PPH]

Dedicated to the ongoing fight against the "show your work" admonitions of math teachers everywhere.

New Math was Horrible!

[Anonymous Coward]

I survived four years of New Math - it was so easy that you couldn't do anything wrong. Straight A's in math.

Then I got to college and met all those equations. Everyone else solved them but me. Two weeks into chem 101 and I was flailing.

It's true - the Bourbaki group *invented* the New Math, and pushed it into classrooms around the world. Millions of adults are now math-illerates because of these oh-so-pure mathematicians.

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.

from Straight Dope

[mbius]

The following examples may help to clarify the difference between the new and old math.

1960: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?

1970 (Traditional math): A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?

1975 (New Math): A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100 and each element is worth $1.

(a) make 100 dots representing the elements of the set M

(b) The set C representing costs of production contains 20 fewer points than set M. Represent the set C as a subset of the set M.

(c) What is the cardinality of the set P of profits?

1990 (Dumbed-down math): A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.

1997 (Whole Math): By cutting down a forest full of beautiful trees, a logger makes $20.

(a) What do you think of this way of making money?

(b) How did the forest birds and squirrels feel?

(c) Draw a picture of the forest as you'd like it to look.

Re:Imagine their meetings!

[KDR_11k]

The first rule of clubs is that whenever anyone mentions any club you claim that its first two rules match those of the Fight Club!

Re:Thees Matematics...

[fugue]

Let zem eet Keeeeek!