Bourbaki, Nicolas

Bourbaki, Nicolas

by Émilie Richer

The Problem

The devastation of World War I presented a unique challenge to aspiring
mathematicians of the mid 1920’s. Among the many casualties of the war
were great numbers of scientists and mathematicians who would at this
time have been serving as mentors to the young students. Whereas other
countries such as Germany were sending their scholars to do scientific
work, France was sending promising young students to the front. A war-time
directory of the école Normale Supérieure in Paris confirms that about
2/3 of their student population was killed in the war.

[

DJ

]

Young men
studying after the war had no young teachers, they had no previous
generation to rely on for guidance. What did this mean? According to Jean
Dieudonné, it meant that students like him were missing out on important
discoveries and advances being made in mathematics at that time. He
explained : “I am not saying that they (the older professors) did not
teach us excellent mathematics (…) But it is indubitable that a 50 year
old mathematician knows the mathematics he learned at 20 or 30, but has
only notions, often rather vague, of the mathematics of his epoch, i.e.
the period of time when he is 50.” He continued : “I had graduated from
the école Normale and I did not know what an ideal was! This gives you
and idea of what a young French mathematician knew in 1930.”

[

DJ

]

Henri Cartan, another student in Paris shortly after the war affirmed :
“we were the first generation after the war. Before us there was a vide,
a vacuum, and it was necessary to make everything new.”

[

JA

]

This is
exactly what a few young Parisian math students set out to do.

The Beginnings

After graduation from the école Normale Supérieure de Paris a group of
about ten young mathematicians had maintained very close ties.

[

WA

]

They had all begun their careers and were scattered across France teaching
in universities. Among them were Henri Cartan and André Weil who were both
in charge of teaching a course on differential and integral calculus at the
University of Strasbourg. The standard textbook for this class at the time was
“Traité d’Analyse” by E. Goursat which the young professors found to be
inadequate in many ways.

[

BA

]

According to Weil, his friend Cartan was
constantly asking him questions about the best way to present a given topic to
his class, so much so that Weil eventually nicknamed him “the grand
inquisitor”.

[

WA

]

After months of persistent questioning, in the winter
of 1934, Weil finally got the idea to gather friends (and former classmates)
to settle their problem by rewriting the treatise for their course. It is at
this moment that Bourbaki was conceived.

The suggestion of writing this treatise spread and very soon a loose circle
of friends, including Henri Cartan, André Weil, Jean Delsarte, Jean
Dieudonné and Claude Chevalley began meeting regularly at the Capoulade,
a café in the Latin quarter of Paris to plan it . They called themselves
the “Committee on the

Analysis

Treatise”

[

BL

]

. According to Chevalley
the project was extremely naive. The idea was to simply write another textbook
to replace Goursat’s.

[

GD

]

After many discussions over what to include in
their treatise they finally came to the

conclusion

that they needed to start
from scratch and present all of essential mathematics from beginning to end.
With the idea that “the work had to be primarily a tool, not usable in some
small part of mathematics but in the greatest possible number of places”.

[

DJ

]

Gradually the young men realized that their meetings were not

sufficient

, and they decided they would dedicate a few weeks in the summer
to their new project. The collaborators on this project were not aware of
its enormity, but were soon to find out.

In July of 1935 the young men gathered for their first congress (as they would
later call them) in Besse-en-Chandesse. The men believed that they would be
able to draft the essentials of mathematics in about three years. They did not
set out wanting to write something new, but to

perfect

everything already
known. Little did they know that their first chapter would not be completed
until 4 years later. It was at one of their first meetings that the young men
chose their name : Nicolas Bourbaki. The organization and its membership
would go on to become one of the greatest enigmas of 20th century mathematics.

The first Bourbaki congress, July 1935. From left to
right, back row: Henri Cartan, René de Possel, Jean Dieudonné, André
Weil, university lab technician, seated: Mirlès, Claude Chevalley, Szolem
Mandelbrojt.

André Weil recounts many years later how they decided on this name. He and
a few other Bourbaki collaborators had been attending the école Normale in
Paris, when a notification was sent out to all first year science students :
a guest speaker would be giving a lecture and attendance was highly
recommended. As the story goes, the young students gathered to hear,
(unbeknownst to them) an older student, Raoul Husson who had disguised himself
with a fake beard and an unrecognizable accent. He gave what is said to be an
incomprehensible, nonsensical lecture, with the young students trying
desperately to follow him. All his results were wrong in a non-trivial way and
he ended with his most extravagant : Bourbaki’s

Theorem

. One student even
claimed to have followed the lecture from beginning to end. Raoul had taken
the name for his theorem from a general in the Franco-Prussian war. The
committee was so amused by the story that they unanimously chose Bourbaki
as their name. Weil’s wife was present at the discussion about choosing a
name and she became Bourbaki’s godmother baptizing him Nicolas.

[

WA

]

Thus was born Nicolas Bourbaki.

André Weil, Claude Chevalley, Jean Dieudonné, Henri Cartan and Jean
Delsarte were among the few present at these first meetings, they were all
active members of Bourbaki until their retirements. Today they are
considered by most to be the founding fathers of the Bourbaki group.
According to a later member they were “those who shaped Bourbaki and
gave it much of their time and thought until they retired” he also claims
that some other early contributors were Szolem Mandelbrojt and René de
Possel.

[

BA

]

Reforming Mathematics : The Idea

Bourbaki members all believed that they had to completely rethink mathematics.
They felt that older mathematicians were holding on to old practices and
ignoring the new. That is why very early on Bourbaki established one its
first and only rules : obligatory retirement at age 50. As explained by
Dieudonné “if the mathematics set forth by Bourbaki no longer correspond
to the trends of the period, the work is useless and has to be redone, this
is why we decided that all Bourbaki collaborators would retire at age 50.”

[

DJ

]

Bourbaki wanted to create a work that would be an essential tool
for all mathematicians. Their aim was to create something logically ordered,
starting with a strong

foundation

and building continuously on it. The
foundation that they chose was

set theory

which would be the first book in a
series of 6 that they named “éléments de mathématique”(with the ’s’
dropped from mathématique to

represent

their underlying belief in the unity
of mathematics). Bourbaki felt that the old mathematical divisions were no
longer valid comparing them to ancient zoological divisions. The ancient
zoologist would classify animals based on some basic superficial similarities
such as “all these animals live in the ocean”. Eventually they realized that
more complexity was required to classify these animals. Past mathematicians
had apparently made

similar

mistakes : “the order in which we (Bourbaki)
arranged our subjects was decided according to a

logical and

rational

scheme.
If that does not agree with what was done previously, well, it means that
what was done previously has to be thrown overboard.”

[

DJ

]

After many
heated discussions, Bourbaki eventually settled on the topics for
“éléments de mathématique” they would be, in order:

I Set theory

II

Algebra

III

Topology

IV

Functions

of one real variable

V

Topological vector spaces

VI Integration

They now felt that they had eliminated all secondary mathematics, that
according to them “did not lead to anything of proved importance.”

[

DJ

]

The following table summarizes Bourbaki’s choices.

What remains after cutting the loose threads

What is
excluded

(the loose threads)

Linear and multilinear algebra

Theory of

ordinals

and

cardinals

A little general topology

the least possible

Lattices

Topological vector Spaces

Most general topology

Homological algebra

Most of

group theory

finite groups

Commutative algebra

Most of number theory

Non-commutative

algebra

Trigonometrical series

Lie groups

Interpolation

Integration

Series of

polynomials

Differentiable manifolds

Applied mathematics

Riemannian geometry

Dieudonné’s metaphorical ball of yarn:

“here is
my picture of mathematics now. It is a ball of wool, a tangled hank where
all mathematics react upon another in an almost unpredictable way. And
then in this ball of wool, there are a certain number of threads coming
out in all directions and not connecting with anything else. Well the
Bourbaki method is very simple-we cut the threads.”

[

DJ

]

Reforming Mathematics : The Process

It didn’t take long for Bourbaki to become aware of the size of their project.
They were now meeting three times a year (twice for one week and once for two
weeks) for Bourbaki “congresses” to work on their books. Their main rule was
unanimity on every point. Any member had the right to veto anything he felt
was inadequate or imperfect. Once Bourbaki had agreed on a topic for a chapter
the job of writing up the first draft was given to any member who wanted it.
He would write his version and when it was

complete

it would be presented at
the next Bourbaki congress. It would be read aloud line by line. According
to Dieudonné “each proof was examined point by point and criticized
pitilessly. He goes on “one has to see a Bourbaki congress to realize the
virulence of this criticism and how it surpasses by far any outside attack.”

[

DJ

]

Weil recalls a first draft written by Cartan (who has unable to attend
the congress where it would being presented). Bourbaki sent him a telegram
summarizing the congress, it read : “union

intersection

partie produit tu
es démembré foutu Bourbaki” (union intersection subset

product

you are
dismembered screwed Bourbaki).

[

WA

]

During a congress any member was
allowed to interrupt to criticize, comment or ask questions at any time.
Apparently Bourbaki believed it could get better results from confrontation
than from orderly discussion.

[

BA

]

Armand Borel, summarized his first
congress as “two or three monologues shouted at top voice, seemingly
independent of one another”.

[

BA

]

Bourbaki congress 1951.

After a first draft had been completely

reduced

to pieces it was the job of
a new collaborator to write up a second draft. This second collaborator
would use all the suggestions and changes that the group had put forward
during the congress. Any member had to be able to take on this task because
one of Bourbaki’s mottoes was “the control of the specialists by the
non-specialists”

[

BA

]

i.e. a member had to be able to write a chapter
in a field that was not his specialty. This second writer would set out on
his assignment knowing that by the time he was ready to present his draft
the views of the congress would have changed and his draft would also be
torn apart despite its adherence to the congress’ earlier suggestions.
The same chapter might appear up to ten times before it would finally be
unanimously approved for publishing. There was an average of 8 to 12 years
from the time a chapter was approved to the time it appeared on a bookshelf.

[

DJ

]

Bourbaki proceeded this way for over twenty years, (surprisingly)
publishing a great number of volumes.

Bourbaki congress 1951.

Recruitment and Membership

During these years, most Bourbaki members held permanent positions at
universities across France. There, they could recruit for Bourbaki, students
showing great promise in mathematics. Members would never be replaced formally
nor was there ever a fixed number of members. However when it felt the need,
Bourbaki would invite a student or colleague to a congress as a “cobaye”
(guinea pig). To be accepted, not only would the guinea pig have to understand
everything, but he would have to actively participate. He also had to show
broad interests and an ability to adapt to the Bourbaki style. If he was
silent he would not be invited again.(A challenging task considering he
would be in the presence of some of the strongest mathematical minds of the
time) Bourbaki described the reaction of certain guinea pigs invited to a
congress : “they would come out with the impression that it was a gathering
of madmen. They could not imagine how these people, shouting -sometimes three
or four at a time- about mathematics, could ever come up with something
intelligent.”

[

DJ

]

If a new recruit was showing promise, he would continue
to be invited and would gradually become a member of Bourbaki without any
formal announcement. Although impossible to have complete anonymity, Bourbaki
was never discussed with the outside world. It was many years before Bourbaki
members agreed to speak publicly about their story. The following table gives
the names of some of Bourbaki’s collaborators.

1

s

⁢

t

generation

(founding fathers)

2

n

⁢

d

generation

(invited after WWII)

3

r

⁢

d

generation

H. Cartan

J. Dixmier

A. Borel

C. Chevalley

R. Godement

F. Bruhat

J. Delsarte

S. Eilenberg

P. Cartier

J. Dieudonné

J.L. Koszul

A. Grothendieck

A. Weil

P. Samuel

S. Lang

J.P Serre

J. Tate

L. Shwartz

3 Generations of Bourbaki

(membership according
to Pierre Cartier)

[

SM

]

. Note: There have been a great number of Bourbaki
contributors, some lasting longer than others, this table gives the members
listed by Pierre Cartier. Different sources list different “official
members” in fact the Bourbaki website lists J.Coulomb, C.Ehresmann,
R.de Possel and S. Mandelbrojt as

1

s

⁢

t

generation members.

[

BW

]

Bourbaki congress 1938, from left to right: S. Weil, C.
Pisot, A. Weil, J. Dieudonné, C. Chabauty, C. Ehresmann, J. Delsarte.

The Books

The Bourbaki books were the first to have such a tight organization, the first
to use an axiomatic

presentation

. They tried as often as possible to start from
the general and work towards the particular. Working with the belief that
mathematics are fundamentally simple and for each mathematical question there
is an optimal way of answering it. This required extremely rigid

structure

and notation. In fact the first six books of “éléments de mathématique”
use a completely

linearly-ordered

reference system. That is, any reference at
a given spot can only be to something earlier in the text or in an earlier
book. This did not please all of its readers as Borel elaborates : “I was
rather put off by the very dry style, without any concession to the reader,
the apparent striving for the utmost generality, the inflexible system of
internal references and the total absence of outside ones”. However,
Bourbaki’s style was in fact so efficient that a lot of its notation and
vocabulary is still in

current

usage. Weil recalls that his granddaughter
was impressed when she learned that he had been personally responsible for
the symbol

∅

for the

empty set

,

[

WA

]

and Chevalley explains
that to “bourbakise” now means to take a text that is considered screwed
up and to arrange it and improve it. Concluding that “it is the notion of
structure which is truly bourbakique”.

[

GD

]

As well as

∅

, Bourbaki is responsible for the

introduction

of
the

⇒

(the

implication

arrow),

ℕ

,

ℝ

,

ℂ

,

ℚ

and

ℤ

(respectively the natural,
real, complex, rational numbers and the integers)

C

A

(

complement

of
a set A), as well as the words

bijective

,

surjective

and

injective

.

[

DR

]

The Decline

Once Bourbaki had finally finished its first six books, the obvious
question was “what next?”. The founding members who (not intentionally)
had often carried most of the weight were now approaching mandatory
retirement age. The group had to start looking at more specialized
topics, having covered the basics in their first books. But was the
highly structured Bourbaki style the best way to approach these topics?
The motto “everyone must be interested in everything” was becoming much
more difficult to enforce. (It was easy for the first six books whose
contents are considered essential knowledge of most mathematicians)
Pierre Cartier was working with Bourbaki at this point. He says “in
the forties you can say that Bourbaki know where to go: his goal was
to provide the foundation for mathematics”.[12] It seemed now that
they did not know where to go. Nevertheless, Bourbaki kept publishing.
Its second series (falling short of Dieudonné’s plan of 27 books
encompassing most of modern mathematics

[

BA

]

) consisted of two
very successful books :

Book VII Commutative algebra

Book VIII Lie Groups

However Cartier claims that by the end of the seventies, Bourbaki’s
method was understood, and many textbooks were being written in its
style : “Bourbaki was left without a task. (…) With their rigid
format they were finding it extremely difficult to incorporate new
mathematical developments”

[

SM

]

To add to its difficulties, Bourbaki was
now becoming

involved in

a battle with its publishing company over royalties
and translation rights. The matter was settled in 1980 after a “long and
unpleasant” legal process, where, as one Bourbaki member put it “both
parties lost and the lawyer got rich”

[

SM

]

. In 1983 Bourbaki published
its last volume : IX

Spectral Theory

.

By that time Cartier says Bourbaki was a dinosaur, the head too far away
from the tail. Explaining : “when Dieudonné was the “scribe of Bourbaki”
every printed word came from his pen. With his fantastic memory he knew every
single word. You could say “Dieudonné what is the result about so and so?”
and he would go to the shelf and take down the book and open it to the right
page. After Dieudonné retired no one was able to do this. So Bourbaki lost
awareness of his own body, the 40 published volumes.”

[

SM

]

Now after
almost twenty years without a significant publication is it safe to say the
dinosaur has become extinct?

1

1

Today what remains is “L’Association
des Collaborateurs de Nicolas Bourbaki” who organize Bourbaki seminars three
times a year. These are international conferences, hosting over 200
mathematicians who come to listen to presentations on topics chosen by
Bourbaki (or the A.C.N.B). Their last publication was in 1998, chapter
10 of book VI commutative algebra.

But since Nicolas Bourbaki never in
fact existed, and was nothing but a clever teaching and research ploy, could
he ever be said to be extinct?

References

BL

L. BEAULIEU: A Parisian Café and Ten Proto-Bourbaki Meetings (1934-1935), The Mathematical Intelligencer Vol.15 No.1 1993, pp 27-35.

BCCC

A. BOREL, P.CARTIER, K. CHANDRASKHARAN, S. CHERN, S. IYANAGA: André Weil (1906-1998), Notices of the AMS Vol.46 No.4 1999, pp 440-447.

BA

A. BOREL: Twenty-Five Years with Nicolas Bourbaki, 1949-1973, Notices of the AMS Vol.45 No.3 1998, pp 373-380.

BN

N. BOURBAKI: Théorie des Ensembles, de la

collection

éléments de Mathématique, Hermann, Paris 1970.

BW

Bourbaki website: [online] at

www.bourbaki.ens.fr

www.bourbaki.ens.fr.

CH

H. CARTAN: André Weil:Memories of a Long Friendship, Notices of the AMS Vol.46 No.6 1999, pp 633-636.

DR

R. DéCAMPS: Qui est Nicolas Bourbaki?, [online] at

http://web.archive.org/web/*/http://faq.maths.free.fr/html-old/faq34.php3

http://faq.maths.free.fr.

DJ

J. DIEUDONNé: The Work of Nicholas Bourbaki, American Math. Monthly 77,1970, pp134-145.

EY

Encylopédie Yahoo: Nicolas Bourbaki, [online] at

http://fr.encylopedia.yahoo.com

http://fr.encylopedia.yahoo.com.

GD

D. GUEDJ: Nicholas Bourbaki, Collective Mathematician: An Interview with Claude Chevalley, The Mathematical Intelligencer Vol.7 No.2 1985, pp18-22.

JA

A. JACKSON: Interview with Henri Cartan, Notices of the AMS Vol.46 No.7 1999, pp782-788.

SM

M. SENECHAL: The Continuing Silence of Bourbaki- An Interview with Pierre Cartier, The Mathematical Intelligencer, No.1 1998, pp 22-28.

WA

A. WEIL: The Apprenticeship of a Mathematician, Birkhäuser Verlag 1992, pp 93-122.

Title

Bourbaki, Nicolas

Canonical name

BourbakiNicolas

Date of creation

2013-03-22 13:33:23

Last modified on

2013-03-22 13:33:23

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

17

Author

Daume (40)

Entry type

Biography

Classification

msc 01A60

Synonym

Bourbaki

Related topic

AlexanderGrothendieckABiographyOf

Generated on Thu Feb 8 19:10:17 2018 by

LaTeXML