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Nicolas Bourbaki
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(Biography)
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by Émilie Richer
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.
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.
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]
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.
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]
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.
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.
generation (founding fathers) |
generation (invited after WWII) |
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 |
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J.P Serre |
J. Tate |
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L. Shwartz |
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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),
,
,
,
and
(respectively the natural, real, complex, rational numbers and the integers) (complement of a set A), as well as the words bijective, surjective and injective. [DR]
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
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?
- 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.
- 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://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.
- 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.
Footnotes
- 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.
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"Nicolas Bourbaki" is owned by Daume.
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Cross-references: translation, weight, obvious, injective, surjective, bijective, complement, integers, rational numbers, complex, arrow, implication, empty set, current, reference, linearly-ordered, rigid, axiomatic, tight, NOR, permanent, volumes, average, reduced, product, subset, intersection, union, line, point, size, geometry, differentiable manifolds, polynomials, interpolation, Lie groups, non-commutative, number theory, commutative, vector, cardinals, ordinals, theory, multilinear, topological vector spaces, variable, real, functions, topology, algebra, similar, similarities, valid, divisions, unity, represent, set theory, foundation, strong, members, even, theorem, right, sufficient, conclusion, project, circle, eventually, class, Calculus, scattered, numbers
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This is version 14 of Nicolas Bourbaki, born on 2003-04-05, modified 2009-08-07.
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Classification:
AMS MSC: | 01A60 (History and biography :: History of mathematics and mathematicians :: 20th century) |
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