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 Elie Cartan 
 1869-1951
Elie Cartan  (1869-1951)

The Many Faces of
Nicolas  Bourbaki
© 2000-2023   Gérard P. Michon, Ph.D.

Bourbaki  recognized only one godfather, Elie Cartan.
Pierre Cartier (b.1932)
The Continuing Silence of Bourbaki 

[We witness]  a monumental exposition of the whole of present day mathematics whose framework won't become obsolete in the near future.  It makes visible the common bond between the various branches and can easily absorb new ideas.
(Edited excerpt; Bulletin of the AMS 59, 1953 p.474)  Emil Artin (1898-1962)

Bourbaki was never meant to present especially interesting mathematics...
It was a service to provide others with useful tools to do interesting things.

 Jean-Pierre Serre (2019-06-30).

 Michon
 

Related articles on this site:

Related Links (Outside this Site)

Nicolas Bourbaki  by  Mike Conlay  (1995-06-11).
Association des Collaborateurs de Nicolas Bourbaki (ENS, 1952-).
Bourbaki and the Foundations of Modern Mathematics (CNRS, 2017-08-14).
25 Years with Nicolas Bourbaki  (1949-1973)  by  Armand Borel.
Nicolas Bourbaki   |   Nicolas Bourbaki  at  www.absoluteastronomy.com.
Bourbaki:   The pre-war years  &  The post-war years   at  MacTutor.
Henri Cartan, interviewed in 1999  (notices of the AMS).
Climbing Mount Bourbaki  by Harvard student  Akhil Mathew  (2010).
Mathematicians Who Never Were  by  Barbara Pieronkiewicz  (2018).
 
Wikipedia   |   Wikipedia Category   |   Alumni of ENS

The Case of the Disappearing Mathematician (24:05, audio)  by  Amir Aczel  (Ira Flatow, NPR, 2006-11-17).
 
Bourbaki's golden age, 1945-75 (1:35:01, French)  J-P. Serre, P. Cartier, J. Dixmier & A. Connes  (2019).
 
Bourbaki, Categories and Structuralism (47:36)  by  Jean Pierre Marquis  (2019-12-08).

 
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The Many Faces of  Nicolas Bourbaki

The French  Ecole Normale Supérieure  (ENS)  has a tradition for pranks.  One such prank was a  fake lecture  given in 1923 to the 21 entering students  (including  Paul Dubreil  and future Bourbakists  Henri Cartan,  Jean Coulomb  and  René de Possel)  by  Raoul Husson  (1901-1967, ENS 1921)  an older peer wearing a false beard to impersonate a distinguished venerable mathematician billed as  "professeur Holmgren".
 
The lecture presented a series of fake results from fictional mathematicians named after famous French generals,  culminating in  Bourbaki's theorem  (named after General  Charles Bourbaki, 1816-1897).  That's how some future ENS alumni took note of the Bourbaki name.
 
According to André Weil  (1992)  the moniker  Nicolas  was picked by his future wife  Eveline de Possel,  who was then still married to fellow founder  René de Possel  (1905-1974).


One of the most famous mathematics textbooks of all time was  Cours d'analyse mathématique  by  Edouard Goursat (1858-1936; ENS 1876).  The first edition was published in two volumes  (1902,1905).  The second edition was published in three volumes  (1910-1913)  which were later updated once or twice  (most recently in 1924, 1925 and 1923, respectively).

Goursat is credited to be the first person who noticed how  Stokes' theorem  could be very beautifully generalized using  differential forms,  as shown below.

Stokes' theorem actually generalizes the fundamental theorem of calculus.  It was first proved by  Poincaré (1899)  well before  Georges de Rham (1903-1990)  put forth a conceptual frame  (1931).  The need for a full elucidation was such a key preoccupation of some Bourbaki founders that it's not an exaggeration to state that  Nicolas Bourbaki  and this general result are significantly  due to each other...

 Border of a Surface
òW   dw     =     ò¶W   w
 Boundary of a Volume

Some Incarnations of  Stokes' Theorem
Gradient Conservativity
ò  b
a 
  grad f . dM     =     f(b) - f(a)
Kelvin-Stokes' Formula òòS   rot U . dS     =     òC   U . dM
Ostrogradsky's Theorem òòòV   div U  dV     =     òòS   U . dS

The birthdate usually quoted for the group which would become known as  Nicolas Bourbaki  is the date of their  second  formal meeting  (1935-01-14).  They had already met one month earlier  (1934-12-10)  and several other preliminary meetings had not been formally  recorded  at all.  The prospective founders were talented young French mathematicians who were all alumni from  ENS,  with the sole  exception  of  Szolem Mandelbrojt (1899-1983).

The first  Congrès Bourbaki  took place from 10 to 20 July 1935 at the  Station biologique universitaire  in the medieval city of  Besse  in central France  (called  Beça  in Occitan)  which was recognized as a  free town  in may 1270 and officially named  Besse-en-Chandesse  between 1961-04-02 and 1973-07-01,  when it merged with  Saint-Anastaise  to become part of  Besse-et-Saint-Anastaise.  The name  Besse-en-Chandesse  is still used on some road signs, road maps and touristic guides.  It's where the ski resort marketed as  Super Besse is located.  On  2003-07-12,  a plaque was unveiled outside the building housing the meeting room identifying it as the  birthplace of Nicolas Bourbaki.  That event followed a  conference  by  Christian Houzel, attended by a crowd of about 120 mathematicians.  It was the brainchild of  Paul-Louis Hennequin (Ulm, 1949)  on behalf of the mathematical department of the  Blaise Pascal University  which had been owning the building since 1901.  Hennequin is the son of  Aimé Hennequin,  from  Taupe Laplace.

Their initial ambition was to put together  (within six months)  a much-needed  Analysis  textbook to replace Goursat's  Cours d'Analyse.  They already had a preliminary agreement with  Hermann  to publish it.

As  André Weil  would later recall,  one key point was the aforementioned urge to elucidate the general validity of the  above formula  (Stokes Theorem)  which unifies great results of vector calculus.  However,  that sparked a search for rigorous settings which would delay by many years the publication of the final presentation by Bourbaki of that particular topic...

Eventually,  they ended up putting on a fresh solid foundation the entire mathematical knowledge of their time.  This took the form of a collection of books entitled  Eléments de Mathématique  (note the militant use of the grammatically incorrect singular form of  mathématiques).

The  Bourbaki  collaboration has been extremely influential in France and elsewhere.  Bourbaki brought about new rigor based on the  logical foundations  of mathematics  (along the way, they also became instrumental in some controversial reforms of mathematical education, dubiously known as  New Math  in the US).

This  can be traced to a 1952 meeting  (in  Melun,  France)  between Swiss psychologist  Jean Piaget (1896-1980),  Bourbakist  Jean Dieudonné (1906-1992)  and two other leading mathematicians concerned with education reform,  Gustave Choquet (1915-2006)  and  André Lichnerowicz (1915-1998).

The  active  founding members of the  Nicolas Bourbaki  group were: 

The official list of founders includes four members who were less active, namely:

Szolem Mandelbrojt  was the uncle and early mentor of the mathematician Benoît Mandelbrot (1924-2010; X1944) of  fractal  fame.

Two other mathematicians had been present at preliminary meetings, before the actual foundation of the Bourbaki group:

Thus,  the number of Bourbaki founders can be argued to be  5,  9  or  11.   Wink

Other noted  bourbakists,  who joined the group later, include:

The guideline is that all members should retire from the group at the age of  50  (Grothendieck and Lang left early, in anger).  Almost all of the above are retired.

It was never a strict rule,  but the level of enforcement is difficult to evaluate because the exact duration of a member's tenure was rarely made public.  Samuel Eilenberg (1913-1988)  prophet of  Category theory,  retired at 53.
 
Rumor has it that a  coconutization  procedure was used,  inspired by an urban legend about tropical tribes:  When the fitness of the chief is questioned,  he has to climb on top of a coconut tree and the whole tribe attempts to shake him down.  If he doesn't fall,  then he's deemed to be still fit for his functions.
 
The mathematical counterpart of that is to throw some utter nonsense during a meeting.  If the unsuspecting target of the coconutization fails to detect it,  then the age-rule will be enforced and he won't get invited to the next meeting.  Nobody has to be rude about it  (kinda).  This was  told  by  Vladimur Arnold (1937-2010)  of  KAM  fame,  who was never a Bourbakist himself.  I heard the same story from a  good friend  who's a former classmate of  Alain Connes.    Salut, Denis!

The semi-mandatory retirement rule was instigated by  Weil  as the founding members were near the critical age:  A letter from him proposing a  gradual disappearance of the founding members  was read by  Cartan  during the Summer conference of 1956,  at the end of a lunch for  Dieudonné's  fiftieth Birthday  (1956-07-01).  It was a drastic way to incite the more recent members to take on full responsibility.  A  necessary condition  for the group's planned immortality.

According to Jean Dieudonné  (interviewed on 1987-06-12)  one of the motivations for the age rule was that many mathematicians seem overly enamored with what they learned when they were young.  The founders wanted the group to remain forever receptive to new ideas.

Retired members were not entirely cut off from active members as they continued to receive the internal newsletter  La Tribu  (The Tribe)  containing the proceedings of the meetings.  (Tribe  is what the original group liked to call itself.)

Pierre Cartier  said that, during his own tenure from 1955 to 1983, the Bourbaki group was holding three yearly meetings  (for a total of about one month per year).

Part of the preliminary recruitment of new Bourbaki members was to invite them to one of those meetings.  Sir  Michael Atiyah  went to a single meeting as a prospect but decided he didn't like the Bourbaki working style.  Only one person who was not formally a member of Bourbaki ever took part in writing up pieces which were eventually published under the Bourbaki name  (after many revision by several members, as usual):  Paul-André Meyer (1934-2003; ENS 1954).  This case is so unique that some notorious Bourbaki members thought he had been formally inducted,  but this ain't so  (video).

The  (nonprofit)  Association  formally incorporated in 1952 seems alive and well,  although it's not nearly as active and/or influential as it once was.  At one point,  there were concerns that the volumes published in 1983 and 1998 would be the last ones of the collection.  However,  one major update was published in 2016.

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Opinions about Nicolas Bourbaki


The hope was that a fairly small simple set of basic structures were the basis of most research.  I think it is also true that, for a while, this was generally believed.  But for the last 20 years, it has been increasingly doubted:  First because Bourbaki began to drown in its own need to be general enough, and they never could be sure when to stop  (e.g., before doing the reals, they need a general theory of topological fields).  Secondly, because math began to be driven by complex theories rather than simple ones  (e.g., the Langlands conjecture at the abstract end, control theory and probability at the applied end).  These theories don't benefit much from the Bourbaki.

David B. Mumford  (b. 1937, Fields medal 1974)  1993-04-21

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Two Glitches in the Popularity of Nicolas Bourbaki


At this writing  (2018-12-08)  this page consistently ranks in Google's top page for queries related to  Nicolas Bourbaki.  Its average rank has been 6.3 over the last three months spanned by the plot below,  where the light-blue curve  (scale at right)  indicates the number of daily  impressions  of whatever Google result-page  (usually the top one)  mentions the page you're now reading.  The dark-blue curve  (scale at left)  gives the number of daily visits our page received from Google.

The latter closely follows the former,  with noise due to the varying location within Google's search results  (on most days, our rank was between 5 and 8).

Either curve is a good way to monitor the number of enquiries about Bourbaki in Google.  That was less than 100 per day before the incident diagnosed below,  which caused a hundredfold peak  (and a tenfold increase in the aftermath).

 Popularity of Nicolas Bourbaki

Whatever happened in late September  (see below)  caused the peak of  11836  impressions observed on Sunday, October 4, 2018  (157 visits to the page you're reading,  whose average position was  5.6  that day).  At first,  I had hoped this had been the result of some nice mathematical online discussion.  Not so.  It turns out that the name of  Nicolas Bourbaki  was just used by songwriter  Tyler Joseph (1988-)  in the third track  (entitled  Morph)  of the  Trench  album  officially released on October 5,  2018  by his  Twenty Øne Piløts  band/duo.

Tyler's  rap lyrics  just pertain to his so-called  Dema  dreamworld,  haunted by  bishops  he named  André  [Weil?]  and  Nico.  He had previously called the latter  Blurryface  before identifying him as  Nicolas Bourbaki.  The  Twenty Øne Piløts  publicity campaign spread doctored bits of iconography from the  original Bourbaki tribe  to send fans on a  wild-goose chase,  which they seem to enjoy.

In the wake of the above,  one such fan produced the following two videos,  which are tedious to watch for those of us who aren't interested in Tyler's ramblings.  At least,  this lady can boast some of the scientific training which Tyler lacks.    Wink

Nearly two years later  (July 2020)  the effectivenes of this page to monitor Bourbaki-related Google searches was hampered by an average rank of 9.4, which no longer puts it consistently on the first page (top 10) thus missing a majority of those searches.  That was still good enough to detect another glitch:

 Popularity of Nicolas Bourbaki

On Sunday July 5, 2020, the observed number of Bourbaki queries was only 112  (the actual number is probably three times larger, for the aforementioned reason).  The next day (Monday, July 6) this rose to 6683.  Then 7188 on Tuesday, 2219 on Wednesday, 2206 on Thursday, 1789 on Friday, 1348 on Saturday, 1104 on Sunday, 988 on Monday July 13.  248 a week later.  242 after another week  (stabilizing to about 150% of the pre-peak level thereafter). So,  what happened on July 6?

Well,  that's the day a well-meaning animated video was released on YouTube:

The greatest mathematician that never lived (5:12)  by  Pratik Aghor  (TED Ed).

The echoes were perfunctory,  resembling one by  Steven Pomeroy (2020-07-07).

Not Even Bullshit  by  Joachim I, KruegerBrown University  (Psychology Today, 2015-12-06).

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A Few Bourbakists :


Samuel Eilenberg
Born: 30 September 30, 1913, in Warsaw  (now Poland)
Died: 30 January 1998, in New-York, NY  (USA)

Topologist.  He invented Category Theory in 1945, with Saunders Mac Lane (1909-2005).

  Sammy Eilenberg
Sammy Eilenberg

 Fields Medal Laurent Schwartz  
Born: 5 March, 1915, in Paris  (France)
Died: 4 July 2002, in Paris  (France)
 Laurent Schwartz
Conceived the  Theory of Distributions  in 1944.  Fields Medalist in 1950.  Noted professor at  Polytechnique,  he taught this writer  Hilbertian Analysis,  in the Fall of 1977.

Laurent schwartz (1915-2002) et la vie collective des mathématiques
by  Anne-Sandrine Paumier  (Doctoral dissertation, Paris VI, 2014-06-30).


 Jean-Pierre Serre    Fields Medal Born: 15 September 1926, in Bâges

Youngest Fields Medalist, in 1954.
First Abel Prize recipient, in 2003.

Britannica   |   MacTutor   |   Wikipedia   |   Collège de France
 
Cohomologie Galoisienne (1:01:52)  Colloque Jean Dieudonné  (Nice, 1996-01).
 
Serre about Grothendieck,  interviewed by  Connes  (Fondation Hugot, 2018-11-27).
 
Interview (55:11)  avec  Jean-Louis Colliot-Thélène  (CIRM, 2015-11-05).


 Alexandre Grothendieck    Fields Medal Born: 28 March, 1928, in Berlin
Died: 13 November, 2014, in Saint-Girons (Ariège).
Once stateless, he carried a UN "Nansen" passport.  He was the most prominent champion of  Category Theory.

Fields Medalist in 1966.  Professor at the French IHES.
Declined the  Crafoord Prize  in 1988.

The Grothendieck Circleby  Leila Schneps.   |   SGA   |   NUMDAM
 
Souvenir (6:53)  by  Michel Demazure  (2015).


Pierre Cartier   Pierre Cartier
Born: 10 June, 1932, in Sedan  (Ardennes, France)

Wikipedia   |   Cartier divisors   |   Cartier-Manin operator   |   Ph.D. 1958 (Cartan & Weil)
 
Pierre Cartier (1:26:18)  interviewed by  Stéphane Dugowson  &  Anatole Khélif
(produced by  Laurence Honnorat, 2014-05-25).
 
Interview (1:08:30)  by Sylvie Paycha  (2014-12-04)
 
Hilbert et l'axiomatisation de la Physique (1:37:48)  Pierre Cartier  (ENS Ulm, 2019-01-14).


 Fields Medal Alain Connes
Born: 1 April, 1947, in Draguignan (Var, France)
Fields Medalist in 1982.
 Alain Connes

www.alainconnes.org   |   Ph.D. 1973 (under Jacques Dixmier)
 
Alain Connes (1:05:59)  by  Stéphane Dugowson  &  Anatole Khélif  (IHES, 2014-02-05).
Quantum Emergence of Time (58:37)  (IHES, 2015-04-09).
Face à la réalité mathématique (7:03)  CdF  (2014).
Parcours d'un mathématicien (1:25:10)  (SAPT, 2017-12-24).


 Jean-Christophe Yoccoz    Fields Medal Born: May 29, 1957, in France
Died: Sept. 3, 2016, in France

Fields Medalist in 1994.

Wikipedia   |   Mathematics Genealogy   |   MacTutor   |   Video (2015)

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Fields Medalists

1936 :

  • Lars Ahlfors (Finland).
  • Jesse Douglas (United States).

1950 :

1954 :

1958 :

  • Klaus Roth (United Kingdom).
  • René Thom (France).

1962 :

  • Lars Hörmander (Sweden).
  • John Milnor (United States).

1966 :

1970 :

  • Alan Baker (United Kingdom).
  • Heisuke Hironaka (United States).
  • Sergei Novikov (USSR).
  • John G. Thompson (United States).

1974 :

  • Enrico Bombieri (Italy).
  • David Mumford (United States).
 Pierre Deligne

1978 :

  • Pierre Deligne (Belgium).
  • Charles Fefferman (United States).
  • Gregori Alexandrovitch Margulis (USSR).
  • Daniel Quillen (United States).

1982 :

1986 :

  • Simon Donaldson (United Kingdom).
  • Gerd Faltings (West Germany).
  • Michael Freedman (United States).

1990 :

  • Vladimir Drinfeld (USSR).
  • Vaughan F. R. Jones (United States).
  • Shigefumi Mori (Japan).
  • Edward Witten (United States).
 Jean Bourgain

1994 :

  • Jean Bourgain (Belgium).
  • Pierre-Louis Lions (France).
  • Jean-Christophe Yoccoz (France).
  • Efim Zelmanov (United States).

1998 :

  • Richard E. Borcherds (UK).
  • William T. Gowers (United Kingdom).
  • Maxim Kontsevich (Russia).
  • Curtis T. McMullen (United States).

2002 :

2006 :

2010 :

2014 :

2018 :

Abel Laureates

2003 :

2004 :

2005 :

2006 :

2007 :

2008 :

2009 :

2010 :

2011 :

2012 :

2013 :

2014 :

2015 :

2016 :

2017 :

2018 :

2019 :

2020 :

2021 :


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