Any Marvel fan – or even most people, really – will know the Avengers – six superheroes who teamed up to save the world. Mathematics, on the other hand, always has the image of being a rather solitary affair. We have written about a modern-day mathematical Polymath Project on the Twin Prime Conjecture in Issue 020, but these collaborations have been happening for a while – almost a whole century ago, in fact.

Meet Nicolas Bourbaki.

Assuming the name of a French general who fought in the Franco-German war (1870-1871), Nicolas Bourbaki, founded 1934-1935, was in fact a collection of mathematicians writing collectively. The name was chosen in jest (founders of the group had a distaste towards hierarchy), and humor is a running theme throughout the group. The group’s founders were all French [1] – although later incarnations involved mathematicians of other nationalities; they were admirers of the German mathematician David Hilbert, and the German school’s emphasis on precision and rigor of proofs, as opposed to the French school’s more handwavy approach based on intuition.

The group’s membership would change throughout the ages – there was an agreement that the current members would be kept secret, although former members discuss their involvement with the group openly. This secrecy is to allow the mathematicians to write as a collective without any individual egos involved and no chance to claim copyright of any sort. Most members agreed to gradually drop out after they turn 50, letting the younger generation take over [2]. In their place, new members are recruited through invitation to their annual conference as “guinea pigs”, and are accepted into the group once all the mathematicians agree to let them join [3].

Bourbaki’s main work was a textbook series on mathematics, known as the Éléments de mathématique (Elements of Mathematics). Originally wanting to write a new textbook for differential calculus [1], the group eventually covered the major areas of mathematics – analysis, algebra and geometry, as well as including more modern areas of active research in later volumes. The group was drawn to abstraction rather than concrete illustration; most of the core six texts (those published before 1954) consist of theorems and proofs written out formally, with little diagrams or examples to illustrate the ideas. The group also wrote articles and hosted a lecture series known as Séminaire Bourbaki (Bourbaki Seminar) in Paris since 1948, adding up to over a thousand lectures as of today.

One of Bourbaki’s main activities was its periodic conferences, usually held in a rural location, where its members met to draft content on certain topics for the Elements. Members would then present those drafts in front of everyone, and the material must be agreed upon unanimously before it can be published under the Bourbaki moniker. As a result, many drafts took years to complete, but they were also guaranteed to be mathematically rigorous after being vetted by all of its members. At these conferences, there were also lively debates and often heated disagreements; Laurent Schwartz reported that André Weil once slapped Henri Cartan on the head with a draft [3]. But almost miraculously, peace was restored within ten minutes, and the group was able to stick together. The group’s very strong belief in the culture of collaboration and lively debate held them together; they would stay together despite personal differences on certain issues, which made Bourbaki able to change its views on certain issues such as which topics to write about, sometimes in a completely opposite direction [3, 4]. As Claude Chevalley discusses in an interview [1], “One never feels like he is talking to a wall when talking to Bourbaki.”

Bourbaki was most active in the middle of the 20th century, during the 1940s and 50s; only one volume of the Elements has been published in the 21st century, on algebraic topology. Nevertheless, the group leaves a sizable legacy on present-day mathematics, especially on symbols used in set theory [5, 6]. The group also had an influence on French structuralism, which prioritizes the study of the relationships between structures before the study of the structures themselves; for example, cultures should be studied in the context of other cultures they exploited in becoming their present day state.

Some also blame Bourbaki for the snobbery towards applied mathematicians in France ever since their foundation, as many see pure mathematics as more superior than applied mathematics. The group’s goal of making mathematics more formal and abstract has made French mathematics students and mathematicians prioritize pure mathematics over applied mathematics, as the former is often expressed in symbols and abstract arguments, while the latter is largely guided by more concrete explanations. Benoit Mandelbrot, a former member who was a geometer, even went as far as to emigrate to the United States in part to escape Bourbaki’s influence in France [3]; in interviews, he has stated that the group was very much against geometry since the field often relied on pictures and diagrams in explanations; the members also scorned against any other mathematicians who disagreed with them on this issue. There was also an air of superiority within the group, with some of the group’s members thinking that they were better than other mathematicians [1, 7].

However, it is undeniable that Bourbaki had a sizable influence on mathematicians, both French and further afield; the Elements remains an encyclopedia of sorts for a number of mathematicians, with certain volumes becoming the standard text for that area [3, 4, 8]. Furthermore, despite all the controversy, perhaps their culture of lively debate and collaboration is one we can all learn from – some of the best work can only be produced after an argument.

References:

[1] Guedj, D. (1985). Nicholas Bourbaki, Collective Mathematician: An Interview with Claude Chevalley. Mathematical Intelligencer, 7(2), 18–22. doi:10.1007/BF03024169

[2] Senechal, M. (1998). The Continuing Silence of Bourbaki: An Interview with Pierre Cartier, June 18, 1997. Mathematical Intelligencer, 20, 22–28. doi:10.1007/BF03024395

[3] Mashaal, M. (2006). Bourbaki: A Secret Society of Mathematicians (A. Pierrehumbert, Trans.). American Mathematical Society.

[4] Borel, A. (1998). Twenty-Five Years with Nicolas Bourbaki, 1949–1973. Notices of the American Mathematical Society, 45(3), 373–380.

[5] Bourbaki, N. (2004). Theory of Sets. Heidelberg, Germany: Springer.

[6] Miller, J. (n.d.). Earliest uses of symbols of set theory and logic. Retrieved from https://jeff560.tripod.com/set.html

[7] Corry, L. (2009). Writing the Ultimate Mathematical Textbook: Nicolas Bourbaki’s Éléments de mathématique. In E. Robson, & J. Stedall (Eds.). The Oxford Handbook of the History of Mathematics (pp. 565–587). Oxford, UK: Oxford University Press.

[8] Aczel, A. D. (2006). The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. New York, NY: Thunder’s Mouth Press.