When observing the recent past of mathematical history, you encounter the name Nicolas Bourbaki quite often. The reason for this is quite simple, actually. Nicolas Bourbaki, born 1935, might be the author of the most math books in history. More importantly, he has written books ranging from mathematical history to algebraic topology, and even calculus. Where it becomes mysterious, however, is that not a single person has seen him. That is because no one by the name of Nicolas Bourbaki has ever lived.
In 1914, a state of emergency was declared in France and nearly everyone was drafted into the military. The consequent blow to the education system of high schools and universities was massive. Students in France were not getting the education they needed and therefore mathematicians were not being raised. Despite this, however, renowned mathematicians like Fermat and Poincare were raised.
In Germany, the opposite was true. Even though the war raged on, brilliant mathematicians like David Hilbert, Helmut Hasse, Emmy Noether, Wolfgang Krull, Emil Artin, and Bartel Leendert Van der Waerden were brought up and learned the fundamental study of math, Algebra.
A group of patriotic mathematicians, who at the time studied at École normale supérieure, decided that France had fallen too far behind Germany and set out to solve the problem through a series of seminars. In time, they started to think more universally and came to the conclusion that Mathematics as a whole must be rewritten. For example, they did not like the Calculus book they were reading, so they decided to rewrite it.
The best way to express their thoughts on modern mathematics was through writing books. However, they decided that it would be best to write as one author, rather than write their individual books. Their most fundamental book, which they frequently revised, Elements de Mathematiques, would leave a massive imprint on contemporary mathematics.
So there it is; the start of Nicolas Bourbaki’s legend, a secret organization. At first, it was only Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonne, and Andre Weil.
They gave themselves three years to publish their first book. For their first book, they relied heavily on German mathematician Van Der Waerden’s famous Algebra book. In one of his pieces, Jean Dieudonne writes about this:
I remember it — I was working on my thesis at that time; it was 1930 and I was in Berlin. I still remember the day that Van der Waerden came out on sale. My ignorance in Algebra was such that nowadays I would be refused admittance to a university. I rushed to those volumes and was stupefied to see the new world which opened before me. At that time my knowledge of algebra went to no further than mathematiques speciales, determinants, and a little on the solvability of equations and unicursal curves. I had graduated from the Ecole Normale and I did not know what an ideal was, and only just knew what a group was! This gives you an idea of what a young French mathematician knew in 1930. So we tried to follow Van der Waerden, but in effect he only covered algebra, and even then just a small part of algebra.
They published their first book under the name Nicolas Bourbaki in 1939 under the address, “Fellow of the Royal Poldavya Academy, currently living in Nancy.” However, there was neither a place named Poldavia nor an academy named Royal Poldavia Academy. The reason for choosing the city of Nancy was because all of their members had graduated high school there. More interestingly, in the town square of Nancy resides a statue of a general named Bourbaki.
Nicolas Bourbaki soon gained recognition. Everyone started wondering who Bourbaki was. Because of the works he published, many started to believe Bourbaki to be a group with more than a thousand members. In reality, however, it was no more than twenty young mathematicians. Bourbaki had started a movement that left its mark in the twentieth-century mathematics trend.
The Bourbaki’s have written more than forty books, about a plethora of topics. While each may seem a simple textbook, they are all quite complex. You cannot simply give a student them and say “read this and learn mathematics.” How were these books written, then? Did each member pick a separate topic and write about it?
The Bourbaki’s followed a very rigorous procedure before publishing their books. Two to three times a year, they would meet and discuss the topic of their next book, which they would then make the responsibility of one of their members. The mathematician would then spend one to two years preparing it and present it at their next meeting. The entire book would be criticized, sometimes painstakingly, sentence by sentence. This process meant that each book would take on average eight to twelve years to publish. One time, Jean Dieudonn published an article by the name Nicolas Bourbaki. However, when he noticed a mistake in it, he published another article under his own name, titled “On One of Monsieur Bourbaki’s Mistakes”, effectively correcting himself.
In the fifties, Nicolas Bourbaki applied to the American Mathematics Society under the name Professor Bourbaki. In their letter, they listed all of the publishings of Bourbaki and even paid the yearly due for the society. However, the society rejects Nicolas Bourbaki, who not a soul had seen. Afterward, dozens of mathematical magazines published their disagreement with the application’s rejection. In a statement on behalf of the society, famous mathematician Ralph Boas says this, “Bourbaki does not exist as a person, it is but an alias. We only give membership to real people, therefore we cannot make Bourbaki a member.”
In response to this statement, the Bourbaki’s quickly released an article in a French magazine saying:
“Ralph Boas does not exist as a person, it is a group of American mathematicians using an alias, an anonymous name. All of the mathematics Boas has made is the making of this twenty-person group.” Interestingly, there are even those who believe this.
Bourbaki’s most important rule was that they did not accept any mathematician over the age of fifty into their group. Once a member reaches fifty, their membership would be withdrawn and their authority revoked. They believed that someone aged fifty would have a hard time adapting to the ideas of someone in their twenties or thirties.
In 1968 the group could no longer rely on publishings and decided to disband. It was published in newspapers that Nicolas Bourbaki had passed away.
If you want to learn about Nicolas Bourbaki you can read Amir D. Aczel’s book, The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. Or you can watch the beuatifully illustrated video about him.
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