The Imaginary Mathematician Who Fixed Math

Nicolas Bourbaki never existed, but he’s been publishing work for over 80 years, making him one of the most important mathematicians of the last century. He was even baptized, threw his daughter a wedding, is known for his quirky humor, fought back against those that questioned his existence, and had an obituary published upon his alleged death.  

“A Dreadful Hecatomb of Young French Scientists”

When the smoke cleared after World War 1, European academia was in disarray. Many scholars had been recruited to the frontlines, where they faced disease, famine, and enemy soldiers. France was especially hit hard, with Aubin and Goldstein claiming that ”from the class entering in 1910, more than 6 out of 10 science graduates never came back from the front.” This left French academia fractured, convoluted, and stagnate for a generation.

Mathematicians in particular struggled for decades to rebuild their field. By the 1930s, lack of coordination in French mathematics—and much of the western world—produced different methods and terminology, and lack of consensus made writing a new textbook infeasible. In fact, a mathematics textbook had not been published since Goursat published his in 1904. The field was languishing and the future of mathematics was hazy. “There was a noticeable impasse and increasing uncertainty around the study of mathematics in France” claims Pieronkiewicz.

Jean Dieudonné thought “This showed a spirit of democracy and patriotism that we can only respect, but the result was a dreadful hecatomb of young French scientists.”

Fortunately, 5 mathematicians (Henri Cartan, Claude Chevalley, Jean Delsarte, André Weil, and Jean Dieudonné) formed a group to not only repair the fractured community but to set the curriculum for the next 2 decades. However, they didn’t realize they were setting in motion a multi-generational endeavor to rethink the foundations of mathematics. They also delighted and annoyed the rest of the mathematical community along the way. 

They would soon comically name their group after Charles-Denis Bourbaki, a French general famous for his spectacular failures during the Franco-Prussian War. Nicolas is thought by some to refer to St. Nicolas, perhaps suggesting the group knew the gifts they were about to bring to the struggling mathematics community.

“A Gathering of Madmen”

“They could not imagine how these people, shouting—some two or three at the same time—about mathematics, could ever come up with something intelligent.” –Jean Dieudonné

The founding members and recent addition, René de Possel, first met at Café Grill-Room A. Capoulade, a small cafe in Paris’ Latin Quarter. Their first project was to find a more rigorous method of handling Stokes’ theorem, a fundamental tool in higher dimensional calculus, as new methods were largely unknown to French mathematicians. Over a lunch of cabbage soup and grilled meats, they also agreed to work as a collective, to have regular conferences, and to incorporate the work of mathematicians from other countries, namely Germany.

In July of 1935, the first formal Bourbaki conference was held, with the addition of 3 well-known French mathematicians. It was at this meeting that they decided to publish all work under the name Bourbaki and to expand the scope of their work to reinventing other mathematical subfields, such as topology, set theory, abstract algebra, and Lie groups. That is, Bourbaki did not set out to invent new mathematics. Rather, they wanted to streamline, organize, create better definitions, incorporate modern techniques, and create a set of fundamental axioms which all of mathematics could be based on.

Jean Dieudonné claimed that “What Bourbaki has done is to define and generalize an idea which already was widespread for a long time.”

Shortly after the meeting, Possel’s wife baptized Nicolas Bourbaki, allowing the group to submit articles under his name. Their first article, “Sur un théorème de Carathéodory et la mesure dans les espaces topologiques,” was submitted and quickly accepted by the mathematics community. 

Within the next few years, the Bourbaki conference was being held 3 times a year and some of the continent’s most prominent mathematicians had gotten involved. The lucky people who have been invited to spectate these spirited and often chaotic meetings would “always come out with the impression that it is a gathering of madmen.” Dieudonné claimed there was no formal structure, and it was common for conferences to involve viscous, unprompted criticism from other members, some of which might be 20 or 30 years apart. He believed that the group offered criticism harsher than anything found on the outside.

In fact, the only official rule was that members had to retire by age 50 because older mathematicians, or so it was thought, would be unlikely to adapt new methods and ways of approaching problems.

These methods resulted in the group creating a plethora of work, which was eventually published as a collected volume in 1939 as Éléments de mathématique. In the following decades, this book became standard reading, expanded over time to 12 books with 6,000 pages, and has been republished numerous times, the last of which was in 2016.

However, in 1940, Germany invaded France, drastically slowing the group’s work. Some members were recruited into the military, while others fled Nazi persecution, leaving French mathematics again at risk of collapse.

“A Center From Which All The Rest Unfolds”

Fortunately, Bourbaki was able to resume its conferences shortly after the liberation of France in August of 1944, and over the next 75 years changed the course of mathematics.

Their greatest achievement was their insistence on rigor over conjecture, prompting some critics to claim Bourbaki sterilized mathematical research. With nothing but hard logic, they developed their own first principles and then reformulated the fields of mathematics. For example, they redefined functions, one of mathematics’ most basic tools. Functions are normally thought of as a machine that takes an input and produces an output. In the function f(x) = x + 1, plugging 1 for x outputs 2. Plugging in 15 produces 16.

Bourbaki, however, saw functions as a way to connect two groups, allowing for them to determine logical relationships between different types. Based on this, they defined the three types of functions: injective, surjective, and bijective. With simple and rigorous definitions such as these, they were able to build a different and more solid foundation.

They also created simplified and more concise definitions to make mathematics more approachable and easy to communicate between both mathematicians and non-mathematicians. Instead of the typical Latin and Greek based names and the abbreviation laden definitions, Bourbaki favored simplicity. Instead of “parallelotopes,” they used “paving stones.” Instead of “hyperspheroids” they used “balls.” Jean Dieudonné said “We think that ink is cheap enough to write things in full, with a well-chosen vocabulary.” 

They also invented new symbols, such as the empty set and the dangerous bend, both of which are still popular today.

Overall, Bourbaki’s purpose was to create a logic-based set of axioms to be the foundation of all mathematics. Jean Dieudonné called this foundation “a center from which all the rest unfolds.”

They also had a lot of fun with their antics over the years. In 1948 and 1950, they formally applied for the American Mathematical Society, but they were rejected as the president was already aware of the group. Some time in the 1950’s Bourbaki defended themselves against Ralph Boas, who wrote an entry for the Encyclopedia Britannica debunking the person-hood of Bourbaki. In fact, they accused Boas himself of being fictional. In 1968, the group killed off Bourbaki in an obituary. Throughout the years, Bourbaki also sent telegrams, greeting cards, suffered from sicknesses, etc., not to mention the many papers published under that name, all in an attempt to make him seem like a real person.

“A Ball of Wool, a Tangled Hank”

The legacy of Bourbaki extends far beyond what the founding members ever thought. Not only are their methods still influencing mathematicians, their terminology and symbols still in use, and Éléments de mathématique still in publication, their work extends into philosophy, psychology, anthropology, among other fields. In particular, Bourbaki’s work helped form the basis of structuralism, a budding movement that pushed for all fields to be reduced down to their elemental pieces, from which more complex structures can be created.

Jean Dieudonné explained that all of mathematics was like “a ball of wool, a tangled hank,” in that pulling a string even a tiny bit caused the entire ball to change. He said this in reference to Bourbaki’s influence over the whole of mathematics, but when he said this in 1968, he didn’t realize that the ball was much bigger and that the influence of the group was felt across all of academia.

Pretty good for a person that never existed.

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