The philosophy of modern mathematics (structures) that swept through American and foreign education systems in the middle of the century was led by the great mathematician and artist Nicolas Bourbaki — who never lived.
Nicolas Bourbaki was one of the influential mathematicians of the 19th century that originated the modern concept of mathematical proof, authored many textbooks, and published dozens of academic papers in international journals.
When that great mathematician applied to the American Mathematical Society in the 1950s, the application was rejected for one reason — Nicolas Bourbaki did not exist.
Who was Nicholas Bourbaki?
Nicholas Bourbaki was not a person but a pseudonym chosen for a secretive math group founded by nine great mathematicians in France in the mid-1930s. Henri Cartan, André Weil, Szolem Mandelbrojt, Frenchmen Claude Chevalley, and Jean Dieudonne were some of the active founders who collectively wrote many academic papers under the Bourbaki pseudonym to represent the essence of a “contemporary mathematician.”
The founders were the admirers of the German mathematician David Hilbert and agreed to retire from the group at age 50. Then the group was driven by new recruits.
That real-life mathematical mystery Nicholas Bourbaki was responsible for the emergence of the new math called “Structural Mathematics.”
According to Nicolas Bourbaki,
“Structures are the weapons of the mathematician.” -
No mathematician in today’s world is free of the influence of Nicolas Bourbaki’s seminaire work.
The Emergence of Nicolas Bourbaki
Two decades ago, mathematics was in chaos. Several practiced mathematicians had lost their lives in the First World War, and the domain had become fragmented. Various branches used a different methodology to seek their own goals. There was a lack of a shared mathematical language, which made it challenging to expand their work.
In 1934, a group of French mathematicians was exhausted. While studying at the École normale supérieure, they discovered the textbook for their calculus class and chose to produce a better one. The small group soon onboarded new members. As the project unfolded, so did their goal.
The outcome was the “Éléments de mathématique,” a treatise that endeavored to create a consistent logical framework uniting every branch of mathematics.
The text opened with a set of simple axioms — laws, and assumptions to build its argument. But to find common ground, the group aspired to discover consistent rules that implemented a broad range of problems. To achieve this, they provided new, clear definitions of some of the most important mathematical objectives, including the function. The group defined functions by how they mapped elements across domains.
Important Contributions Under Bourbaki’s Name
The band of great mathematicians wanted to revolutionize the field of mathematics. They decided to publish a plethora of research under Bourbaki’s name.
The group introduced:
- Éléments de mathématique
- The null set symbol
- Functions, axioms, and ubiquitous terms injective, surjective, bijective
- Generalizations of theorems, such as the Bourbaki-Witt theorem, the Jacobson-Bourbaki theorem, and the Bourbaki-Banach-Alaoglu theorem.
- Elements of Mathematics that swelled to above 6,000 pages and formed a solid foundation for modern mathematics in terms of basic structural components.
Nicolas Bourbaki described “The Architecture of Mathematics” as a hierarchy of structures as follows:
“From the axiomatic point of view, mathematics appears as a storehouse of abstract forms — the structures; and it so happens — without our knowing how — that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation.”
The Apparent Death Of Nicolas Bourbaki
Across the next two decades, Bourbaki’s papers became standard references. Their invented mathematician “Nicolas Bourbaki” claimed to be a recursive Russian genius who would only engage with his chosen partners. They forwarded telegrams under Bourbaki’s name, announced his daughter’s wedding, and openly mocked anyone who questioned his presence. In 1968, when they could no longer uphold the trick, the group ceased their joke by printing Bourbaki’s obituary, concluded with mathematical puns.
Despite his apparent death, the group exhibiting Bourbaki’s name lives on today.
Though he’s not affiliated with any particular major discovery, Bourbaki’s influence apprises much current research. And the current emphasis on formal proofs owes a sweeping deal to his meticulous methods.
Nicolas Bourbaki may have been imaginary — but the legacy is real.