When Nicolas Bourbaki applied  to the American Mathematical Society in the 1950s, he was already one of the most influential mathematicians of his time. He’d published articles  in international journals and his textbooks were required reading. Yet his application was firmly rejected  for one simple reason— Nicolas Bourbaki did not exist. Two decades earlier,  mathematics was in disarray. Many established mathematicians had lost  their lives in the first World War, and the field had become fragmented. Different branches used disparate  methodology to pursue their own goals. And the lack of a shared  mathematical language made it difficult to share  or expand their work. In 1934, a group of French mathematicians  were particularly fed up. While studying at the prestigious  École normale supérieure, they found the textbook  for their calculus class so disjointed that they decided to write a better one. The small group  quickly took on new members, and as the project grew,  so did their ambition. The result was  the "Éléments de mathématique," a treatise that sought to create  a consistent logical framework unifying every branch of mathematics. The text began  with a set of simple axioms— laws and assumptions it would use  to build its argument. From there, its authors derived  more and more complex theorems that corresponded with work  being done across the field. But to truly reveal common ground, the group needed to identify  consistent rules that applied to a wide range of problems. To accomplish this, they gave new,  clear definitions to some of the most important  mathematical objects, including the function. It’s reasonable to think of functions  as machines that accept inputs and produce an output. But if we think of functions  as bridges between two groups, we can start to make claims about  the logical relationships between them. For example, consider a group of numbers and a group of letters. We could define a function where  every numerical input corresponds to the same alphabetical output, but this doesn’t establish  a particularly interesting relationship. Alternatively, we could define a function  where every numerical input corresponds to a different  alphabetical output. This second function sets up  a logical relationship where performing a process on the input  has corresponding effects on its mapped output. The group began to define functions by how they mapped elements across domains. If a function’s output came  from a unique input, they defined it as injective. If every output can be mapped  onto at least one input, the function was surjective. And in bijective functions, each element  had perfect one to one correspondence. This allowed mathematicians to establish  logic that could be translated across the function’s domains  in both directions. Their systematic approach  to abstract principles was in stark contrast to the popular  belief that math was an intuitive science, and an over-dependence on logic  constrained creativity. But this rebellious band of scholars  gleefully ignored conventional wisdom. They were revolutionizing the field,  and they wanted to mark the occasion with their biggest stunt yet. They decided to publish  "Éléments de mathématique" and all their subsequent work  under a collective pseudonym: Nicolas Bourbaki. Over the next two decades, Bourbaki’s  publications became standard references. And the group’s members took their prank  as seriously as their work. Their invented mathematician claimed  to be a reclusive Russian genius who would only meet  with his selected collaborators. They sent telegrams in Bourbaki’s name,  announced his daughter’s wedding, and publicly insulted anyone  who doubted his existence. In 1968, when they could  no longer maintain the ruse, the group ended their joke  the only way they could. They printed Bourbaki’s obituary, complete with mathematical puns. Despite his apparent death, the group  bearing Bourbaki’s name lives on today. Though he’s not associated  with any single major discovery, Bourbaki’s influence informs  much current research. And the modern emphasis on formal proofs owes a great deal to his rigorous methods. Nicolas Bourbaki may have been imaginary—  but his legacy is very real.