The greatest mathematician that never lived - Pratik Aghor

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.