LinkedIn wouldn’t let me write more than 2000 characters in my old “co-mathematician” job description, so I’m posting it here instead.
Okay, so what’s a co-mathematician? What exactly was I doing for a year at MIT in 2015?
Now, it is commonly known (from Erdős) that a mathematician is a device for turning coffee into theorems, i.e. Mathematician : Coffee → Theorems. But in category theory, every morphism f : X → Y has a dual, co(f), which is a morphism co(f) : co(Y) → co(X). Therefore, a co(Mathematician) must be a map from co(Theorems) → co(Coffee). Since coffee is involutive—co(co(X)) = X—we have Co-mathematician : Cotheorems → Ffee. Now, by the Curry-Howard correspondence, a theorem or proposition is just an object in a category where the morphisms are proofs. Such categories often have a terminal object, True, such that for all objects X there is a unique morphism X → True, and an initial object, False, such that for all objects X there is a unique morphism False → X. Since co(True) = False and a theorem is a true conclusion (i.e. terminal), a cotheorem should be an initial premise in a proof; namely, it is just an axiom. Supposing that f is idempotent, i.e. f = f∘f, we thus have Co-mathematician : Axioms → Fee.
In other words, a co-mathematician turns things that everyone believes into money.
Trust me, this is a very funny joke in category theory.
P.S. It’s not necessary for the job description, but arguably co-mathematicians should live inside a *-autonomous category, i.e. the category for linear logic, where resources are constrained. 🤔 This means that a co-mathematician *spends* axioms (as a mathematician consumes coffee; once consumed the coffee is gone) in order to make money. More subtly, it means that a co-mathematician is spending down the consistency of mathematics itself… like in this joke.
P.P.S. If we think of co(-) as the typical involution operator * in a *-autonomous category, then co(Y) = Y* is usually interpreted as the set of obstructions to Y, i.e. the refutations or counter-examples to Y, the demonstrations that Y is false. This feels like a more natural interpretation of cotheorem, and it suggests that a co-mathematician is a device that removes obstructions in order to make money. So give me the pesky counter-example to that theorem you want to prove, and I’ll make it disappear… for a price 😉