Ha, you thought this was going to be about Y Combinator, the famous startup accelerator, didn't you? Well, I tricked you, this post is actually gonna be about math. (But don't worry, at the end it is loosely related to the business).

The "Y Combinator" is a term originating from lambda calculus, which, roughly speaking is the "language" mathematicians use to talk about functions. The term originates from Haskell Curry, though similar ideas were already in circulation by the time he popularized it.

The Y Combinator is a function that finds the fixed point of any other function. That is, the Y combinator is defined with the following relation:

$YF=F(YF)$

In words, "Y applied to F" is the same as "F applied to (Y applied to F)".

Why is this useful? Without getting in to too much detail, it is what allows the lambda notation to describe functions that refer to themselves. In other words, it allows for recursive functions, which can be quite useful in computation.

It can also be used to generate a lot of paradoxes, because it is what lets a function reference itself, and many paradoxes arise from self referentiality. In fact, Curry called his Y Combinator the "paradoxical combinator".

This self-referential property is what inspired Paul Graham, founder of Y Combinator the startup accelerator, to choose its name. From the Y Combinator FAQ:

Q: Why did you choose the name Y Combinator?

A: The Y combinator is one of the coolest ideas in computer science. It's also a metaphor for what we do. It's a program that runs programs; we're a company that helps start companies.

To be mildly pedantic, the Y Combinator isn't just a program that runs other programs. It's actually a program that allows another program to run itself. In a way, that's even a better description, because an accelerator is a company that helps other companies become self sustaining.

Pretty cool history huh? Hope this helps you sound smart at parties.

Economist bonus: Fixed points are vitally important in economics. Not just economics, but any dynamical system. In fact, every equilibrium is actually a fixed point.

Imagine that $F$ describes a rule to transform the state of the world from the current state to the next state, i.e.:

$X_{t+1}=F{X}_t$

The world is in equilibrium if $X$ is a fixed point of $F$. That is, if:

$X=FX$

This means that $X$ is a world state that stays fixed even after the transformation is applied.

And guess what? $YF$ is the fixed point of $F$. So the Y-Combinator by definition lets us find the stable point of any dynamical system. Or a market equilibrium in economics.

Cool stuff.