Baking the Y Combinator from Scratch

It's a pedagogical pet peeve of mine when technical concepts are simply listed in their fully developed state, with little in the way of motivation, derivation, or historical context. This, I think, is like publishing a cookbook filled with beautiful pictures of cakes, but no recipes. My aim with the "Baking from scratch" series is to describe not only what, but how and why, with the hope that this will lead to a richer and more durable understanding. The Famous Y combinator This post is about the Y combinator. Not Y Combinator the accelerator, but the mathematical construct that it's named after - the Y combinator. The Y combinator looks like this: $$ Y = \lambda f. \enspace (\lambda x. \enspace f(x \enspace x)) \enspace (\lambda x. \enspace f(x \enspace x)) $$ Confronted with this definition, you might ask yourself: Why does it look like that? What is it used for? Why do people still talk about it? (What the heck is a combinator?) The one-line answer to all of these is: " ... Read full article.