Fundamentals of Lambda Calculus for People Who Love Birds

This (beautifully formatted and well-paced-and-delivered and surprisingly sparsely attended) talk by Gabriel Lebec on the fundamentals of Lambda Calculus is one of my favorite talks ever.

As Lebec explains, the lovely bird names come from this book called “To Mock a Mockingbird” by mathematician and logician Raymond Smullyan. The naming is simply delightful. As Matthew Gilliard explains:

The premise is that there are enchanted forests which contain many (or sometimes very few) talking birds. Smullyan dedicated the book to Haskell Curry - an early pioneer in combinatory logic and an avid bird-watcher. The birds, which I suppose represent the combinators, have an interesting characteristic:
Given any two birds A and B, if you call out the name of B to A it will respond by calling out the name of some bird to you.

This bird whose name A calls when you call B is denoted as AB. Once you have several birds in place, a single call can cascade around the forest with each call following rules depending on who produces it.

The very first bird we are introduced to is the Mockingbird whose characteristic behaviour is that whatever name you call to the Mockingbird, it will reply as if it is the bird whose name you called. This is denoted:

Mx = xx

For any bird x we can say that Mx (the result of calling x to a Mockingbird) is the same as xx (the result of calling x to a bird of type x). It really does mock other birds! And what’s more, the existence of the Mockingbird, in combination with various others, unlocks some really fascinating group behaviour from these birds.


Soon we discover that birds have certain properties: The can be fond of other birds, they can be egocentric if they are fond of themselves. The can be hopelessly egocentric if they only ever talk about themselves. There are happy birds, normal birds, agreeable birds and many others. We also meet other types of birds with specific properties - the Lark, the Kestrel, Sage birds, Bluebirds, aristocratic birds, Eagles, the list goes on and on. Luckily there is a Who’s Who list of birds in the back to keep track.