Lambda the Ultimate

Subtitled "A Graphical Notation for the Lambda Calculus with Animated Reduction." A visual introduction to the lambda calculus. Scroll slowly down the page and watch the animations. (Source: Lambda Calculus <-> Category Theory Discussion)

As pointed out by Raymond Smullyan the eminent logician and author of several puzzle books, the theory of combinators is an abstract science dealing with objects whose only important property is how they act upon each other. We are free to choose other properties of these objects in any way we like. In his delightful book "To mock a mockingbird", Smullyan (1985) chooses birds for his combinators, in memory of Haskell Curry, an early pioneer in the theory of combinators (1958) and an avid bird-watcher.

The story begins, "A certain enchanted forest is inhabited by talking birds. Given any birds A and B, if you call out the name of B to A, then A will respond by calling out the name of some bird to you; this bird we designate AB. Thus AB is the bird named by A upon hearing the name of B." ... In the hope of making it even easier I introduce the following graphical notation by extending Smullyan's bird metaphor. We will go inside the birds' heads and see how to draw diagrams of the internal plumbing which connects their ear to their throat in order to produce the correct song in response to each song that they hear.