'x ^: 'y ^: 'x ! 'y
'x ^: 'y ^: 'x ! ('x ! 'y)
'x ^: 'y ^: 'x ! ('x ! ('x ! 'y))
which translates to:
λx.λy.xy
λx.λy.x(xy)
λx.λy.x(x(xy))
(You might recognize these as the Church numerals one through three.)
To digest this, you first have to imagine abstraction as an infix operator rather than a prefix operator. If you squint a little, you can see a caret as a lambda (the colon is necessary to get Scala to make the operator right-associative, as is conventional).
The exclamation point is the application operator by analogy with the notation used in Scala Actors. It takes only a wee bit of mental squinting to see passing an argument to a function as sending a message to an actor.
The single quote prefix in Scala turns an identifier into a Symbol, which is like a String but with only half the quoting. Scala Symbols are intended for purposes that are, uh, symbolic—I think a lambda calculus variable qualifies.
As cute as dancing kittens? You be the judge.
