One can define operators
a ^ b := b(a) -- Application in inverse.
(a * b)(x) := b(a(x)) -- Function composition in inverse.
(a + b)(x) := a(x) * b(x)
O(x) := I -- Constant function returning identity.
I(x) := x -- Identity.
and use them to define lambda calculus (suffices with the first four; Church reverses the order of "*").
The simple elegance of writing this encoding just increased my already infinite love of Haskell by another cardinality.
a .^ b = b a
(a .* b) x = b (a x)
(a .+ b) x = a x .* b x
o x = i
i x = x
toNat x = x (+1) 0
fromNat n = foldr (.) id . replicate n
Luke