Re: [Haskell-cafe] Origins of '$'

On 8 Dec 2008, at 19:36, Dan Piponi wrote:

On Sun, Dec 7, 2008 at 2:05 AM, Hans Aberg wrote:

...

As for the operator itself, it appears in Alonzo Church, "The Calculi of Lambda-Conversion", where it is written as exponentiation, like x^f

That's reminiscent of the notation in Lambek and Scott where (roughly speaking) the function converting an element of an object A^B to an arrow B->A (something Haskellers don't normally have to think about) is written as a superscript integral sign. Presumably this comes from the same source. Both $ and the integral sign are forms of the letter 's'. Don't know why 's' would be chosen though.

In set theory, and sometimes in category theory, A^B is just another notation for Hom(B, A), and the latter might be given the alternate notation B -> A. And th reason is that for finite sets, computing cardinalities result in the usual power function of natural numbers - same as Church, then. And the integral sign comes from Leibnitz: a stylized "S" standing for summation. Also, it is common to let "s" or sigma stand for a section, that is, if given functions s: A -> B pi: B -> A such that the composition pi o s: A -> B -> A is the identity on A, then s is called a section and pi a projection (as in differential geometry). Hans