[Haskell-cafe] Origins of '$'

[Hans Aberg]

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

> On Sun, Dec 7, 2008 at 2:05 AM, Hans Aberg <haberg at math.su.se> 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