[Haskell-cafe] Origins of '$'
Hans Aberg
haberg at math.su.se
Mon Dec 8 14:29:30 EST 2008
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
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