Re: (*) -> 1

[<tomas@tuxteam.de>]

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On Wed, Jan 18, 2023 at 08:50:07AM +0100, Óscar Fuentes wrote:
> Michael Heerdegen <michael_heerdegen@web.de> writes:
> 
> > Óscar Fuentes <ofv@wanadoo.es> writes:
> >
> >> The languages that use the concept of partially applied function usually
> >> have no support for variadic functions, so the duality problem you refer
> >> to is not an issue.
> >
> > Interesting.  I do not know many other languages.
> >
> > I see your point now: while I wrote about the procedure of finding an
> > interpretation [of the technical or mathematical semantics of a formula]
> > in the real world [this is what had been ongoing: Ferraris etc], you
> > mention that even the technical/theoretical semantics of a formula like
> > (*) can be different.  This is an interesting point, especially since
> > terms like "right" and "wrong" had been used.
> >
> > Although I think the "meaning" of the expression (*) in Elisp is clear,
> > it describes a mathematical term, so the question, asked specifically
> > for Elisp, has to be answered using the mathematical background.  In my
> > understanding the OP asked specifically about the empty algebraic
> > product.
> 
> I was prompted to enter the discussion when I saw your reference to
> Mathematics. As almost every other math-related thing in computers,
> Elisp's + is a toy representation of Sigma. And then the relevant
> characteristics of Sigma for this discussion are a convention among
> practitioners, not a proper mathematical fact.

This goes a bit deeper: whenever you have an associative operator
[i.e. (a + b) + c == a + (b + c)],

 1. the "variadic extension" is straightforward, since you can leave
    out the parentheses in (a + (b + (c + ...))) without losing info
 2. the extension to zero arguments is also straightforward *provided*
    you have a neutral element, since 0 + a == a.

This works for products, logical and, or, set union and intersection,
you name it (in mathematical logic, you often see the or/and cousins
of Sigma and Pi; in set theory likewise the union/intersection things).

> Although it is possible that the implementors were inspired by Sigma, I
> think it is more probable that they made + variadic because s-exps like
> (+ (+ 1 2) 3) are awkward and then extended the function with support
> for 0 and 1 arguments because they are convenient when defining macros.

I think a mathematician doesn't really distinguish between "Sigma" and
just "sum". Sigma is just a notation. No magic, just maths.

Cheers
-- 
t

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