% is not an operator [was Re: Verbose and flexible args and kwargs syntax]

[Jussi Piitulainen]

Steven D'Aprano writes:
> On Mon, 12 Dec 2011 09:29:11 -0800, Eelco wrote:
> 
> [quoting Jussi Piitulainen <jpiitula at ling.helsinki.fi>]
> >> They recognize modular arithmetic but for some reason insist that
> >> there is no such _binary operation_. But as I said, I don't
> >> understand their concern. (Except the related concern about some
> >> programming languages, not Python, where the remainder does not
> >> behave well with respect to division.)
> 
> I've never come across this, and frankly I find it implausible that
> *actual* mathematicians would say that. Likely you are
> misunderstanding a technical argument about remainder being a
> relation rather than a bijunction. The argument would go something
> like this:

(For 'bijunction', read 'function'.)

I'm not misunderstanding any argument. There was no argument. There
was a blanket pronouncement that _in mathematics_ mod is not a binary
operator. I should learn to challenge such pronouncements and ask what
the problem is. Maybe next time.

But you are right that I don't know how actual mathematicians these
people are. I'm not a mathematician. I don't know where to draw the
line.

A Finnish actual mathematician stated a similar prejudice towards mod
as a binary operator in a Finnish group. I asked him what is wrong
with Knuth's definition (remainder after flooring division), and I
think he conceded that it's not wrong. Number theorists just choose to
work with congruence relations. I have no problem with that.

He had experience with students who confused congruences modulo some
modulus with a binary operation, and mixed up their notations because
of that. That is a reason to be suspicious, but it is a confusion on
the part of the students. Graham, Knuth, Patashnik contrast the two
concepts explicitly, no confusion there.

And I know that there are many ways to define division and remainder
so that x div y + x rem y = x. Boute's paper cited in [1] advocates a
different one and discusses others.

[1] <http://en.wikipedia.org/wiki/Modulo_operation>

But I think the argument "there are several such functions, therefore,
_in mathematics_, there is no such function" is its own caricature.

> "Remainder is not uniquely defined. For example, the division of -42
> by -5 can be written as either:
> 
>     9*-5 + 3 = -42
>     8*-5 + -2 = -42
> 
> so the remainder is either 3 or -2. Hence remainder is not a bijection 
> (1:1 function)."

Is someone saying that _division_ is not defined because -42 div -5 is
somehow both 9 and 8? Hm, yes, I see that someone might. The two
operations, div and rem, need to be defined together.

(There is no way to make remainder a bijection. You mean it is not a
function if it is looked at in a particular way.)

[The square root was relevant but I snipped it.]

> Similarly, we can sensibly define the remainder or modulus operator
> to consistently return a non-negative remainder, or to do what
> Python does, which is to return a remainder with the same sign as
> the divisor:
...
> There may be practical or logical reasons for preferring one over
> the other, but either choice would make remainder a bijection. One
> might even define two separate functions/operators, one for each
> behaviour.

Scheme is adopting flooring division, ceiling-ing division, rounding
division, truncating division, centering division, and the Euclidean
division advocated by Boute, and the corresponding remainders. There
is no better way to bring home to a programmer the points that there
are different ways to define these, and they come as div _and_ rem.