[swift-evolution] protocol-oriented integers (take 2)

[Xiaodi Wu]

On Sun, Jan 15, 2017 at 6:42 PM, David Sweeris <davesweeris at mac.com> wrote:

>
>
>
> Sent from my iPhone
> On Jan 15, 2017, at 18:02, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>
> "Mathematically correct" integers behave just like Int in that there is
> not a multiplicative inverse. What we're trying to do here is to determine
> how much of what we know about mathematics is usefully modeled in the
> standard library. The answer is not zero, because there is more than just
> counting that people do with integers.
>
>
> It's an interesting problem... When I was in school, "integer" division
> "returned" a "quotient and remainder", a "fraction" (which, occasionally,
> could be simplified to just an integer), or a "real". We never talked about
> division in the context of "(Int, Int) -> Int", though. OTOH, I never took
> any math classes past Differential Equations or Linear Algebra, either...
> I'm *aware* of areas of math where you formally restrict yourself to the
> kind of "(Int, Int) -> Int" operations we're doing here, but I don't
> really know much about it. Is division even well-defined in that context?
>
> - Dave Sweeris
>

I'm no mathematician, and I'm not sure how to tackle the question of
"well-defined." Hopefully someone who is more knowledgable can chime in
here.

But I'll have a go at replying to your point as it relates to the practical
issue here. Two Int values can be "divided" to produce another Int, and
that gives a predictable and well-understood result. It's an operation
that's always going to be there--first, because it'd be insane to remove it
since much working code relies on it, and second, because we're only
re-designing integer protocols and not the concrete types. However, it _is_
true that such an operation has very different semantics from division as
you learned it in math.

This is why I'm advocating for perhaps another look at the top of this
integer protocol hierarchy. At the moment, `Arithmetic` offers reasonable
semantic guarantees for a lot of things, but `/` has different semantics
for integer types and floating point types and is really closer to just
syntax. Other mathematical types--which certainly the stdlib doesn't have
to offer, but the stdlib protocol hierarchy shouldn't preclude their
conformance to relevant protocols if it's possible--such as fractions and
complex numbers, share with floating point types the semantics of `/` that
qualify these types as fields. Dave A's question as to practical uses can
probably best be answered in this way: to the extent that any generic
algorithm relies on `/` having semantics and can be applied to fractions
and real numbers, it would be useful to distinguish such an operation from
integer "division."
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