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

[David Sweeris]

> On Jan 15, 2017, at 17:19, Jacob Bandes-Storch via swift-evolution <swift-evolution at swift.org> wrote:
> 
> 
>> On Sun, Jan 15, 2017 at 2:42 PM, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>>> On Sun, Jan 15, 2017 at 3:29 PM, Jacob Bandes-Storch via swift-evolution <swift-evolution at swift.org> wrote:
>> 
>>> [ proposal link: https://gist.github.com/moiseev/62ffe3c91b66866fdebf6f3fcc7cad8c ]
>>> 
>>> 
>>>> On Sat, Jan 14, 2017 at 4:55 PM, Dave Abrahams via swift-evolution <swift-evolution at swift.org> wrote:
>>>> 
>>>> Responding to both Jacob and Xiaodi here; thanks very much for your
>>>> feedback!
>>>> 
>>>> on Sat Jan 14 2017, Xiaodi Wu <swift-evolution at swift.org> wrote:
>>>> > I think, in the end, it's the _name_ that could use improvement here. As
>>>> > the doc comments say, `Arithmetic` is supposed to provide a "suitable basis
>>>> > for arithmetic on scalars"--perhaps `ScalarArithmetic` might be more
>>>> > appropriate? It would make it clear that `CGVector` is not meant to be a
>>>> > conforming type.
>>>> 
>>>> We want Arithmetic to be able to handle complex numbers.  Whether Scalar
>>>> would be appropriate in that case sort of depends on what the implied
>>>> field is, right?
>>> 
>>> I think "scalar" is an appropriate term for any field. The scalar-ness usually comes into play when it's used in a vector space, but using the term alone doesn't bother me.
>>>  
>>>> It's true that CGPoint and CGVector have no obvious sensible
>>>> interpretation of "42", and that's unfortunate.  The problem with
>>>> protocols for algebraic structures is that there's an incredibly
>>>> complicated lattice (see figures 3.1, 3.2 in
>>>> ftp://jcmc.indiana.edu/pub/techreports/TR638.pdf) and we don't want to
>>>> shove all of those protocols into the standard library (especially not
>>>> prematurely) but each requirement you add to a more-coarsely aggregated
>>>> protocol like Arithmetic will make it ineligible for representing some
>>>> important type.
>>> 
>>> Yep, it is quite complicated, and I understand not wanting to address all that right now; calling it ScalarArithmetic seems appropriate to clarify the limitations. FieldArithmetic might also be appropriate, but is less clear (+ see below about quaternions).
>>> 
>>> Daves Sweeris and Abrahams wrote:
>>> 
>>> > > I was under the impression that complex numbers are scalar numbers... although maybe not since once you get past, I think quaternions, you start losing division and eventually multiplication, IIRC. (I hate it when two of my recollections try to conflict with each other.) 
>>> >
>>> > Well, you can view them as 2d vectors, so I'm not sure.  We need more of a numerics expert than I am to weigh in here.
>>>  
>>> But complex numbers have multiplication and division operations defined (they form a field), unlike regular vectors in R². Meaning you can have a vector space over the field of complex numbers.
>>> 
>>> You still have multiplication and division past quaternions, but the quaternions are not commutative. This isn't really a problem in Swift, since the compiler never allows you to write an expression where the order of arguments to an operator is ambiguous. This means they are not a field, just a division ring (a field is a commutative division ring). (I believe you can't technically have a vector space over a non-commutative ring; the generalization would be a module. That's probably an argument for the name ScalarArithmetic over FieldArithmetic.)
>> 
>> Hmm, the issue is that the integers are not a field. So, if we're going to have it all modeled by one protocol, maybe neither is the best term.
> 
> Eurgh. That's true. Appropriate mathematical terms go out the window when "division" doesn't actually produce a multiplicative inverse.
> 
> BasicArithmetic?

I was thinking something similar... Could we just rename Int/UInt to Counter/UnsignedCounter, and leave all these "mathematically correct" protocols and types to mathematically correct numeric libraries?

- Dave Sweeris
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