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

Xiaodi Wu xiaodi.wu at gmail.com
Sun Jan 15 18:02:52 CST 2017

"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.

On Sun, Jan 15, 2017 at 17:54 David Sweeris <davesweeris at mac.com> wrote:

> 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 <https://en.wikipedia.org/wiki/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 <https://en.wikipedia.org/wiki/Module_%28mathematics%29>.
> 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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