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

[Xiaodi Wu]

On Mon, Jan 16, 2017 at 12:44 AM, Dave Abrahams via swift-evolution <
swift-evolution at swift.org> wrote:

>
> on Sun Jan 15 2017, Stephen Canon <swift-evolution at swift.org> wrote:
>
> > Responding to the thread in general here, not so much any specific email:
> >
> > “Arithmetic” at present is not a mathematically-precise concept, and
> > it may be a mistake to make it be one[1]; it’s a
> > mathematically-slightly-fuzzy “number” protocol. FWIW, if I had to
> > pick a mathematical object to pin it to, I would make it a
> > [non-commutative] ring, which give it:
> >
> >       addition, subtraction, multiplication, IntegerLiteralConvertible,
> and an inverse: T?
> >       property.
> >
> > Note that this would make division go out the window, but we would
> > *preserve* int literal convertible (because there’s a unique map from
> > Z to any ring — Z is the initial object in the category of rings). I
> > think that it’s quite important to keep that for writing generic code.
> >
> > Vectors and points are not “numbery” in this sense, so they should not
> > conform to this protocol, which is why it’s OK that multiplication and
> > int literals don’t make sense for them. They’re property a vector
> > space or module built over a scalar type.
> >
> > I would be OK with moving division out of the Arithmetic protocol, as
> > the division operator is fundamentally different for integers and
> > floating-point, and you want to have separate left- and right-
> > division for quaternions and matrices. I would be pretty strongly
> > opposed to removing integer literals; they rightfully belong here.
> >
> > I think the name “Arithmetic” is sound. It is deliberately *not* one
> > of the standard mathematical abstractions, reserving those names for
> > folks who want to build a precise lattice of algebraic
> > objects. Vectors don’t belong in this protocol, though.
>
> OK, suppose we move division, and state in the documentation that models
> of Arithmetic should have the mathematical properties of a
> (non-commutative) Ring?
>

Unless I'm mistaken, after removing division, models of SignedArithmetic
would have the mathematical properties of a ring. For every element a in
ring R, there must exist an additive inverse -a in R such that a + (-a) =
0. Models of Arithmetic alone would not necessarily have that property.

> – Steve
> >
> > [1] We don’t want to make a semester course in modern algebra a
> > prerequisite to using Swift, as much as I would enjoy building a tower
> > of formalism as a mathematician.
>
> Indeedy.
>
> --
> -Dave
>
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