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

[Dave Abrahams]

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?

> – 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