What is Expressiveness in a Computer Language

[Marshall]

Chris Smith wrote:
> Marshall <marshall.spight at gmail.com> wrote:
> > Ouch; I have a really hard time understanding this.
> >
> > I can't see how you'd call + on a and b if you think they might
> > not be numbers. If they could be something other than numbers,
> > and you're treating them as if they are, is that sort of like
> > doing a case analysis and only filling in one of the cases?
> > If so, wouldn't you want to record that fact somehow?
>
> The obvious answer -- I don't know if it's what Pascal meant -- is that
> they might be 4x4 matrices, or anything else that behaves predictably
> under some operation that could be called addition.  As a mathematical
> analogy, the entire mathematical field of group theory comes from the
> idea of performing operations on values without really knowing what the
> values (or the operations) really are, but only knowing a few axioms by
> which the relationship between the objects and the operation is
> constrained.
>
> [As an interesting coincidence, group theory papers frequently use the
> addition symbol to represent the (arbitrary) binary relation of an
> Abelian group.  Not that this has anything to do with choosing "+" for
> the example here.]

Sure. This is what abstract algebra is all about; group theory is
just an example. In particular, much of Stepanov's life work
has been developing abstract algebra within the framework
of a static type system.

http://www.stepanovpapers.com/
In particular, see the paper "Greatest Common Measure, the
Last 2500 Years."


> Programming languages do this all the time, as well.  The most popular
> example is the OO sense of the word polymorphism.  That's all about
> being able to write code that works with a range of values regardless of
> (or, at least, a range that less constraining than equlity in) types.

Exactly.


Marshall