[Haskell-cafe] Re: Numeric type classes

[Henning Thielemann]

On Mon, 11 Sep 2006, Ross Paterson wrote:

> On Mon, Sep 11, 2006 at 04:26:30PM +0200, Henning Thielemann wrote:
> > On Sat, 9 Sep 2006, Ross Paterson wrote:
> > > I think that a finer grain numeric hierarchy, while retaining Num, etc,
> > > is feasible without changing the language: unlike the case of monads,
> > > the people who will be defining instances of numeric classes are the
> > > very ones who are inconvenienced by the current hierarchy.  The main
> > > impact on clients of the classes is that some functions would have
> > > more general types.
> > 
> > There are many Num instances around in libraries where people wrap to
> > external libraries: functionalMetapost, CSound wrapper in Haskore,
> > SuperCollider (GSL too?). What about Num (algebraically Ring) instances of
> > polynomials, residue classes and other such advanced mathematical objects?
> 
> And what do abs and signum mean for Haskore's orchestra expressions,
> polynomials, residue classes, vectors, matrices, functions, etc?

For clarification: Haskore does not define any arithmetic for music, but
CSound provides some arithmetic and Haskell wraps it with Num instances.

> The people who define those wish they were defining Ring, but they must
> define Num.

It seems we are at a point, where we have to define what is a 'number'.  
More precisely: Can you tell me the difference between numbers and "more
complex mathematical objects"? Is a complex number a number? Is a
quaternion a number? Is a residue class a number? We can calculate with
integers modulo some other integer like with integers - is that considered
computation with numbers? Shall we distinguish between matrices of numbers
and matrices of more complex mathematical objects? In signal theory
matrices of polynomials are very common.