Revamping the numeric classes

[Dylan Thurston]

On Thu, Feb 08, 2001 at 11:24:49AM +0000, Jerzy Karczmarczuk wrote:
> First, a general remark which has nothing to do with Num.
> 
> PLEASE WATCH YOUR DESTINATION ADDRESSES
> People send regularly their postings to haskell-cafe with
> several private receiver addresses, which is a bit annoying
> when you click "reply all"...

Yes, apologies.  The way the lists do the headers make it very easy to
reply to individuals, and hard to reply to the list.

> And, last, but very high on my check-list:
> 
> The implicit coercion of numeric constants: 3.14 -=->>  (fromDouble
> 3.14)
> etc. is sick. (Or was; I still didn't install the last version of GHC,
> and with Hugs it is bad). The decision is taken by the compiler
> internally,
> and it doesn't care at all about the fact that in my prelude 
> I have eliminated the Num class and redefined fromDouble, fromInt, etc. 

Can't you just put "default ()" at the top of each module?

I suppose you still have the problem that a numeric literal "5" means
"Prelude.fromInteger 5".  Can't you define your types to be instances
of Prelude.Num, with no operations defined except Prelude.fromInteger?

> Dylan Thurston terminates his previous posting about Num with:
> 
> > Footnotes:
> > [1]  Except for the lack of abs and signum, which should be in some
> > other class.  I have to think about their semantics before I can say
> > where they belong.
> 
> Now, signum and abs seem to be quite distincts beasts. Signum seem to
> require Ord (and a generic zero...).
> 
> Abs from the mathematical point of view constitutes a *norm*. Now,
> frankly, I haven't the slightest idea how to cast this concept into
> Haskell class hierarchy in a sufficiently general way...

This was one thing I liked with the Haskell hierarchy: the observation
that "signum" of real numbers is very much like "argument" of complex
numbers.  abs and signum in Haskell satisfy an implicit law:
   abs x * signum x = x      [1]
So signum can be defined anywhere you can define abs (except that it's
not a continuous function, so is not terribly well-defined).  A
default definition for signum x might read
   signum x = let a = abs x in if (a == 0) then 0 else x / abs x
(Possibly signum is the wrong name.  What is the standard name for
this operation for, e.g., matrices?)  [Er, on second thoughts, it's
not as well-defined as I thought.  Abs x needs to be in a field for
the definition above to work.]

> I'll tell you anyway that if you try to "sanitize" the numeric
> classes, if you separate additive structures and the multiplication,
> if you finally define abstract Vectors over some field of scalars,
> and if you demand the existence of a generic normalization for your
> vectors, than *most probably* you will need multiparametric classes
> with dependencies. 

Multiparametric classes, certainly (for Vectors, at least).
Fortunately, they will be in Haskell 2 with high probability.  I'm not
convinced about dependencies yet.

> Jerzy Karczmarczuk
> Caen, France

Best,
	Dylan Thurston

Footnotes: 
[1]  I'm not sure what I mean by "=" there, since I do not believe
these should be forced to be instances of Eq.  For clearer cases,
consider the various Monad laws, e.g.,
   join . join = join . map join
(Hope I got that right.)  What does "=" mean there?  Some sort of
denotational equality, I suppose.