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.