Dylan Thurston wrote (on 31-05-01 18:50 -0400):
On Thu, May 31, 2001 at 04:27:14PM -0600, Alastair David Reid wrote:
While pondering what else we'd put in mathematics other than numeric stuff, I hit upon the crazy idea of adding:
Mathematics.Monad
instead of putting Monad in Control (which is the current proposal, I believe).
I'd vote against this. Monads are not terribly prominent in mathematics. They're used in category theory, but the reason they're in Haskell is because of their usefulness as a control structure.
I would also oppose this, but for different reasons. First, I think it is too early to put anything into Mathematics. There ought to be some more formal standards for what goes in and what doesn't and how its innards are arranged and named. For example, I can imagine several different ways of encoding mathematical constructs in Haskell: as in Monad, where you use a Curry-Howard-et al.-style correspondence---so, you have sets/objects/propositions as types and functions/arrows/proofs as terms; as in a metacircular interpreter or symbolic manipulation package, where you collapse these two levels; as above, but for a three-level structure with type classes or kinds encoding the third level (say 2-categories). And then, as Jan pointed out, there are often many algorithms, especially for the less foundational, continuous stuff. Second, as everybody knows, there is nothing in Monad which forces user-defined instances to be monads in the mathematical sense. Whether such things belong in Mathematics may be open to argument. For example, you might want to put the class definition outside Mathematics, but the standard instances in (to `certify' them).
Hmm. Reading this list, categories don't seem to fit that well. On reflection, "Mathematics" is extremely broad; perhaps too broad. Maybe "Algebra" is better?
I think it's too broad also; what parts of mathematics are "blessed" enough to be put in Mathematics? By the C-H correspondence, every program is a piece of mathematics in some sense. I have no better suggestion. Maybe someone can give a list of examples of things that are intended to go into Mathematics...? -- Frank Atanassow, Information & Computing Sciences, Utrecht University Padualaan 14, PO Box 80.089, 3508 TB Utrecht, Netherlands Tel +31 (030) 253-3261 Fax +31 (030) 251-379