Am Mittwoch, 21. Dezember 2005 16:28 schrieb David Barton:
Wolfgang Jeltsch writes: ----- Original Message -----
Am Mittwoch, 21. Dezember 2005 13:15 schrieb Creighton Hogg:
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Monads, I believe, can be just thought of as containers for state.
I would say that you are talking especially about the I/O monad here. A monad as such is a rather general concept like a group is in algebra.
While this is correct, I'm afraid that for most of us it is a flavorless answer. I wish I had the mathematical mind that made the word "group" in this context instantly intuitively recognizable, but I don't.
The point is that the concept of a monad is a rather abstract one. Maybe we should take vector spaces instead of groups. A vector space is just a set together with two operations, namely vector addition and scalar multiplication whereby the operations have to fulfill a couple of laws. One law, for example, would be that vector addition is commutative. The point is that this vector space concept is very general. There are a lot of concrete vector spaces which all fit that general idea. For example, R^2 together with the commonly defined vector addition and scalar multiplication is a vector space. But there are many others, lots of which seem to have nothing to do with R^2 and its operations from the first sight. So we have a general concept "vector space" and a lot of concrete vector spaces. Now, "monad" is a general concept like "vector space". In Haskell terms a monad is a type of kind * -> * (i.e., a type which needs to be applied to one type parameter in order to be able to be used as a type of expressions) together with operations (>>=) and return whereby (>>=) and return have to have specific types and have to fulfill certain laws. IO together with its (>>=) and return is an example of a concrete monad. [] together with concatMap and \x -> [x] is another one. This may seem very surprising at first ("What do lists have to do with I/O computations?") but it's really so.
I think Phil Wadler said it best when he said that a monad is a *computation*.
No, a value of type IO <something> is a computation. The type IO plus (>>=) plus return is a monad. So is [] plus concatMap plus \x -> [x].
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Dave Barton
Best wishes, Wolfgang