Question

Okay, so I am not a Haskell programmer, but I am absolutely intrigued by a lot of the ideas behind Haskell and am looking into learning it. But I’m stuck at square one: I can’t seem to wrap my head around Monads, which seem to be fairly fundamental. I know there are a million questions on SO asking to explain Monads, so I’m going to be a little more specific about what’s bugging me:

I read this excellent article (an introduction in Javascript), and thought that I understood Monads completely. Then I read the Wikipedia entry on Monads, and saw this:

A binding operation of polymorphic type (M t)→(t→M u)→(M u), which Haskell represents by the infix operator >>=. Its first argument is a value in a monadic type, its second argument is a function that maps from the underlying type of the first argument to another monadic type, and its result is in that other monadic type.

Okay, in the article that I cited, bind was a function which took only one argument. Wikipedia says two. What I thought I understood about Monads was the following:

1. A Monad’s purpose is to take a function with different input and output types and to make it composable. It does this by wrapping the input and output types with a single monadic type.
2. A Monad consists of two interrelated functions: bind and unit. Bind takes a non-composable function f and returns a new function g that accepts the monadic type as input and returns the monadic type. g is composable. The unit function takes an argument of the type that f expected, and wraps it in the monadic type. This can then be passed to g, or to any composition of functions like g.

But there must be something wrong, because my concept of bind takes one argument: a function. But (according to Wikipedia) Haskell’s bind actually takes two arguments! Where is my mistake?

Answer 1

You are not making a mistake. The key idea to understand here is currying – that a Haskell function of two arguments can be seen in two ways. The first is as simply a function of two arguments. If you have, for example, (+), this is usually seen as taking two arguments and adding them. The other way to see it is as a addition machine producer. (+) is a function that takes a number, say x, and makes a function that will add x.

(+) x = \y -> x + y
(+) x y = (\y -> x + y) y = x + y

When dealing with monads, sometimes it is probably better, as ephemient mentioned above, to think of =<<, the flipped version of >>=. There are two ways to look at this:

(=<<) :: (a -> m b) -> m a -> m b

which is a function of two arguments, and

(=<<) :: (a -> m b) -> (m a -> m b)

which transforms the input function to an easily composed version as the article mentioned. These are equivalent just like (+) as I explained before.