Haskell Monad bind operator confusion

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.

Answer 2

Allow me to tear down your beliefs about Monads. I sincerely hope you realize that I am not trying to be rude; I'm simply trying to avoid mincing words.

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.

Not exactly. When you start a sentence with "A Monad's purpose", you're already on the wrong foot. Monads don't necessarily have a "purpose". Monad is simply an abstraction, a classification which applies to certain types and not to others. The purpose of the Monad abstraction is simply that, abstraction.

A Monad consists of two interrelated functions: bind and unit.

Yes and no. The combination of bind and unit are sufficient to define a Monad, but the combination of join, fmap, and unit is equally sufficient. The latter is, in fact, the way that Monads are typically described in Category Theory.

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.

Again, not exactly. A monadic function f :: a -> m b is perfectly composable, with certain types. I can post-compose it with a function g :: m b -> c to get g . f :: a -> c, or I can pre-compose it with a function h :: c -> a to get f . h :: c -> m b.

But you got the second part absolutely right: (>>= f) :: m a -> m b. As others have noted, Haskell's bind function takes the arguments in the opposite order.

g is composable.

Well, yes. If g :: m a -> m b, then you can pre-compose it with a function f :: c -> m a to get g . f :: c -> m b, or you can post-compose it with a function h :: m b -> c to get h . g :: m a -> c. Note that c could be of the form m v where m is a Monad. I suppose when you say "composable" you mean to say "you can compose arbitrarily long chains of functions of this form", which is sort of true.

The unit function takes an argument of the type that f expected, and wraps it in the monadic type.

A roundabout way of saying it, but yes, that's about right.

This [the result of applying unit to some value] can then be passed to g, or to any composition of functions like g.

Again, yes. Although it is generally not idiomatic Haskell to call unit (or in Haskell, return) and then pass that to (>>= f).

-- instead of
return x >>= f >>= g
-- simply go with
f x >>= g

-- instead of
\x -> return x >>= f >>= g
-- simply go with
f >=> g
-- or
g <=< f