Excerpts from Paolo Donadeo's message of Mon Jul 21 23:28:36 +0200 2008:
> I'm disappointed with myself and my incredibly low IQ. Late this
> evening I decided -- and this is the third time -- to be enlightened
> by the concept of monad.
> 
> I like functional programming, but monads [1] must be too little to be
> grabbed by my mind. This time the interest in monads was aroused by
> the interesting article of David Teller, Arnaud Spiwack and Till
> Varoquaux [2] about the error monad, but for using the library they
> wrote I need at least some knowledge about monads and the do-notation.
> 
> I tried with some tutorials found around, but I still cannot catch the
> point: what the hell is a monad?

Two key points that helped me:

* Monads help to separate some plumbing from your code.

* Monads provide a way to abstract code over some "let" construct.

I will note that specific "let" construct "let!", it's somewhat
like the do-notation but more atomic.

Monads also come with "return", but that's not the essence of them.

Think about that example:

  val div : int -> int -> int option
  val square : int -> option

  let f x y =
    match div x y with
    | None -> None (* here 'y' was equal to 0 *)
    | Some z ->
        match square z with
        | None -> None
        | Some x2 -> Some x2

In the previous example the plumbing is error handling (where errors are
represented by None), and the "let!" construct is:

  let! x = e1 in e2   ===>   match e1 with None -> None | Some x -> e2

And "return" is "Some".

Another similar example:

  type ('a,'b) either = Left of 'a | Right of 'b

  val div : int -> int -> (string, int) either
  val square : int -> (string, int) either

  let f x y =
    match div x y with
    | Left error_msg -> Left error_msg
    | Right z ->
        match square z with
        | Left error_msg -> Left error_msg
        | Right x2 -> Right x2

Here the plumbing is still there for "error handling", but is somewhat
different. The "let!" construct is:

  let! x = e1 in e2   ===>   match e1 with Left m -> Left m | Right x -> e2

And "return" is "Right".

Finally one could have used the "let!" construct
abstractly (and also "return").

  let f x y =
    let! z = div x y in
    let! x2 = square z in
    return x2

One can obtain the two previous versions by choosing
which "let!"/"return" we want, namely choosing the monad.

Hope that helps,

-- 
Nicolas Pouillard aka Ertai