Why is the second operation of a Monad called bind?

Question

I understand how monadic computation works, I am just wondering where does the name come from. I cannot relate the thing that the bind operator actually does (i.e. unbox the monadic value, and then apply a function to produce another monadic value) and the name bind (which, to me, means associate two things together).

I understand why we have the name return (I suspect it has to do with the Haskell do-notation), although IMHO it isn't a good name either. Return indicates a change in control flow but it really isn't, all it does is produce a monadic value.

Answer 1

In Haskell the syntactic sugar for bind e₁ (λ x . e₂) is

do x ← e₁
   e₂

That is, the result of the computation e₁ is bound to x, after which e₂ is computed.

In ML-like languages the same thing is written as

let x = e₁ in e₂ 

which again is a form of binding a variable to a value.

As for return being a control-flow mechanism – that's precisely what it is. You are probably thinking of control mechanisms for branching, looping, raising exceptions etc. Well, return is the one that ends a computation and yields a result.

Answer 2

In λ-calculus, the term $\lambda x. y$ binds $x$ in $y$. Once a value is given to $x$, any well-scoped reference to $x$ in $y$ will be replaced by that same value.

Function composition $(f\circ g)$ can be seen as an operator that binds the result of $g$ to $f$'s argument. In a similar way, you can view bind : Monad m => m a -> (a -> m b) -> m b as an operator that binds the contents (the value underneath the monadic wrapper) of its first argument to the function supplied as its 2nd argument (a -> m b).

What if the 1st argument of type m a is the result of some previously applied function? In general, you can make consecutive applications of bind to mimic something like function composition, but under a monad.