[Haskell-beginners] confusing type signature with sections

[Patrick Redmond]

Thank you both!

> The key is that typeclasses are open. You could write a Fractional instance
> for (a -> a), in which case it would be possible to do _something_ with this
> code. Would it be useful? Even Haskell can't guarantee that.

Yes, this is important! Thanks.

> Now, because we are writing ((+ 1) / 2) and we know that (/) takes two
> arguments that must be of the same type, we know that the type (Num a')
> => a' -> a' and the type (Num a'') => a'' have to be the same type, so
> it must be that a' = a -> a, so now we have:
>
>     (+ 1) :: (Num a, Num (a -> a)) => a -> a
>     2 :: (Num a, Num (a -> a)) => a -> a

I'm still a little confused here. How can passing "2" into "(+ 1) /"
cause its type to be mangled? "2" has a type "(Num a) => a". How can
the presence of "(+ 1)" force the type of "2" to suddenly accept an
argument? How come it doesn't happen the other way around? (Meaning
"2" forces the type of "(+ 1)" to become simply "(Num a) => a".)

    Prelude> :t 2 / (+ 1)
    2 / (+ 1) :: (Fractional (a -> a), Num a) => a -> a

Thank you,
Patrick