Why is function definition for all types at once not allowed in Haskell?

Question

This is probably a very basic question, but ...

A function that\'s defined as, say

foo :: a -> Integer

denotes a function from <

Answer 1

The type a -> Integer does not really mean "function from any type to Integer" as you're describing it. When a definition or expression has type a -> Integer, it means that for any type T, it is possible to specialize or instantiate this definition or expression into a function of type T -> Integer.

Switching notation slightly, one way to think of this is that foo :: forall a. a -> Integer is really a function of two arguments: a type a and a value of that type a. Or, in terms of currying, foo :: forall a. a -> Integer is a function that takes a type T as its argument, and produces a specialized function of type T -> Integer for that T. Using the identity function as an example (the function that produces its argument as its result), we can demonstrate this as follows:

-- | The polymorphic identity function (not valid Haskell!)
id :: forall a. a -> a
id = \t -> \(x :: t) -> x

This idea of implementing polymorphism as a type argument to a polymorphic function comes from a mathematical framework called System F, which Haskell actually uses as one of its theoretical sources. Haskell completely hides the idea of passing type parameters as arguments to functions, however.