A better way to understand the logic of nested functions (Currying)

Question

I am learning Jason Hickey's Introduction to Objective Caml. Just have a question about nested functions (currying).

there is a existing question How to understand the "Currying" in Haskell?, but I guess I am seeking for the answer of slightly different question.

It says that we can write let sum = fun i j -> i + j;; as let sum = fun i -> fun j -> i + j;;

My question is simple:

Can I understand the above definition in this way: let sum = fun i -> i + fun j -> j;;?

I know it won't pass the compiler, but I just try to map this kind of OCaml function definition to the mathematics functions.

In my above imagination, we can write the function easily in mathematics, f(i) = i + g(j); and g(j) = j.

Should I always do this kind of logic mapping for easy understanding?

gasche · Accepted Answer · 2012-12-03 12:39:50Z

Your imagination is not really correct: f(i) = i + g(j) doesn't mean much as j is undefined here.

A correct way to understand fun i j -> foo is to consider it some convenient syntactic sugar for the more explicit notation fun i -> fun j -> foo.

All the definitions below are exactly equivalent :

let sum i j = i + j let sum i = fun j -> i + j let sum = fun i -> (fun j -> i + j) let sum = fun i -> fun j -> i + j let sum = fun i j -> i + j let sum i = let add_i = fun j -> i + j in add_i

Mathematically this could be written (i ↦ (j ↦ i+j)), as an element of the function space (ℕ → (ℕ → ℕ)).

winitzki · Accepted Answer · 2012-12-04 09:00:05Z

OCaml functions can be directly mapped to and from mathematical notation. However, you need to realize that mathematical notation is ambiguous. There is not a clear distinction between the value of the function, f(i), on a particular value i, and the function f itself. Often one writes f(i) when one means the function f itself. ("Let us consider a function f(i)=i+1... This function is ...") To write correct code in OCaml, you have to see clearly whether you are working with a function itself or with the value of a function.

When you say in mathematical notation, "Consider the function f(i,j)=i+g(j) where g(j)=j, you are writing the values of the functions. In OCaml, this is translated to

 let f i j = let g j = j in i + g j;;

or

 let f = let g = fun j -> j in fun i j -> i + g j;;

If you are trying to write let sum = fun i -> i + fun j -> j;;, then in mathematical notation you are saying "Consider the function sum such that sum(i) = i + g, where g is a function defined by g(j)=j." This is mathematically incorrect: you cannot add an integer value i and a function g. You can only add an integer i and the value of the function g on some other integer j. The expression "i+g" is undefined, strictly speaking. Either you wanted to write i + g(i), or i+g(j), but not i+g. This is so in mathematics, and this is so in OCaml.

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