What is 'Currying'?

Currying is when you break down a function that takes multiple arguments into a series of functions that each take only one argument. Here's an example in JavaScript:

function add (a, b) {
  return a + b;
}

add(3, 4); // returns 7

This is a function that takes two arguments, a and b, and returns their sum. We will now curry this function:

function add (a) {
  return function (b) {
    return a + b;
  }
}

This is a function that takes one argument, a, and returns a function that takes another argument, b, and that function returns their sum.

add(3)(4);

var add3 = add(3);

add3(4);

The first statement returns 7, like the add(3, 4) statement. The second statement defines a new function called add3 that will add 3 to its argument. This is what some people may call a closure. The third statement uses the add3 operation to add 3 to 4, again producing 7 as a result.

In an algebra of functions, dealing with functions that take multiple arguments (or equivalent one argument that's an N-tuple) is somewhat inelegant -- but, as Moses Schönfinkel (and, independently, Haskell Curry) proved, it's not needed: all you need are functions that take one argument.

So how do you deal with something you'd naturally express as, say, f(x,y)? Well, you take that as equivalent to f(x)(y) -- f(x), call it g, is a function, and you apply that function to y. In other words, you only have functions that take one argument -- but some of those functions return other functions (which ALSO take one argument;-).

As usual, wikipedia has a nice summary entry about this, with many useful pointers (probably including ones regarding your favorite languages;-) as well as slightly more rigorous mathematical treatment.

Here's a concrete example:

Suppose you have a function that calculates the gravitational force acting on an object. If you don't know the formula, you can find it here. This function takes in the three necessary parameters as arguments.

Now, being on the earth, you only want to calculate forces for objects on this planet. In a functional language, you could pass in the mass of the earth to the function and then partially evaluate it. What you'd get back is another function that takes only two arguments and calculates the gravitational force of objects on earth. This is called currying.