Why is `($ 4) (> 3)` equivalent to `4 > 3`?

Question

I noticed as I was playing around with Haskell today that it is possible to do something like

($ 4) (> 3)

which yields True

Answer 1

You can partially apply an infix operator from either side. For commutative operators such as +, it doesn't matter if you say (+ 1) or (1 +), but for example for division you can supply either the dividend (5 /) or the divisor (/ 5).

The function application operator takes a function as the left-hand operand and a parameter as the right-hand operand (f $ x), so you can partially apply it either with a function (f $) or with a parameter ($ x). So given

($ 4) (> 3)

You first partially apply the $-operator with the parameter 4 and supply it with the function (> 3). So what this essentially becomes is

(> 3) $ 4

Which is the same as (4 > 3).

Answer 2

($ 4) is the function that takes a function and applies 4 to it.

(> 3) is the function that takes a number and checks if it is greater than 3.

So by giving the latter function to the former, you are essentially applying 4 to the function that checks if its input is greater than 3, and thus you get True.

Answer 3

($ 4) is what's called a section. It's a way of partially applying an infix operator, but providing the right-hand side instead of the left. It's exactly equivalent to (flip ($) 4).

Similarly, (> 3) is a section.

($ 4) (> 3)

can be rewritten as

(flip ($) 4) (> 3)

which is the same as

flip ($) 4 (> 3)

which is the same as

(> 3) $ 4

And at this point, it should be clear that this boils down to (4 > 3).