Frege uses the blank character - the most silent one in the character set - to denote function application. Function application has the highest precedence of all operators. It binds more tightly than all others. It is also left associative such that f x y is really two function applications: first f x, which returns a function that then takes y as an argument.

When a function application like g x appears as an argument to f then it must be written in parentheses like f (g x) because otherwise, g would become an argument to f, which is not what we want. This can get unwieldy when the nesting gets deeper and therefore there is an alternative using the dollar sign.

Interestingly, the $ operator is not a language construct. It is an ordinary function that is declared as an infix operator. The definition of $ is f $ x = f x . The trick is that $ has very low operator precedence and therefore all operators to its right (and analogously to the left as we will see later) are applied before its result is finally applied to f.

Now, even if we love to see dollars, the $ operator is certainly not very silent in its appearance, but the dot-operator is. It it the operator for function composition: f(g(x)) can - just like in school math - be written as (f • g) x.

We said that function application has the highest precedence - even higher than function composition. Therefore, we need the parentheses above to prevent the binding between buz and x. But, hey, we learned that dollar has lowest precedence, so putting that in between should also work, right? Yes it does.

Now, can we shave the x off such that we can drop dollars and parentheses as well?

Yes, but that needs a bit of background.

Let’s assume we have a method inc that just adds 1 to its argument. There are two equivalent ways of writing this:

Now if we want a function f to work like inc, we can use inc in the definition of f and relay the argument.

But we can even cancel down the x such that f looks like an alias for inc. And here is proof that this is a valid transformation:

This finally leaves us with four ways of defining our function:

Keeping the above equivalence in mind should help you to become confident with these options when writing your own code but even more so when reading other developer’s code.

You will find all four styles in any reasonably sized program. For example, we have used the point-free style in the previous post, when we said f = reverse and I’m sure you guessed the meaning even though it was not explained. We could have said f x = reverse x but there is a subtle distinction - not in what the code does but in what it communicates:

The prototypical adoption cycle for newcomers is: they are first puzzled by the point-free style, then excited, then they overuse it, then they settle at a reasonable dose.