We have seen that functions are a method of abstraction that describe compound operations independent of the particular values of their arguments. That is, in square,

>>> def square(x):
        return x * x

we are not talking about the square of a particular number, but rather about a method for obtaining the square of any number. Of course, we could get along without ever defining this function, by always writing expressions such as

>>> 3 * 3
9
>>> 5 * 5
25

and never mentioning square explicitly. This practice would suffice for simple computations such as square, but would become arduous for more complex examples such as abs or fib. In general, lacking function definition would put us at the disadvantage of forcing us to work always at the level of the particular operations that happen to be primitives in the language (multiplication, in this case) rather than in terms of higher-level operations. Our programs would be able to compute squares, but our language would lack the ability to express the concept of squaring.

One of the things we should demand from a powerful programming language is the ability to build abstractions by assigning names to common patterns and then to work in terms of the names directly. Functions provide this ability. As we will see in the following examples, there are common programming patterns that recur in code, but are used with a number of different functions. These patterns can also be abstracted, by giving them names.

To express certain general patterns as named concepts, we will need to construct functions that can accept other functions as arguments or return functions as values. Functions that manipulate functions are called higher-order functions. This section shows how higher-order functions can serve as powerful abstraction mechanisms, vastly increasing the expressive power of our language.