(define (square x) (* x x)) | | | | | | To square something, multiply it by itself.
The importance of this decomposition strategy is not simply that one is dividing the program into parts. After all, we could take any large program and divide it into parts—the first ten lines, the next ten lines, the next ten lines, and so on. Rather, it is crucial that each procedure accomplishes an identifiable task that can be used as a module in defining other procedures.
In becoming an expert photographer, for example, one must learn how to look at a scene and know how dark each region will appear on a print for each possible choice of exposure and development conditions. Only then can one reason backward, planning framing, lighting, exposure, and development to obtain the desired effects.
Our programs would be able to compute cubes, but our language would lack the ability to express the concept of cubing.
We now come to the decisive step of mathematical abstraction: we forget about what the symbols stand for. … [The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for.
In a system intended to perform many operations on many rational numbers, such bookkeeping details would clutter the programs substantially, to say nothing of what they would do to our minds.
We are using here a powerful strategy of synthesis: wishful thinking.
In general, the underlying idea of data abstraction is to identify for each type of data object a basic set of operations in terms of which all manipulations of data objects of that type will be expressed, and then to use only those operations in manipulating the data.
Consider the notion of a pair, which we used in order to define our rational numbers. We never actually said what a pair was, only that the language supplied procedures cons, car, and cdr for operating on pairs. But the only thing we need to know about these three operations is that if we glue two objects together using cons we can retrieve the objects using car and cdr.
It is no exaggeration to regard this as the most fundamental idea in programming: The evaluator, which determines the meaning of expressions in a programming language, is just another program.