It is unworthy of excellent people to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used.
Gottfried Wilhelm von Leibniz, 1685
The first five chapters focused on ways to use language to describe procedures. Although finding ways to describe procedures succinctly and precisely would be worthwhile even if we did not have machines to carry out those procedures, the tremendous practical value we gain from being able to describe procedures comes from the ability of computers to carry out those procedures astoundingly quickly, reliably, and inexpensively. As a very rough approximation, a typical laptop gives an individual computing power comparable to having every living human on the planet working for you without ever making a mistake or needing a break.
This chapter introduces computing machines. Computers are different from other machines in two key ways:
Whereas other machines amplify or extend our physical abilities, computers amplify and extend our mental abilities.
Whereas other machines are designed for a few specific tasks, computers can be programmed to perform many tasks. The simple computer model introduced in this chapter can perform all possible computations.
The next section gives a brief history of computing machines, from prehistoric calculating aids to the design of the first universal computers. Section 6.2 explains how machines can implement logic. Section 6.3 introduces a simple abstract model of a computing machine that is powerful enough to carry out any algorithm.
We provide only a very shallow introduction to how machines can implement computations. Our primary goal is not to convey the details of how to design and build an efficient computing machine (although that is certainly a worthy goal that is often pursued in later computing courses), but to gain sufficient understanding of the properties nearly all conceivable computing machines share to be able to predict properties about the costs involved in carrying out a particular procedure. The following chapters use this to reason about the costs of various procedures. In Chapter 12, we use it to reason about the range of problems that can and cannot be solved by any mechanical computing machine.