What is computation? The question deserves careful thought.
To understand and reshape the world, people have created sets of rules inspired by nature. Many of these rules, including counting systems, have become widely accepted.
We readily understand ideas such as one apple or two oranges. We also have an intuitive sense of what “once” and “twice” mean because we learn these rules from childhood. But what if you had to define those concepts? If an alien arrived one day, how would you explain the idea of “once”?
- Something that appears simple is not necessarily self-evident.
- We may say, “That is just how it is; it is common sense.” The idea is so familiar that we struggle to explain it scientifically.
What is a counting system for? It lets us count. Suppose we have already established some basic concepts. Setting aside the seemingly tedious question above, we can make a few obvious observations.
A counting system tells us that two apples can feed two people. If a third person arrives, each person can no longer receive one whole apple on average.
We can perform operations within this counting system. Addition tells us that two apples plus one apple makes three apples. Division tells us that two apples divided among three people cannot give each person a whole apple. An operation produces new information. Its rules are also conventions created and accepted by people. In this sense, an operation derives one piece of information from another.
Taken together, a counting system and its operations allow us to obtain more information.
These operations usually take place in the human mind. When they become too complicated, we need aids: symbols such as Arabic numerals and +-×÷, computational rules such as column addition, and recording tools such as pen and paper. The symbols and rules can be regarded as a model. Computation produces new results according to deterministic rules.
This general system can produce stable results, but its range may be limited. Addition and multiplication tell us that one apple plus another makes two apples, yet we often want more complex information. We cannot immediately calculate whether it will rain tomorrow. A characteristic human strategy is to break a complicated problem into a sequence of simple steps.
We can build more sophisticated models for more difficult problems. The more complex the problem becomes, however, the more complex its computation will be.
Eventually, the human mind and paper calculations are no longer enough, so we need machines to perform the operations for us. Just as people build vehicles because they do not want to walk everywhere, we build machines when we do not want to calculate by hand or cannot keep up with the calculations.
How can such a machine calculate? It first needs a set of basic operations that physical devices can implement.
The essence of a computer, therefore, is the use of physical devices to realize a computational process.
Before computation begins, a concrete problem must also be transformed into something those rules can process. The question “What will the weather be tomorrow?” must become a collection of additions and multiplications. Those operations produce numerical results, which are then converted back into specific information about the weather.
Because we lack a perfect model that directly describes the weather, we represent it with a numerical model. The numbers can then be processed using our existing computational rules.
Why is computation so important? At the most basic level, modern electronic computers perform calculations like 1+1=2. How can such simple calculations solve an ever-growing range of human problems?
- Can every problem be transformed into calculations of this kind?
- If not, is there a better approach?
Consider brain-inspired computing as a simple example. Suppose I care only about the result rather than the exact computation that produces it. I want to travel from point A to point B but do not know how to plan the route, so I turn to one of nature’s most capable navigators—the ant—for help. The most direct approach would be