Natural Numbers and Integers

Abstract

In this chapter, we recall how natural numbers and integers may be constructively defined, and how to prove the basic rules of computation we learn in school. The purpose is to give a quick example of developing a mathematical theory from a set of very basic facts. The idea is to give the reader the capability of explaining to her/his grandmother why, for example, 2 times 3 is equal to 3 times 2. Answering questions of this nature leads to a deeper understanding of the nature of integers and the rules for computing with integers, which goes beyond just accepting facts you learn in school as something given once and for all. An important aspect of this process is the very questioning of established facts that follows from posing the why,which may lead to new insight and new truths replacing the old ones.

“But”, you might say, “none of this shakes my belief that 2 and 2 are 4”. You are right, except in marginal case... and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition “2 and 2 are 4” is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arrive in which you are doubtful whether two of them are dogs. “Well, at any rate there are four animals” you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. “Well, then living organisms,” you may say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: “Two entities and two entities are four entities”. When you have told me what you mean by “entity” I will resume the argument. (Russell)