Arithmetic

Abstract

The beauty and fascination of numbers can be summed up by one simple fact: anyone can count 1, 2, 3, 4, but no one knows all the implications of this simple process. Let me elaborate. We all realize that the sequence 1, 2, 3, 4, continues 5, 6, 7, 8, and that we can continue indefinitely adding 1. The objects produced by the counting process are what mathematicians call the natural numbers. Thus if we want to say what it is that 1, 2, 3, 17, 643, 100097801, and 4514517888888856 have in common, in short, what a natural number is, we can only say that each is produced by the counting process. This is slightly troubling when you think about it: the simplest, and most finite, mathematical objects are defined by an infinite process. However, the concept of natural number is inseparable from the concept of infinity, so we must learn to live with it and, if possible, use it to our advantage.