The basic idea of counting is a very old and foundational idea for humans. It started with simple matching of the objects of interest with tracking markers such as pebbles to be kept in a bag. From these humble origins, more and more extensive and permanent means of tracking emerged, along with predictive means.

We will not be learning how to count here. Most people naturally pick it up from those around them. But we will look at and analyze the emergence of the natural numbers from that perspective.

We start with the concept of number as correspnding to groups of objects. This leads to not only a notion of equality, but also what it means for there to be more in one group than another.

From this, arises the ideas of using symbols for each number. Eventually, we run out of our ability to know the symbols representing all of this and we need a system of compacting the numbers. This gives rise to using combinations of the symbols (digits) to represent larger numbers.

We then delve into tagging numbers with what they are representing. It turns out that this is an extremely important notion that mathematics often neglects, much to the detriment of the intuition of the mathematical learning.

What happens if we have a different number of base symbols? Conceptually, this is simple. In practice, it is head spinning. So we will just gingerly step down that road a little.

We conclude this chapter with the unanswerable question "Is there a largest number?" We explore some reasonable answers that one might have to this question.