Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.10.COUNTING BY THE ORDINAL PRINCIPLE

The ability to do things systematically in some order is a prerequisite for counting. It is already present in primates and is probably connected with what one can do with one's hands. When manipulating several objects at a time, it becomes necessary to do this in a certain order, thereby avoiding doing something at the wrong time or unnecessarily twice. Indeed, primates can be observed to proceed in a very systematic and orderly way when harvesting fruits from the branches of a tree or when searching each other's fur in mutual grooming activities.

More directly related to counting is the observation that objects in a finite set can always be brought in a certain order. This can be done, for example, by arranging them in a row or by sorting them according to size, weight, or some other property. The ordinal principle for counting states that even in case of an apparently disordered set like the one in figure 1.2, we have to take the objects in a certain order, as first, second, third, and so on. In figures 1.4 and 1.5, this order is symbolized by the chain of arrows leading from one object to the next.

But the set of objects to be counted need not have any predefined order. Indeed, the set of pebbles in figure 1.6 is a typical set without natural predefined ordering. During counting, we associate each object in the chain with a unique counting tag. The counting tags are always used in a fixed order; that is, they must be from an ordered set whose elements are in a predefined, strict, and invariable order. Counting tags are useful only if they are always used in the same order. Then the last tag given to the last object in the chain would describe the cardinality of the set.

In mathematics, a set with this kind of strict ordering is called a sequence. In a sequence there is a unique “first element,” and every element in a sequence has a unique successor.

The most natural counting tags are, of course, the familiar symbols (1, 2, 3, 4, 5,…). They have a natural, predefined order. The set of natural numbers used for counting starts with 1, and then every number has a unique successor:

1 < 2 < 3 < 4 < 5 <…(natural order of natural numbers).

The strict order of the cardinal numbers makes them suitable as counting tags, and hence the (finite) natural numbers serve as ordinal numbers and as cardinal numbers at the same time. In the process of counting, each ordinal number is at the same time the cardinal number of the set of already-counted objects.

Obviously, the bijection principle and the ordinal principle closely work together in the process of counting. The counting tags (number words or number symbols) form an ordered sequence with a unique first element, and each counting tag has a unique successor. And when we reach a certain number word during counting, we have actually recited all number words that were first in the sequence. For example, reaching “six” as a result of counting means that we have counted “one, two, three, four, five, six.” The counting procedure thus establishes a one-to-one correspondence (or bijection) between a given set of six elements and the set of number words up to six, as shown, for example, in figure 1.7. We learn from this that a sentence like

“This box contains six items”

is actually a very brief account of the activity of counting. This statement actually means something like

“I have just found a one-to-one correspondence between the set of objects in the box and the following set of number words {one, two, three, four, five, six}.”

And this just means that the set of objects in the box contains exactly as many elements as the initial section of the sequence of number words, which (in virtue of the strict ordering) is uniquely determined by the word six.