Numbers

Counting things is an extraordinarily useful and important part of our everyday thinking. Suppose you are taking the ducklings Quacky, Beaky, Tailfeather, Downhead, and Webfoot on an outing. At several points — getting off the bus, distributing worms, returning home — you're anxious to make sure that none of the ducklings has slipped away or been inadvertently left behind. You can do this by looking for and locating each of the ducklings, or by having them respond when you call out their names. However, if you don't know the ducklings personally and don't have a list of their names it may be quite difficult to keep track of them this way. You can then adopt the expedient of simply counting the ducklings at each check, making sure there are five of them. This way, you don't need to look for Quacky, Beaky, etc. individually, and it doesn't matter if you don't know them or their names.

There is a price to pay for this greater convenience. Counting the ducklings means abstracting from, or in other words disregarding, their particular identities. The only information noted is their number. Thus it is quite compatible with there being at all times five ducklings in your charge that Quacky has absconded and his place been taken by another duckling. If, as is to be assumed, such an event is undesirable, counting will replace individual inspection only when the risk that a substitution will occur is thought to be negligible.

In other cases, it doesn't matter if a substitution takes place, as long as the item substituted is recognizably of the proper kind. This is the case since ancient times when people count coins, sacks of grain or other commercial items, and when generals count soldiers.

The great usefulness of counting amply justifies and explains the occurrence in language of names and notations for the counting numbers 1,2,3,and the very concept of “number” associated with them. “Three soldiers”, “three sacks of grain”, “three days”, are recognized as having in common that there are three of them. When counting, the items counted are tallied against the names of the numbers in a memorized order, and thus a total is arrived at. These totals can be checked and compared with other totals, giving useful information of various kinds.