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.

Inexhaustibility: the positive side of Incompleteness

In 1931, Kurt Gödel presented his famous incompleteness theorem, which has since had a great and continuing impact on logic and the philosophy of mathematics, and has also like no other result in formal logic caught the interest and imagination of the general public.

Gödel presented two results in his 1931 paper, usually referred to as the first and the second incompleteness theorem. The proof of the first incompleteness theorem shows that for every consistent formal axiomatic theory in a wide class of such theories, there is at least one statement which can be formulated in the language of the theory but can neither be proved nor disproved in the theory. Such a statement is said to be undecidable in the theory. By a consistent formal theory is meant one in which no logical contradiction — both a statement A and its negation not-A — can be proved. The second theorem states that for a wide class of such theories T, if T is consistent, the consistency of T cannot be proved using only the axioms of the theory T itself.

In discussions of the meaning and implications of these theorems, their negative or limiting aspects are most often kept in the foreground: a formal theory of arithmetic cannot be complete, a theory cannot be proved consistent using only the resources of that theory. But the second incompleteness theorem also has a positive aspect, which was emphasized by Gödel (Collected Works, vol. III, p. 309, italics in the original):