The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Preface for the Second Edition

The most primitive of herdsman used a pouch of stones to keep track of the number of sheep he had in the field. As each sheep would enter the field, the herdsman would place a stone in a pile. As the sheep would leave the field, the herdsman would place the stones back into the pouch. If there were stones left on the ground, then some sheep were missing. If there were no stones left, and no sheep left then all was well with the herd. And if there were no more stones but there were more sheep, then somehow the herdsman had picked up an ewe or two.

This correspondence between pouch stones and sheep is one of the most primitive forms of counting known. In today's language, this is known as a one-to-one correspondence, or a bijection between pouch stones and sheep. This kind of counting is continued today when we make an attendance sheet. Each name on the sheet corresponds to exactly one child in the class, and we know some child is missing if he or she does not respond to his or her name. A more important correspondence is found in the grocery store. There we associate a certain number called a price with each item we put in our cart. The items in the cart correspond to a number called the total price of the cart. When we compare our receipt with the objects in the cart, we are imitating the sheep herdsman's pouch stones.

Believe it or not, mathematicians count like the primitive herdsmen. The number 1 is all sets that match up in an exact manner to ...