Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting



According to the bijection principle, any act of counting establishes a bijection between the objects to be counted and a set of counting tags. Typically, the counting tags are number words, but, as far as the bijection principle is concerned, they could be anything, even objects of another set. And for small collections, instead of number words one could use letters of an alphabet or the words of a counting-out rhyme, such as “eeny meeny miney mo.” Japanese people, for example, while having a perfectly logical system of number words, occasionally also use a particularly poetic alternative for enumerating up to forty-seven items. They use the syllables of a famous poem, the Iroha, as counting words. The Iroha is an ingenious piece of poetry from the Heian period (794–1185), in which every possible syllable, and thus every sound of the Japanese language, occurs exactly once. Usually written in hiragana (a Japanese writing system), it starts as follows:


















and it goes on to use every single hiragana character once, and only once, without repetitions. The Iroha is a poem about the transience of all being. In English, this line roughly means: “Although the colors (of blossoms) smell, the paint scatters away.” The poem is sometimes used, even today, for teaching the Japanese syllabary. But the fixed order of syllables defined by this poem and the uniqueness of the syllables makes it also suitable for counting (see the bijection principle and the ordinal principle, described above). Indeed, it is sometimes used, for example, to number the seats in a theater:

i, ro, ha, ni, ho, he, do,…su
1, 2, 3, 4, 5, 6, 7,…47

The bijection principle enabled people to deal with numerosities even in a time before number words were invented. In ancient times, a shepherd counted his flock by putting a pebble in a bag for each animal leaving shelter. Therefore, he established a one-to-one correspondence between pebbles in the bag and animals in the herd. Although the shepherd could not count, he would know exactly the number of eventually missing animals. When he took a pebble out of the bag for each animal returning to shelter, pebbles remaining in the bag would precisely indicate the number of lost animals.

Another prehistoric method of counting involved the marking of notches on a tally stick. This created a bijection between marked notches and things to count. This method could also be used to count events. Events are “objects” that lack the property of permanence. Once an event has occurred, it is gone and exists only in memory. In order to count events, they have to be remembered, which is difficult because of the limited capacity of our memory. Therefore, it is a good idea to create a permanent record of each event. For example, we can create a tally mark for each passing day. The oldest tally sticks probably used in that way are some thirty thousand years old. Even more recently, prisoners counted days with scratches on the wall, thereby creating a one-to-one correspondence between days in prison and tally marks.

A common way of counting in prehistoric times was by using body parts. They used not only fingers, but also wrists, elbows, shoulders, then toes, ankles, knees, and hips, all in a fixed order.5 Probably the earliest counting tags were simply the names of the corresponding body parts spoken together with appropriate gestures. Prehistoric people developed concrete procedures with pebbles, tally marks, or body counting, or combinations of these, in order to trade goods or fix the dates of religious festivals. They did all this even before they had developed any abstract knowledge of numbers or sufficient vocabulary for counting in the sense that we understand it today.