EVOLUTION OF NUMERALS - Numbers and Psychology - Numbers: Their Tales, Types, and Treasures

Numbers: Their Tales, Types, and Treasures.

Chapter 2: Numbers and Psychology


Counting is a cultural invention. The beginning of systematic counting can already be seen at a very primitive level, based on the number two. It is reported that at the beginning of the twentieth century, some indigenous tribes in Australia, South America, and South Africa still had a number-word limit of two but were able to express numbers up to four using the scheme one, two, two-one, two-two. It would be easy to go beyond four with that method, such as two-two-one for five, and two-two-two for six. However, there seems to be no real need for this in a foraging society; hence, it is usually not done. It is commonly assumed that the language for counting was developed after people became sedentary. A nonterminating progression of counting words was not invented overnight. Rather, it was an intricate and long-lasting process, which took place in times of unrecorded history. Number words belong to the oldest parts of the vocabulary, and many languages still reflect some of the early encountered difficulties, which give indirect evidence of this development process. For example, the English words eleven and twelve are related to the Gothicain-liv and twa-lif, which mean “one-left” and “two-left.” This hints at an early stage in the development of a Proto-Germanic language, where ten was the upper limit of number words and people faced a situation in which, after counting to ten, one or two objects still remained.

The numbers one, two, three, and four play a special role in many languages. In social life, they correspond to the elementary ideas of “me/alone,” “you/pair,” “someone else,” and “two pairs.” Thus, they belong to the oldest words in any language, and they are the only number words that are occasionally changed according to the gender and case of the objects to which they belong. In Latin, the first four numbers (unus, duos, tres, quattuor) are declined like adjectives, while beginning with five (quinque) the Latin numerals are invariable. Even in today's German, onewould be ein in the masculine form and eine if it refers to a feminine noun. Two and three were also inflected like real adjectives in Old and Middle-High German, but they have lost their variability in modern German. An old word for the masculine form of two in German is zween, which survived in English as twain and twenty. Today's German word zwei was originally the neutral gender, while the old feminine form zwo is still occasionally heard today in counting, but only for clarity of speech. Also, bear in mind that more than half of the words in the English language stem from the German language.

In English, the special role of one, two, and three is still seen in the ordinal words that are usually created by adding a th (like fourth, fifth, etc.). But the ordinal number words corresponding to one, two, and three are first, second, and third.

The special role of numerals up to three or four might well be related to the subitizing limit mentioned earlier. It is also visible in writing. In most writing systems, the numerals 1, 2, and 3 are derived from a symbol, which can be thought of representing one, two, or three fingers or counting sticks: The corresponding Roman numerals are I, II, and III. In Chinese, it is 一, 二, and 三. Beyond three, the Chinese have symbols of different origin—四, 五, 六, 七, 八, 九. It would be too difficult to distinguish and recognize at a glance symbols made of four, five, or six parallel lines. Therefore, other types of symbols, more useful for practical purposes, came into use for the representation of higher numerals. It is most likely that two and three parallel lines, which over time became connected in writing, also were the origin of our digits 2 and 3 (see figure 2.6).


Figure 2.6: Our digits 1, 2, and 3 probably evolved from the corresponding number of lines.