Chapter 10
Primitive Numerals

82. Gesture Symbols. There is little doubt that primitive counting was done on the fingers, that the earliest numeral symbols were groups of the fingers formed by associating a single finger with each individual thing in the group of things whose number it was desired to represent.

Of course the most immediate method of representing the number of things in a group—and doubtless the method first used—is by the presentation of the things themselves or the recital of their names. But to present the things themselves or to recite their names is not in a proper sense to count them; for either the things or their names represent all the properties of the group and not simply the number of things in it. Counting was first done when a group was used to represent the number of things in some other group; of that group it would represent the number only and, therefore, be a true numeral symbol, which it is the sole object of counting to reach.

Counting ignores all the properties of a group except the distinctness or separateness of the things in it and presupposes whatever intelligence is required consciously or unconsciously to abstract this from its remaining properties. On this account, that group serves best to represent numbers, in which the individual differences of the members are least obtrusive. The naturalness of finger-counting, therefore, lies not only in the accessibility of the fingers, in their being always present to the counter, but in this: that the fingers are so similar in form and function that it is almost easier to ignore than to take account of their differences.

But there is other evidence than its intrinsic probability for the priority of finger-counting over any other. Nearly every system of numeral notation of which we have any knowledge is either quinary, decimal, vigesimal, or a mixture of these;³⁰ that is to say, expresses numbers which are greater than 5 in terms of 5 and lesser numbers, or makes a similar use of 10 or 20. These systems point to primitive methods of reckoning with the fingers of one hand, the fingers of both hands, all the fingers and toes, respectively.

Finger-counting, furthermore, is universal among uncivilized tribes of the present day, even those not far enough developed to have numeral words beyond 2 or 3 representing higher numbers by holding up the appropriate number of fingers.³¹

83. Spoken Symbols. Numeral words—spoken symbols—would naturally arise much later than gesture symbols. Wherever the origin of such a word can be traced, it is found to be either descriptive of the corresponding finger symbol or—when there is nothing characteristic enough about the finger symbol to suggest a word, as is particularly the case with the smaller numbers—the name of some familiar group of things. Thus in the languages of numerous tribes the numeral 5 is simply the word for hand, 10 for both hands, 20 for “an entire man” (hands and feet); while 2 is the word for the eyes, the ears, or wings.³²

As its original meaning is a distinct encumbrance to such a word in its use as a numeral, it is not surprising that the numeral words of the highly developed languages have been so modified that it is for the most part impossible to trace their origin.

The practice of counting with numeral words probably arose much later than the words themselves. There is an artificial element in this sort of counting which does not appertain to primitive counting³³ (see § 5).

One fact is worth reiterating with reference to both the primitive gesture symbols and word symbols for numbers. There is nothing in either symbol to represent the individual characteristics of the things counted or their arrangement. The use of such symbols, therefore, presupposes a conviction that the number of things in a group does not depend on the character of the things themselves or on their collocation, but solely on their maintaining their separateness and integrity.

84. Written Symbols. The earliest written symbols for number would naturally be mere groups of strokes—-$|$, $\left|\right|$, $\left|\right||$, etc. Such symbols have a double advantage over gesture symbols: they can be made permanent, and are capable of indefinite extension—there being, of course, no limit to the numbers of strokes which may be drawn.