2. The numeration, taking its development as a whole, appears to sub-divide into three principal periods: concrete numeration, as studied above, in animals and children; concrete-abstract numeration, with which we are now occupied; purely abstract numeration, which we shall examine later, as translated into organised arithmetic.
We have seen that speech at its origin was so humble as to need gesture to complete and to elucidate it. During its concrete-abstract period, numeration is in an analogous position. At first its extension is very limited: it progresses slowly and painfully from unity. Further, it can operate only when sustained by the concrete; it must have a material accompaniment. Counting is accomplished by the enunciation of words, with the aid of enumerated objects, as perceived at the same time, or with that of the fingers: which, let it be remarked, is the first essay in substitution. There is simultaneously concrete or digital, and verbal numeration.[74]
We know that many Australian and South American tribes can count verbally to two only; some say two-one = three; two-two = four; others by the same process arrive at six (two-three = five, three-three = six): everything else is “much.” For the most part they count without words, with the aid of fingers or of articulation; even when they employ words, the two numerations—digital and verbal—are performed simultaneously.[75]
This manner of counting is in first degree concrete; the concrete-abstract form is only there in embryo. A great advance, made early enough in many tribes, consisted in counting by five, taking the hand (five fingers) as a new unit, superior to the simple unit. Then: one hand = 5; two hands or half a man = 10; two hands, one foot = 15; two hands, two feet, or a man = 20. Such is the evident origin of the quinary, decimal, and vigesimal numerations. Sometimes fingers, as instruments of numeration, have been replaced by objects of a typical number. Ex.: 1 = moon or sun; 2 = the eyes or legs, etc.
However varied these processes (of which only a few have been mentioned) in different races and periods, they are fundamentally identical to the psychologist. They may be reduced to this; numeration is performed more particularly with the aids of sensible perceptions; words are but an insignificant accompaniment, a superfluity—existing only as a proliferation—of so little utility that they are for the most part neglected.
Though it is less often spoken of, we may remark that the measure of continuous quantity passed through the same concrete-abstract phase; and here it appeared at a somewhat early stage, owing to practical and social wants. Hence we find at the outset, the foot, the finger, the thumb (inch = Fr. pouce), the palestra (four fingers’ length), span, cubit (arm’s reach = coudée), fathom, etc., the stadium (distance a good runner could cover without stopping).[76] The concrete character of all these measures is obvious. Again, there are survivals in certain current locutions, such as a day’s journey. More than this; they have a human character, their standard and starting-point being, at least at the outset, certain parts of the body, or a determined sum of muscular movements. Little by little they lost their original significance, progressing through centuries towards our metrical system the type of a scientific, deliberate, rationally abstract system, as far as possible liberated from anthropomorphism.
The reader will probably obtain a more definite idea of the nature of these lower forms by recapitulating the examples cited, than from any long dissertation. Is their intellectual level very superior to that of the generic image? This question is doubtful. At times the only distinction between them is the presence of the word: at the present stage it makes but a poor figure,—yet with all its modesty, it augurs a new world wherein it is to be of prime importance.
II.
We now pass to a study of transition. In ascending from the lower to the higher forms of abstraction, we traverse the intermediate region between the states directly superposed upon generic images, and the higher concepts. In fact, we shall to some extent have to penetrate into this extreme region, before the close of the chapter.
At the risk of repetition, we must first indicate the characteristics by which the general notions we are at present concerned with are distinguished from the abstractions above and below them. To recapitulate briefly: In the concrete-abstract phase (which we are leaving) the general notion—so-called—is constituted by concrete elements, plus words, whose substitutory office is weak or null.