The notion of unity must however find its point of departure in experience, at first under a concrete form; Although it may enter consciousness by several doors, some psychologists, with no legitimate reason, have attributed its origin to one definite mode of external, or even internal, perception which they have chosen to the exclusion of all others.
For some, it is the primordial sense, the sense par excellence; touch. The child regards as a unity the object which it can hold in its hand (a ball, a glass), or follow uninterruptedly in all its boundaries. Wherever his operations are interrupted, where there are breaks of surface continuity, he perceives plurality. In other terms unity is the continuous, plurality is the discontinuous. Numerous observations prove that children actually have a far more exact and precocious notion of continuous quantity (extension), than of discontinuous, discrete quantity (number).[91]
For others, it is sight, for which all that was said above may obviously be repeated. The retina replaces the cutaneous surface: an image clearly perceived without discontinuity is the unity; the perception of simultaneous images leaving intermediate lacunæ in the field of vision, gives plurality.
The same may be said of the acoustic sensations. Preyer, in a work on “Arithmogenesis,” claims that “hearing takes first rank in the acquisition of the concept of number.” Number must be felt before it is thought. Ideas of number and of addition have to be acquired, and this, according to him, takes place in the child when it hears and compares sounds. Subsequently, touch and sight complete this first outline. It is known that Leibnitz assimilated music to an unconscious arithmetic. Preyer reverses the proposition and says: Arithmetica est exercitium musicum occultum nescientis se sonos comparare animi.[92]
As against those who seek the origin of the idea of unity in external events, others attribute it to internal experience.
Thus it has been maintained that consciousness of the ego as a monad which knows itself, is the prototype of arithmetical unity. Obviously this assertion is open to numerous objections. To wit, the late formation of the notion of the ego, the fruit of reflexion; its instability,—still more, this unity, like all the preceding, is concrete, complex; it is a composite unity.
The thesis of W. James is very superior: “Number seems to signify primarily the strokes of our attention in discriminating things. These strokes remain in the memory in groups, large or small, and the groups can be compared. The discrimination is, as we know, psychologically facilitated by the mobility of the thing as a total.... A globe is one if undivided; two, if composed of hemispheres. A sand heap is one thing, or twenty thousand things, as we may choose to count it.”[93] This reduction to acts of attention brings us back definitely to the essential and fundamental conditions of abstraction.
Save this last, the hypotheses enumerated (and internal sensation might also have been invoked; e. g., a localised pain as compared with several scattered pains) give only percepts or images, i. e., the raw material of abstract unity. This is itself a subjective notion. We said above ([Chapter II]) that the question whether consciousness starts from the general or the particular is a misstatement, because it applies to the mind which is in process of formation, categories valid only for the adult intelligence. So here. At the outset there is no clear perception of primary unity and subsequent plurality, or vice versa: neither observation nor reasoning justifies an affirmation. There is a confused, indefinite state, whence issues the antithesis of continuous and discontinuous, the primitive equivalents of unity and plurality. It took centuries to arrive at the precise notion of abstract unity as it exists in the minds of the first mathematicians, and this notion is the result of a decomposition, not of any direct and immediate act of postulation. It was necessary to decompose an object or group into its constituent parts, which are or appear to be irreducible. Then a new synthesis of these parts was required to reconstitute the whole, in order that the notion of relation between unity and plurality should be perceived clearly. It cannot be doubted that for the lesser numbers two, three, four, the successive perception of each separate object, and then of the objects apprehended together at a single glance, has aided the work of the mind in the conception of this relation. We have seen that many human races never passed beyond this phase. The abstract notion of unity is that of the indivisible (provisory). It is this abstract quality of the indivisible, fixed by a word, that gives us the scientific idea of unity as opposed to the vulgar notion. Perceived unity is a concrete, conceived unity is a quality, an abstract; and in one sense it may be said that unity, and consequently all abstract number, is a creation of the mind. It results like all abstraction from analysis—dissociation. Like all abstraction, it has an ideal existence; yet this in no way prevents it from being an instrument of marvellous utility.
II. It is owing to this that the sequence of numbers, homogeneous in material, can be constituted; for the identity of unities is the sole condition in virtue of which they can be counted, and the scant numerations of the concrete-abstract period transcended. The sequence is constituted by an invariable process of construction, which may be reduced to addition or subtraction. “Thus the number 2, simplest of all numbers, is a construction in virtue of which unity is added to itself; the number 3 is a construction in virtue of which unity is added to the number 2, and so on in order. If numbers are composed by successively adding unity to itself, or to other numbers already formed by the same process, they are decomposed by withdrawing unity from the previously constructed sums; and thus, to decompose is again to compose other numbers. For example, if 3 is 2 + 1, it is also 4 - 1. Addition and subtraction are two inverse operations whose results are mutually exclusive: they are the sole primitive numerical functions.”[94]
The simplicity and solidity of this process result from its being always identical with itself, and although the series of numbers is unlimited, some one term of the sequence is rigorously determined, because it can always be brought back to its point of departure, unity. In this labor of construction by continuous repetition, two psychological facts are to be noted: