1. No sooner is unity passed, in the elaboration of numbers, than intuition fails altogether. Directly we reach 5, 6, 7, etc, (the limit varies with the individual), objects can no longer be perceived or represented together: there is now no more in consciousness than the sign, the substitute for the absent intuition: each number becomes a sum of unities fixed by a name.
2. For our unity-type we substitute higher unities, which admit of simplification. Thus in the predominating decimal system, ten and a hundred are unities ten and a hundred times larger than unity, properly so called. They may be of any given magnitude: the Hindus, whose exuberant imagination is well known, invented the koti, equivalent to four billions three hundred and twenty-eight millions of years, for calculating the life of their gods; each koti represents a single day of the divine life.[95]
Inversely, we may consider the unity-type as a sum of identical parts, and represent 1 = 10/10 or 100/100, etc. A tenth, a hundredth, are unities ten times, a hundred times, smaller than unity properly so called, but they obey the same laws in the formation of fractional numbers.
It is well for the psychologist to note the privileged position of what we term the unity-type, or simply 1. It originates in experience, because unity, even when concrete, and apprehended by gross perception, appears as a primitive element, special and irreducible. So long as the mind confines itself to perceiving or imagining, there is in the passage from one object to two, three, or four objects, or inversely in the passage from four objects to three, two, or only one, an augmentation or diminution. But below unity in the first case, and above unity in the second, there is no longer any mental representation; unity seems to border on nonentity and to be an absolute beginning.
From this privileged point the mind can follow two opposite directions, by an identical movement: the one towards the infinitely great, with constant augmentation; the other towards the infinitely small, with constant diminution—but in one sense or the other, infinity is a never exhausted possibility. Here we reach the much disputed question of infinite number: psychology is not concerned with this. For some, infinite number has an actual existence. For others, it only exists potentially, i. e., as an intellectual operation which may, as was said above, add or subtract, without end or intermission.[96]
III. The importance of signs, as the instruments of abstraction and generalisation, is nowhere so well shown as in their multiple applications to discrete or continuous quantity. The history of the mathematical sciences is in part that of the invention, and use of symbols of increasing complexity, whose efficacy is clearly manifested in their theoretical or practical results. In the first place, words were substituted for the things that were held to be numerable; next, particular signs, or figures; later still, with the invention of algebra, letters took the place of figures, or at any rate assumed their function and part in the problem to be solved; later still, the consideration of geometrical figures was replaced by that of their equations; finally, the use of new symbols corresponded with calculations for infinitesimal quantities, negative quantities, and imaginary numbers.
These symbols are such a powerful auxiliary to the labor of the mathematicians that those among them who affect philosophy have gladly discoursed upon their nature and intrinsic value. They seem to be divided into two camps.
One faction attribute reality to the symbols, or at least incline that way. It is the introduction of the nomina numina into mathematics. They maintain that these pretended conventions are only the expression of necessary relations which the mind is obliged, on account of their ideal nature, to represent by arbitrary signs, but which are not invented by caprice, or by the necessity of the individual mind—since this contents itself with laying hold of that which is offered by the nature of the things. Do we not see moreover that the labor accomplished by their aid is, with necessary modifications, applicable to reality?
To the other, symbols are but means, instruments, stratagems. They mock at those who “look upon relations once symbolised as things which have in themselves an à priori scientific content, as idols, which we supplicate to reveal themselves” (Renouvier). Signs, whatever they may be, are nothing more than conventions: negative quantities represent a change in the direction of thought. Imaginary numbers “represent important relations under a simple and abridged form.” Symbols are an aid in surmounting difficulties, as, empirically, the lever and its developments serve for the lifting of weights. “It is not calculation,” said Poinsot, “that is the secret of this art which teaches us to discover; but the attentive consideration of things, wherein the intellect seeks above all to form an idea of them, endeavoring by analysis properly so called to decompose them into other more simple ideas, and to review them again subsequently as if they had been formed by the union of those simpler things of which it had full knowledge.”[97]