ANALYSIS OF THE IDEA OF NUMBER IN ITSELF AND IN ITS RELATIONS WITH SIGNS.
58. In order clearly to conceive the idea of number, and the way it is engendered in our mind, let us study its formation in a deaf and dumb person.
We have no better way of giving such a one an idea of unity than by presenting an object to him. Now, if we would convey to him the idea of two, we show him two fingers, then two oranges, then two books, and in each of these operations make a sign which must be always the same. If we repeat this operation a number of times, the deaf and dumb person will associate the idea of two with that of the sign, and one will suggest the others; and he will endeavor to show us that he has seen two objects of some kind, by uniting the expression of the object with the sign of two. The same will take place with three, or four. When we reach higher numbers, the sign becomes more indispensable; since the less easily the idea of number is represented, the more necessary is the sign to secure it. But what we do to convey an idea of number to the deaf and dumb person, what he himself must do to express the number which he conceives, we must all do if we would obtain the idea.
59. Numeration is a repetition of operations; and the art of facilitating it consists in instituting signs which recall to our memory what we have done. It is an exceedingly complicated labyrinth, and we cannot trust ourselves to its windings with any expectation of finding our wayout again, if we do not take care to mark the path we have followed.
It is to the admirable simplicity of the decimal system, united to its inexhaustible variety, that the facility and fecundity of our arithmetic are due. Algebra, going a step beyond, expresses without determining numbers, and presents the results of its operations without effacing its footsteps on the road travelled, is far superior to arithmetic, and has made the human mind take gigantic strides. But how? Solely by aiding the memory. Thus, the very principle that enables the child to say four and one, five, instead of adding unity five times to unity, the dumb man to express five by a hand, a hundred by a grain, enables the algebraist to express the result of his longest operations by a formula easy of retention by the memory. Both attain their object simply by aiding the memory. A grain of wheat denotes to the dumb man the idea of hundred, and this he applies to all similar collections; a few letters combined in a simple manner designate to the mathematician a property of certain quantities, and this he applies to all which are found in the same case.
60. Numeration is only an aggregation of formulas; and the more easy these are of mutual transformation with a slight modification, the more perfect will be the numeration. The better one knows the relations of these formulas and the manner of transforming them, the better will he know how to count. The greater a person's intellectual power of fixing simultaneously the attention upon many formulas, and of composing them, the more perfect arithmetician will he be, because the simultaneous comparison of many, leads to the perception of new relations.
61. What is our idea of hundred? The union of the units composing it, a union which we have made more or less frequently when learning to count. But how do we knowthat it is the same union? Because we have a formula called a hundred, expressed by a sign 100. This formula is so easily recollected that we have no difficulty in recollecting the idea of hundred and all the properties connected with it. We may be asked if a hundred is more than ninety. Were we under the necessity counting one and one and one, we should be bewildered, and never succeed in distinguishing the greater; but knowing as we do that to reach the formula hundred, we must pass by another formula ninety, and that this was in ascending, we know, once for all, that hundred expresses ninety and something more, that is, a hundred is more than ninety. And if it be further inquired what is the excess, we shall not undertake to ascertain this by adding units, but by the two formulas ninety and ten which compose the formula hundred.
62. By generalization we unite many similar things in one idea. The general idea is a kind of formula. Numeration unites in one sign many things contained in a general idea, but this sign has, at the same time, its own distinctive character. Thus the general idea belongs as a predicate to each of its particular objects; number belongs to no one in particular, but to all joined. We perceive in abstraction a common property, and lay aside all the particular objects which it presents; in numeration, we perceive similarity, but always with distinction. Abstraction is the result of comparison, but not comparison. Numeration implies a permanent comparison, or the recollection of it.
63. The idea of number is not conventional; a hundred is always a hundred with all its properties and relations, and this, too, prior to all convention and even to all human perception. The sign, and the sign only, is conventional. Were there no intellectual creature, and a hundred beings distinct among themselves were to exist, there would reallybe this number. The number three exists in the august mystery of the Trinity, from all eternity, and of absolute necessity. Number requires only the existence of distinct things; since, however unlike they may be, they always have something in common being, which may be included in a general idea, and consequently they fulfil the two conditions necessary to number.
64. The perception of being and of distinction, that is, of substantive being and of relative not-being, is the perception of number. The science of the relations of every collection, with its measure, which is unity, is the science of numbers.