54. It may here be asked if the actual system be the most perfect possible? And if facility depend upon the distribution of collections in signs, can there be any thing more perfect than this distribution? Either there is question of new signs to denote new collections, or of the combination of signs. There can be no number which we cannot express with our present system, and consequently there is no need of inventing any thing to denote new collections. New signs might perhaps be invented for these collections, and these collections might possibly be distributed in a simpler and more convenient manner. In this case we admit an amelioration to be possible, though very difficult; but none in the former. In a word, the only possible progress would be in expressing better, not in expressing more.
55. The sign connects many ideas which, without it, would be isolated; hence its necessity in many cases, its utility in all cases. With the word hundred, or its numerical representative, 100, we know that we have one repeated a hundred times. Were this help to fail, we could not speak of a hundred, base calculations upon it, or even form it. It is, however, well said that we do not succeed in forming it except by tens, by repeating the calculation ten ten times.
56. Let it not, therefore, be thought that the idea of the number is the idea of the sign; for evidently the same idea of ten corresponds to the word ten, whether written, spoken, or numerically represented by the figures 10, although these three signs are very different. Every language has a word of its own to express ten, and all people have the same idea of it.
57. This last remark creates a difficulty as to what the idea of ten consists in. We cannot say that it is the recollection of the repetition of one ten times; first, because we do not think of this recollection when thinking of ten; and second, because, according to what has already been said, a clear recollection of this repetition is impossible. Neither is it the idea of the sign, for the idea signified existed before the sign was invented, otherwise the invention would have had no object, and would even have been impossible. There can be no sign where there is nothing to signify.
The idea of number includes more difficulties than Condillac ever imagined; who, if he had, after his close analysis of what facilitates numeration, profoundly meditated upon the idea itself, would not so readily have censured St. Augustine, Malebranche, and the whole Platonic school, for having said that numbers perceived by the pure understanding are something superior to those perceived by the senses.
[CHAPTER VII.]
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.