CONNECTION OF THE IDEAS OF NUMBER WITH THEIR SIGNS.
51. The connection of ideas and impressions, in a sign, is a most wonderful intellectual phenomenon, and at the same time of the greatest help to our mind. Were it not for this connection, we could scarcely reflect at all upon objects somewhat complex, and above all our memory would be exceedingly limited.[26]
52. Condillac made some excellent remarks upon this matter: in his opinion, we cannot, unaided by signs, count more than three or four. If, indeed, we had no sign but that of unity, we could readily count two, saying one and one. Having only two ideas, we could easily satisfy ourselves that we had twice repeated one. But it is not so easy to be certain of the exactness of our repetition when we have to count three, by saying one and one and one; still, this is not difficult. It is more so to count four, and next to impossible to go as far as ten. If we undertake to abstract the signs, we shall find that it is impossible toform an idea of ten by repeating one; and that it will be alike impossible, if we employ no sign, to make sure that we have repeated one exactly ten times.
53. Suppose the sign two, and one half of the difficulty is obviated; thus it will be much easier to say two and one, than one and one and one. In this supposition four will be no more difficult than was two, since, just as we before said, one and one, two; we now say, two and two, four. The attention before divided four times by the repetition of one, is now only divided twice. Six was before a hard number to count, but, in the present supposition, it is as easy as three was before; for, if we repeat two and two and two, we shall have six. The attention before distracted by six signs, is now distracted only by three. Evidently, if we continue to form the numbers three, four, and so on, expressive of distinct collections, we shall gradually facilitate numeration, until we attain the decimal simplicity now in use.
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.