Fundamental Philosophy, Vol. 1 (of 2)

by Jaime Luciano Balmes

The synthesis of the schoolmen consists in the union of conceptions, and does not refuse to admit as analytical the total conceptions, from the decomposition of which results the knowledge of the relations of the partial conceptions.

If Kant had stopped with the judgments of experience, there would be no objection to his doctrine. But extended to the purely intellectual order, it is either inadmissible, or at least expressed without much exactness.

260. Kant says all mathematical judgments are analytic, and that this truth which in his opinion "is certainly incontestible and important on account of its consequences, seems to have hitherto escaped the sagacity of the analysts of human reason, causing very contrary opinions." We think it is the sagacity of his Aristarchus, and not that of the analysts, that is at fault.

"One would certainly think at first sight that the proposition, 7 + 5 = 12, is a purely analytic proposition, which follows from the conception of a sum of seven and five, according to the principle of contradiction. But if we examine it more closely, we find that the conception of the sum of seven and five contains nothing farther than the union of both numbers in one, from which it cannot by any means be inferred what this other number is which contains them both."[25]

Were we to say that whoever hears seven plus five, does not always think of twelve, because he does not see clearly enough that one conception is the same as the other, although it is under a different form, it would be true. But from this it does not follow that the conception is not purely analytic. The mere explanation of both suffices to show their identity.

That this may be better understood, we will invert the equation thus: 12 = 7 + 5. It is evident that if any one does not know that 7 + 5 = 12, he will not know that 12 = 7 + 5. Now, in examining the conception 12, we certainly see 7 + 5 contained in it. Therefore, the conception of 12 is identical with the conception of 7 + 5; and just as, because he who hears 12, does not always think of 7 + 5, we cannot thence infer that 12 does not contain 7 + 5; so, also, we cannot, because he who hears 7 + 5, does not always think of 12, thence infer that the first conception does not contain the second.

The cause of the equivocation is, that the two identical conceptions are presented to the intellect under different forms; and until we have the form, and look to what is under it, we shall not discover the identity. This is not, strictly speaking, reasoning but explanation.

What Kant adds concerning the necessity of recurring, in this case, to an intuition, with respect to one of the numbers, adding five to seven on the fingers, is exceedingly futile. First, in whatever way he adds the five, there will never be anything but the five that is added, and it will neither give more nor less than 7 + 5. Secondly, the successive addition on the fingers is equivalent to saying 1 + 1 + 1 + 1 + 1 = 5. This transforms the expression, 7 + 5 = 12, into this other, 7 + 1 + 1 + 1 + 1 + 1 = 12; but the conception, 1 + 1 + 1 + 1 + 1, has the same relation to 5, as 7 + 5 to 12; therefore, if 7 + 5 are not contained in 12, neither are 7 + 1 + 1 + 1 + 1 + 1 contained in it. It may be replied that Kant does not speak of identity, but of intuitions. This intuition, however, is not the sensation, but the idea; and if the idea, it is only the conception explained. Thirdly, we know this method of intuition not to be even necessary for children. Fourthly, this method is impossible in the case of large numbers.

281. Kant adds that this proposition, "a right line is the shortest distance between two points," is not purely analytic, because the idea of shortest distance is not contained in the idea of right line. Waiving the demonstrations which some authors give, or pretend to give, of this proposition, we shall confine ourselves to Kant's reasons. He forgets that here the right line is not taken alone, but compared with other lines. The idea of right line alone neither does nor can contain the ideas of more or less; for these ideas suppose a comparison. But from the moment the right line and the curve are compared, with respect to length, the relation of superiority of the curve over the right line is seen. The proposition is then the result of the comparison of two purely analytic conceptions with a third, which is length.