Kant, then, plainly held that each unit must be added separately, and that the same process might be applied to large numbers as to small. Thus, if we desired to find the sum of 25 and 27, we might do so by direct counting, by saying 25 and 1 are 26, 25 and 2 are 27, 25 and 3 are 28, &c. Theoretically, this method is always applicable, though never used in practice, except for small numbers; but on Mr. Mahaffy's principles it is absolutely and altogether impossible beyond a certain limit. We are not to suppose that Kant fell into Mansel's error in reference to large numbers any more than into Mr. Mahaffy's. In p. 63 he says: "Thus our enumeration-and this is more observable in large numbers-is a synthesis according to conceptions, because it takes place according to a common basis of unity-for example the decade."

The more closely we consider the first passage quoted from Kant the more we find it opposed to Mr. Mahaffy's theory. Kant is arguing against the possibility of basing Arithmetic on the mere analysis of concepts, and says that we must have recourse to an intuition. He does not say an intuition in space, and he seems to consider that external objects are required only for the purpose of indicating a definite number of distinct units. We may have recourse, he says, to five points, or our five fingers, and by gradually adding the units so indicated, at length we see the number 12 arise. It does not by any means appear that we must take in the whole collection of objects corresponding to the sum at one simultaneous glance. In order to do this, the objects added ought to be similar, and capable of being compactly arranged in space. There are no objects similar to my five fingers under my control except the five fingers of my other hand, and in the example considered by Kant five has to be added to seven, besides which, Kant says five points will do as well. Moreover-and this is decisive against Mr. Mahaffy-Kant does not think that there need be any intuition corresponding to 7 at all. "We must," says Kant, "go beyond these conceptions, and have recourse

to an intuition which corresponds to one of the two." If there is no intuition of 7 there can be none of 12. The theory of Kant is confirmed by the ordinary practice of inexperienced arithmeticians. Such a person, if desirous of knowing the sum of 7 and 5, would not count one, two, three, four, &c., up to seven, but would say at once seven and one are eight, seven and two are nine, &c., and finally, seven and five are twelve. The process indicated by Kant is precisely similar, and differs only in this, that the separate units of one number are indicated by objects in space instead of by sounds.

If Mr. Mahaffy's theory were correct, a person who forgot anything in his addition table would have to perform a complicated experiment requiring extraordinary rapidity and acuteness of vision in order to obtain the required information, except, indeed, he had recourse to a cogitatio caeca sive symbolica. I would not wish to cast any imputation on Mr. Mahaffy's character, yet I fear that he, as well as Mr. Mill, exhibits unmistakeable symptoms of having been "debauched" by philosophy. Were I criticizing Mr. Mahaffy's theory in itself, and not merely considering whether he is correct in attributing it to Kant, it would be necessary to take notice of the remarks on Mansel: for here Mr. Mahaffy seems to admit that intuitions in time, if we only could get them, would be sufficient for everything.

In the note to p. 15 of Fischer's Commentary, Mr. Mahaffy mentions the passages in Kant on which he relies. The first is that in p. 10 of the Critique, which I have already considered at length. Mr. Mahaffy then observes, that in the transcendental exposition of time Kant makes no mention of arithmetic, and concludes that it must be founded on space. This argument does not appear to be worth much, for arithmetic is not mentioned in the transcendental exposition of space any more than in that of time. Kant mentions in each case only general self-evident propositions, and he thought, as appears from the Critique, p. 124, that

Q

there are no such propositions relating to numbers. He plainly regarded Geometry as the most important branch of Mathematics, and the one which most readily exhibited the truth of his philosophical principles. His account of the method of Algebra (pp. 437, 447) is extremely brief, compared with his account of that of Geometry. In p. 442, he almost identifies Geometry with Mathematics in general.

It is not surprising that the importance of the commutative and distributive laws was not noticed by Kant. He naturally took for granted that the fundamental principles of Mathematics were those recognised by the professed masters of the science. When Mansel and Mr. Mill, contemporaries of Professor Boole, with the Calculus of Operations an actually existing science, make no mention of the commutative and distributive laws, can we wonder that Kant did not anticipate the discoveries of posterity?

Mr. Mahaffy next refers to pp. 177 and 180 of the Critique. The passage in p. 177 is as follows: "With the same ease can it be demonstrated that the possibility of things as quantities, and consequently the objective reality of the category of quantity, can be grounded only in external intuition, and that by its means alone is the notion of quantity appropriated by the internal sense." In this passage there is a mistranslation. The latter part should be, "and by its means alone hereafter applied also to the internal sense."

There is not a word about "the notion of quantity" in the original. It is the possibility of things as quantities that Kant is considering, and it is the objective reality of the category of quantity which cannot be applied to the internal sense, except by means of external intuition. The meaning of the passage will be better understood if it be taken in connexion with what goes before. In p. 176, Kant observes "that to understand the possibility of things according to the categories, and thus to demonstrate the objective reality of the latter, we require not merely intuitions but external intuitions." He then goes on to show the truth of this re

mark in the case of several categories, and among them that of quantity. The whole discussion is intimately connected with Kant's refutation of Idealism, and rests on the general principle that neither time itself, regarded as the permanent substratum of events nor its objective reality as a quantum, nor any determination of time, can be cogitated without external intuition. "We cannot," says Kant, p. 94, "cogitate time unless, in drawing a straight line (which is to serve as the external figurative representation of time), we fix our attention on the act of the synthesis of the manifold, whereby we determine successively the internal sense, and thus attend also to the succession of this determination. Motion, as an act of the subject (not as a determination of an object), consequently the synthesis of the manifold in space, if we make abstraction of space, and attend merely to the act by which we determine the internal sense according to its form, is that which produces the conception of succession." Time, being merely the form of the internal sense, and having in itself no content, cannot be in itself an object of perception, nor be cogitated except by the apprehension of something in time. A right line being perfectly simple, homogeneous, and continuous, and being, moreover, an a priori representation made up of parts whose synthesis, starting from a given point, is possible continuously in only one way, is the most perfect image of time which is possible, and, therefore, in order to cogitate time, we generate a right line in time, or, in other words, successively apprehend or contemplate its parts. This successive contemplation of the parts of space is what Kant means by motion as an act of the subject. In the note, p. 95, he observes, that it belongs not only to geometry but even to transcendental philosophy. It is on this account that geometry must (as we have already seen at length) be in accordance with the laws of time.

In order to ascertain whether the laws of quantity rest ultimately on space or time, we must ask what is the condition which an object must fulfil in order to be a quan

tum. It is not difficult to determine the answer which Kant would give to this question. In p. 181 he says, "The conception of quantity cannot be explained except by saying that it is the determination of a thing whereby it can be cogitated how many times one is placed in it. But this how many times is based on successive repetition; consequently upon time and the synthesis of the homogeneous therein.” In p. 128 we are told that "when the synthesis of the manifold of a phenomenon is interrupted, there results merely an aggregate of several phenomena, and not properly a phenomenon as a quantity." From these two passages we may collect that, in order to be a quantum, an object must be capable of being generated by a continuous synthesis in time. This is the only condition required; there is no reference to space.

This result is confirmed by the proof of the Anticipations of Perception. Mr. Mahaffy observes, in the note p. 96 of Fischer's Commentary, that we consider reality as a quantum "by regarding it as the result of a gradual increase of degrees of sensation generated in successive moments of time from 0 upwards." The possibility then of generating a phenomenon by means of a continuous synthesis in time renders it a quantum without any reference to space. In p. 134 Kant informs us that the two Mathematical Principles of the Understanding "instruct us how phenomena, as far as regards their intuition or the real in their perception, can be generated according to the rules of a mathematical synthesis. Consequently numerical quantities, and with them the determination of a phenomenon as a quantity, can be employed in the one case as well as in the other. Thus, for example, out of 200,000 illuminations by the moon, I might compose and give a priori, that is construct the degree of our sensations of the sun light." Were the science of quantity dependent on space, the laws of quantity could not be applied to the intensity of a sensation which does not occupy space. Objects in space must indeed exhibit the laws of quantity, but such objects are