for the intuition of the composite (matter) one must find the intuition of the simple. This is by the laws of our sensibility, and, hence, in the case of objects of our senses, wholly impossible."

Here Kant takes a double position, if I may so express it. In the closing words of the extract he falls back upon the assertion that the "laws of our sensibility" make it impossible that the absolutely simple should be given in intuition. That is, he simply invokes the magic of an "intuition" in the second sense of the word. But he has admitted, as we have seen, that the simple may apparently be given in intuition. He accepts the minimum sensibile recognized by Berkeley and Hume before him, merely arguing that mathematics furnishes proof that this is a false and deceitful minimum, a composite masquerading in the attire of simplicity. Kant thus maintains: (1) That what is given in intuition must be composite, for, by the law of our sensibility, nothing can be given in intuition that is not composite-which statement, if we accept it as true, ought to close the whole question; and (2) he argues that it is subversive of mathematics to deny the infinite divisibility of what is given in intuition. These positions may be met by maintaining: (1) That the statement that it is a law of our sensibility that the simple cannot be given in intuition is either a baseless assumption, or it is based upon the mathematical reasonings to which Kant refers; and (2) that the opposing doctrine is seen to be by no means subversive of mathematical reasonings, when their significance is clearly understood. What may be said upon these points will be considered later. Before passing on to this I wish to make clear the difficulties above alluded to, which attach to the Kantian doctrine, and which should be honestly faced by those who elect to become its adherents. It will not do to give them a perfunctory glance, call them logical puzzles, and straightway forget them. As we shall see, they are deserving of most serious consideration.

CHAPTER XI

DIFFICULTIES CONNECTED WITH THE KANTIAN DOCTRINE OF SPACE

MORE than two thousand years ago, it was argued by Zeno of Elea that motion is impossible, on the ground that, since space is infinitely divisible, no space, however small, can be passed over by a moving body. To go from one place to another, a body would have to pass through an unlimited number of intermediate spaces. That is, it would have to reach the last term of an unlimited series, which is absurd.

The more clearly this problem is stated, the more evident it seems to become that the difficulty is insurmountable. It appears to arise out of the very notion of space and of motion in space as continuous. "The idea expressed by that word 'continuous,"" says Professor Clifford,1 "is one of extreme importance; it is the foundation of all exact science of things; and yet it is so very simple and elementary that it must have been almost the first clear idea that we got into our heads. It is only this: I cannot move this thing from one position to another, without making it go through an infinite number of intermediate positions. Infinite; it is a dreadful word, I know, until you find out that you are familiar with the thing which it expresses. In this place it means that between any two positions there is some intermediate position; between that and either of the others, again, there is some other intermediate; and so on without any end. Infinite means without any end. If you went on with that work of counting forever, you would never get any further than the beginning of it. At last you would only have two positions very close together, but not the same; and the whole process might be gone over again, beginning with those as many times as you like."

In this extract Professor Clifford plays directly into the hand of Zeno, although it is no part of his purpose to support the con

tention of that philosopher. He is merely trying to make quite clear what we mean by calling space continuous; and is it not generally admitted that space is continuous? But, then, how can anything move through space? The difficulties that beset a moving point Clifford has himself admirably exhibited, and again without the slightest intention of unduly emphasizing these difficulties or of denying the possibility of motion. He writes: 1_

"When a point moves, it moves along some line; and you may say that it traces out or describes the line. To look at something definite, let us take the point where this boundary of red on paper is cut by the surface of water. I move all about together.

Now you know that between any two positions of the point there is an infinite number of intermediate positions. Where are they all? Why, clearly, in the line along which the point moved. That line is the place where all such points are to be found."

". . . It seems a very natural thing to say that space is made up of points. I want you to examine very carefully what this means, and how far it is true. And let us first take the simplest case, and consider whether we may safely say that a line is made up of points. If you think of a very large number-say, a million of points all in a row, the end ones being an inch apart, then this string of points is altogether a different thing from a line an inch long. For if you single out two points which are next one another, then there is no point of the series between them; but if you take two points on a line, however close together they may be, there is an infinite number of points between them. The two things are different in kind, not in degree." 2

"... When a point moves along a line, we know that between any two positions of it there is an infinite number (in this new sense 3) of intermediate positions. That is because the motion is continuous. Each of those positions is where the point was at some instant or other. Between the two end positions on the line, the point where the motion began and the point where it stopped, there is no point of the line which does not belong to that series. We have thus an infinite series of successive positions of a continuously moving point, and in that series are included all the points

1 Op. cit., pp. 143–144.

2 Ibid., pp. 146-147.

* Professor Clifford has used the word "number" in two senses, a quantitative and a qualitative. By number in the latter sense he means simply "unlimited

units."

of a certain piece of line-room. May we say, then, that the line is made up of that infinite series of points?

"Yes; if we mean no more than that the series makes up the points of the line. But no, if we mean that the line is made up of those points in the same way that it is made up of a great many very small pieces of line. A point is not to be regarded as a part of a line, in any sense whatever. It is the boundary between two parts."

Surely Zeno would have welcomed all this as directly establishing his position. "When a point moves along a line, we know that between any two positions of it there is an infinite number . . . of intermediate positions." "Infinite means without any end." The positions with which we are dealing are "the successive positions of a continuously moving point." Hence, to complete its motion over any given line whatever, the moving point must pass, one by one, an endless series of positions, and must finish with the end position. If the moral of this is not that a point cannot move along a line, there is no validity in human reasonings.

Again: The moving point must take, one by one, the "successive positions" in the series. Even the (conscious or unconscious) Kantian has his preference in absurdities, and rejects some rather than others. Clifford does not conceive the point as in two positions at once, or as making some ingenious flank movement by means of which it can " scoop in " a whole stretch of line simultaneously. It must move along the line, from end to end, taking one position at a time, and taking them in their order. It cannot make jumps, and are not the positions "successive"? Its path seems clearly marked out for it a smooth road, and without turnings. Alas! the line is "continuous." The point cannot take successive positions, for have we not seen that no position can immediately succeed any other on a continuous line? "Between any two positions there is some intermediate position; between that and either of the others, again, there is some other intermediate; and so on without any end." Can any living soul conceive the gait that must be adopted by a point, which must move continuously (without jumps?) over a line, and yet is debarred from passing from any one position to the next in the series? It cannot pass first to some position which is not the next, and then get around to the next after a while. That is palpably absurd. And 1 Op. cit., pp. 149-150.

it cannot pass to the next at once, for there is no next. I can imagine the shade of Zeno rubbing its hands over this development of his doctrine. "The way for a point to get on," says Clifford, "is for it never to take the next step." "Of course that means," adds Zeno, with ghostly laughter, "that a point cannot get on at all."

And what shall we say to the statement that, although "all the points of a certain piece of line-room" are included in the "infinite series of successive (sic) positions of a continuously moving point," yet the line is not made up of these points, but is made up "of a great many very small pieces of line"? What are these small pieces of line, which are to be distinguished from the whole series of points? They are not material things, for we are not now discussing a bit of string or a chalk-mark, but we are discussing a geometrical line, an aspect of space. What lies between any two points on the line? More points for one thing. What else?

Bits of line. But what are bits of line? When a point has moved over a line, has it done anything but pass through a series of successive positions? It seems reasonable, at first sight, to assume that such a series of positions is what we mean by a line. We are informed, however, that a point is not to be regarded as part of a line in any sense whatever. It is "the boundary between two parts." Does the assumption of these bits of line, which are not positions, but lie between positions, make more comprehensible the motion of a point over a line?

Manifestly not. If the bits of line could be supposed to take up some of the line-room in such a way as to reduce the number of points, they might be of some help, but no one supposes them to do this. Bits of line or no bits of line, the moving point must Occupy successively all the positions in an infinite series. And if we turn our attention from the points, and confine it to the bits of line, we are no better off. If the number of points is endless, so is the number of bits of line, for these separate the points, which are only their boundaries, and we are forced to ask ourselves how an endless series of bits of line can come to an end in a last bit which completes the line. It is not a whit easier to conceive of a given finite line as composed of bits of line, than it is to conceive of it as composed of points, if we once admit that the line in question is infinitely divisible. We have only added a new element of mystification. What do we mean by these mys