An Introduction to Philosophy by George Stuart Fullerton (bookreader TXT) 📖

To move at all, it must begin somewhere; it cannot begin with what has no halves, for then it is not moving over any part of the line, as all parts have halves; and it cannot begin with what has halves, for that is not the beginning. What does the point do first? that is the question. Those who tell us about points and lines usually leave us to call upon gentle echo for an answer.

The perplexities of this moving point seem to grow worse and worse the longer one reflects upon them. They do not harass it merely at the beginning and at the end of its journey. This is admirably brought out by Professor W. K. Clifford (1845-1879), an excellent mathematician, who never had the faintest intention of denying the possibility of motion, and who did not desire to magnify the perplexities in the path of a moving point. He writes:—

"When a point moves along a line, we know that between any two positions of it there is an infinite number . . . 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 of a certain piece of line-room." [1]

Thus, we are told that, when a point moves along a line, between any two positions of it there is an infinite number of intermediate positions. Clifford does not play with the word "infinite"; he takes it seriously and tells us that it means without any end: "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."

But really, if the case is as stated, the point in question must be at a desperate pass. I beg the reader to consider the following, and ask himself whether he would like to change places with it:—

(1) If the series of positions is really endless, the point must complete one by one the members of an endless series, and reach a nonexistent final term, for a really endless series cannot have a final term.

(2) The series of positions is supposed to be "an infinite series of successive positions." The moving point must take them one after another. But how can it? Between any two positions of the point there is an infinite number of intermediate positions. That is to say, no two of these successive positions must be regarded as next to each other; every position is separated from every other by an infinite number of intermediate ones. How, then, shall the point move? It cannot possibly move from one position to the next, for there is no next. Shall it move first to some position that is not the next? Or shall it in despair refuse to move at all?

Evidently there is either something wrong with this doctrine of the infinite divisibility of space, or there is something wrong with our understanding of it, if such absurdities as these refuse to be cleared away. Let us see where the trouble lies.

26. WHAT IS REAL SPACE?—It is plain that men are willing to make a number of statements about space, the ground for making which is not at once apparent. It is a bold man who will undertake to say that the universe of matter is infinite in extent. We feel that we have the right to ask him how he knows that it is. But most men are ready enough to affirm that space is and must be infinite. How do they know that it is? They certainly do not directly perceive all space, and such arguments as the one offered by Hamilton and Spencer are easily seen to be poor proofs.

Men are equally ready to affirm that space is infinitely divisible. Has any man ever looked upon a line and perceived directly that it has an infinite number of parts? Did any one ever succeed in dividing a space up infinitely? When we try to make clear to ourselves how a point moves along an infinitely divisible line, do we not seem to land in sheer absurdities? On what sort of evidence does a man base his statements regarding space? They are certainly very bold statements.

A careful reflection reveals the fact that men do not speak as they do about space for no reason at all. When they are properly understood, their statements can be seen to be justified, and it can be seen also that the difficulties which we have been considering can be avoided. The subject is a deep one, and it can scarcely be discussed exhaustively in an introductory volume of this sort, but one can, at least, indicate the direction in which it seems most reasonable to look for an answer to the questions which have been raised. How do we come to a knowledge of space, and what do we mean by space? This is the problem to solve; and if we can solve this, we have the key which will unlock many doors.

Now, we saw in the last chapter that we have reason to believe that we know what the real external world is. It is a world of things which we perceive, or can perceive, or, not arbitrarily but as a result of careful observation and deductions therefrom, conceive as though we did perceive it—a world, say, of atoms and molecules. It is not an Unknowable behind or beyond everything that we perceive, or can perceive, or conceive in the manner stated.

And the space with which we are concerned is real space, the space in which real things exist and move about, the real things which we can directly know or of which we can definitely know something. In some sense it must be given in our experience, if the things which are in it, and are known to be in it, are given in our experience. How must we think of this real space?

Suppose we look at a tree at a distance. We are conscious of a certain complex of color. We can distinguish the kind of color; in this case, we call it blue. But the quality of the color is not the only thing that we can distinguish in the experience. In two experiences of color the quality may be the same, and yet the experiences may be different from each other. In the one case we may have more of the same color—we may, so to speak, be conscious of a larger patch; but even if there is not actually more of it, there may be such a difference that we can know from the visual experience alone that the touch object before us is, in the one case, of the one shape, and, in the other case, of another. Thus we may distinguish between the stuff given in our experience and the arrangement of that stuff. This is the distinction which philosophers have marked as that between "matter" and "form." It is, of course, understood that both of these words, so used, have a special sense not to be confounded with their usual one.

This distinction between "matter" and "form" obtains in all our experiences. I have spoken just above of the shape of the touch object for which our visual experiences stand as signs. What do we mean by its shape? To the plain man real things are the touch things of which he has experience, and these touch things are very clearly distinguishable from one another in shape, in size, in position, nor are the different parts| of the things to be confounded with each other. Suppose that, as we pass our hand over a table, all the sensations of touch and movement which we experience fused into an undistinguishable mass. Would we have any notion of size or shape? It is because our experiences of touch and movement do not fuse, but remain distinguishable from each other, and we are conscious of them as arranged, as constituting a system, that we can distinguish between this part of a thing and that, this thing and that.

This arrangement, this order, of what is revealed by touch and movement, we may call the "form" of the touch world. Leaving out of consideration, for the present, time relations, we may say that the "form" of the touch world is the whole system of actual and possible relations of arrangement between the elements which make it up. It is because there is such a system of relations that we can speak of things as of this shape or of that, as great or small, as near or far, as here or there.

Now, I ask, is there any reason to believe that, when the plain man speaks of space, the word means to him anything more than this system of actual and possible relations of arrangement among the touch things that constitute his real world? He may talk sometimes as though space were some kind of a thing, but he does not really think of it as a thing.

This is evident from the mere fact that he is so ready to make about it affirmations that he would not venture to make about things. It does not strike him as inconceivable that a given material object should be annihilated; it does strike him as inconceivable that a portion of space should be blotted out of existence. Why this difference? Is it not explained when we recognize that space is but a name for all the actual and possible relations of arrangement in which things in the touch world may stand? We cannot drop out some of these relations and yet keep space, i.e. the system of relations which we had before. That this is what space means, the plain man may not recognize explicitly, but he certainly seems to recognize it implicitly in what he says about space. Men are rarely inclined to admit that space is a thing of any kind, nor are they much more inclined to regard it as a quality of a thing. Of what could it be the quality?

And if space really were a thing of any sort, would it not be the height of presumption for a man, in the absence of any direct evidence from observation, to say how much there is of it—to declare it infinite? Men do not hesitate to say that space must be infinite. But when we realize that we do not mean by space merely the actual relations which exist between the touch things that make up the world, but also the possible relations, i.e. that we mean the whole plan of the world system, we can see that it is not unreasonable to speak of space as infinite.

The material universe may, for aught we know, be limited in extent. The actual space relations in which things stand to each other may not be limitless. But these actual space relations taken alone do not constitute space. Men have often asked themselves whether they should conceive of the universe as limited and surrounded by void space. It is not nonsense to speak of such a state of things. It would, indeed, appear to be nonsense to say that, if the universe is limited, it does not lie in void space. What can we mean by void space but the system of possible relations in which things, if they exist, must stand? To say that, beyond a certain point, no further relations are possible, seems absurd.

Hence, when a man has come to understand what we have a right to mean by space, it does not imply a boundless conceit on his part to hazard the statement that space is infinite. When he has said this, he has said very little. What shall we say to the statement that space is infinitely divisible?