1/2, 1/4, 1/8, 1/16, . . . [Greek omicron symbol]
Now, it is quite true that the motion of the point can be described in a number of different ways; but the important thing to remark here is that, if the motion really is uniform, and if the line really is infinitely divisible, this series must, as satisfactorily as any other, describe the motion of the point. And it would be absurd to maintain that _a part_ of the series can describe the whole motion. We cannot say, for example, that, when the point has moved over one half, one fourth, and one eighth of the line, it has completed its motion. If even a single member of the series is left out, the whole line has not been pa.s.sed over; and this is equally true whether the omitted member represent a large bit of line or a small one.
The whole series, then, represents the whole line, as definite parts of the series represent definite parts of the line. The line can only be completed when the series is completed. But when and how can this series be completed? In general, a series is completed when we reach the final term, but here there appears to be no final term. We cannot make zero the final term, for it does not belong to the series at all.
It does not obey the law of the series, for it is not one half as large as the term preceding it--what s.p.a.ce is so small that dividing it by 2 gives us [omicron]? On the other hand, some term just before zero cannot be the final term; for if it really represents a little bit of the line, however small, it must, by hypothesis, be made up of lesser bits, and a smaller term must be conceivable. There can, then, be no last term to the series; _i.e._ what the point is doing at the very last is absolutely indescribable; it is inconceivable that there should be a _very last_.
It was pointed out many centuries ago that it is equally inconceivable that there should be a _very first_. How can a point even begin to move along an infinitely divisible line? Must it not before it can move over any distance, however short, first move over half that distance? And before it can move over that half, must it not move over the half of that? Can it find something to move over that has no halves? And if not, how shall it even start to move? 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 hara.s.s 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 pa.s.s. 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 s.p.a.ce, 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 s.p.a.cE?--It is plain that men are willing to make a number of statements about s.p.a.ce, 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 s.p.a.ce is and must be infinite. How do they know that it is? They certainly do not directly perceive all s.p.a.ce, 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 s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce? They are certainly very bold statements.
A careful reflection reveals the fact that men do not speak as they do about s.p.a.ce 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 s.p.a.ce, and what do we mean by s.p.a.ce? 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 s.p.a.ce with which we are concerned is real s.p.a.ce, the s.p.a.ce 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 s.p.a.ce?
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 pa.s.s our hand over a table, all the sensations of touch and movement which we experience fused into an undistinguishable ma.s.s. 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 const.i.tuting 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 _s.p.a.ce_, the word means to him anything more than this system of actual and possible relations of arrangement among the touch things that const.i.tute his real world? He may talk sometimes as though s.p.a.ce 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 s.p.a.ce should be blotted out of existence. Why this difference? Is it not explained when we recognize that s.p.a.ce 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 _s.p.a.ce_, _i.e._ the system of relations which we had before.
That this is what s.p.a.ce means, the plain man may not recognize explicitly, but he certainly seems to recognize it implicitly in what he says about s.p.a.ce. Men are rarely inclined to admit that s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce must be infinite. But when we realize that we do not mean by s.p.a.ce 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 s.p.a.ce as infinite.
The material universe may, for aught we know, be limited in extent.
The actual s.p.a.ce relations in which things stand to each other may not be limitless. But these actual s.p.a.ce relations taken alone do not const.i.tute s.p.a.ce. Men have often asked themselves whether they should conceive of the universe as limited and surrounded by void s.p.a.ce. 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 s.p.a.ce. What can we mean by void s.p.a.ce 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 s.p.a.ce, it does not imply a boundless conceit on his part to hazard the statement that s.p.a.ce is infinite. When he has said this, he has said very little. What shall we say to the statement that s.p.a.ce is infinitely divisible?
To understand the significance of this statement we must come back to the distinction between appearances and the real things for which they stand as signs, the distinction discussed at length in the last chapter.
When I see a tree from a distance, the visual experience which I have is, as we have seen, not an indivisible unit, but is a complex experience; it has parts, and these parts are related to each other; in other words, it has both "matter" and "form." It is, however, one thing to say that this experience has parts, and it is another to say that it has an infinite number of parts. No man is conscious of perceiving an infinite number of parts in the patch of color which represents to him a tree at a distance; to say that it is const.i.tuted of such strikes us in our moments of sober reflection as a monstrous statement.
Now, this visual experience is to us the sign of the reality, the real tree; it is not taken as the tree itself. When we speak of the size, the shape, the number of parts, of the tree, we do not have in mind the size, the shape, the number of parts, of just this experience. We pa.s.s from the sign to the thing signified, and we may lay our hand upon this thing, thus gaining a direct experience of the size and shape of the touch object.
We must recognize, however, that just as no man is conscious of an infinite number of parts in what he sees, so no man is conscious of an infinite number of parts in what he touches. He who tells me that, when I pa.s.s my finger along my paper cutter, _what I perceive_ has an infinite number of parts, tells me what seems palpably untrue. When an object is very small, I can see it, and I cannot see that it is composed of parts; similarly, when an object is very small, I can feel it with my finger, but I cannot distinguish its parts by the sense of touch. There seem to be limits beyond which I cannot go in either case.
Nevertheless, men often speak of thousandths of an inch, or of millionths of an inch, or of distances even shorter. Have such fractions of the magnitudes that we do know and can perceive any real existence? The touch world of real things as it is revealed in our experience does not appear to be divisible into such; it does not appear to be divisible even so far, and much less does it appear to be infinitely divisible.
But have we not seen that the touch world given in our experience must be taken by the thoughtful man as itself the sign or appearance of a reality more ultimate? The speck which appears to the naked eye to have no parts is seen under the microscope to have parts; that is to say, an experience apparently not extended has become the sign of something that is seen to have part out of part. We have as yet invented no instrument that will make directly perceptible to the finger tip an atom of hydrogen or of oxygen, but the man of science conceives of these little things as though they could be perceived.
They and the s.p.a.ce in which they move--the system of actual and possible relations between them--seem to be related to the world revealed in touch very much as the s.p.a.ce revealed in the field of the microscope is related to the s.p.a.ce of the speck looked at with the naked eye.
Thus, when the thoughtful man speaks of _real s.p.a.ce_, he cannot mean by the word only the actual and possible relations of arrangement among the things and the parts of things directly revealed to his sense of touch. He may speak of real things too small to be thus perceived, and of their motion as through s.p.a.ces too small to be perceptible at all.
What limit shall he set to the possible subdivision of _real_ things?
Unless he can find an ultimate reality which cannot in its turn become the appearance or sign of a further reality, it seems absurd to speak of a limit at all.
We may, then, say that real s.p.a.ce is infinitely divisible. By this statement we should mean that certain experiences may be represented by others, and that we may carry on our division in the case of the latter, when a further subdivision of the former seems out of the question. But it should not mean that any single experience furnished us by any sense, or anything that we can represent in the imagination, is composed of an infinite number of parts.
When we realize this, do we not free ourselves from the difficulties which seemed to make the motion of a point over a line an impossible absurdity? The line as revealed in a single experience either of sight or of touch is not composed of an infinite number of parts. It is composed of points seen or touched--least experiences of sight or touch, _minima sensibilia_. These are next to each other, and the point, in moving, takes them one by one.
But such a single experience is not what we call a line. It is but one experience of a line. Though the experience is not infinitely divisible, the line may be. This only means that the visual or tactual point of the single experience may stand for, may represent, what is not a mere point but has parts, and is, hence, divisible. Who can set a limit to such possible subst.i.tutions? in other words, who can set a limit to the divisibility of a _real line_?
It is only when we confuse the single experience with the real line that we fall into absurdities. What the mathematician tells us about real points and real lines has no bearing on the const.i.tution of the single experience and its parts. Thus, when he tells us that between any two points on a line there are an infinite number of other points, he only means that we may expand the line indefinitely by the system of subst.i.tutions described above. We do this for ourselves within limits every time that we approach from a distance a line drawn on a blackboard. The mathematician has generalized our experience for us, and that is all he has done. We should try to get at his real meaning, and not quote him as supporting an absurdity.
[1] "Seeing and Thinking," p. 149.
CHAPTER VII
OF TIME
27. TIME AS NECESSARY, INFINITE, AND INFINITELY DIVISIBLE.--Of course, we all know something about time; we know it as past, present, and future; we know it as divisible into parts, all of which are successive; we know that whatever happens must happen in time. Those who have thought a good deal about the matter are apt to tell us that time is a necessity of thought, we cannot but think it; that time is and must be infinite; and that it is infinitely divisible.
These are the same statements that were made regarding s.p.a.ce, and, as they have to be criticised in just the same way, it is not necessary to dwell upon them at great length. However, we must not pa.s.s them over altogether.
As to the statement that time is a _necessary_ idea, we may freely admit that we cannot in thought _annihilate_ time, or _think it away_. It does not seem to mean anything to attempt such a task. Whatever time may be, it does not appear to be a something of such a nature that we can demolish it or clear it away from something else. But is it necessarily absurd to speak of a system of things--not, of course, a system of things in which there is change, succession, an earlier and a later, but still a system of things of some sort--in which there obtain no time relations?
The problem is, to be sure, one of theoretical interest merely, for such a system of things is not the world we know.
And as for the infinity of time, may we not ask on what ground any one ventures to a.s.sert that time is infinite? No man can say that infinite time is directly given in his experience. If one does not directly perceive it to be infinite, must one not seek for some proof of the fact?
The only proof which appears to be offered us is contained in the statement that we cannot conceive of a time before which there was no time, nor of a time after which there will be no time; a proof which is no proof, for written out at length it reads as follows: we cannot conceive of a time _in the time_ before which there was no time, nor of a time _in the time_ after which there will be no time. As well say: We cannot conceive of a number the number before which was no number, nor of a number the number after which will be no number. Whatever may be said for the conclusion arrived at, the argument is a very poor one.