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

literature of the subject know that it has long been customary to make regarding space certain other statements to which the plain man does not usually make serious objection when he is introduced to them. Thus it is said:—

(1) The idea of space is necessary. We can think of objects in space as annihilated, but we cannot conceive space to be annihilated. We can clear space of things, but we cannot clear away space itself, even in thought.

(2) Space must be infinite. We cannot conceive that we should come to the end of space.

(3) Every space, however small, is infinitely divisible. That is to say, even the most minute space must be composed of spaces. We cannot, even theoretically, split a solid into mere surfaces, a surface into mere lines, or a line into mere points.

Against such statements the plain man is not impelled to rise in rebellion, for he can see that there seems to be some ground for making them. He can conceive of any particular material object as annihilated, and of the place which it occupied as standing empty; but he cannot go on and conceive of the annihilation of this bit of empty space. Its annihilation would not leave a gap, for a gap means a bit of empty space; nor could it bring the surrounding spaces into juxtaposition, for one cannot shift spaces, and, in any case, a shifting that is not a shifting through space is an absurdity.

Again, he cannot conceive of any journey that would bring him to the end of space. There is no more reason for stopping at one point than at another; why not go on? What could end space?

As to the infinite divisibility of space, have we not, in addition to the seeming reasonableness of the doctrine, the testimony of all the mathematicians? Does any one of them ever dream of a line so short that it cannot be divided into two shorter lines, or of an angle so small that it cannot be bisected?

24. SPACE AS NECESSARY AND SPACE AS INFINITE.—That these statements about space contain truth one should not be in haste to deny. It seems silly to say that space can be annihilated, or that one can travel "over the mountains of the moon" in the hope of reaching the end of it. And certainly no prudent man wishes to quarrel with that coldly rational creature the mathematician.

But it is well worth while to examine the statements carefully and to see whether there is not some danger that they may be understood in such a way as to lead to error. Let us begin with the doctrine that space is necessary and cannot be "thought away."

As we have seen above, it is manifestly impossible to annihilate in thought a certain portion of space and leave the other portions intact. There are many things in the same case. We cannot annihilate in thought one side of a door and leave the other side; we cannot rob a man of the outside of his hat and leave him the inside. But we can conceive of a whole door as annihilated, and of a man as losing a whole hat. May we or may we not conceive of space as a whole as nonexistent?

I do not say, be it observed, can we conceive of something as attacking and annihilating space? Whatever space may be, we none of us think of it as a something that may be threatened and demolished. I only say, may we not think of a system of things—not a world such as ours, of course, but still a system of things of some sort—in which space relations have no part? May we not conceive such to be possible?

It should be remarked that space relations are by no means the only ones in which we think of things as existing. We attribute to them time relations as well. Now, when we think of occurrences as related to each other in time, we do, in so far as we concentrate our attention upon these relations, turn our attention away from space and contemplate another aspect of the system of things. Space is not such a necessity of thought that we must keep thinking of space when we have turned our attention to something else. And is it, indeed, inconceivable that there should be a system of things (not extended things in space, of course), characterized by time relations and perhaps other relations, but not by space relations?

It goes without saying that we cannot go on thinking of space and at the same time not think of space. Those who keep insisting upon space as a necessity of thought seem to set us such a task as this, and to found their conclusion upon our failure to accomplish it. "We can never represent to ourselves the nonexistence of space," says the German philosopher Kant (1724-1804), "although we can easily conceive that there are no objects in space."

It would, perhaps, be fairer to translate the first half of this sentence as follows: "We can never picture to ourselves the nonexistence of space." Kant says we cannot make of it a Vorstellung, a representation. This we may freely admit, for what does one try to do when one makes the effort to imagine the nonexistence of space? Does not one first clear space of objects, and then try to clear space of space in much the same way? We try to "think space away," i.e. to remove it from the place where it was and yet keep that place.

What does it mean to imagine or represent to oneself the nonexistence of material objects? Is it not to represent to oneself the objects as no longer in space, i.e. to imagine the space as empty, as cleared of the objects? It means something in this case to speak of a Vorstellung, or representation. We can call before our minds the empty space. But if we are to think of space as nonexistent, what shall we call before our minds? Our procedure must not be analogous to what it was before; we must not try to picture to our minds the absence of space, as though that were in itself a something that could be pictured; we must turn our attention to other relations, such as time relations, and ask whether it is not conceivable that such should be the only relations obtaining within a given system.

Those who insist upon the fact that we cannot but conceive space as infinite employ a very similar argument to prove their point. They set us a self-contradictory task, and regard our failure to accomplish it as proof of their position. Thus, Sir William Hamilton (1788-1856) argues: "We are altogether unable to conceive space as bounded—as finite; that is, as a whole beyond which there is no further space." And Herbert Spencer echoes approvingly: "We find ourselves totally unable to imagine bounds beyond which there is no space."

Now, whatever one may be inclined to think about the infinity of space, it is clear that this argument is an absurd one. Let me write it out more at length: "We are altogether unable to conceive space as bounded—as finite; that is, as a whole in the space beyond which there is no further space." "We find ourselves totally unable to imagine bounds, in the space beyond which there is no further space." The words which I have added were already present implicitly. What can the word "beyond" mean if it does not signify space beyond? What Sir William and Mr. Spencer have asked us to do is to imagine a limited space with a beyond and yet no beyond.

There is undoubtedly some reason why men are so ready to affirm that space is infinite, even while they admit that they do not know that the world of material things is infinite. To this we shall come back again later. But if one wishes to affirm it, it is better to do so without giving a reason than it is to present such arguments as the above.

25. SPACE AS INFINITELY DIVISIBLE.—For more than two thousand years men have been aware that certain very grave difficulties seem to attach to the idea of motion, when we once admit that space is infinitely divisible. To maintain that we can divide any portion of space up into ultimate elements which are not themselves spaces, and which have no extension, seems repugnant to the idea we all have of space. And if we refuse to admit this possibility there seems to be nothing left to us but to hold that every space, however small, may theoretically be divided up into smaller spaces, and that there is no limit whatever to the possible subdivision of spaces. Nevertheless, if we take this most natural position, we appear to find ourselves plunged into the most hopeless of labyrinths, every turn of which brings us face to face with a flat self-contradiction.

To bring the difficulties referred to clearly before our minds, let us suppose a point to move uniformly over a line an inch long, and to accomplish its journey in a second. At first glance, there appears to be nothing abnormal about this proceeding. But if we admit that this line is infinitely divisible, and reflect upon this property of the line, the ground seems to sink from beneath our feet at once.

For it is possible to argue that, under the conditions given, the point must move over one half of the line in half a second; over one half of the remainder, or one fourth of the line, in one fourth of a second; over one eighth of the line, in one eighth of a second, etc. Thus the portions of line moved over successively by the point may be represented by the descending series:

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 passed 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 space 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?