An Introduction to Philosophy

by George Stuart Fullerton

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? 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.