“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.