What, we shall now ask, can it be that binds together these sense qualities so that we speak of them as a thing? There are two elements that seem to enter into everything whatever that comes into our experience, and which it seems to us would remain if everything in the universe were annihilated. These are space and time. Are they reality? Here we are met with a new kind of difficulty. It was possible to dismiss material substance as a false idea, an idea of something whose existence is impossible; but space and time are certainly not false ideas. The difficulty about them is that we cannot make our thought of them consistent--they are ideas that contain a self-contradiction, or at least that lead to a self-contradiction when we affirm them of reality. With the ideas of space and time are closely linked the ideas of change, of movement, of causation, of quality and quantity, and all of these exhibit this same puzzling characteristic, that they seem to make us affirm what we deny and deny what we affirm. I might fill this little book with illustrations of the paradoxes that are involved in these ordinary working ideas. Everyone is familiar with the difficulty involved in the { 34} idea of time. We must think there was a beginning, and we cannot think that there was any moment to which there was no before. So also with space, it is an infinite extension which we can only think of as a beyond to every limit. This receding limit of the infinitely extensible space involves the character of infinite divisibility, for if there are an infinite number of points from which straight lines can be drawn without intersecting one another to any fixed point there is therefore no smallest space that cannot be further divided. The contradictions that follow from these demonstrable contents of the idea of space are endless. The relation of time to space is another source of contradictory ideas. I shall perhaps, however, best make the meaning of this self-contradictory character of our ordinary ideas clear by following out a definite illustration. What is known as the antinomy of motion is probably familiar to everyone from the well-known paradox of the Greek philosopher Zeno. The flying arrow, he said, does not move, because if it did it would be in two places at one and the same time, and that is impossible. I will now put this same paradox of movement in a form which, so far as I know, it has not been presented before. My illustration will involve the idea of causation as well as that of movement. If we suppose a space to be fully occupied, we shall agree that nothing within that space can move without thereby displacing whatever occupies the position into which it moves. That is to say, the movement of any occupant of one position must cause the displacement of the occupant of the new position into which he moves. But on the other hand it is equally clear that the displacement of the occupant of the new position is a prior condition of the possibility of the movement of { 35} the mover, for nothing can move unless there is an unoccupied place for it to move into, and there is no unoccupied place unless it has been vacated by its occupant before the movement begins. We have therefore the clear contradiction that a thing can only move when something else which it causes to move has already moved. Now if we reflect on it we shall see that this is exactly the position we occupy in our three-dimensional space. The space which surrounds us is occupied, and therefore we cannot move until a way is made clear for us, and nothing makes way for us unless we move. We cannot move through stone walls because we cannot displace solid matter, but we can move through air and water because we are able to displace these. The problem is the same. My movement displaces the air, but there is no movement until the air is displaced. Can we escape the contradiction by supposing the displacement is the cause and the movement the effect. Are we, like people in a theatre queue, only able to move from behind forward as the place is vacated for us in front? In that case we should be driven to the incredible supposition that the original cause or condition of our movement is the previous movement of something at the outskirts of our occupied space, that this somewhat moving into the void made possible the movement of the occupant of the space next adjoining, and so on until after a lapse of time which may be ages, which may indeed be infinite, the possibility of movement is opened to us. In fact we must believe that the effect of our movement--namely, the displacement of the previous occupants from the positions we occupy in moving--happened before it was caused.