(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?