If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.
If things are a many, they will be infinite in number; for there will always be other things between them, and others again between these. And so things are infinite in number. R. P. 133.[[885]]
The unit.
161. If we hold that the unit has no magnitude—and this is required by what Aristotle calls the argument from dichotomy,[[886]]—then everything must be infinitely small. Nothing made up of units without magnitude can itself have any magnitude. On the other hand, if we insist that the units of which things are built up are something and not nothing, we must hold that everything is infinitely great. The line is infinitely divisible; and, according to this view, it will be made up of an infinite number of units, each of which has some magnitude.
That this argument refers to points is proved by an instructive passage from Aristotle’s Metaphysics.[[887]] We read there—
If the unit is indivisible, it will, according to the proposition of Zeno, be nothing. That which neither makes anything larger by its addition to it, nor smaller by its subtraction from it, is not, he says, a real thing at all; for clearly what is real must be a magnitude. And, if it is a magnitude, it is corporeal; for that is corporeal which is in every dimension. The other things, i.e. the plane and the line, if added in one way will make things larger, added in another they will produce no effect; but the point and the unit cannot make things larger in any way.
From all this it seems impossible to draw any other conclusion than that the “one” against which Zeno argued was the “one” of which a number constitute a “many,” and that is just the Pythagorean unit.
Space.
162. Aristotle refers to an argument which seems to be directed against the Pythagorean doctrine of space,[[888]] and Simplicius quotes it in this form:[[889]]
If there is space, it will be in something; for all that is is in something, and what is in something is in space. So space will be in space, and this goes on ad infinitum, therefore there is no space. R. P. 135.