would be valid if the traditional contradictions in infinite numbers were insoluble, which they are not.
We may conclude, therefore, that Zeno’s polemic is directed against the view that space and time consist of points and instants; and that as against the view that a finite stretch of space or time consists of a finite number of points and instants, his arguments are not sophisms, but perfectly valid.
The conclusion which Zeno wishes us to draw is that plurality is a delusion, and spaces and times are really indivisible. The other conclusion which is possible, namely, that the number of points and instants is infinite, was not tenable so long as the infinite was infected with contradictions. In a fragment which is not one of the four famous arguments against motion, Zeno says:
“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.”[1]
This argument attempts to prove that, if there are many things, the number of them must be both finite and infinite, which is impossible; hence we are to conclude that there is only one thing. But the weak point in the argument is the phrase: “If they are just as many as they are, they will be finite in number.” This phrase is not very clear, but it is plain that it assumes the impossibility of definite infinite numbers. Without this assumption, which is now known to be false, the arguments of Zeno, though they suffice (on certain very reasonable assumptions) to dispel the hypothesis of finite
- ↑ Simplicius, Phys., 140, 28 D (R.P. 133); Burnet, op. cit., pp. 364-365.
