iyo SCIENTIFIC METHOD IN PHILOSOPHY 



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 con- 

 clude 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 



1 Simplicius, Pkys., 140, 28 D (R.P. 133) ; Burnet, op. cit., pp. 364-365. 



