178 SCIENTIFIC METHOD IN PHILOSOPHY 



many C's, and yet cannot have passed more than one each 

 instant. Hence the number of instants since the motion 

 began is twice the number of A's passed, though we pre- 

 viously found it was equal to this number. From this 

 result, Zeno's conclusion follows. 



Zeno's arguments, in some form, have afforded grounds 

 for almost all the theories of space and time and infinity 

 which have been constructed from his day to our 

 own. We have seen that all his arguments are valid 

 (with certain reasonable hypotheses) on the assumption 

 that finite spaces and times consist of a finite number of 

 points and instants, and that the third and fourth almost 

 certainly in fact proceeded on this assumption, while the 

 first and second, which were perhaps intended to refute 

 the opposite assumption, were in that case fallacious. 

 We may therefore escape from his paradoxes either by 

 maintaining that, though space and time do consist of 

 points and instants, the number of them in any finite 

 interval is infinite ; or by denying that space and time 

 consist of points and instants at all ; or lastly, by denying 

 the reality of space and time altogether. It would seem 

 that Zeno himself, as a supporter of Parmenides, drew 

 the last of these three possible deductions, at any rate in 

 regard to time. In this a very large number of philoso- 

 phers have followed him. Many others, like M. Bergson, 

 have preferred to deny that space and time consist of 

 points and instants. Either of these solutions will meet 

 the difficulties in the form in which Zeno raised them. 

 But, as we saw, the difficulties can also be met if infinite 

 numbers are admissible. And on grounds which are 

 independent of space and time, infinite numbers, and 

 series in which no two terms are consecutive, must in 

 any case be admitted. Consider, for example, all the 

 fractions less than i, arranged in order of magnitude. 



