THE PROBLEM OF INFINITY 171 



indivisibles, do not suffice to prove that motion and 

 change and plurality are impossible. They are not, how- 

 ever, on any view, mere foolish quibbles : they are 

 serious arguments, raising difficulties which it has taken 

 two thousand years to answer, and which even now are 

 fatal to the teachings of most philosophers. 



The first of Zeno's arguments is the argument of the 

 race-course, which is paraphrased by Burnet as follows : * 



" You cannot get to the end of a race-course. You 

 cannot traverse an infinite number of points in a finite 

 time. You must traverse the half of any given distance 

 before you traverse the whole, and the half of that again 

 before you can traverse it. This goes on ad infinitum ^ 

 so that there are an infinite number of points in any 

 given space, and you cannot touch an infinite number 

 one by one in a finite time." 2 



Zeno appeals here, in the first place, to the fact that 



1 op. cit. y p. 367. 



2 Aristotle's words are : " The first is the one on the non-existence of 

 motion on the ground that what is moved must always attain the middle 



point sooner than the end-point, on which we gave our opinion in the 

 earlier part of our discourse." Phys., vi. 9. 939B (R.P. 136). Aristotle 

 seems to refer to Phys., vi. 2. 223AB [R.P. 136A] : "All space is continu- 

 ous, for time and space are divided into the same and equal divisions. . . . 

 Wherefore also Zeno's argument is fallacious, that it is impossible to go 

 through an infinite collection or to touch an infinite collection one by one 

 in a finite time. For there are two senses in which the term 'infinite' 

 is applied both to length and to time, and in fact to all continuous things, 

 either in regard to divisibility, or in regard to the ends. Now it is not 

 possible to touch things infinite in regard to number in a finite time, but 

 it is possible to touch things infinite in regard to divisibility : for time 

 itself also is infinite in this sense. So that in fact we go through an infinite, 

 [space] in an infinite [time] and not in a finite [time], and we touch infinite 

 things with infinite things, not with finite things." Philoponus, a sixth- 

 century commentator (R.P. 136A, Exc. Paris Philop. in ArisL Phys., 

 803, 2. Vit.), gives the following illustration : "For if a thing were moved 

 the space of a cubit in one hour, since in every space there are an infinite 

 number of points, the thing moved must needs touch all the points of 

 the space : it will then go through an infinite collection in a finite time, 

 which is impossible." 



