indivisibles, do not suffice to prove that motion and change and plurality are impossible. They are not, however, 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:[1]
“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
- ↑ Op. cit., p. 367.
- ↑ 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 continuous, 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 Arist. 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.”
