ZENO’S ARGUMENTS ABOUT MOTION
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in the Dichotomy. He observes that time is divisible in
exactly the same way as a length; if therefore a length is
infinitely divisible, so is the corresponding time; he adds
‘ this is why {816} Zeno’s argument falsely assumes that it is
not possible to traverse or touch each of an infinite number of
points in a finite time V thereby implying that Zeno did not
regard time as divisible act infinitum like space. Similarly,
when Leibniz declares that a space divisible ad infinitum
is traversed in a time divisible ad infinitum, he, like Aristotle,
is entirely beside the question. Zeno was perfectly aware that,
in respect of divisibility, time and space have the same
property, and that they are alike, always, and concomitantly,
divisible ad infinitum. The question is how, in the one as
in the other, this series of divisions, by definition inexhaustible,
can be exhausted; and it must be exhausted if motion is to
be possible. It is not an answer to say that the two series
are exhausted simultaneously.
The usual mode of refutation given by mathematicians
from Descartes to Tannery, correct in a sense, has an analogous
defect. To show that the sum of the infinite series 1 + \ + % + ...
is equal to 2, or to calculate (in the Achilles) the exact moment
when Achilles will overtake the tortoise, is to answer the
question when ? whereas the question actually asked is how ?
On the hypothesis of divisibility ad infinitum you will, in the
Dichotomy, never reach the limit, and, in the Achilles, the
distance separating Achilles from the tortoise, though it con
tinually decreases, will never vanish. And if you introduce
the limit, or, with a numerical calculation, the discontinuous,
Zeno is quite aware that his arguments are no longer valid.
We are then in presence of another hypothesis as to the com
position of the continuum; and this hypothesis is dealt with
in the third and fourth arguments. 2
It appears then that the first and second arguments, in their
full significance, were not really met before G. Cantor formu
lated his new theory of continuity and infinity. On this I
can only refer to Chapters xlii and xliii of Mr. Bertrand
Russell’s Principles of Mathematics, vol. i. Zeno’s argument
in the Dichotomy is that, whatever motion we assume to have
taken place, this presupposes another motion ; this in turn
1 Ih. vi. 2, 2B8 a 16-23. 2 Brochard, Joe. cit., p. 9.