From the Beginning to Plato

The Arguments about Motion
FROM THE BEGINNING TO PLATO 143
(a)
Aristotle’s evidence
There is only one certain primary source for the content of Zeno’s arguments
about motion: Aristotle, who states and discusses them in Physics VI and VIII (VI
2, 233a21–30; VI 9, 239b5–240a18; VIII 8, 263a4–b9). Aristotle’s source is not
known; no book of Zeno that might have contained them is recorded. It is
perfectly possible that they reached Aristotle by oral tradition. In any case, while
there is no reason to doubt that they are substantially authentic, there is also no
reason to suppose that Zeno’s own formulations have been faithfully preserved.
(A source possibly independent of Aristotle is mentioned in (d) below.)
The four individual arguments, as Aristotle reports them, derive contradictions
from the supposition that something moves. Three of them purport to show that
what moves, does not move. They are ‘dramatized’, in so far as they introduce
particular supposed moving things: a runner; two runners; an arrow; three
moving and stationary masses. Aristotle presents the arguments as designed to be
mutually independent. 38
(b)
The ‘Stadium’ and the ‘Achilles’
Suppose a runner is to run along a running-track. The stretch to be traversed (call
it S) may be considered as divided up into substretches in various ways. Given the
starting and finishing points we understand what is meant by ‘the first half of S’,
‘the third quarter of S’ and so on. It seems that however short a substretch is
specified in this way, it will always have positive length and may be thought of
as divided into two halves. 39
Going on in this way we can specify a division of S into substretches which
will be such that the runner runs through a well-ordered but infinite series of
substretches. First the runner traverses the first half, then half of what remains,
then half of what remains, and so on. In this way, for any positive integer n, at
the end of the nth substretch the runner has covered
of S, and the nth substretch is ½n of the whole length of the track. However large a finite number n becomes, the
fraction is never equal to 1; there are infinitely many substretches.
With such a division, the series of substretches is well-ordered, and the runner
who traverses S has been through all of the substretches in order: for every finite
number N, the runner has traversed the Nth substretch. Hence the runner has
traversed an infinite series of substretches, in a finite time; but this is impossible.
This is an expansion of Aristotle’s formulations (Physics VI 2, 233a21–23; VI
9, 239b11–14) of the ‘Stadium’ argument. 40 The ‘Achilles’ (Physics VI 9,
239b14–29) makes the same point more dramatically, pitting a very fast runner