From the Beginning to Plato

144 PYTHAGOREANS AND ELEATICS
against a very slow one. The slow runner is given a start. The stretch covered by
the faster runner is divided up in such a way that it appears the faster can never
catch the slower within any finite time. This drives home the point that speed is
irrelevant. No limit of speed is prescribed or needed by the argument; the speed
of the fast runner could increase without limit without removing the problem.
(c)
The ‘Arrow’
Another way of looking at things supposedly in motion throughout a time-stretch
is to select any one moment during that stretch. Say an arrow is in flight.
1 At any one moment the arrow must be ‘in one place’. No part of it can be in
two places at once; so it must occupy ‘a space equal to itself (i.e. of the same
shape and size).
2 The arrow must be at rest at this moment. There is no distance through
which it moves, in a moment; hence it does not move at a moment, so it
must be at rest at that moment.
3 But the moment chosen was an arbitrary moment during the flight of the
arrow. It follows that the arrow must be at rest at all moments during its
flight.
4 Hence, since the arrow during its flight is never not at a moment of its flight,
the arrow is always at rest during its flight; so it never moves during its flight.
The above argument cannot claim to be more than a plausible filling-out of
Aristotle’s abbreviated report (Physics VI 9, 239b5–9 and 30–3). 41 Aristotle
himself is interested only in step (4), where he thinks to find the fallacy; he gives
the only briefest sketch of (1), (2) and (3). 42
(d)
The ‘Moving Rows’
Aristotle (Physics VI 9, 239b33–240a18) reports this argument in terms of
unspecified ‘masses’ on a racecourse; to make it easier for a modern reader, the
masses may be thought of as railway trains. 43
Consider three railway trains of the same length, on three parallel tracks. One
of the trains is moving in the ‘up’ direction, another is moving at the same speed
in the ‘down’ direction, and the third is stationary. As may be easily verified,
either of the moving trains takes twice as long to pass the stationary train as it
does to pass the other moving train.
Just how Zeno derived a contradiction from this fact, is uncertain. According
to Aristotle, Zeno simply assumed that the passing-times must be equal, since the
speeds are equal and the two masses passed are equal in length. Then it follows
that the time is equal to twice itself. The assumption, though, has often been