Paradoxes
The
fifth in Francis Moorcroft's series looking at some the classic
philosophical paradoxes.
No.
5 Zeno's Paradox
Francis Moorcroft
The
four Paradoxes of Zeno, which attempt to show that motion is impossible,
are most conveniently treated as two pairs of paradoxes. The reasons
for this will hopefully become clearer later. The first two paradoxes
are as follows.
The
Racecourse or Stadium argues that an athlete in a
race will never be able to start. The reason for this is that before
the runner can complete the whole course they have to complete half
the course; and before they can complete half the course they have
to complete a quarter; and before they can complete a quarter they
have to complete an eighth; and so on. Therefore the runner has
to complete an infinite amount of events in a finite amount of time
 assuming that the race is to be run in a finite amount of time.
As it is impossible to do an infinite amount of things in a finite
amount of time, the race can never be started and so motion is impossible!
The
second paradox is that of Achilles and the Tortoise, where
in a race, Achilles gives the Tortoise a head start. The argument
attempts to show that even though Achilles runs faster than the
Tortoise, he will never catch her. The argument is as follows: when
Achilles reaches the point at which the Tortoise started, the Tortoise
is no longer there, having advanced some distance; when Achilles
arrives at the point where the Tortoise was when Achilles arrived
at the point where the Tortoise started, the Tortoise is no longer
there, having advanced some distance; and so on. Hence Achilles
can never catch the Tortoise, no matter how much faster he may run!
The
diagram below may help to understand this argument.
The
race starts at t0 with the Tortoise having a head start over Achilles.
By time t1, when Achilles has reached the point at which the tortoise
started, the tortoise has moved on; by t2 Achilles has reached the
point where the tortoise was at t1 but the tortoise has moved on;
by t3 Achilles has reached the point where the tortoise was at t2
but the tortoise has moved on; and so on. To be sure, the distance
between Achilles and the tortoise is getting less and less each
time but Achilles never catches up with  far less overtakes  the
Tortoise.
Zeno,
it seems, believed quite seriously that motion did not exist and
that arguments such as these established it. What do we, who believe
that races can be run and slow objects can be overtaken by faster
moving ones, say in response?
One
common reply is that Zeno has misunderstood the nature of infinity.
Modern mathematics, it is said, has shown that the infinite sequences
that Zeno generates do have a finite sum. In particular, to take
the Racecourse example, the sequence 1/2 + 1/4 + 1/8 + 1/16 + .
. . is equal to 1.
This
reply, however, misunderstands what modern mathematics has shown.
Mathematicians do use sequences such as 1/2 + 1/4 + 1/8 + 1/16 +
. . . but they say that they have a limit of 1, or tend
to 1. That is, we can get nearer and nearer towards 1 by adding
on more and more members of the sequence, but not actually arrive
at 1  this would be impossible because we are considering an infinite
sequence. So far from providing an argument against Zeno, mathematics
is actually agreeing with him!
Further,
this reply seems to miss the point of Zeno's argument: simply pointing
out that there is a branch of mathematics that deals with the infinite
does not reduce the puzzling aspects of the Paradoxes. We know
that races can be run and that Achilles will overtake the Tortoise,
what we want to know is what is wrong with the arguments that show
that these things can't happen.
The
first two Paradoxes of Zeno attempt to find contradictions in the
idea that motion is continuous and space can be infinitely
subdivided. But motion may not be continuous: space may be discrete
and motion be a series of tiny jumps. On this view there would
be a finite  but very large  number of steps between the beginning
of the race and its end. So the Paradox of the Racecourse could
be avoided by saying that there is some first, incredibly small,
step that can be taken, where there is no step of half the size.
Similarly, there is some small, and indivisible, last step that
Achilles can take which will allow him to catch the Tortoise and
then overtake her. This possibility is criticized by the other two
Paradoxes of Zeno to be considered next time.
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The
paradoxes series will be updated towards the end of December
Previous
articles in the Paradox series
2.
3.
4.
