The Race Course: Part 2
1.
Our look at the plurality argument suggests that Zeno may have thought that to run all the
Z
runs would be to run a distance that is
infinitely long
. If this is what he thought, he was
mistaken.
The reason the sum of all the
Z
intervals is not an infinitely large distance is that there is
no smallest
Z
interval. And Zeno does not establish that there is some smallest
Z
run. (If
there were a smallest
Z
run, he wouldn’t have been able to show that R had to make
infinitely many
Z
runs.)
2.
What about Aristotle’s understanding of Zeno? Here is what he says [RAGP
8
]:
“
Zeno’s argument makes a false assumption when it asserts that it is impossible to
traverse an infinite number of positions or to make an infinite number of contacts
one by one
in a finite time
” (
Physics
233a2124).
3.
Aristotle points out that there are two ways in which a quantity can be said to be infinite:
in
extension
or in
divisibility
. The race course is infinite in divisibility. But, Aristotle
goes on, “the time is also infinite in this respect.”
Hence, there is a sense in which
R
has an infinite number of distances to cross. But in that
sense he also has an infinite amount of time to do it in. (If a finite distance is infinitely
divisible, then why isn’t a finite time also infinitely divisible?)
4.
So Zeno cannot establish (2) for either of the first two reasons we considered: to make all
the
Z
runs,
R
does not have to run infinitely far. Nor does
R
have to keep running forever.
1.
On this reading, Zeno’s argument attempts to show that it is
logically impossible
for
R
to
reach
G
. That is, Zeno’s puzzle is not that the runner has to run
too far
, or that the runner
has to run for
too long a time
, but that the claim that the runner has completed all the
Z

runs
leads to a contradiction
.
2.
Following James Thomson [“Tasks and SuperTasks,” on reserve], let us define a
super
task
as an infinite sequence of tasks. Can one perform a supertask? Bertrand Russell
thought that one could, as Thomson explains [“Tasks and SuperTasks,” p. 93]:
“Russell suggested that a man’s skill in performing operations of some kind might increase so fast
that he was able to perform each of an infinite sequence of operations after the first in half the time
he had required for its predecessor. Then the time required for all of the infinite sequence of tasks
would be only twice that required for the first. On the strength of this Russell said that the
performance of all of an infinite sequence of tasks was only medically impossible.”
But Thomson argues that to assume that a supertask has been performed in accordance
with Russell’s “recipe” leads to a logical contradiction.
a.
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 Fall '06
 Buechner
 Philosophy, Set Theory, Philosophy of mathematics, Indian mathematics

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