significance. Any quantity of space, say the space enclosed
within a circle, must either be composed of ultimate indi-
visible units, or it must be divisiblead infinitum. If it is
composed of indivisible units, these must have magnitude,
and we are faced with the contradiction of a magnitude
which cannot be divided. If it is divisiblead infinitum, we
are faced with the contradiction of supposing that an in-
finite number of parts can be added up and make a finite
sum-total. It is thus a great mistake to suppose that Zeno’s
stories of Achilles and the tortoise, and of the flying arrow,
are merely childish puzzles. On the contrary, Zeno was the
first, by means of these stories, to bring to light the essen-
tial contradictions which lie in our ideas of space and time,
and thus to set an important problem for all subsequent
philosophy.
{56}
All Zeno’s arguments are based upon the one argument de-
scribed above, which may be called the antinomy of infinite
divisibility. For example, the story of the flying arrow. At
any moment of its flight, says Zeno, it must be in one place,
because it cannot be in two places at the same moment.
This depends upon the view of time as being infinitely di-
visible. It is only in an infinitesimal moment, an absolute
moment having no duration, that the arrow is at rest. This,
however, is not the only antinomy which we find in our
conceptions of space and time. Every mathematician is ac-
quainted with the contradictions immanent in our ideas of
infinity. For example, the familiar proposition that parallel
straight lines meet at infinity, is a contradiction. Again,
a decreasing geometrical progression can be added up to
infinity, the infinite number of its terms adding up in the
sum-total to a finite number. The idea of infinite space
itself is a contradiction. You can say of it exactly what
Zeno said of the many. There must be in existence as much
space as there is, no more. But this means that there must
be a definite and limited amount of space. Therefore space
is finite. On the other hand, it is impossible to conceive a
limit to space. Beyond the limit there must be more space.
Therefore space is infinite. Zeno himself gave expression
to this antinomy in the form of an argument which I have
not so far mentioned. He said that everything which ex-
ists is in space. Space itself exists, therefore space must be
in space. That space must be in another space and soad
infinitum. This of course is merely a quaint way of saying
that to conceive a limit to space is impossible.But to return to the antinomy of infinite divisibility, {57}
on which most of Zeno’s arguments rest, you will perhaps
expect me to say something of the different solutions which
have been offered. In the first place, we must not forget
Zeno’s own solution. He did not propound this contradic-
tion for its own sake, but to support the thesis of Par-
menides. His solution is that as multiplicity and motion
contain these contradictions, therefore multiplicity and mo-
tion cannot be real. Therefore, there is, as Parmenides
said, only one Being, with no multiplicity in it, and ex-
cludent of all motion and becoming. The solution given
by Kant in modern times is essentially similar. According
to Kant, these contradictions are immanent in our concep-
tions of space and time, and since time and space involve
these contradictions it follows that they are not real beings,