phy, and is used for that type of reasoning which seeks
to develop the truth by making the false refute and contradict itself.
The conception of dialectic is especially important in Zeno, Plato,
Kant, and Hegel.

All the arguments which Zeno uses against multiplicity and motion are
in reality merely variations of one argument. That argument is as
follows. It applies equally to space, to time, or to anything which
can be quantitatively measured. For simplicity we will consider it
only in its spatial significance. Any quantity of space, say the space
enclosed within a circle, must either be composed of ultimate
indivisible units, or it must be divisible _ad 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 divisible _ad infinitum_, we are faced with the contradiction
of supposing that an infinite 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 essential 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 described 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 divisible. 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 acquainted 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