de, and as such, is
divisible into parts. These parts still have magnitude, and are
therefore further divisible. However far we proceed with the division
the parts still have magnitude and are still divisible. Hence the many
is divisible _ad infinitum_. It must therefore be composed of an
infinite number of parts, each having magnitude. But the smallest
magnitude, multiplied by infinity, becomes an infinite magnitude.
Therefore the many is infinitely large. (2) The {54} many must be, in
number, both limited and unlimited. It must be limited because it is
just as many as it is, no more, no less. It is, therefore, a definite
number. But a definite number is a finite or limited number. But the
many must be also unlimited in number. For it is infinitely divisible,
or composed of an infinite number of parts.

_Zeno's arguments against motion_.

(1) In order to travel a distance, a body must first travel half the
distance. There remains half left for it still to travel. It must then
travel half the remaining distance. There is still a remainder. This
progress proceeds infinitely, but there is always a remainder
untravelled. Therefore, it is impossible for a body to travel from one
point to another. It can never arrive. (2) Achilles and the tortoise
run a race. If the tortoise is given a start, Achilles can never catch
it up. For, in the first place, he must run to the point from which
the tortoise started. When he gets there, the tortoise will have gone
to a point further on. Achilles must then run to that point, and finds
then that the tortoise has reached a third point. This will go on for
ever, the distance between them continually diminishing, but never
being wholly wiped out. Achilles will never catch up the tortoise. (3)
This is the story of the flying arrow. An object cannot be in two
places at the same time. Therefore, at any particular moment in its
flight the arrow is in one place and not in two. But to be in one
place is to be at rest. Therefore in each and every moment of its
flight it is at rest. It is thus at rest throughout. Motion is
impossible.

{55}

This type of argument is, in modern times, called "antinomy." An
antinomy is a proof that, since two contradictory propositions equally
follow from a given assumption, that assumption must be false. Zeno is
also called by Aristotle the inventor of dialectic. Dialectic
originally meant simply discussion, but it has come to be a technical
term in philoso