are not units. Since they are indivisible they can have no
magnitude, for that which has magnitude is divisible. The
many, therefore, is composed of units which have no magni-
tude. But if none of the parts of the many have magnitude,
the many as a whole has none. Therefore, the many is in-
finitely small. But the many must also be infinitely large.
For the many has magnitude, 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 divisiblead infinitum. It must therefore be
composed of an infinite number of parts, each having mag-
nitude. But the smallest magnitude, multiplied by infinity,
becomes an infinite magnitude. Therefore the many is in-
finitely 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 num-
ber. 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 “anti-
nomy.” An antinomy is a proof that, since two contradic-
tory propositions equally follow from a given assumption,
that assumption must be false. Zeno is also called by Aris-
totle the inventor of dialectic. Dialectic originally meant
simply discussion, but it has come to be a technical term
in philosophy, 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 mea-
sured. For simplicity we will consider it only in its spatial