ZENO of Elea
Elea was Parmenides' favourite pupil and accompanied him to Athens in about
448. He is famous for his so-called
paradoxes, which were seen by Plato [Parmenides,
128c] as an attempt to show the incoherence of the views on plurality held by
Pythagoras and other opponents of the Eleatic philosophers. Aristotle called him "the founder of
dialectic" because of his skill in formulating arguments logically and
systematically, by means of which fallacies might be revealed.
 Zeno constructed a
number of dialectical
arguments [a] against the view that the universe is made up of many things [b]. His paradoxes of
magnitude are typical.
(a) According to a later
commentator, Simplicius [Commentary on
Aristotle's Physics, 139/140], Zeno said that each of the "many things" must have no magnitude because "each is the same as itself and one" [fr. 2] Nothing more is known of his argument, but
"each is one" suggests that he was referring to the common view that things are
units, ultimate and indivisible. If they had magnitude they would
be divisible and therefore not ultimate [c]. By "each is
the same as itself" he probably meant that a thing is homogeneous, that is, the
same throughout, lacking internal distinction. If it were not homogeneous it would not be a genuine unity.
(b) However, Zeno also
argues that each of the many things has unlimited magnitude [fr. 1]. (i) That which has no magnitude would not
exist, for if it were added to or subtracted from an existing thing that thing
would not be made respectively bigger or smaller; what is added or taken away
would already be nothing. [Cf. fr. 2: Existing things are not nothing, so they
must have magnitude, as must their parts.] (ii) Why then is each thing unlimited in magnitude? Zeno's argument is that a magnitude can be divided into
and these parts into further parts which have no magnitude, and so on;
therefore there must be an infinite number of parts with magnitude; hence the
original magnitude is unlimited. The conclusions to (a) and (b) clearly
contradict each other. The notion of a
plurality must be rejected.
 Zeno also worked out
some interesting arguments to show that motion and change cannot be coherently described and are
therefore, together with space and time, illusions of sense [a]. rather than belonging to what is real. We owe our knowledge of these to Aristotle
[see Physics Ζ 9, 239ff].
(a) The Stadium paradox
[Ζ9, 239b11]. Suppose a man is
running in a race from A to B. Let the
distance be, say, 2 kilometres. He must
first traverse half the distance from A to B. But at this point he then has to run half this distance, thus 1 km. And so on ad infinitum: 1 + 1/2 + 1/4 + ... So to get to the end the runner has to traverse an
infinite number of distances. This is
impossible, either as a matter of logic or because a person could not perform
an infinite number of tasks. Because the
distance AB can be any length we wish, he cannot get anywhere at all. So motion is impossible.
(b) A variation on this
argument is a paradox which later came to be known as the 'Achilles' paradox or
'Achilles and the Tortoise'
[Ζ9 239b14] [a]. Achilles, challenged to race with a tortoise,
gives him a start of half the course. By
the time Achilles has reached the tortoise's starting point the tortoise has
moved on further. And so on. It would seem then that the tortoise must
always be in front, although by an ever decreasing distance. To overtake the tortoise Achilles would have
to go through an infinite number of distances. This being impossible, motion must be illusory.
(c) Another argument is
the 'Arrow' paradox. Aristotle's account
[Ζ9 239b 30-3, 5-9] is not very clear, but the essentials seem to be as
follows. Take any instant in an arrow's
flight. The arrow occupies its own
volume and not a greater one. It is not
therefore traversing a distance and is therefore not at rest. But during its flight it must always be at
some instant. Hence it must be at rest
throughout its flight; it does not really move.
Zeno's paradoxes (only a
few of which have been summarized above) have engendered a great deal of
philosophical debate and a variety of responses. Much of his reasoning is regarded as
fallacious though it is not always easy to pin down precisely what his errors
are. One might question, say, his claim
(argument 1a) that 'ultimates' cannot have magnitude. The electrons, quarks, and so on of modern
quantum physics are arguably counter-examples. His inference (in 1b) from the assertion that an object possesses an
infinite number of parts with magnitude to the conclusion that the original
object has unlimited is also certainly dubious. As for the arguments concerning motion (2a and b), we may note:
(i) The traversing of an
infinite number of positions in space (and indeed instants of time, time and
space being inseparable) does not entail that the finite distance over which the race is run cannot be
completed. Geometrical series of the
type 1 + 1/2 + 1/4 + ... tend to a definite limit (in this particular
case to 2) as the number of terms approaches infinity.
(ii) It follows that if the
course is completed, what is expected of the runner (in terms of physical
'tasks') must have been achieved 'task' here being defined as what is needed
for the runner to traverse each point or instant.
It is of course an
ongoing dispute as to whether or not Zeno's attacks on pluralism are
tenable. Some scholars have argued that
Parmenides' monism is equally susceptible to his arguments [see Kirk, Raven and
Schofield]. However, his importance as a
philosopher is not in doubt. He remains
significant for his introduction into philosophy of dialectical argument, and
for the stimulus he gave to the examination by later philosophers of the
concepts of space , time, and infinity.
Kirk, J. E. Raven, & M. Schofield, The Presocratic Philosophers, ch.