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ZENO of Elea

(c. 490 — 440? B.C.)



Zeno of 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.



[1] 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 parts [c], 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.

[2] 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.



G. S. Kirk, J. E. Raven, & M. Schofield, The Presocratic Philosophers, ch. IX.

R. D. McKirahan, Philosophy before Socrates, ch. 12.

W. C. Salmon (ed.), Zeno's Paradoxes.

See also essays by H. Frnkel, G. E. L. Owen, and G. Vlastos (all in Furley and Allen)





[1a] Dialectic argument








[1a 2c]






[1b 2a] One and many; change and permanence; space, time and motion illusory










[1a c]


[1a d]


[3a 3c 5e]

[12b c]





[1c] In/divisibility (influence by virtue of treatment of topic)




[1d 1d]

[1a 1a]