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Born: C. 490 B.C., Elea, Italy

Died: C. 430 B.C.

Major Works: No extant works, but the following famous paradoxes are attributed to Zeno: Paradox of Achilles and the Tortoise, Paradox of the Arrow, Paradox of the Race Course, Paradox of the Stadium, Paradox of Plurality

Major Ideas

The Paradox of the Race Course argues that motion would require a process of infinite division.

The Paradox of Achilles and the Tortoise concludes that a swifter runner cannot overtake a slower.

The Paradox of the Arrow argues that an arrow cannot move in flight.

The Paradox of the Stadium maintains that double the time is sometimes equal to half the time.

The Paradox of Plurality argues that if space and time are composed of discrete units, the number of such units is both finite and infinite.

Zeno of Elea was a disciple of Parmenides, who argued that Reality is One--an unmoving, solid, homogeneous sphere. In his support of Parmenides and against the Pythagoreans, Zeno composed a number of paradoxes designed to show that if one accepts a pluralistic account of space and time, a view that space is made up of discrete and divisible units of space, and time of discrete and divisible units of time, then one runs into difficulties, if not contradictions.

Very little is known of Zeno's life. He was born about 490 B.C. in Elea (now Velia, southern Italy). His father was Teleutagoras. It is reported (in Plato's Parmenides) that Zeno was "tall and fair to look upon," and that at the time of a purported discussion involving Zeno, Socrates, and Parmenides, Zeno was forty years old and Parmenides sixty-five. Zeno appears to have been politically active and to have been critical of a Sicilian tyrant and to have refused, even under torture, to reveal the name of his political associates. (One account tells us that, rather than reveal anything that might be damaging to his

associates, Zeno bit off part of his tongue and threw it at his torturers.)

In various ancient works, reference is made to Zeno's "writings," but unfortunately none of Zeno's works is extant. However, in a discussion of the continuity of motion, space, and time, Aristotle (in his Physics) gives an account of four paradoxes of motion attributed to Zeno and comments critically on them.

A fifth paradox--which has been called the Paradox of Plurality--may be found in the Physics of Simplicius.

The Paradox of the Race Course

The first of Zeno's arguments against the possibility of motion is given by Aristotle in these words (translation by W. D. Ross in the Oxford Translation of Aristotle):

The first [of the arguments about motion] asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the halfway stage before it arrives at the goal.

The paradox becomes clearer if one attends to John Burnet's rendering of the argument in his Early Greek Philosophy:

You cannot cross a race-course. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you can traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum. . .

There are two ways of considering this paradox dramatically, both of them mind-challenging. According to Zeno, any attempt to cover any finite distance (say, to make a trip or to run a race) is frustrated by the circumstance that before finishing the whole distance, there is still half the remaining distance to go. Even if that half were covered, there would still be half the remaining distance. There is always half the remaining distance to go, no matter how many halves one manages to cover. So it appears that a trip or a race (or any passage from one place to another) can never be finished.

Another way of looking at it is this: One cannot even get started in the effort to move from one place to another. For before covering the first half of the distance, one would have to cover half of that first half. But be fore covering that half, one would have to cover half of it. But there is never a first half by covering which one could get started. The frustration is complete before the trip (or race) even gets started!

Aristotle criticizes this paradox by contending that anything continuous (such as motion from one place to another) is infinitely divisible. (A half is itself divisible, and the quarters are divisible, and so forth, ad infinitum.) But to say that a distance is "infinite" in that it is infinitely divisible, he explains, does not mean that it is composed of an infinite number of points that would have to be covered one by one. "Accordingly," Aristotle writes, "Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time" (since anyone in motion traverses a course that is infinitely divisible and can do so in a finite time). After all, Aristotle points out, if space is "infinite" in that it is infinitely divisible, then so is time.

One notes that the adverse criticism of the paradox is not after all a criticism of Zeno. For Zeno was apparently attempting to show, by way of his paradox, that if a distance is composed of an infinite "many," it could never be covered in a finite amount of time by covering that infinite many one by one. His argument against the conception of a distance as composed of discrete units of space, either finite or infinite, is sound. Zeno's argument is a reduction ad absurdum (a bringing-out of the inconsistencies) in the Pythagorean conception of space as composed of a plurality of discrete units.

The Paradox of Achilles and the Tortoise

The second of the arguments is the famous paradox known as The Paradox of Achilles and the Tortoise (as translated from Aristotle's Physics, book 6, chapter 9, by Philip H. Wicksteed and Francis M. Cornford):

The second [of Zeno's paradoxes of motion] is what is known as "the Achilles," which purports to show that the slowest will never be overtaken in its course by the swiftest, inasmuch as, reckoning from any given instant, the pursuer, before he can catch the pursued, must reach the point from which the pursued started at that instant, and so the slower will always be some distance in advance of the swifter.

Imagine the race between Achilles and the tortoise as follows: The tortoise has taken the lead. By the time Achilles starts, the tortoise has reached a point we'll call A. By the time Achilles reaches point A, then (no matter how fast Achilles runs--since the tortoise is moving steadily forward toward the goal) the tortoise has reached point B. But by the time Achilles reaches B, the tortoise has reached C. Even if Achilles comes closer and closer to the tortoise, it appears that since Achilles is forever reaching points (and there is an infinite number of such points) where the tortoise was, the tortoise is forever moving some distance ahead. Achilles (the swifter runner) cannot overtake the tortoise (the slower)!

Aristotle's criticism here is that the Achilles argument depends, like the previous bisection argument, on a mode of division that never terminates. Although the tortoise is never overtaken while it holds a lead, it does not follow, of course, that it is never overtaken. In other words, although the process of division (concentrating on successive cases of Achilles's reaching points at which the tortoise was) is never terminated, the race (or the overtaking of the tortoise by Achilles) certainly can be.

If one now drops the mode of analysis that is interminable (a concentration on points where the tortoise was) and turns to a consideration of what happens when Achilles reaches a point where the tortoise is, one realizes that once Achilles has covered all the points at which the tortoise was, Achilles has succeeded in overtaking the tortoise (at which point Achilles can take the lead, if he wishes, since he is the swifter).

But how can Achilles cover all the points at which the tortoise was if the number of such points is infinite? By simply closing the gap between himself and the tortoise, of course. As Aristotle points out, it is a fallacy to suppose that one cannot cover an infinite number of points in a finite number of time. After all, the points are not units of space but are simply what might be called nondimensionless "cuts," envisaged by a mode of analysis, of a conceived line.

The Paradox of the Arrow

The third argument is The Paradox of the Arrow (Burnet's translation):

The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move.

The force of the argument comes from the assumption that at any point in its flight, an arrow is somewhere. To be somewhere is to occupy a space, and to occupy a space is to be at rest. Hence, at any point in its flight, an arrow is at rest. But motion could hardly be a sequence of rests. Therefore, the arrow cannot move.

Aristotle argues that the Paradox of the Arrow depends on the assumption that time is composed of moments, and he contends that if the assumption is not granted, the argument collapses.

This criticism is perhaps correct, but it is vague. The argument involves the premise that the arrow "at any given moment.., occupies a space equal to itself." If we are not willing to grant that at an instant (that is, at some "cut" in time--like high noon, which has no duration) the arrow "occupies" a space, that is, is for some time in a space defined by its dimensions, then we need not grant that the arrow is "resting" there, even for an instant. To "occupy" a space, to be at rest, involves being somewhere for some time, that is, for some duration of time; since an instant is not a duration, an arrow cannot rest anywhere in an instant.

(We are not disputing the premise that "at any point in its flight, an arrow is somewhere." At a given instant, the arrow is, indeed, somewhere: Its coordinates could be determined. But it does not occupy" that space; it does not rest there. And we cannot say where the arrow is in "the next instant," for there is no next instant even though, after the given instant, there are any number of instants.)

The Paradox of the Stadium

The fourth argument is The Paradox of the Stadium (as given by Burnet):

Half the time may be equal to double the time. Let us suppose three rows of bodies, one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions. By the time they are all in the same part of the course, B will have passed twice as many of the bodies in C as in A. Therefore the time which it takes to pass C is twice as long as the time it takes to pass A. But the time which B and C take to reach the position of A is the same. Therefore double the time is equal to the half.

According to Aristotle (Ross translation), "The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size which is not; which is false."

Aristotle's point is, of course, well taken: It would take longer to pass a line of standing horses (or chariots) than it would take at the same rate to pass the same number of bodies moving in the opposite direction. It is because of our realization of this point that Zeno's paradox does not strike us as being paradoxical. It might very well take only half the time to pass a moving object as it would take to pass that object were it not moving; we would not say, on that account, that half the time is equal to twice the time (which, taken out of context, is nonsense).

However, if one understands Zeno to be making the point that if the Pythagoreans are right in maintaining that there are units of space and time, and if a unit of time is relative to a unit of space, then if while passing say two stationary objects, one is passing four objects moving in the opposite direction then two units of time (relative to the two objects) are equal to four units of time (relative to the four objects) hence (on the Pythagorean assumption) half the time is equal to twice the time.

Zeno's argument is. effective and interesting only in relation to the Pythagorean conception of space and time as made up of discrete units.

The Paradox of Plurality

The Paradox of Plurality was briefly stated by Simplicius in his Physics (as translated by Burnet):

If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.

If things are a many, they will be infinite in number; for there will always be other things between them, and: others again between these. And so things are infinite in number.

The conclusion, then, is that "if things are a many," they will be both finite and infinite in number, which is impossible.

Zeno may appear to make two mistakes in this argument. In the first place, a set consisting of an infinite number of entities would have "just as many as they are," namely, an infinite number: Hence, the set, although having just as many as there are, would not necessarily be finite in number. Second, even if a set were finite in number and each of the units were infinitely divisible, it would not follow that "things are infinite in number" if one restricted the use of the term "thing" to the units.

However, if the argument is related to the Pythagorean conception of space and time, the difficulties that follow from assuming that space and time are composed of units of some magnitude are brought out. If points and instants are of no magnitude, they are nothing, and space and time are illusions. If, on the other hand, points and instants are of some magnitude, the above paradox would seem to apply (if one regards the parts of units as themselves units). The paradox calls into question the very conception of a "unit." In Simplicius's Physics, Eudemos, referring to Zeno, reports (as translated by Burnet): "... If anyone could tell him what the unit was, he would be able to say what things are."

Zeno's Accomplishment

Critics differ as to what Zeno attempted and what he accomplished with his paradoxes. Some have maintained that Zeno was indirectly defending Parmenides by simply arguing against the Pythagoreans, who maintained that there are "many" units of time and space, not a single continuity; Zeno constructed his paradoxes by assuming his opponents' premises and showing that they led to contradictions. Other critics say that Zeno directly defended Parmenides by showing that the idea of a reality of space, time, and motion is absurd.

As far as the accomplishments are concerned, most critics agree in thinking that Zeno failed to show that motion is impossible or that time and space are unreal. Some critics, however, think that Zeno did succeed in bringing out the contradictions that are generated by assuming the Pythagorean idea of discrete units of time and space. Bertrand Russell wrote (in Our Knowledge of the External World) that "Zeno's arguments, in some form, have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own." Zeno's principal accomplishment, according to Russell, was to show that if one assumes that covering an infinite number of points would take an in finite amount of time, paradoxes result that suggest that motion is impossible. The remedy. Russell goes on to say, is not to deny motion but to realize that covering an infinite number of nondimensional points can be accomplished in a finite amount of time.

Some critics have insisted that Zeno's arguments are valid, given the assumptions that Zeno set out to undermine; others have judged the arguments to be fallacious, either in the assumptions or in the lines of reasoning involved. Anyone interested in pursuing the history of the criticism of Zeno's arguments will find a great deal of material for intensive consideration, some of it involving technical mathematical and physical theory, but most of it challenging and all of it, either directly or indirectly, a tribute to Zeno.

Further Reading

Burnet, John. Early Greek Philosophy. 4th ed. London: Adam & Charles Black, 1930, reprinted 1963. The Zeno material is in chapter 8, "The Younger Eleatics." Burnet not only presents the sources but also elucidates the texts, places Zeno's work in context, and credits Zeno with establishing continuity by discrediting the Pythagorean idea that spatial and temporal quantities are discrete.

McGreal, Ian P. Analyzing Philosophical Arguments. San Francisco: Chandler, 1967. In Chapter 4, McGreal analyzes Zeno's Paradox of Achilles and the Tortoise. The argument is reconstructed and analyzed line by line. Ross, W D. Aristotle Selections. New York: Charles Scribner's Sons, 1927, 1938. Section 32 contains Aristotle's accounts and criticisms of Zeno's paradoxes of motion. The translation used is the Oxford Translation.

Salmon, Wesley C., ed. Zeno's Paradoxes. Indianapolis and New York: Bobbs-Merrill, 1970. An excellent collection of some of the most illuminating and provocative criticisms of Zeno's paradoxes, with essays by Salmon, Bertrand Russell, Henri Bergson, Max Black, J. 0. Wisdom, James Thomson, Paul Bencerraf, G. E. L. Owen, and Adolf Grunbaum. Contains a very useful bibliography.

Wicksteed, Philip H., and Francis M. Cornford, trans. The Physics. Cambridge, Mass. and London: Harvard University Press and William Heinemann, 1932. A clear, accurate, and lively translation of Aristotle's text that contains his version of Zeno's arguments.
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Publication:Great Thinkers of the Western World
Article Type:Biography
Geographic Code:4EUIT
Date:Jan 1, 1999
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