The Paradoxes of Zeno

First transcribed: c. 465 b.c.e. (Zeno of Elea: A Text, 1936; commonly known as The Paradoxes of Zeno)

Type of Philosophy: Logics, metaphysics

Context

Zeno of Elea is one of the most amazing figures in the history of philosophy. No fragments remain of what he wrote, and yet two and a half millennia after his time, professional philosophers and mathematicians continue to argue, as they have ever since he first propounded his paradoxes, about the point and validity of his arguments. Numerous historians and commentators, as well as philosophers, have written of Zeno’s thought, and a review of Zeno’s paradoxes may reasonably consist of constructions that represent what the consensus appears to be concerning the lines of argument that Zeno devised.

Speaking of Zeno, philosopher Bertrand Russell wrote that “by some care in interpretation it seems possible to reconstruct the so-called sophisms’ which have been refuted’ by every tyro from that day to this.” Both Aristotle and Plato have something to say about Zeno. Simplicius, the scholarly historian of philosophy and commentator on Aristotle, tells us something of Zeno’s arguments; but he was writing in the sixth century c.e., and although he may have had some reliable information about Zeno’s views, no one knows for sure. In any case, what he writes is fascinating and promising, and he is often quoted as a source of Zeno’s arguments. Diogenes Laërtius gives some information. Although these scattered passages do not bear evidence of authenticity, the arguments that emerge have a certain genius and integrity about them, leading scholars to be reasonably confident that if they do not thereby know Zeno, at least they know the shadow on the wall.

Before turning to the paradoxes themselves, however, we can hope to gain better understanding and appreciation of them by considering the probable point of their delivery. Zeno of Elea was a defender of Parmenides, also of Elea, who in his Peri physeōs (fifth century b.c.e., only fragments exist; The Fragments of Parmenides, 1869; commonly known as On Nature), developed a thoroughgoing monistic theory of reality to the effect that reality is a homogeneous sphere, eternal and indestructible; neither motion nor change is possible. Although sense experiences lead us to suppose that we observe a world of change and motion, such experiences are misleading. It is reasonable to suppose, then, that Zeno, as a disciple of Parmenides, used his paradoxical arguments to defend monism and to deny the pluralism of the Pythagoreans and other schools of his day.

The paradoxes as stated below are not in Zeno’s words, of course; and it is entirely possible that the arguments do not correspond even in sense to Zeno’s arguments, but the fragments of secondhand reports suggest arguments along the lines given below.

The Paradoxes of Plurality and of Space

Presuming (for the sake of argument) that there is a plurality of things, if the many are of no magnitude, then if one thing were added to another, there would be no change in size of the thing receiving the addition. Hence, what would be added would be nothing. If a thing of no magnitude were subtracted from another thing, the other would not at all be diminished; hence, what would be subtracted would be nothing. Hence, if there is a plurality, then each thing is of some magnitude.

However, if each thing in a plurality is of some magnitude, it is divisible into parts ad infinitum. Because there would be no end to its parts, such a thing would be infinitely large. Hence, because if there is a plurality, everything that exists is either of some magnitude or not, and if things of no magnitude are nothing, and if things of some magnitude are infinite in size, then, if there is a plurality, things are either so small as to be nonexistent or so large as to be infinitely large. Hence, it is false that there is a plurality of things.

Second, if there is a plurality, the number of things would have to be definite, but if the number is definite, then it could not be infinite because the infinite has no limit and is not definite. On the other hand, if there is a plurality, by the division of parts, one would arrive at an infinite number of things. Hence, if things are many, they are both finite and infinite in number, which is impossible. Hence, it is false that there is a plurality of things.

According to the paradox of space, if there is space, then everything that exists is in something—namely, space. However, then space would be in space, ad infinitum. Therefore, there is no space.

The Paradoxes of Motion

There are five paradoxes of motion, the dichotomy paradox, the paradox of Achilles and the tortoise, the arrow paradox, the stadium paradox, and the millet seed paradox.

According to the paradox of dichotomy (the race course), before going the whole of any distance, a runner would first have to go half that distance. However, before going the half, the runner would have to cover the first half of that half (a quarter of the original distance). However, before going the first quarter of the course, the runner would have to go the first eighth. The number of requirements is infinite and cannot be satisfied in a finite amount of time. Therefore, the runner cannot even get started.

In the paradox of Achilles and the tortoise, in a race between Achilles, the swifter runner, and a tortoise that has taken the lead, Achilles cannot overtake the tortoise. Considering the situation at any instant in the race while the tortoise is in the lead, by the time Achilles reaches the point where the tortoise was at that instant, the tortoise (who keeps moving) will be some distance ahead. By the time Achilles reaches that point, the tortoise will have moved some distance ahead. Hence, because no matter how many points Achilles reaches at which the tortoise was, the tortoise will be some distance ahead. Achilles will never overtake the tortoise.

According to the arrow paradox, if an arrow is in flight, at any given instant, it is somewhere, occupying a space equal to its dimensions. However, anything occupying a space is at rest, and anything at rest is not in motion. Hence, an arrow in flight (a moving arrow) could not move. The arrow cannot move where it is; neither can it move where it is not.)

In the stadium paradox, in the middle of a stadium, there is a line of three stationary chariots (line B). A line of three moving chariots, A, approaching from one end of B, reaches a point at which the chariots are side by side with those of B; another line of three chariots, C, approaches at the same time from the other end of B and reaches a similar side by side position relative to B. (The lines of moving chariots accomplish this task at the same velocity.) Note that in the time it takes Chariot A-1 to move from opposite B-1 to opposite B-3, it passes only one chariot in B, namely B-2, but it passes two chariots in C, namely, C-1 and C-2. Hence, in passing B, line of chariots A travels at half the speed with which it passes line C. If time is relative to motion, then half the time is equal to the whole. (See the accompanying table, “The Stadium Paradox.”)

If T represents the amount of time it takes a chariot in A to pass a chariot, then T (the amount of time taken to pass B-2) is equal to 2T (the amount of time taken to pass C-1 and C-2).

The paradox of the millet seed looks at a single millet seed, which makes no noise when it falls. It would be impossible, then, for a number of seeds falling together (say, a bushelful) to make a noise (because the sum of any number of silences could hardly be a sound).

These are the principal paradoxes traditionally attributed to Zeno. Whatever the inaccuracies of the various accounts, the genius of Zeno comes through. Accordingly, philosophical critics—including geniuses who would not bother with trivial arguments—have given a great deal of thought over the centuries to Zeno’s arguments: Some have supported Zeno, calling his arguments valid (at least as interpreted in their manner); others have found his arguments to be fallacious. Among those who support him are those who do so semantically, logically, or metaphysically; and among those who find him at fault are those who do so semantically, logically, or metaphysically. The literature of commentary has acquired much of the fascination and inspired something of the genius of Zeno’s paradoxes as we know them.

Principal Ideas Advanced

•Reality cannot be a plurality, a many; reality must be one.

•Motion is impossible.

•Space cannot consist of points; time cannot consist of instants.

•The paradoxes of plurality: If the many have being and magnitude, they must be both large and small; if the many have no magnitude, they cannot be.

•If the units of the many be indivisible, they are nothing; if things are a many and divisible, they are both finite and infinite in number.

•The paradox of dichotomy: One cannot traverse an infinite number of points in a finite amount of time.

•The paradox of Achilles and the tortoise: A swifter runner cannot overtake a slower runner.

•The paradox of the arrow: An arrow in flight cannot move.

•The paradox of the stadium: If passage is relative to points, one set of moving things moving at the same velocity as another set can pass twice as many points as the other in the same amount of time: Double the time is equal to half the time.

Bibliography

Boardman, John, Jasper Griffin, and Oswyn Murray, eds. The Oxford History of the Classical World. Oxford: Oxford University Press, 1986. Martin West’s article on “Early Greek Philosophy” is useful in placing Zeno in his historical and cultural context.

Copleston, Frederick. A History of Philosophy: Greece and Rome. Garden City, N.Y.: Doubleday, 1962. Copleston provides an instructive introduction in a brief chapter on Zeno and his famous paradoxes about time and motion.

Curd, Patricia, ed. A Presocratics Reader: Selected Fragments and Testimonia. Translations by Richard D. McKirahan, Jr. Indianapolis, Ind.: Hackett, 1996. Contains important texts and commentary that are important for understanding Zeno.

Faris, J. A. The Paradoxes of Zeno. Brookfield, Vt.: Avebury, 1996. A helpful study of the main logical paradoxes advanced by Zeno.

Grünbaum, Adolf. Modern Science and Zeno’s Paradoxes. Middletown, Conn.: Wesleyan University Press, 1967. Grünbaum brings the resources of contemporary mathematics and physics to bear on paradoxes having to do with motion and time.

Hussey, Edward. The Pre-Socratics. Indianapolis, Ind.: Hackett, 1995. Includes a sympathetic analysis of Zeno, which focuses on concepts such as time, change, diversity, separation, and completeness.

McGreal, Ian Philip. Analyzing Philosophical Arguments. Corte Madera, Calif.: Chandler, 1967. McGreal offers a detailed analysis of Zeno’s argument about Achilles and the tortoise, which concentrates on time and motion.

McKirahan, Richard D., Jr. Philosophy Before Socrates: An Introduction with Texts and Commentary. Indianapolis, Ind.: Hackett, 1994. An excellent study of pre-Socratic philosophy that includes a discussion of Zeno.

Salmon, Wesley C., ed. Zeno’s Paradoxes. Indianapolis, Ind.: Bobbs-Merrill, 1970. This useful and intellectually entertaining anthology contains important articles by leading philosophers who deal with Zeno’s paradoxes. Includes an excellent bibliography.

Schrempp, Gregory Allen. Magical Arrows: The Maori, the Greeks, and the Folklore of the Universe. Madison: University of Wisconsin Press, 1992. A comparative analysis that includes discussion of Zeno’s reflections on time and motion.