Zeno of Elea c. 490–430 BCE
ZENO OF ELEA
c. 490–430 BCE
According to Plato (Parmenides (127A–C), Zeno was born around 490 BCE. He was a citizen of Elea, a Greek city in southern Italy with which Parmenides was also associated. Little is known about his life. The setting of Plato's Parmenides is a visit Zeno and Parmenides made to Athens in Socrates' youth (around 450 BCE), but since the conversation in that dialogue between Parmenides and Socrates certainly did not take place, there is no strong reason to believe that the visit did either. According to tradition, Zeno died heroically defying a tyrant in Elea. Philosophically he was a follower of Parmenides, whose doctrines he defended by arguing against opposing views; hence Aristotle called him the father of dialectic. Although Zeno wrote a book containing forty arguments against plurality. very little of his writing remains; approximately twenty lines of quotations, supplemented by relatively scanty testimonia. We have information about a dozen of his arguments. Under these circumstances, Zeno's immense influence on the history of philosophy is all the more remarkable.
Plato, our earliest witness, depicts Zeno as defending Parmenides' views against people who ridiculed Parmenides on the grounds that his views have absurd consequences. Zeno paid them back in their own coin, pursuing implications of the opposing views, which he showed have consequences even more absurd than those the opponents claimed to follow for Parmenides (Parmenides 128C–D). Zeno's book comprised a series of polemical arguments that employed the strategy reductio ad absurdum against the claim that there exists more than one thing (ibid. 128B–D).
Although Plato's account fits some of Zeno's arguments, it does not hold for them all. Several argue that motion cannot exist, another that the senses fail to discern the truth, another that things do not have locations. And so it is unclear how reliable Plato (whose reports of some other early philosophers are unreliable) is as a source on Zeno. Some scholars deny that Parmenides was a monist at all, or in the relevant sense, and some have held that some of Zeno's arguments tell as strongly against Parmenides' monism as they do against his opponents' pluralism. If this is correct, then Plato's account of Zeno's arguments is wholly misguided. Others have also argued that Zeno is better defined as a proto-sophist, a paradox-monger who constructed ingenious arguments with perverse conclusions, without any philosophical commitments at all.
Despite these concerns, the text of this encyclopedia follows the traditional view that Parmenides believed that there exists only one entity, which is motionless and changeless (it has other attributes as well); that the human senses are entirely deceptive as a source of knowledge of reality; and that Zeno defended this theory through arguments that derive absurd consequences not only from the assumption that there exists a plurality of entities but also from the assumptions that motion and change exist, and other assumptions that humans make about the world. Plato's account is taken to be essentially correct; when it states that Zeno defended Parmenides' view that there is just one thing, it is quoting this core Parmenidean thesis as a shorthand method of referring to the entire theory.
Scholars have disputed the identity of Parmenides' opponents against whom Zeno directed his arguments. Some held that they were the Pythagoreans, but the case collapsed for lack of evidence. Others have suggested that the opponents were not actual objectors but any possible objectors, that Zeno constructed a series of arguments that systematically refuted all possible alternative theories—for example, the theory that motion is continuous and also the theory that motion is discrete—but that interpretation failed for the same reason. What remains is the most natural interpretation, that Parmenides' opponents were people (ordinary folk and philosophers as well) who found Parmenides' views obviously, radically, and amusingly wrong because they conflict so strongly with humankind's most deeply held beliefs about the world.
The Zenonian legacy is a number of arguments known as paradoxes because of their implausible conclusions. Many of them have the form of an antinomy, which is a special kind of reductio argument. Zeno proves a thesis by demonstrating that its contradictory has incompatible consequences. Since the consequences cannot both be true, the contradictory of the original thesis is false, so the thesis itself is true. As a matter of fact, Zeno's arguments do not contain the final move, which is characteristic of reductio arguments: they stop when they have shown that the contradictory of the thesis is false and do not draw the inference that the thesis itself is true. It has therefore been claimed that the arguments are not reductio arguments at all. But this criticism affects only the form, not the intent of the argument; they are reductio arguments in spirit if not in letter.
Arguments against Plurality
Several of Zeno's arguments against plurality survive. They include the argument of both like and unlike; the argument of both large and small; and the argument of both limited and unlimited.
argument of both like and unlike
The first argument against plurality is as follows: (a) If things are many, they must be both like and unlike; but (b) what is like cannot be unlike and what is unlike cannot be like; therefore (c) there cannot be many things (Parmenides 127D). The meaning of (a) is unclear: In what way are many things both like and unlike? One attempt to explicate it as follows. If there are many things, each of them is like itself in that everything that is true of it is true of it. This is trivially true. But if one thing (A) is counted as like another (B) only if everything that is true of A is true of B and/or vice versa, and if A is unlike B if and only if A is not like B (i.e., "like" and "unlike" are contradictories, as (b) indicates), then any two things are unlike one another.
For example, even if A and B are as alike as two peas in a pod, A will be unlike B because it is true of A that it is A but it is not true of B that it is A. Following this interpretation, which places a very strong condition on things being "like," (a) is true but (b) is false, so it follows that the alleged impossibility is not impossible at all. For impossibility to occur, the things would have to be both like and unlike the same thing (whereas here A is like one thing (A) and unlike something else (B)). Further, they would have to be both like and unlike the same thing in the same respect (since A can be like B in color but unlike B in weight) and at the same time (since A can be like B in color at one time but not another). The paradox fails on the interpretation given. It also fails if one admits a weaker condition for one thing being like another. For example, if one counts A as like B if at least one thing true of A is also true of B, so that A will be unlike B only if nothing true of A is true of B, the alleged impossibility again proves perfectly possible, since the only way something can be both like and unlike is by being like one thing and unlike something else. Other attempts to reconstruct the argument have been proposed, but none has yet succeeded in making it plausible, so it seems likely that Zeno's first argument is fallacious.
argument of both large and small
Two of Zeno's surviving five fragments contain parts of a different and more complex argument against plurality. The argument claims that if things are many, they are both large and small: (a) so large that they are infinite and (b) so small that they have no size. The argument consists of two separate parts, one showing that things are large and one that they are small. It is an antinomy, but in this case Zeno argues that each branch of the antinomy is subject in its own right to a serious objection.
The entire argument for (a) has survived, but only part of the argument for (b). The proof of (b) came first, and the part that is reported is as follows: "Nothing has size because each of the many things is the same as itself and one." Zeno then argues that anything without size, thickness, or bulk does not exist: "If it is added to something else that exists, it will not make it any larger. For if it were of no size and were added, what it is added to cannot increase in size. It follows immediately that what is added is nothing. But if when it is subtracted the other thing is no smaller, and it is not increased when it is added, clearly the thing added or subtracted is nothing." (DK 29B2) This argument holds for three-dimensional bodies (though not for other kinds of things: I do not become larger by becoming happier, though one might say that happiness is added to me), so it is reasonable to take Zeno as arguing against the kind of pluralism that supposes that there exists a plurality of bodies (physical pluralism). What is missing is a reason to hold that "nothing has size because each of the many things is the same as itself and one."
The argument for (a) states: "If it exists, each thing must have some size and thickness, and a part of it must be apart from the rest. And the same reasoning holds for the part that is in front: that too will have size and part of it will be in front. Now to say this once is the same as to keep saying it forever. No such part of it will be last, nor will there be one part unrelated to another. Therefore, if there are many things they must be both small and large; so small as not to have size, but so large as to be unlimited." (DK 29B1) The first claim follows from (b). Zeno proceeds on the assumption that size implies divisibility: any body can be divided into spatially distinct parts, each of which is itself a body. This in turn entails divisibility without limit: the process of subdividing never reaches an end, so the parts are so large as to be unlimited.
Most scholars believe that the argument claims to prove that the size of the totality of the parts is infinitely large. If so, it is fallacious. All it proves is that number of the parts is infinitely large, and as the series 1/2 + 1/4 + 1/8 + … (whose sum is 1) shows, the sum of an infinite series need not be infinite. In the present case, the size of the totality of the parts remains equal to the size of the original whole. But if we adopt another interpretation the argument is valid. Since the argument focuses not on the size of the parts, but on the process and the products of division, the problem it raises concerns not the size of the totality of the parts but the possibility of completing the division. According to this interpretation, Zeno is demonstrating a difficulty in ordinary notions of physical bodies and spatial extension. People think that bodies are divisible and Zeno points out there is no reason to postulate that divisibility is impossible beyond some minimum size. It follows that bodies are infinitely divisible: even a small body is large enough to have an infinite number of parts. This conclusion is surprising enough to be worthy of Zeno.
The account of division just given suggests a way to supply the missing step in the argument for (b). How does the innocuous fact that something is the same as itself and one imply that it has no size? Perhaps it is because being "one" entails having no parts—otherwise it would be many. Since (as the process of division shows) anything with size can be divided into parts, only something without size will have no parts and so be "one." And then the argument for (b) comes into play.
argument of both limited and unlimited
The argument is: "(a) If there are many things, they must be just as many as they are, neither more nor less. But if they are just as many as they are, they must be limited. (b) If there are many things, the things that exist are unlimited, since between things that exist there are always others, and still others between those. Therefore the things that exist are unlimited." (DK 29B3) Branch (a) of this antinomy amounts to the claims that any plurality of things consists of a definite number of things and that any definite number is limited. The latter of these is equivalent to the claim that there is no such thing as a definite unlimited number. It has been objected that this last claim is false, since some infinite collections are pluralities that in a relevant way are definite and yet not "just as many as they are." But Zeno did not have the modern understanding of the infinite available to him, and the notion of "unlimited" with which he was working (in which the word means "inexhaustible" or "endless") makes it reasonable, even truistic, to say that an unlimited collection of things has no definite number. The former claim, that every plurality contains a definite number of things, as at least superficially plausible, which is enough to launch the paradox. Whether or not it is true will depend on how an individual counts the things in question (and perhaps their parts as well—see the paradox of both large and small), which Zeno does not specify.
Branch (b) can be interpreted in several ways, some of them anachronistic (for example, that the plurality in question is not three-dimensional objects but mathematical points on a line) and some open to obvious objections (for example, if Zeno is talking about three-dimensional objects that can touch one another, it is just false that there are always other objects in between). The source for this argument suggests a more interesting approach, saying: "In this way he proved the quantity unlimited on the basis of bisection." (Simplicius, In Physica 140, 33). Simplicius need not have quoted all the Zenonian text he had access to, and since he quoted part of the argument he could very well have known the rest of it. In the kind of division referred to, an object is first cut in half, then one of the halves is cut in half, and so on ad infinitum. If there are two adjacent objects A and B, this argument can be used to prove not that A and B have other objects in between them, but that there is no part of A nearest to B. If A is adjacent to B on the left, then the right half of A (which is itself a part of A) is in some sense nearer to B than A is, and so is the right half of that half, and so on. The point is the same as that of the argument discussed above: when A is divided in this way it turns out to have an unlimited number of parts. And again, this conclusion follows validly if one assumes certain views about physical bodies and spatial extension.
Arguments against Motion
Four of Zeno's arguments against motion were particularly difficult to refute, according to Aristotle, who summarized them and offered solutions. They are the Dichotomy (or the Stadium); the Achilles; the Flying Arrow; and the Moving Rows. The following exposition is based mainly on Aristotle's penetrating discussion.
the dichotomy (or the stadium)
This paradox argues that motion does not exist because it requires something impossible to happen. In order to cross a stadium from the starting line (A) to the finish line (B), after setting out one must reach A1, the midpoint of the interval AB, before reaching B, then A2, the midpoint of the interval A1B, and so on. Each time one reaches the midpoint of an interval one still has another interval to cross with a midpoint of its own. There is an infinite number of intervals to cross. But it is impossible to cross an infinite number of intervals. Therefore one cannot reach the finish line.
The backbone of the argument lies in the following claims. (a) To move any distance one must always cross half the distance; (b) there is an infinite number of half-distances; (c) it is impossible to get completely through an infinite number of things one by one in a finite time; therefore (d) it is impossible to move any distance.
Aristotle, the primary source for the paradox, discusses the paradox several times in Physics (233a21, 239b9, 263a4). He rejects the inference to (d) on the grounds that the time of the motion is not finite, but infinite. Not that he supposes that every motion takes an infinite length of time; rather, as he has argued elsewhere in the Physics (6, 1–2), time is divisible in the same way that the distance traversed is divisible. If it takes a minute to cross the whole distance, it takes half a minute to cross the first half-distance, a quarter of a minute to cross the second half-distance, and so on. As the distances become smaller so does the time required to cross them, and the time interval required for the whole movement can be divided into the same number of subintervals as the number of subintervals into which the distance of the whole movement can be divided. So the time (just like the distance) is infinite in one respect (Aristotle calls this "infinite by division") and finite in another ("in extent").
Aristotle, however, does not stop here. He observes, "This solution is sufficient to use against the person who raised the question … but insufficient for the facts of the matter and the truth" (Physics 263a15), and then proceeds to discuss a deeper issue that the paradox raises: whether it is possible at all to perform an infinite number of acts, even the acts of getting through the sequence of decreasing time intervals. Granted that if one can do it, it will take a finite time, but can we do it at all?
Aristotle's solution to this stronger version of the paradox relies on his distinction between the actual infinite and the potential infinite. It is impossible to complete an actually infinite number of tasks, but possible to complete tasks that are potentially infinite. A line or a time-interval contains a potentially infinite number of points or instants. A point is actualized by stopping there; an instant is actualized by stopping then. Crossing the distance by making a single continuous movement does not actualize any of the midpoints. Hence, according to Aristotle's analysis, motion is possible because it does not involve completing an infinite number of tasks. Aristotle's final position on the paradox is that (d) does not follow from (a) (b) and (c), and Zeno committed an elementary blunder in supposing that it does, and moreover that (b) is true only if taken to claim that there is a potentially infinite number of half-distances, whereas (c) is true only if taken to refer to an actually infinite number of things and additionally if the proviso "in a finite time" is deleted.
This paradox too argues against the possibility of motion. The swiftest runner (Achilles) gives the slowest (traditionally a tortoise, although no mention of the reptile occurs in Aristotle's account) a head start. But then he cannot catch up. He must first reach the tortoise's starting point (A), by which time the tortoise will have moved ahead some distance, however small, to another point (A1). Getting to A proves to be only the first stage of a longer race. In the second stage of the race Achilles must reach A1, but by then, the tortoise will have gone ahead an even smaller distance, to A2, and so on. Each time Achilles reaches the point from which the tortoise has started, the tortoise is no longer there, so Achilles never catches up.
Aristotle observes, "This is the same argument as the Dichotomy, but it differs in not dividing the magnitude in half ": Achilles runs more than twice as fast as the tortoise. Therefore, on the basis of his analysis of the Dichotomy argument, Aristotle thinks that the Achilles goes as follows: (a) To catch up with the tortoise, Achilles must always reach the point from which the tortoise started; (b) There is an infinite number of such starting points; (c) It is impossible to get completely through an infinite number of things one by one in a finite time; Therefore (d) Achilles cannot catch up with the tortoise. Unlike the Dichotomy, this argument does not conclude with the statement that motion is impossible. However, since the nature of motion implies that a faster runner will eventually catch up with a slower one, Zeno's conclusion that this cannot happen entails that motion cannot exist. According to Aristotle's analysis, though (which remains the dominant interpretation), the Achilles is fallacious since it commits the same mistake as the Dichotomy.
However, Aristotle's own statement of the Achilles (Physics 239b14) suggests that this interpretation is mistaken. The passage reads: "The slower will never be caught by the swiftest. For the pursuer must first reach the point from which the pursued departed, so that the slower must always be some distance in front." This summary says nothing about there being an infinite number of starting points or about the impossibility of performing an infinite number of tasks, or performing them in a finite time. Rather, the paradox turns on the words "always" and "never," which points to a different interpretation of the argument: (a) Achilles catches up with the tortoise when he reaches the point where the tortoise then is; (b) each time, before catching the tortoise, Achilles must reach the point from which the tortoise started; (c) when Achilles reaches the point from which the tortoise started, the tortoise has moved ahead; therefore, (d) the tortoise is always some distance ahead of Achilles [from (b and c)]; therefore (e) Achilles never catches up [from (d)].
This argument is different from the Dichotomy argument and is not open to the same objection. Where the Dichotomy is based on the impossibility of performing an infinite number of tasks, the Achilles turns on the words "always" and "never." The Achilles challenges the existence of motion if (e) is taken to assert that there is no time at which is it true that Achilles reaches the point where the tortoise then is; and this is in fact is the natural way to understand (e). But if in (e) "never" means "there is no time at which is it true that… " then in order for the argument to go through, (d) "always" must correspondingly mean "at all times is it true that …." So, (d) must be taken to claim that the tortoise is ahead of Achilles at all times. In faact, this is a valid inference: If the tortoise is always (in this sense of "always') ahead, then Achilles never (in the corresponding sense of "never") catches up. But (d) appears obviously false, since faster things do in fact catch up with slower things. In the argument, (d) follows from (b) and (c), but these premises do not entail that the tortoise is ahead of Achilles at all times (as is needed for the argument to go through to (e)), only that the tortoise is still ahead at every time during the race. For example, if the tortoise's head start is nine miles and its speed is 1 m.p.h. while Achilles' speed is 10 m.p.h, then Achilles catches up with the tortoise at the end of one hour. During the race—before the hour is over—Achilles is always catching up and the tortoise is always ahead. But the scope of "always" is restricted to the time during which Achilles has not yet caught up; it does not have unrestricted scope ("at all times") as is needed for (d) to entail (e). As was noted above, (e) will follow only if there is no time at which it is true that Achilles has caught up, and the argument—in particular (b) and (c)—has given no reason to believe this.
the flying arrow
Aristotle's summary is as follows: If everything is always at rest when it is in a space equal to itself, and what is moving is always at an instant, the moving arrow is motionless" (Physics 239b5). The argument is incomplete as it stands and has been completed in various ways, one of which is the following: (a) Whenever something is in a space equal to itself, it is at rest (from Aristotle's summary); (b) an arrow is in a space equal to itself at each instant of its flight (supplemented); therefore (c) an arrow is at rest at each instant of its flight [from (a) and (b)]; (d) what is moving is always at an instant (from Aristotle's summary); therefore (e) during the whole of its flight the arrow is at rest [from (c) and (d)].
Aristotle objects: the argument "follows from assuming that time is composed of instants; if this is not conceded, the deduction will not go through" (Physics 239b31). This fastens on the move from (d) to (e). Aristotle's view of time, that it is not composed of instants, defeats the paradox. It can also be objected (again on Aristotelian grounds, see Physics 6, 3) that rest and motion take place over time intervals, not at instants. Motion requires occupying different places at different times; it is measured by the distance covered in an interval of time; nothing can move in an instant or for an instant. Likewise, rest is properly understand as the absence of motion: something is at rest during a time interval when it is not in motion. It makes no more sense to speak of rest in an instant or for an instant than it does to say that it is moving in or for an instant. This constitutes an objection to (c).
Another objection concerns (a), which implies that something in motion is not in a space equal to itself. But what does this mean? When is the moving thing not occupying a space equal to itself, and in what way? Two possible answers to the first of these questions are that it does not occupy a space equal to itself over the entire duration of its motion and (ii) at an instant during its motion. On interpretation (i) the idea is that in its motion the arrow occupies different positions at different instants and the sum (in some sense of the word) of those positions is larger than any of the individual positions. If the arrow initially occupies position AB (extending from point A to point B) and ends up at position CD (where the distance from C to D is equal to the distance from A to B), then the distance from A to D is equal to the space the arrow is in during the whole of its flight, and the distance from A to D is larger than the distance from A to B. Conversely, during any period when the arrow is at rest, it will be in a space equal to itself. Interpretation (i) makes sense of (a), but if make the argument invalid. Because (a) concerns motion and rest over the duration of the motion, which is an interval of time, not at an instant, and it is in general illegitimate to infer a conclusion about the behavior of something at individual instants in an interval from its ehavior during the interval as a whole, or vice versa. Consquently the inferences to (c) and (e) are invalid. On interpretation (ii) the move to (c) is valid, but there is no obvious reason why Zeno should have thought or should have expected anyone to agree that things change size during their motion, so that at any instant of its flight an arrow is larger or smaller than when it is at rest. Thus the argument fails: On one interpretation (i) it is invalid and on another (ii), although valid, it contains an unacceptable premise.
the moving rows
Aristotle reports this argument as follows: "The fourth argument concerns equal bodies moving in a stadium alongside equal bodies in the opposite direction, the one group moving from the end of the stadium, the other from the middle, at equal speed. [Zeno] claims in this argument that it follows that half the time is equal to the double. … Let A's represent the equal stationary bodies, B's the bodies beginning from the middle, equal in number and size to the A's, and C's the bodies beginning from the end, equal in number and size to these and having the same speed as the B's. It follows that the first B is at the end at the same time as the first C, as the B's and C's move alongside one another, and the first C has come alongside all the B's but the first B has come alongside half the A's. And so the time is half. For each of them is alongside each thing for an equal time. It follows simultaneously that the first B has moved alongside all the C's, for the first C and the first B will be at the opposite ends simultaneously, because both have been alongside the A's for an equal amount of time" (Physics 239b33).
In discussing this passage, Simplicius, in Physics 1016, 19, provides diagrams to illustrate the starting position and the finish:
The kernel of the argument is as follows: (a) The time it takes the first B to have come alongside four C's is equal to the time it takes the first B to have come alongside two A's; (b) the first B is alongside each A and also alongside each C for the same amount of time; (c) but during its motion B is alongside two A's and B is alongside four C's; therefore (d) the total time B is alongside the A's is half the total time B is alongside the C's [from (b) and (c)]; therefore (e) half the time is equal to the double [from (a) and (d)]. Here "the double" refers to the time taken in being alongside the four C's; it means "the double of half the time," not "the double of the whole time." (Another iteration of the argument will yield the conclusion that half the whole time equals double the whole time.)
Aristotle claims the argument is based on an elementary mistake: "The mistake is in thinking that an equal magnitude moving with equal speed takes an equal time in moving alongside something in motion as it does in moving alongside something at rest" (Physics 240a1). Thus, (b) is false, and consequently so are (d) and (e). Aristotle's analysis is correct if Zeno is treating the motion of extended bodies over a continuous magnitude. But could Zeno have committed so gross a blunder?
An influential interpretation acquits Zeno of this charge. Zeno is arguing not against the ordinary view of time (and perhaps space and motion as well) as being continuous, but against another possible view, that they are discrete: there are "atoms" of time and space, and motion proceeds in atomic "jumps," going from one atomic location to the next from one atomic instant to the next. Either something is moving or it is not; if it is moving, it is in successive locations at successive instants, if it is not, it is in the same location at successive instants. By hypothesis the B's and the C's are moving. One instant after the instant they occupy the starting position (Diagram 1) they will occupy the position illustrated in Diagram 3:
One instant later, they will occupy the position illustrated in Diagram 2. And contrary to what happens if space and time are continuous, there is no instant at which the lead B is next to the lead C (as in Diagrams 4 and 5).
Those who hold this interpretation have claimed the Moving Rows argument to be Zeno's most sophisticated argument and one that tells decisively against the view that time and space are atomic. But there are two obstacles to it. First, it conflicts with Aristotle's statement of the argument, which states that "each of them is alongside each thing for an equal time"; as just noted the lead B is never alongside the lead C. Second, there is no evidence in favor of it. Our sources give no hint that the bodies are atomic bodies or the times are atomic instants and there is no reason to think that such a theory of space and time had been considered by anyone as early as Zeno. The only reason given to support this interpretation is that Zeno was too clever to make the mistake that Aristotle finds—an assessment that is refuted by the equally elementary mistake diagnosed in the paradox of like and unlike.
Two More Paradoxes
Zeno did not limit himself to arguments against the existence of plurality and motion. Two other arguments—the Millet Seed and the Place of Place—survive that challenge other deeply held beliefs.
the millet seed
This argument apparently criticizes the senses, therefore supporting Parmenides' view that the senses are radically unreliable. It is preserved in the form of a dialogue between Zeno and Protagoras (Simplicius, In Physica, 1108.18). In essence it states: (a) One millet seed or one ten-thousandth of a millet seed does not make a sound when it falls; (b) a bushel of millet seeds makes a sound when it falls; (c) there is a ratio between the bushel of millet seeds and one millet seed or one ten-thousandth of a millet seed; (d) the sounds made by the bushel, the millet seed, and the ten-thousandth of a millet seed have the same ratios as the ratios identified in (c); therefore (e) a millet seed makes a sound when it falls, and so does one ten-thousandth of a millet seed [from (b) and (d)]. (e) contradicts (a), which depends on the evidence of hearing. Therefore, hearing is unreliable.
Aristotle rebuts the paradox by saying that a threshold of force is needed to produce sound, and that the force of one millet seed falling is below the threshold. Other solutions suggest themselves as well.
the place of place
This argument is reported in several sources, including Aristotle's Physics (209a23, 210b22) and Simplicius's In Physica. Its essence is as follows: (a) Everything that exists is in a place; therefore (b) place exists; therefore (c) place is in a place [from (a) and (b)]; (d) but this goes ad infinitum. Therefore (e) place does not exist.
Aristotle and his followers rebutted the argument by denying (a): not everything that exists is in a place, "for no one would say that health or courage or ten thousand other things were in a place" (Eudemus, quoted in Simplicius In Physica 563.25); and "nothing prevents the first place from being in something else, but not in it as in a place" (Aristotle, Physics 210b24). One can grant that a three-dimensional object has a place without conceding that its place is the kind of thing that can have a place. Alternatively one might accept the reasoning through (d) but deny that (d) entails (e). Not all infinite regresses are vicious.
The present treatment has offered versions of the most important of Zeno's surviving arguments and has suggested ways to refute them. This follows the tradition in discussing Zeno and the other Eleatic philosophers that has been dominant since Plato (Sophist 258B–D). Aristotle employed this practice and not just as a matter of historical interest. His philosophical method required him to take his predecessors' views into account and find solutions for puzzles and problems they presented, and his views on place, time, motion, and the infinite were framed with Zeno's paradoxes in mind. Philosophical interest in Zeno was renewed (notably by Bertrand Russell) after the modern conception of the infinite had been elaborated; once again contemporary philosophical tenets were employed to refute the paradoxes (principally the Dichotomy, the Achilles, and the Flying Arrow) and the challenge they present to ordinary views of space, time, and motion, and once again the discussion went beyond what Zeno proposed and encompassed related puzzles that his paradoxes suggested.
This astonishing ability to invent exciting and fruitful paradoxes is not Zeno's only contribution to philosophy. If Parmenides was the first pre-Socratic philosopher to employ deductive arguments, Zeno was the first to do so in prose, and his fragments show that he made great advances over Parmenides in the clarity of his reasoning and the complexity of his arguments. Also noteworthy is his use of deductions to point out the danger of maintaining familiar beliefs without examining them. These contributions easily outweigh any errors one may (frequently by employing concepts, distinctions and proof techniques that were not developed for centuries or millennia after Zeno's time) detect in his arguments.
See also Aristotle; Dialectic; Infinity in Mathematics and Logic; Logic, History of; Logical Paradoxes; Melissus of Samos; Motion; Parmenides of Elea; Plato; Russell, Bertrand Arthur William; Set Theory; Simplicius; Socrates.
Caveing, Maurice. Zénon d'Élée. Prolégomènes aux doctrines du continu. Paris: Vrin, 1982.
Diels, Hermann, and Walther Kranz. Die Fragmente der Vorsokratiker. 5th ed. Dublin and Zurich: Weidmann, 1951.
Untersteiner, Mario. Zenone. Testimonianze e Frammenti. Florence: La Nuova Italia, 1963.
Barnes, Jonathan. The Presocratic Philosophers. 2nd ed. Boston: Routledge & Kegan Paul, 1982.
Black, Max. "Achilles and the Tortoise." Analysis 11 (1951): 91–101.
Dillon, John. "More Evidence on Zeno of Elea?" Archiv für Geschichte der Philosophie 58 (1976): 221–222.
Dillon, John. "New Evidence on Zeno of Elea?" Archiv für Geschichte der Philosophie 56 (1974): 127–131.
Fränkel, Hermann. "Zeno of Elea's Attacks on Plurality." American Journal of Philology 63 (1942): 1–25, 193–206.
Furley, David. Two Studies in the Greek Atomists. Princeton, NJ: Princeton University Press, 1967.
Grünbaum, Adolf. Modern Science and Zeno's Paradoxes. Middletown, CT: Wesleyan University Press, 1967.
Guthrie, W. K. C. A History of Greek Philosophy, Vol. 2: The Presocratic Tradition from Parmenides to Democritus. Cambridge, U.K.: Cambridge University Press, 1965.
Hinton, J. M., and C. B. Martin. "Achilles and the Tortoise." Analysis 14 (1954): 56–68.
Kirk, G. S., and J. E. Raven. The Presocratic Philosophers. Cambridge, U.K.: Cambridge University Press, 1957.
Kirk, G. S., J. E. Raven, and M. Schofield. The Presocratic Philosophers. 2nd ed. Cambridge, U.K.: Cambridge University Press, 1982.
Lear, Jonathan. "A Note on Zeno's Arrow." Phronesis 26 (1981): 91–104.
McKirahan, Richard D. "La dichotomie de Zénon chez Aristote." In Qu'est-ce que la philosophie présocratique, edited by André Laks and Claire Louguet, 465–496. Villeneuve d'Ascq (Nord): Presses Universitaires du Septentrion, 2002.
McKirahan, Richard D. "Zeno." In The Cambridge Companion to Early Greek Philosophy, edited by A. A. Long, 134–158. Cambridge, U.K.: Cambridge University Press, 1999.
Owen, G. E. L. "Zeno and the Mathematicians." Proceedings of the Aristotelian Society 58 (1957–58): 199–222.
Russell, Bertrand. Our Knowledge of the External World. Chicago: Open Court, 1914.
Ryle, Gilbert. Dilemmas. Cambridge, U.K.: Cambridge University Press.
Salmon, Wesley, ed. Zeno's Paradoxes. New York: Bobbs-Merrill, 1970.
Solmsen, Friedrich. "The Tradition about Zeno of Elea Re-Examined." Pronesis 16 (1971): 116–141.
Tannery, Paul. Pour l'histoire de la science. Hellène. Paris: Alcan, 1887.
Thomas, L. E. "Achilles and the Tortoise." Analysis 12 (1952): 92–94.
Vlastos, Gregory. "A Note on Zeno's Arrow." Phronesis 11 (1966): 3–18.
Vlastos, Gregory. "Plato's Testimony concerning Zeno of Elea." Journal of Hellenic Studies 95 (1975): 136–162.
Vlastos, Gregory. "A Zenonian Argument against Plurality." In Essays in Ancient Greek Philosophy, edited by John P. Anton and George L. Kustas, 119–144. Albany: State University of New York Press, 1971.
Vlastos, Gregory. "Zeno of Elea." In Encyclopedia of Philosophy, 1st ed., edited by Paul Edwards, 369–379. New York: Macmillan, 1967.
Vlastos, Gregory. "Zeno's Race Course: With an Appendix on the Achilles." Journal of the History of Philosophy 4 (1966): 95–108.
Richard McKirahan (2005)