Ancient History & Civilisation


T1 gives us both a summary of one of Zeno’s arguments (on similarity and dissimilarity) and an influential account of their purpose. There were said (by Proclus, Commentary on Plato’s ‘Parmenides’ 694.23–4 Cousin) to be forty arguments in his original treatise, all with the same purpose—to defend Parmenides’ thesis that all is one, by demonstrating the absurdity of the consequences of the assumption that there is a plurality. What we have extant are the reports of a number of arguments which, by and large, pursue this aim. They do not necessarily pursue this aim directly, but if Parmenides’ monism outraged common sense, then Zeno’s paradoxes constitute an assault on common sense, and so offer at least indirect support for Parmenides. The surviving arguments fall into several categories: there are arguments against the possibility of plurality, motion, and place; and one miscellaneous argument whose original purpose is unclear. However, the original forty also contained, for instance, an argument aiming to prove that if there is a plurality, every member of that plurality is both similar and dissimilar (Plato, Phaedrus 261d).

The most famous are the arguments against the possibility of motion,1 in which he claims to show that the assumption of motion leads to paradoxical consequences, and so that there can be no such thing (compare Parmenides F8. ll. 26–33 on p. 60). These arguments are summarized and criticized by Aristotle in T2 and T3. Aristotle’s paraphrases are for the most part perfectly clear. There are four arguments, known respectively as the Dichotomy, the Achilles, the Arrow, and the Stadium (or the Moving Rows). The Dichotomy states that in order to complete any process of motion, the moving object first has to cross half of the space on the way to its goal; it then has to cross half the remaining space, and then again half the remaining space, and so on ad infinitum. So it has an infinite number of tasks to perform in a finite time (see T2); but this is absurd, and so the whole notion of motion is absurd. The solution, according to Aristotle in T2, is to appreciate that time is just as liable to infinite division as space. It has been objected that Zeno was perfectly well aware of the notion of infinite divisibility, but it is hard to find it in the extant evidence (above all when Zeno talks of the possibility of dividing something down into nothing), and I think Aristotle’s criticism is fair here, as far as it goes.2 As Aristotle himself admits at Physics 263a–3, this reply is effective against the paradox as formulated by Zeno, but does not address the potential importance and interest of Zeno’s argument.3 It would perhaps have been more relevant, then, for Aristotle to have argued against Zeno’s assumption that an infinite series of tasks has to be performed. For while it is true that an infinite series of tasks would have to be performed were the runner to mark every successive half-way point that he reaches, the conditional form of this sentence is important: it is true, logically, that it is always possible for another mark to be made, however many marks have already been made, but it is not true that any runner need make an infinite series of marks.

The Achilles is probably the best-known of Zeno’s paradoxes. As Aristotle says, it depends on the same fallacy as the Dichotomy, and therefore its solution is the same. The puzzle states that in a race in which Achilles has a handicap and starts behind a slower runner (e.g. a tortoise), in order to overtake the tortoise he has first to reach the place where the tortoise started from; but by then the tortoise has moved on, so Achilles has next to cross the (shorter) distance to where the tortoise is now; but by the time he gets there the tortoise has moved on … and so on ad infinitum. As Aristotle protests, Zeno must grant the evidence of his senses, that Achilles does catch up with and overtake the tortoise—that a finite distance can be traversed. One can complete an infinite series of tasks, provided it is understood that the infinitude comes in this case from infinite divisibility, not infinite extension. No one doubts that Achilles cannot mark his traversal of an infinite series of decreasing distances, but equally, Aristotle says, no one doubts that Achilles can traverse an infinitely divisible distance. Zeno needs, then, to distinguish which kinds of infinite tasks are not completable, and which are.

The third paradox of motion is the Arrow. This states that at any given moment an arrow in flight is occupying a space equal to its own size. But this is by definition what it is to be at rest: it is to be occupying a space which is, as one might put it, opposite another space of equal size. Therefore an arrow cannot move, since at every given moment it is at rest. Aristotle’s solution is to suggest that time is more fluid than Zeno supposes: it does not consist (cinematographically, so to speak) of a series of discrete units of time,4 and from the fact that an arrow is not moving at any given instant, it does not follow that it does not move in the overall stretch of time involved. In any case, the concepts of motion and being at rest implicitly import the concept of a stretch of time: motion entails speed, and speed is a measure of distance covered at a certain time; by the same token we call a thing at rest if it does not cover any distance in a given period of time. Therefore, Aristotle implies, Zeno was wrong to talk of motion and being at rest in an instant (seePhysics 234a24–b9).

The fourth paradox, the Stadium, is the most controversial. It will help to have a diagrammatic representation of the puzzle. The starting-point is this:


Now, apparently Zeno’s ‘paradox’ is simply that by the time the Bs have reached the end of the As, having traversed two As, they have also reached the end of the Cs, having traversed four Cs. How can the Bs traverse two As and four Cs in the same time, when the As and the Cs are the same size? As Aristotle remarks, the solution is simple: it takes longer to pass a stationary body than it does to pass a body which is coming towards you.

It may well be that this was all the Stadium stated, and that it was that straightforward. There are signs of equal ‘naïvety’ in others of Zeno’s arguments. It is just as likely that Zeno supported Parmenides’ monism by sheer weight of the number of his arguments, as that he made each and every argument a deep paradox. However, many scholars think that Zeno could not have been so naïve, and so that Aristotle misunderstood his argument. They generate a more profound argument out of the elements given by Aristotle. Suppose that each of the blocks of As, Bs, and Cs is an atomic unit of space, and suppose that it takes one atomic unit of time (let’s call it a ‘click’) for one atomic unit of space to pass another atomic unit of space. In one click, then, the leading B has moved from being opposite the second A to being opposite the third A. But in the same click it has moved from being opposite none of the Cs, to being opposite the second C. When, then, was it opposite the first C? It looks as though the click, which is by definition atomic (that is, indivisible), has to be subdivided, and by the same token so do the supposedly atomic blocks. (The solution, I suppose, is to insist that there was no time when the first B-block was opposite the first C-block.) The advantage of this interpretation is that it gives Zeno a more interesting argument; the main problems with it are that it departs from what Aristotle says, and there is no evidence that in Zeno’s time there was a theory of atomic units of space and time.

In a somewhat roundabout way, F1 gives us the bare bones of a series of Zenonian arguments against the possibility of plurality. If we need to find a particular target for these arguments, the theories of Anaxagoras are the best bet. Restoring its parts to their natural order, the argument would have gone somewhat as follows:

1. If there are many things, each of them is both infinitely small (i.e. non-existent) and infinitely large. Any thing, X, is the same as itself; if anything were added to it, it would not be X, but X + Y. But everything is divisible into parts (this is as close as Zeno comes to the notion of infinite divisibility). Everything has magnitude, which is to say that there is distance between one part of it and another; wherever you divide it there will always be an extra, protruding part yet to be divided. The possession of magnitude is an essential property of existence, because if something had no magnitude, it would make no difference were it to be added to or subtracted from something, which is to say that it would have no existence. But if the possession of magnitude is an essential property of existence, and if every magnitude is divisible into parts, then every existing thing is X+ Y, and if anything were just X it would not exist. Therefore, if there are many things, they are either self-identical, which is to say that they have no parts, which is to say that they are infinitely small, which is to say that they do not exist; or they are infinitely large, because they are divisible into infinite parts, and infinite parts do not add up to anything of merely finite size.

2. If every existing thing is infinitely divisible into parts, then either nothing exists or everything is one. For either division ends at an infinite number of atomic minimal parts (but anything made up of infinite parts has infinite magnitude), or it ends when the division of the last two parts leaves nothing (but this is inconceivable). But the concept of a plurality of existing things stands or falls with the concept of infinite divisibility into parts. Therefore, since the concept of infinite divisibility into parts is absurd, there is no plurality, only unity.

3. If there are many things they are both infinite and finite in number. They are either just as many as they are, in which case they are finite in number, or, given infinite divisibility into parts, they are infinite in number. But this is absurd, and so there cannot be many things, only unity.

This must have seemed a pretty devastating series of arguments to Zeno’s contemporaries. The arguments are flawed, of course: Zeno appears to assume, for instance, that anything made up of infinite parts must be infinitely huge. But the solution to the puzzles requires some fairly complex thinking about infinity, and in particular the recognition of the possibility of infinite division: this is effectively the challenge Zeno set his successors.

It must also be noticed that in arguing that anything without magnitude does not exist, Zeno is arguing against the existence of Parmenides’ ‘what-is’, just as much as he is arguing against common sense.5 What, then, of Plato’s statement, in T1, that Zeno’s purpose was to defend Parmenidean monism? On the whole, this seems to fit Zeno’s arguments well, but for someone like Zeno there are no sacred cows. He demands that we think about all our assumptions, whether they are derived from common sense or from the authority of Parmenides; and he delights in the argumentative methods he polished: the infinite regress, the reductio ad absurdum.

T4 is a good example of an infinite regress, by which Zeno attempted to reduce to absurdity the idea of place. Since pluralism requires the existence of places, the argument can again be seen as supportive of Parmenidean monism; also, if existence is conceived of as corporeal, and corporeality as requiring space or place, then Zeno may be seen as attacking the notion that all existence is corporeal. Aristotle’s solution in T5 is to point out that ‘in’ can mean different things. There need be no infinite series of containing places, because you can say that one thing is ‘in’ another without meaning that it is ‘in a place’. This is a good argument as far as it goes, but it is still not clear how it stops the regress, rather than simply providing a different perspective on how to describe any member of the regress. Perhaps Aristotle means that we can say that the duvet is ‘in’ the cover, in the sense that the cover is the place of the duvet; but in saying this we are not attaching the property of ‘being in a place’ to the duvet, so much as attaching the property of ‘being a place’ to the cover. This would stop the regress immediately, because it would take a fresh argument to claim that the cover itself was in a place. Alternatively, one might argue that the place of the place of anything was just the place of that thing; this too effectively stops the regress.

Zeno’s argument in T6 is perfectly clear and straightforward. It is not clear how it serves his overall purpose of defending Parmenidean monism (or at least assaulting common sense), but one can see how it might fit in with his general concerns to argue that the smallest part of anything (here each individual seed in a bushel of millet seeds) has magnitude. Alternatively, it may simply have been an argument against reliance on the senses: the senses tell us that a single seed makes no sound as it falls, but reason, more reliably, informs us that it must, otherwise the whole bushel would not make a sound.

Aristotle, so important in preserving accounts of Zeno’s arguments, may have the last word. In T7 he describes Zeno as the founder of dialectic. In this context, ‘dialectic’ means a polemical method of arguing which shows the falsity of an opponent’s premisses and assumptions. This is how Zeno earns his place in the history of philosophy, for a similar argumentative method was to flourish in Plato’s dialogues and give rise to the origins of logic in Aristotle.

T1 (DK 29A12; KRS 314, 327) [Part of a discussion between Socrates, Parmenides, and Zeno] After Socrates had listened to Zeno reading his treatise, he asked him to repeat the first hypothesis of the first argument. After it had been read through he said, ‘What do you mean by this, Zeno? If there are many things, they must be both like and unlike one another, and this is impossible, because dissimilar things cannot be similar and similar things cannot be dissimilar. Is that what you mean?’*

‘Yes,’ said Zeno.

‘So if it is impossible for dissimilars to be similar and similars to be dissimilar, it is also impossible for there to be a plurality of things, because if there were a plurality of things, they would be liable to impossibilities. Is this the point of your arguments? Isn’t it precisely to insist, contrary to everything that is said, that there is no plurality? And don’t you think that each of your arguments proves just this same point, with the result that you think that you have come up with as many proofs that there is no plurality as you have written arguments? Is this what you mean, or have I misunderstood you?’

‘No,’ said Zeno. ‘You have an excellent grasp of the point of the whole treatise.’

‘Parmenides,’ Socrates said, ‘I see that Zeno’s treatise is another means he uses, along with his general friendship, to get close to you. In a sense his work is the same as yours, but he has made it look different as a way of trying to fool us into thinking that he is saying something different. I mean, in your poem you say that everything is one, and you come up with excellent arguments to demonstrate this, while he says that there is no plurality, and again comes up with a huge number of arguments to prove this at enormous length. So, with the one of you saying “One” and the other saying “Not many”, and with each of you speaking in such a way as to make it seem as though there is nothing remotely the same in what you’re saying, although in fact what you’re saying is more or less identical, it looks as though the rest of us have missed the point of what you’ve been saying.’

‘Yes, Socrates,’ said Zeno, ‘but in certain respects the true facts about my treatise have escaped your notice … The truth is that it is a kind of reinforcement of Parmenides’ argument against those who try to mock it by arguing that, if there is only unity, the argument entails many absurd and even self-contradictory consequences. My treatise, then, responds to those who argue in favour of a plurality, paying them back what is due to them and then more besides. My intention is to demonstrate that their assumption of plurality, when followed through far enough, is even more absurd than the assumption of unity.’ (Plato, Parmenides 127d6–128d6 Burnet)

T2 (DK 29A25; KRS 320; L 19) That is why Zeno’s argument makes a false assumption, that it is impossible to traverse what is infinite or make contact with infinitely many things one by one in a finite time. For there are two ways in which distance and time (and, in general, any continuum) are described as infinite: they can be infinitely divisible or infinite in extent. So although it is impossible to make contact in a finite time with things that are infinite in quantity, it is possible to do so with things that are infinitely divisible, since the time itself is also infinite in this way. And so the upshot is that it takes an infinite rather than a finite time to traverse an infinite distance, and it takes infinitely many rather than finitely many nows to make contact with infinitely many things. (Aristotle,Physics 233a21–31 Ross)

T3 (DK 29A25–8; KRS 317, 318, 322, 323, 325; L 19, 26, 28, 29, 35) Zeno’s reasoning is invalid. He claims that if it is always true that a thing is at rest when it is opposite to something equal to itself, and if a moving object is always in the now, then a moving arrow is motionless. But this is false, because time is not composed of indivisible nows, and neither is any other magnitude.

Zeno came up with four arguments about motion which have proved troublesome for people to solve. The first is the one about a moving object not moving because of its having to reach the half-way point before it reaches the end. We have discussed this argument earlier [T2].

The second is the so-called Achilles. This claims that the slowest runner will never be caught by the fastest runner, because the one behind has first to reach the point from which the one in front started, and so the slower one is bound always to be in front. This is in fact the same argument as the Dichotomy, with the difference that the magnitude remaining is not divided in half. Now, we have seen that the argument entails that the slower runner is not caught, but this depends on the same point as the Dichotomy; in both cases the conclusion that it is impossible to reach a limit is a result of dividing the magnitude in a certain way. (However, the present argument includes the extra feature that not even that which is, in the story, the fastest thing in the world can succeed in its pursuit of the slowest thing in the world.*) The solution, then, must be the same in both cases. It is the claim that the one in front cannot be caught that is false. It is not caught as long as it is in front, but it still is caught if Zeno grants that a moving object can traverse a finite distance.

So much for two of his arguments. The third is the one I mentioned a short while ago, which claims that a moving arrow is still. Here the conclusion depends on assuming that time is composed of nows; if this assumption is not granted, the argument fails.

His fourth argument is the one about equal bodies in a stadium moving from opposite directions past one another; one set starts from the end of the stadium, another (moving at the same speed) from the middle. The result, according to Zeno, is that half a giventime is equal to double that time. The mistake in his reasoning lies in supposing that it takes the same time for one moving body to move past a body in motion as it does for another to move past a body at rest, where both are the same size as each other and are moving at the same speed. This is false. For example, let AA … be the stationary bodies, all the same size as one another; let BB … be the bodies, equal in number and in size to AA …, which move from the middle of the stadium; and let CC … be the bodies, equal in number and in size to the others, which start from the end of the stadium and move at the same speed as BB … Now, it follows that the first B and the first C, as the two rows move past each other, will reach the end of each other’s rows at the same time. And from this it follows that although the first C has passed all the Bs, the first B has passed half the number of As; and so (he claims) the time taken by the first B is half the time taken by the first C, because in each case we have equal bodies passing equal bodies. And it also follows that the first B has passed all the Cs, because the first C and the first B will be at opposite ends of the As at the same time, since (according to Zeno) the first C spends the same amount of time alongside each B as it does alongside each A, because both theCs and the Bs spend the same amount of time passing the As. Anyway, that is Zeno’s argument, but his conclusion depends on the fallacy I mentioned. (Aristotle, Physics 239b5–240a18 Ross)

F1 (DK 29B1–3; KRS 315, 316; L 2, 9–12) In his treatise, however, which contains many arguments, he shows in each case the contradictory consequences of the assertion that there is a plurality. One of these arguments is the one in which he demonstrates that if there are many things they are both large and small—large enough to be infinite in magnitude, and small enough to have no magnitude at all.

In the following argument he demonstrates that anything which has no magnitude, solidity, or bulk does not exist. After all, he says, ‘If such a thing were added to anything else, it would not make it larger; for if (despite the fact that it has no magnitude) it is added, no increase with respect to magnitude can take place. And therefore the thing which is added is bound to be nothing. If when it is subtracted the other item becomes no smaller and when it is added the other item does not increase, obviously what was added or subtracted is nothing.’ Now, the point of this argument of Zeno’s is not to reject singularity,* but to claim that each member of a plurality has magnitude—and so that the many are infinitely many, by virtue of the fact that, on account of infinite divisibility, there is always something in front of any given thing. But his demonstration of this point is preceded by his demonstration that no member of the plurality has magnitude because each member of the plurality is the same as itself and is one …

Porphyry believes that it was Parmenides who made use of the argument from dichotomy, in an attempt to show that what exists is one. Porphyry writes as follows: ‘Parmenides had another argument which used dichotomy to prove, apparently, that what-is is only one, and that it has no parts and is indivisible. For supposing it to be divisible, he says, let it be divided into two, and then let each of the parts be further divided into two. Once this has gone on and on happening, it is obvious, he says, that either there will remain certain ultimate magnitudes, which are minima and are indivisible, but infinite in number, in which case the whole will be composed of numerically infinite minima; or else it will vanish and be dissolved into nothing, in which case it will be composed of nothing. Both of these outcomes are absurd, and therefore it is indivisible, and remains one. Or again, since it is everywhere alike, then if it is divisible, it will be equally divisible everywhere, rather than being divisible in one place but not in another. So let it be divided everywhere. Again, it is obvious that nothing will remain and that the whole will vanish, and that (supposing it to be a compound) it is composed of nothing. For as long as anything remains, it will not yet have been divided everywhere. And the upshot of these considerations is, he says, that what-is will be indivisible, without parts, and one.’ [Simplicius goes on to argue, rightly, that the attribution of this argument to Parmenides is incorrect, and that the argument stems from Zeno] …

Then again, in demonstrating that if there is a plurality, the same things are both finite and infinite, Zeno writes as follows (I quote his exact words): ‘If there are many things, they are bound to be as many as they are, neither more nor less; but if they are as many as they are, they are finite in number. If there are many things, there are infinitely many things, since there are always other things between any two given things, and others again between any two of those, and so things are infinite in number.’

As for infinity with respect to magnitude, he demonstrated that earlier in his book by the same kind of argument. He first demonstrates that anything without magnitude does not exist, and then he goes on: ‘But if there is a plurality, it is necessary for each thing to have a certain magnitude and solidity, and for there to be distance between one part of it and another. And the same goes for the part of it that protrudes: it too will have magnitude and some part of it will protrude. And it makes no difference whether one says this once or goes on and on saying it, since the item will have no such thing as a last part, and there will not be a part that does not stand in relation to another part. And so, if there are many things, they are bound to be both small and large—small enough to have no magnitude and large enough to be infinite.’ (Simplicius, Commentary on Aristotle’s ‘Physics’, CAG IX, 139.5–141.8 Diels)

T4 (DK 29A24; L 15) Zeno’s argument seemed to do away with the existence of place. It raised the following puzzle: If there is a place, it will be in something, because everything that exists is in something. But what is in something is in a place. Therefore the place will be in a place, and so on ad infinitum. Therefore, there is no such thing as place. (Simplicius, Commentary on Aristotle’s ‘Physics’, CAG IX, 562.3–6 Diels)

T5 (DK 29A24; L 14) We can see, then, that it is impossible for something to be in itself in the primary sense of the expression. Nor is it difficult to find a solution to Zeno’s puzzle that if there is such a thing as place, it must be in something. For it is perfectly plausible for the immediate place to be in something else, as long as ‘in’ is not understood as implying location within a place, but is taken in the sense in which health is ‘in’ hot things (because it is a state of hot things) and in which heat is ‘in’ the body (because it is an affection of the body). This avoids the infinite regress. (Aristotle, Physics 210b21–7 Ross)

T6 (DK 29A29; L 37) The fact that a given power as a whole has moved an object such-and-such a distance does not mean that half the power will move it any distance in any amount of time. If it did, one man could move a ship, since the power of the haulers and the distance which they all moved the ship together are divisible by the number of haulers. That is why Zeno is wrong in arguing that the tiniest fragment of millet makes a sound; there is no reason why the fragment should be able to move in any amount of time the air which the whole bushel moved as it fell. (Aristotle, Physics 250a16–22 Ross)

T7 (DK 29A10; KRS 328) In his Sophist Aristotle describes Empedocles as the discoverer of rhetoric and Zeno as the discoverer of dialectic. (Aristotle [fr. 65 Rose] in Diogenes Laertius, Lives of Eminent Philosophers 8.57. 1–2 Long)

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—— ‘A Note on Zeno’s Arrow’, in [26], ii. 184–200, and in [33], 205–18 (first pub. Phronesis, 11 (1966)).

—— ‘A Zenonian Argument against Plurality’, in [21], 119–44, and in [33] 219–40.

—— ‘Plato’s Testimony Concerning Zeno of Elea’, in [33], 264–300 (first pub. Journal of Hellenic Studies, 95 (1975)).

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