by A. A. Long
Fourth, the argument is an antinomy. In this particular form of reductio ad impossibile, the impossibility inferred from the premise is a logical contradiction of the form “p and not-p.” This kind of argument is typically Zenonian. We hear of other arguments that show that the same things are one and many and are moving and at rest (Plato, Phaedrus 261d). One argument is preserved that argues that if there are many things, the same things are limited (peperasmenon) and unlimited (apeiron) (DK 29 B3). Another argument concludes that each of the many is both small and large (part of this argument is found in B1-2). The arguments on motion can be construed as antinomies too.
ANOTHER PARADOX OF PLURALITY
“Zeno stated that if anyone could make clear to him what the one is, he would be able to speak of existing things” (Eudemus, Phys. fr. 7, quoted in Simplicius, In phys. 97.12-13). This was Zeno’s challenge to pluralists: give me a coherent account of what it is to be one of your many things and I will grant you your pluralism. He then proceeded to demonstrate the impossibilities that result from various conceptions of pluralism.12
One of these arguments, apparently directed against the view that there are three-dimensional bodies, involves the antinomy that if things are many, they are both small and large, specifically (a) so small as to have no size and (b) so large as to be unlimited (apeiron). Zeno holds not only that (a) and (b) are mutually inconsistent but also that each of them presents serious difficulties in its own right.
The reasoning for (a) is incompletely preserved. We are told only that Zeno argued that each of the many is “the same as itself and one” and from that he concluded that each has no size. He then argued that “anything with no size, thickness or bulk does not exist,” as follows.
For if it should be added to something that exists, it would not make it any bigger. For if it were of no size and was added, it [the thing it is added to] cannot increase in size. And so it follows immediately that what is added is nothing. But if when subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (B2)
He then argued for (b).
But if it exists [or, if they (the many things) exist], each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part (of any such part) not related 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. (B1)
If there are many things, then according to B2 each of them has size. Consider any one of them. We can distinguish one part of it from the rest. This part has size (otherwise, by B2, it would not exist), so we can distinguish a part of it from the rest. This part too has size, and so on forever: we never reach a last subpart.
Zeno concludes: “if there are many things, they must be … so large as to be unlimited.” It is commonly held that the argument shows that anything with positive (finite) size has an infinite number of parts, each with positive size, and that Zeno erred in thinking that the sum of an infinite number of positive magnitudes must be infinite.13 But the argument does not point in this direction, and the conclusion can be taken otherwise. The problem is not how to reconstitute the original thing once it has been divided into an infinite number of parts, but how to complete the division. If the division stops somewhere, so that at some point we reach the minimal building-blocks out of which the original thing is composed, the regress is blocked; these are the ones out of which our many is made. But Zeno shows that there is no good reason for stopping the division. Anything extended in space can be divided into parts that are themselves extended in space, so we can in principle never finish the division. He concludes that each of the many things is so large that it has an unlimited number of parts – without committing himself to a view on the question of whether anything with an unlimited number of parts can have a limited size.
It is important to note that the argument does not require matter to be infinitely divisible. Clearly, if there are smallest units of matter (as in ancient atomism, for example), physical division comes to an end at a certain point. But the argument applies even to individual atoms. We can mentally distinguish the right half from the left half of an atom, and likewise distinguish the right half of the right half from the left half of the right half, and this process of mental or “theoretical” division never reaches an end.14 All the argument requires is the assumption that spatial extension is continuous.15
Apeiron AND INFINITY
Zeno’s deployment of apeiron and related notions in the argument just discussed and more famously in some of the paradoxes of motion, has given rise to most of the excitement that the paradoxes have generated, especially in the twentieth century.16 In certain contexts apeiron can be rendered as “infinite,” and many of Zeno’s arguments involve infinite regresses. Moreover, certain arguments, especially the Dichotomy, the Achilles, and the Flying Arrow, raise issues that could not be properly dealt with before the theory of the mathematical infinite was developed in the nineteenth century. Much of the most important work on the paradoxes in this century has been a matter of interpreting them in terms of this theory and its possible physical applications. I shall return to these matters, but now I want to point out that in an important sense this work is anachronistic and wrong-headed.
It is one thing to ask what Zeno or any other philosopher meant by what he said and another to ask what a philosopher’s words mean to us. Similarly, what counted for Zeno as a problem or a solution may not be the same as what counts as such for us. The philosopher who takes Zeno’s paradoxes seriously and addresses the problems they raise is doing something different from the historian of philosophy who aims to understand what Zeno meant by the paradoxes, and what would be satisfactory solutions for him. Since at least Aristotle’s time, philosophers have regarded the paradoxes as puzzles demanding solution, and their solutions have typically involved theories, concepts, and distinctions unknown to Zeno. It is remarkable that Zeno could formulate puzzles that go to the heart of our conceptions of space, time, and motion; this is a good reason to examine them in the light of our own theories. But we also need to keep in mind the distinctions drawn above, and the frequent failure to do so by historians of philosophy as well as philosophers has made Zeno a much misunderstood man.
Regarding the key word apeiron, it is safe to say that it did not mean “infinite” in Zeno’s time. It is a compound of a-, meaning “not,” and either the nounperas (limit, boundary), so that it means “unlimited,” “boundless,” “indefinite,” or the root per- (through, beyond, forward), so that it means “unable to be got through,” or “what cannot be traversed from end to end.” Zeno contrasts apeiron with peperasmenon, “limited” (B3): In Aristotle these words have the meanings “infinite” and “finite.” Aristotle worked out a theory of the infinite in some technical detail and mobilized this theory against Zeno, but the fifth century was innocent of such technical meanings. In that age, something that was apeiron was “inexhaustible,” “vast,” “endless,” such as the “boundlessly high air” (Euripides, fr. 941), and “a plain stretching away without limit as far as the eye can see” (Herodotus, I.204). In particular, by definition anything that is apeiron has no limits; in this way what is apeiron for Zeno is crucially different from what we regard as infinite, in particular certain infinite sequences. We are accustomed to think that the infinite sequence 1/2, 1/4, 1/8,… has no last term but does have a limit o, and that the infinite sequence of partial sums 1/2, 3/4, 7/8 … likewise has no last term but has a finite limit, 1; for Zeno the corresponding thoughts framed in terms of apeiron would have been self-contradictory. He claims it is impossible to perform an apeiron sequence of tasks, one that by definition has no limit. To say that mathematicians
prove that this is possible by defining the sum of the original infinite sequence as the limit of the partial sums deserves a contemptuous retort: “Stipulating a definition doesn’t make anything possible, especially when the definition is a contradiction! What is apeiron has no limit, and simply declaring that some apeiron things have limits doesn’t make it so.”
This is not the last word on the subject – clearly – but it shows that since Zeno’s conception of apeiron is not identical with our notion of the infinite, to state the paradox in terms of the infinite and solve it in those terms is to state and solve a different paradox. This is not to claim that there is no place at all for this approach, only to call attention to what we are doing when we tackle old paradoxes with modern tools. In fact, the modern notion of the infinite is superior to Zeno’s conception of the apeiron. For example, unlike Zeno it distinguishes among different sizes of infinities, it enables us to do mathematical operations involving infinite quantities, to compare infinite quantities in a precise way, and to specify different respects in which a single thing can be infinite. Where he bluntly asserted that it is impossible to perform an apeiron sequence of tasks, we now know that some infinite sequences of tasks can be completed, though some cannot, and we can explain why. How these observations apply to the paradoxes of motion will be apparent in what follows.
THE ARGUMENTS AGAINST MOTION
“Four of Zeno’s arguments concerning motion cause difficulties for those who try to solve them,” says Aristotle (Phys. VI.9 239b9), reporting the quartet of paradoxes that have caused difficulties right up to the present time and show every sign of continuing to do so. Whether these arguments were among the forty against plurality is disputed. If they were not, then not all of Zeno’s paradoxes appeared in the book Plato mentions; if they were, then Plato’s statement that all the paradoxes in the book were directed against pluralism is rendered dubious – though not as dubious as some think, since motion involves a plurality of places and times. Moreover we have seen that one of the paradoxes that is certainly directed against pluralism relies on assumptions about space.
Typically, Aristotle sees the paradoxes as puzzles that need to be solved; he is not out to understand them in Zeno’s terms. He summarizes them in bare-bones fashion and presents his own solutions, most of which are based on concepts that he himself developed and that were not available to Zeno. I shall discuss three of the four paradoxes,17 beginning with the Dichotomy and the Achilles, which Aristotle declares amount to the same thing.
The Dichotomy
There is no motion, because what is moving must reach the midpoint before the end (Phys. VI.9 239b11). It is always necessary to cross half the distance, but these are infinite (apeiron), and it is impossible to get through things that are infinite (Phys. VIII.8 263a5)
The Achilles
The slowest as it runs will never be caught by the fastest. For the pursuer must first reach the point from which the pursued departed, so that the slower must always be some distance in front (Phys. VI.9 239b14).
As Aristotle sees it, the Achilles “is the same argument as the Dichotomy, but it differs in not dividing the given magnitude in half” (Phys. VI.9 239b18-20). Aristotle solves them both by means of his distinction between infinite in extent or quantity and infinite by division:
It is impossible to come into contact with things infinite in quantity in a finite time, but it is possible to do so with things that are infinite in division. For time too is infinite in this way [i.e., infinite in division]. And so it follows that it crosses the infinite in an infinite, not a finite time, and comes into contact with infinite things in infinite, not finite times. (phys. VI.2 233a26-31).
Although the paradoxes can both be treated this way, even in Aristotle’s summaries they are importantly different. The Dichotomy explicitly turns on an alleged property of the infinite, while the Achilles does not mention the infinite, but turns on the terms “always” and “never.” As we have them, they are subject to different analyses; I shall take them up separately.
THE DICHOTOMY
Following is an expanded reading of the Dichotomy:
There is no motion. Motion involves going from one place to another. Consider, for example, motion across a stadium. To get from the starting line (A) to the finish line (B), we must first reach A1, the midpoint of the interval AB. But in order to get from A1 to B, we must first reach A2, the midpoint of A1 B, and so on. Each time we reach the midpoint of an interval we still have another interval to cross, which has 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 we cannot reach the finish line.
On an alternative reading, Zeno argues that in order to reach A1, we must first reach the midpoint of the interval AA1, and so on. The difference between these interpretations can be put with rhetorical effectiveness as follows: on the first reading you cannot complete a motion, on the second reading you cannot begin one. Either way the point is the same: motion is proved impossible because any motion involves an endless sequence of tasks.
Zeno attacks the view that there is motion. We can imagine that the Dichotomy constituted part of an antinomy: (a) If there is motion, then the motion from A to B takes a limited number of steps. (This is our ordinary view. For example, we can cover 100 m. in 100 steps of 1 m. each.) (b) If there is motion, then the motion from A to B takes an unlimited number of steps. (This follows from the description of motion presented in the Dichotomy.)18
Whether or not this imaginary reconstruction is correct, the Dichotomy argues that a belief contrary to one of Parmenides’ views, here the Parmenidean view that what is does not move, involves a logical impossibility. Anyone who believes that there is motion is committed to the belief that it is possible to get to the end of an endless series of submotions. (In other words, it is possible to complete an uncompletable series, to reach the limit of an unlimited series or the boundary of an unbounded series). But this is flatly impossible. If the series is endless (or uncompletable or unlimited or boundless), it has no end (limit, and so on), so there can be no way to reach its end.
To resist Zeno’s conclusion, we must show that motion does not involve the impossible. One response is that of Antisthenes the Cynic who, “since he could not contradict Zeno’s arguments against motion, stood up and took a step, thinking that a demonstration through what was obvious was stronger than any opposition in arguments” (Elias, In cat. 109.20-22) – a comically inadequate refutation, since Zeno did not deny that our senses tell us that there is motion. (The Eleatics consequently rejected the senses as unreliable.) Simply providing one more instance of apparent motion whose reality Zeno would deny means that Antisthenes either completely misunderstood Zeno’s point or felt the need to prove (to himself if not to Zeno) that he could still move.
Another way to avoid Zeno’s conclusion is to show that he misdescribes the situation. The paradox does not arise if it is not true that we must reach the midpoint before reaching the end, and that each time we reach a midpoint what remains is an interval with a midpoint that must be reached before the remaining interval is crossed. But his description is unobjectionable: In order to go the whole distance we must go half the distance, three-quarters of the distance, and so on. As long as space is continuous, as we (along with Zeno’s opponents, we may suppose) intuitively think and as modern physics does not contradict, there is no end to this sequence.
A third way of avoidance is to show that although Zeno does not actually misdescribe the situation, his description is not helpful; a more helpful description would be that in order to reach the finish line we must take some definite number of steps – a task we can complete without difficulty. The idea behind this objection is that motion is possible if there is some description that does not involve impossibility. But this approach stacks the deck unfairly against Zeno. He need not claim that every correct description of motion leads to contradiction, only that at least one correct description does.<
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Further, if we point out that motion can be correctly described without contradiction, we actually help Zeno’s cause. He can now claim that the existence of motion entails the antinomy mentioned above, that it is both possible (as we are pointing out) and impossible (as the Dichotomy proves) to cross any given distance – a contradiction that refutes the premise that there is motion.
In any case, Zeno can accept that alternative description. If it takes 100 steps to reach the finish line we must first take 50, then 25, then 12 1/2, and so on. The opponent will object: fractional steps are not allowed in his description. But Zeno can agree that a fractional step is not a step (12 1/2 steps is not twelve full steps plus one shorter step, thirteen steps in all), but still maintain that to make one step is to move your foot from A to B, and in doing this your foot moves first to the midpoint between A and B and so on, so the regress still rears its ugly head. Zeno’s challenge to pluralists to provide a coherent account of what it is to be one of their many things applies here too. It is not enough to say that the motion can be described as 100 steps, where the step is the unit of motion. Zeno can fairly press his point against this unit, and when he does so, the opposition collapses.
Another move is to accept that there is an unlimited number of intervals to cross in getting from A to B, but to object that Zeno errs in assuming that it takes an unlimited amount of time to cross them all. Here we may substitute “infinite” for “unlimited” without affecting the argument. Clearly, if it takes the same length of time to cross each of an infinite number of intervals, the total time will be infinite. This is how Aristotle interpreted the paradox – and he solved it by distinguishing being infinite in division from being infinite in extent. The Dichotomy relies on the infinite divisibility of distance and motion, and does not imply that the total distance is infinite in extent. There is no reason to suppose that the time taken is infinite in extent either; like distance and motion, time is infinite in division. If it takes 1/2 minute to go 1/2 the distance, it will take 1/4 minute to go 1/4 the distance, and so on. So just as the total distance moved is finite, the total time elapsed is also finite. However, the Dichotomy says nothing about taking an infinite extent of time to cover the distance. It turns simply on the alleged impossibility of getting through an infinite number of things, not of getting through them in a finite time. As a result, Aristotle’s objection (as well as his solution) misfires since it attributes to Zeno an error that there is no reason to suppose he made.