A Note on Zeno's ArrowAuthor(s): Jonathan LearSource: Phronesis, Vol. 26, No. 2 (1981), pp. 91-104Published by: BRILLStable URL: http://www.jstor.org/stable/4182117Accessed: 02/09/2009 13:34

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A Note on Zeno's Arrow

JONATHAN LEAR

? 1. Zeno's paradox of the arrow is reconstructed from two condensed passages in Aristotle. Physics Z9 begins:

"Zeno argues fallaciously; for he says that if everything always rests when it is against what is equal (xar& To 'iaov) and what is moving is always in the now (EV TrO

vivv) the moving arrow is motionless. But this is false: for time is not composed from indivisible nows, just as neither is any other magnitude." (Physics Z9, 239b5-9).

Later in Z9 he says

"The third <Zenonian> argument, the one just stated, is that the moving arrow is standing. This follows from assuming that time is composed from nows; for if one does not grant this, there will be no syllogism." (239b30-33).1

The paradox, I conjecture, had the following form: (1) Anything that is occupying a space just its own size is at rest. (2) A moving arrow, while it is moving, is moving in the present. (3) But in the present the arrow is occupying a space just its own size. (4) Therefore, in the present the arrow is at rest. (5) Therefore a moving arrow, while it is moving, is at rest. Two items about the reconstruction deserve mention. First, I have inter- preted the phrase 'is against what is equal' (xtTo T61L'oov) as 'is occupying a space just its own size'. The Greeks notoriously had difficulty working out a conception of space and the interpretation, I think, preserves the sense of the Greek while sparing us its artificial ring. Second, the phrase, 'in the now' (EV -v viv) is probably Aristotelian.2 But it does, I shall argue, capture a concept crucial to Zeno's argument which has been overlooked by modem commentators: the concept of the present instant. Commentators tend to interpret Zeno as saying that in a moment the arrow occupies a space its own size.3 And yet much of the strength of the paradox - and of Aristotle's response - depends on the fact that the moment of travel with which Zeno is concerned is the present moment.4

? 2. Aristotle attacks the paradox on two broad fronts. First, as one can see from the quoted passages, he denies that time is composed of nows. The now is simply a division of past and future, itself having no duration. When he talks of a collection of nows (T& viv), he is speaking of durationless

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instants, each of which either is, was or will be present. Since each now has no temporal magnitude, a collection of nows cannot together compose a temporal magnitude.5 Therefore, even if Zeno was right in saying that in each now the moving arrow is stationary, it does not follow that the arrow was stationary throughout the duration of its flight. For the duration should not be thought of as composed of nows. So even if all the premisses are true, there is no paradox, for the argument is invalid. The argument depends on the assumption that if a property 0 holds of an object at every present instant of a given period of time, then 0 holds of the object throughout the period. This assumption, according to Aristotle, is not valid. The appearance of validity depends on a misconception of the nature of time.

Second, Aristotle also denies the truth of the premisses. He argues that it is a mistake to speak of the arrow either as moving or as being at rest in a now. For motions occur at different velocities and velocity is a matter of distance travelled over time elapsed. So for an object to be moving at any given velocity it must be in the process of moving over a certain distance in a given period of time.6 It follows that it does not make sense to speak of an object moving in a now:

"That nothing is moving in the now is evident from what follows; for if <objects could be moving in the now> then it is possible that one object be moving faster and another object be moving slower. Let the now be N, and let the faster object have moved the distance AB in N. Then the slower object will be moved less than AB, let us say AC, in the same time. Since the slower object has moved AC in the whole time, the faster object will have moved AC in less time than this, so that the now will be divided. But the now is indivisible. Therefore it is not possible to be moving in the now." (Physics Z3, 234a24-3 1)7

The idea is that moving objects move at varying speeds, so that if objects could be moving in a now, one could use the varying speeds to divide the indivisible now. For, as in the above example, one could ask how long it took the faster object to move the distance AC (the distance travelled by the slower object in the now) and the answer would have to be some time that is less than the now.8

But, Aristotle continues, neither does it make sense to speak of an object being at rest in a now.

"Neither is it possible that there be resting in a now. For we say to be resting that which naturally moves, but which is not moving when, where and as it naturally does.9 Since nothing naturally is moving in the now, it is evident that neither is it resting.

Further, if the same now is in two periods of time <e.g. the past and future>, and it is possible that an object be moving throughout the whole of one period, and resting

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throughout the whole of the other period; <and since> a moving object will be moving in whatever part of the whole time in which it is naturally moving, and the resting object will similarly be resting <in whatever part of the whole period in which it is naturally resting>; it will follow that the same object will be moving and resting at the same time. For the same now is the extreme of both periods of time.

Again, we say to be resting that which is in a similar state, both itself and its parts, now and earlier; but in a now there is no earlier <moment>, so that neither is there resting. Therefore, necessarily the moving object moves and the resting object rests in time." (Physics Z3, 234a3 I-b9)10

Rest, like motion, must occur over a period of time: for just as motion requires that an object be at different places at different times, so rest demands that the object be at the same place at different times.

So on this front Aristotle attacks the premisses of Zeno's argument. (1) is false because in a now an object will occupy a space just its own size and yet neither be in motion nor at rest.'1 Indeed Aristotle denies that one can precisely locate a moving object, in the sense of saying exactly what it is up against (xOr& Ti. pCyTov) (239a23-b26). One can locate the object precisely and thus be able to say that the object occupies a spacejust its own size only if the object is resting or if one is speaking of its position in a now (239a26-b 1). Premiss (2) is false if by 'the present' Zeno meant, as Aristotle takes him to mean, the present instant. For the arrow is not moving in the present instant. If, however, one should interpret 'the present' to be a duration of time, so that (2) becomes uncontentiously true, then (3) be- comes false: during a present period of time, the moving arrow is not occupying a space just its own size.

Aristotle's two lines of attack are intimately related. For if time were composed of nows then either one would have to explain the motion that occurs over a period of time in terms of the motions that occur in the nows or one would have to deny that there is continuous motion and admit that all that happens is that an object is at different places at different nows.'2 Precisely because time is not composed of nows, one does not have to think of the motion that occurs over a period of time as dependent upon anything that happens in a now.

? 3. Compelling as Aristotle's response may initially appear, there is probably no other of his arguments that has come in for such criticism. Owen, for example, has argued that the question of whether or not time is composed of instants is completely irrelevant to solving Zeno's paradox.13 Aristotle, according to Owen, attacks Zeno's assumption that

(*) what is true of the arrow at each moment of its flight is true of it throughout the whole period.

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The basis of Aristotle's attack is that time is not made up of moments. But, Owen argues, Zeno's fallacy cannot lie in this assumption.14 Zeno, it is alleged, may remain agnostic as to whether time is composed of moments or not. However time is composed, all one needs to admit is that the inference from

'x is moving at any moment in period p' to

'x is moving throughout the whole period p' is valid; and this, says Owen's 'Zeno', "is ordinary sense and not bad logic."'15 The reason that the inference is valid and Zeno's assumption (*) is justified is, according to Owen, that they embody a type of rule that gives a sense to the expression 'x is moving at a moment'.16 It is constitutive of the sense of that expression that when one can say 'x is moving throughout a period p' one can also say 'x is moving at any moment t in p' and vice versa.

Aristotle's mistake, according to Owen, is his assumption that the in- ference can be valid only if the expression 'x is moving' has exactly the same sense in the premiss and conclusion. For it to have the same sense, the arrow would actually have to traverse some distance in a moment, and then the whole period would be thought of as composed of these moments. By pointing out thAt time cannot be composed of moments, Aristotle mis- takenly thinks he has revealed the fallacy. But, says Owen, Aristotle has overlooked the fact that though the two uses of 'x is moving' are not strictly synonymous, they are systematically related.'7 By ruling out as illegitimate one use of 'x is moving', Aristotle unjustly deprives us of a common and useful expression.

This naturally leads one to criticize Aristotle's belief that an object cannot be moving in an instant. Both Owen and Vlastos distinguish be- tween the notion of an object moving at an instant and an object moving in an instant.18 To say that the object is moving in an instant is to say that the object actually traverses some distance during the instant; that is, it is to construe the instant as a very small duration of time. Aristotle correctly dismisses as absurd the notion of an object moving in an instant. But this does not show that an object cannot be moving at an instant. For to say that an object is moving at a moment is only to say that that moment is contained in a period during which the object is moving. Indeed, Zeno's fallacy lies not in the assumption (*) but in the assumption

(**) at each moment of its flight the arrow must be stationary since evidently it has no time to move.

This assumption (**) is fallacious because one can say that the object is moving at each moment of its flight even though it has in any moment no

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time to move. On this interpretation the arrow is moving at each moment of its flight, even though it covers no ground in any moment, and thus Zeno's argument collapses.

? 4. Unfortunately, Owen's interpretation does justice neither to Zeno nor to Aristotle. The arrow is alleged to be moving at an instant if and only if it is moving in a period that contains the instant. But Zeno would not be at all happy about our simply helping ourselves to the asumption that there exists a period in which the arrow is moving. "For surely," he would say, "if the arrow is moving at all, there is no time it could be moving other than the present. And yet you have admitted that the arrow is not moving in the present, in the sense that it is not actually traversing any distance in the present. You want to say that the arrow really is moving at the present, in the sense that the present is part of a period of time in which the arrow does traverse some distance. However, you should have admitted that there is no time the arrow could be moving other than at the present. So it is absurd for you to say that the arrow is moving at the present in virtue of its moving in some other time!"'19

It is Vlastos who most eloquently and explicitly disputes the claim that Zeno's Arrow can have anything to do with the notion of the present. His defense is that

". . .['T6 vv' is one of Aristotle's favourite technical terms. He used it commonly as a name for the durationless instant, but occasionally, in controversial contexts, he also allowed himself to denote the atomic quantum of duration integral multiples of which would make up aH larger intervals, if time were discontinuous. Since neither of these two uses of vvv have any known precedent, it would be most unsafe to assume that Zeno had anticipated one of them across a gap of a hundred years or more. Its presence here is explicable as an Aristotelian plant: by sticking it into his account of the puzzle Aristotle makes it all the easier for his readers to feel the appositeness of his refutation *. "s20

Even if the premiss of Vlastos' argument is correct, it does not establish the conclusion for which he is arguing. It does not matter if a particular technical use of "T6 vivv' was not in evidence before Aristotle's time. Zeno's paradox, as I have construed it, does not depend on any technical use of 'vvv', but indeed on a highly general notion of the present - which perhaps by its very generality makes it difficult to see how to refute the paradox. A general notion of the present certainly was in evidence in Zeno's time. Parmenides, Zeno's alleged mentor, says, "Nor was it ever, nor will it be: since now (vvv) it is, all together'".2 It is precisely the Parmenidean assumption that something can only be in the present that gives Zeno's Arrow its point.

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By construing the claim that the arrow cannot be moving ?v -r4 viv as the claim that the arrow cannot be moving at an instant, commentators have unwittingly diminished Zeno's paradox. For then it is too easy to go on to say, perhaps invoking concepts derived from calculus, that the arrow can be moving at an instant in virtue of its moving in a period that contains the instant. But since Zeno was concerned with the special case of the present, one cannot answer him with the notion of motion at an instant. Indeed such a notion will either be inapplicable or superfluous. If we do not assume the existence of a period of time in which the arrow is moving - a period that can be divided such that some of it is past, the rest of it is future, - then the notion of motion at an instant is inapplicable. For one can only say that the arrow is moving at an instant if it is moving in a period of time that contains the instant. If, however, one follows Aristotle by assuming the existence of a period of time in which the arrow is moving, 'the now' merely being an instantaneous division of this period, then the appeal to the notion of motion at an instant is superfluous: one has already assumed all that is needed to show that the arrow is moving.

Barnes attempts to dissolve Zeno's Arrow by arguing that premiss (1) is false: i.e. that it is simply false that if an object occupies a space just its own size, then it is not moving. According to Barnes, "objects do, at every instant of their temporal careers, occupy a space exactly equal to their volume at that instant. And they do even if they are in motion throughout their temporal careers. Why should anyone find that puzzling?"22

Barnes has not dissolved Zeno's puzzle, though it is difficult to see why not if one assumes that '1v -rC vvv' means 'at an instant'.23 For it is certainly true that at an instant an object occupies a space just its own size. One can then imagine oneself, say, on an express train from Cambridge to London: at every instant of the journey one occupies a space just one's own size and yet one is moving throughout the entire journey. Once one begins thinking in these terms it becomes difficult to see why anybody could have thought that premiss (1) is plausible. However, the paradox now looks so unap- pealing that one might become suspicious that one's victory over Zeno has been too easy. Such suspicion is, I think, justified. If one cannot uncon- tentiously assume the existence of a period of time in which an object is moving, then one cannot go on to say that the object is moving at any instant contained in that period. Zeno's use of the present is designed to make contentious our assumption that there exists a period of time that both contains the present instant and is a period in which the object is moving. For, as Aristotle pointed out in his very first &nopL'op about time, of any period of time that contains the present instant, some of it is past, the

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rest is future.24 "6How," Zeno might ask, "can one say that the arrow is moving in virtue of things that have happened to it or will happen to it?"

It is now possible to see the problem with Barnes' solution. Of course, if we can say that we are on a train moving from Cambridge to London, then we can also say that we are moving at any instant during our journey; even though at that instant we occupy a space just our own size. But, Zeno wants to ask, how can we say there is any time during which the train is moving? Have we not helped ourselves to the notion of a period of time when an object is moving? And yet we have already conceded both that the only time an object can be moving is in the present (premiss (2)) and that in the present it does not actually traverse any distance (premiss (3)). There does not seem to be any time during which we can say that the train - or the arrow - is moving, in virtue of which we can say that it is moving at an instant. Premiss (1) looks obviously false only if one begs the question by assuming that there is a period of time in which an object is moving.

One response that does not beg the question is to deny premiss (2): i.e. deny that for an object to be moving it needs to be moving in the present. One can say that an object is moving during a period of time p solely in virtue of its being at different places at different times in p. Then one can go on to say that an object is moving at the present instant if that instant is contained in a period in which the object is moving. To Zeno's incredulous question, "So you think that an object can be moving solely in virtue of positions it has occupied in the past and will occupy in the future?" one would simply answer, "yes".25 This would be the response of someone who did not wish to incorporate the notion of a present duration into his scientific theory of time. Theorists of time who do not wish to give the present a special status may prefer this solution to Aristotle's.

However, it is worth noting that this strategy gives a victory to Zeno. For we primarily use the continuous present tense 'x is moving' to talk about an event that requires a duration of time all of which is considered present. We may also talk about an object moving at an instant, but this use is derivative of the primary use. But the theorists we are considering deny that any period of time can, strictly speaking, be treated as present. So these theorists ought to concede to Zeno that, as we ordinarily use the terms, the arrow is not moving during the course of its flight. One can concede this and nevertheless maintain that the arrow is at a different position at any moment from its position at any other moment.

Zeno has scored a victory on this analysis that is more than verbal. For anyone who adopts this analysis will come to think that some of his previous beliefs about motion were merely perspectival; that is, dependent

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on the human point of view. From the human perspective, an arrow seems to be moving due to changes that are occurring in a period of time that can legitimately be conceived to be present. Advocates of this analysis urge that on the impersonal scientific conception of the world there is no period of time that is present. The flight of an arrow consists solely in its being at different positions at different times.

Still, one might argue that Zeno has not shown very much. For he has not shown that the arrow cannot be in different positions at different times. And even if we have to grant Zeno that the arrow is not moving in the sense that we originally thought it was, we can nevertheless revise our usage of'x is moving' to accommodate the scientific conception of motion. Zeno's Arrow does not, one might complain, really show that motion is impos- sible. One can imagine Zeno's response: "With that paradox I was only trying to show that there was no time for the arrow to be moving. Now if you want to see that the arrow cannot occupy different positions in the flight, consider that before it reaches the target it first has to get half way, but before it gets half way . . ."

If one does not wish to grant any victory to Zeno, the first line of attack should be premiss (3): that in the present the arrow is occupying a space just its own size.26 One can do this not by pointing out any obvious fallacy, but by developing a theory of time in which the present can be conceived of as a period of time. That this is part of Aristotle's strategy is revealed by his doctrine that time is not composed of nows. Any period of time can only be thought of as composed of smaller periods of titpe. Having developed a theory of time that construes the present as a period of time, one can then proceed, as Aristotle did not, to give a sense to the notion of an object moving at an instant or at the present instant. It is only then that one is entitled to say, with Barnes, that an object can be moving at an instant even though it only occupies a space its own size. This is a far from trivial truth, depending as it does upon a theory of time that treats the present as a period of time.

? 5. It is often said that Zeno's paradox was puzzling to the Greeks only because they lacked the modern concepts of the calculus, in particular the notion of motion at an instant. By now it should be clear that such a claim is unjustified. It is also commonly said that Aristotle's argument that there cannot be motion at an instant dramatically retarded the development of dynamics. Of course it is possible for a good argument to have stultifying effects and this may have been Aristotle's legacy. But it is commonly thought that Aristotle presented a fallacious argument which because of its

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fallaciousness badly influenced all those who believed it. This belief is, I think, unjustified: Aristotle's argument is valid and there is no intrinsic reason why it should have had any negative influence on the progress of dynamics.

Aristotle's argument, as we have seen, is that a moving object must be moving at some velocity and velocity is a matter of distance travelled over time elapsed. It would therefore be absurd to ask the velocity of an object at an instant; for an instant is not a duration of time and a fortiori not a duration of time in which any distance can be travelled. But since it is not moving at any velocity at an instant, it is not moving at an instant.

No discovery of the calculus or of dynamics reveals any flaw in this argument. Rather it has been discovered that there is a use in dynamics for taking the limit of velocities at which an object is moving during suc- cessively shorter periods of time which converge on a given instant. Each of the achieved velocities of which one is taking the limit is of course cal- culated by dividing the distance covered in a period of time by the length of the period of time. Traditionally this limit has been called the 'instan- taneous velocity' of a moving object or the velocity at which the object is 'moving at an instant'. This does not show that there is any mistake in Aristotle's argument, only that there is a use of the expressions 'instan- teneous velocity' and 'moving at an instant' that he did not envisage, i.e. as designating the limit of velocities. Of course modern dynamics surpasses Aristotelian dynamics in part due to the fact that we understand the concept of a limit much better than he did; but this admission differs significantly from the claim that Aristotle fallaciously argued that there cannot be motion in an instant.

It might, however, be objected that one cannot think of motion as occurring only over periods of time. For if one considers an object that is constantly accelerating, the most natural way to express this phenomenon is to say that at every moment the object is moving at a different velocity.27 And if the object is at each moment moving at a different velocity from any other moment, this means that there is no period of time during the acceleration that the object is travelling at any fixed velocity.

This objection will not stand up to scrutiny. For the instantaneous velocity of an object at t is calculated by determining the velocities of the object during temporal intervals which converge on t. So in order to determine any instantaneous velocity one must be able to determine the velocities of the object during certain periods of time. During a period of constant acceleration the following phenomenon will occur: if one takes any two instants during that period, no matter how close together, and

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calculates the respective limits of the velocities achieved over periods which converge on those instants, the results will be different. If one wishes to describe this phenomenon by saying 'at every instant the object's velocity is different' that is all right, provided that one is not led to believe that something special is happening in an instant. For that is to be misled by one's vocabulary.

? 6. Since the calculus is impotent to solve Zeno's Arrow, there is reason to go back and re-examine Aristotle's proposed solution. This will shed some light both on what constitutes a reply to a sceptical argument and on why Zeno's paradox continues, and will continue, to fascinate. Aristotle, as we have seen, attacks both the validity of the argument and the truth of the premisses. The argument is invalid, he says, because even if one grants that the arrow is stationary in each present instant, it does not follow that the arrow was stationary throughout the period of its flight. The reason is that a period of time is not composed of present instants or nows. But Aristotle does not prove that time is not composed of nows; rather in Physics A 10-14 he develops a theory of time in which a period of time can be said to be composed only of smaller periods of time and not of instantaneous nows. While he does argue that the premisses of Zeno's argument are false, the argument depends upon his theory of the structure of time which is not so much proved as rigorously presented. Of course every proof must rely ultimately upon premisses that are not themselves proved, so Aristotle is hardly at fault for not proving every assumption. Indeed he repeatedly insists that one must distinguish that which needs proof from that which does not, and prove the former by the latter.28

Aristotle, I think, began with the belief that an arrow obviously does move during the course of its flight, a belief based on the testimony of sensory experience.29 Zeno and Aristotle agreed on the testimony of the senses, but differed on its significance. Zeno, in Eleatic fashion, took his argument to show that sensory experience must give a misleading picture of the nature of reality; Aristotle, by contrast, took the sensory experience to show that there must be something wrong with any argument that leads to such a drastically conflicting conclusion. In Physics F-Z he constructs a theory of space, time and motion which purports to describe abstractly how the motion he evidently saw to occur actually does occur. The problem is that arguments in physics, to a much greater degree than pure mathematical proofs, may depend on assumptions that even upon mature consideration do not appear self-evident or forever beyond reproach. For example, Aristotle took time to be a measure of motion.30 The existence

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and nature of time was taken to be derivative of the existence and nature of motion. Further, Aristotle reasoned that since time is a measure or number of motion, it depends for its existence upon the existence of a soul or mind which does the numbering or measuring.31

We, like Aristotle, tend to begin with the belief that objects do actually move and that there must be some theory which explains how such motion is possible. That time is not composed of nows seems plausible enough in the context of Aristotle's theory of time, and if we are gripped by that theory we will know how to answer Zeno. But it is important to be aware that it is we who have been persuaded of Zeno's fallacy, not Zeno.32 Zeno would think that Aristotle's theory begs the question by assuming that there is a period of time; a period which can be represented either as divided into past and future with the present a durationless instant or as a period that is entirely present. However, this does not reveal a fundamental flaw in Aristotle's response. To assume that it does is to assume that one must always be able to answer the sceptic using no assumptions at all or assumptions that are blindingly self-evident and incontrovertible. For most interesting sceptical arguments, such as Zeno's Arrow, no such refu- tation is available. At best, one can follow Aristotle and answer Zeno with arguments based on premisses one sincerely believes to be true. Given that one does sincerely believe the premisses, the sceptical paradox will cease to be problematicfor oneself: it will no longer be a genuine a&ropit. That is the best, indeed the only, way to meet as ingenious an argument as Zeno offers. The sceptical puzzle is not refuted, in the sense of being dismissed on the basis of absolutely incontrovertible assumptions; it goes away. However, sincere beliefs, no matter how sincerely believed, are not guaranteed to be stable over time for an individual or a community. Should the assumptions of a theory used to answer a sceptical paradox come in question, the puzzle which one may have thought buried forever will be resurrected. One may be able to construct another theory which will answer the paradox, but there is no theory which can guarantee that one will forever be able to keep a good puzzle down.33

In the case of Zeno's arrow, there is a more specific reason for suspecting that one's response might be undermined. This is that we do not have clear or precise intuitions about the phenomenology of our temporal experience. We know that there is a temporal dimension to our experience, but the attempt to explain our experience of the present, the period of time in which we are self-consciously existing, tends to induce vertigo. Such vertigo motivates the desire for a theory which will explain our experience to us. We do not have clear intuitions which dictate the shape of an adequate

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theory; rather we seek a theory that will help us conceptualize our in- tuitions. Thus one can both believe that time is not composed of nows and also believe that that belief could be seriously undermined. Because of this basic doubt, one must admit that Zeno's Arrow may again emerge as a serious challenge to those who believe in motion.34

Clare College, Cambridge.

NOTES

There is also a non-Aristotelian report from Epiphanius:

"(Zeno) also argues thus: what is moving moves either in the place in which it is or in the place in which it is not. And it moves neither in the place in which it is nor in that which it is not. Therefore nothing moves." (Adversus Haereticos 111.11, H. Diels Dox. Graeci 590)

Cf. also Diogenes' similar report, DK 29B4 = Vitae Philos. IX,72. There is a disagree- ment over whether this argument is truly Zenonian. Vlastos, for example, incorporates it in his reconstruction of Zeno's arrow. Cf. "A note on Zeno's Arrow" in R. E. Allen and D. J. Furley (eds.) Studies in Presocratic Philosophy, v.2, London: Routledge and Kegan Paul 1975. But Owen, for example, omits it and Barnes expresses scepticism about its authenticity. (Cf. G. E. L. Owen, "Zeno and the Mathematicians", in Allen and Furley, op. cit., p. 157: J. Barnes, The Presocratic Philosophers v. 1, London: Routledge and Kegan Paul, 1979, p. 276). I am inclined to agree with Barnes that the dilemma may well be due to Diodorus Cronus. Cf. Sextus Empiricus, Adv. Math. X.86-90, X. 1 12; and David Sedley, "Diodorus Cronus and Hellenistic Philosophy", Proceedings of the Cambridge Philological Society 23, 1977. The translation of 239b6 above reflects Ross's suggestion that e kineitai should be bracketed. 2 Cf. Vlastos, op. cit. pp. 187, 192; Owen "Zeno", op. cit. p. 165 note 38. But cf. Parmenides DK 28B8 line 5. 3 Cf. e.g. Owen, "Zeno" op. cit. p. 157; Vlastos op. cit. p. 192; Barnes op. cit. pp. 276-285. This interpretation is odd, since in other contexts one clearly cannot interpret 'T6 viv' as meaning merely 'an instant'. Indeed Owen himself says elsewhere that the doctrine of the now conflates two distinct ideas, that of the present and that of an instant. (G. E. L. Owen, "Aristotle on Time", in P. Machamer and R. Turnbull, Motion and Time, Space and Matter, Ohio State University Press, 1967, pp. 15-16.) I am not now concerned with Owen's charge of conflation, but with his absolutely correct claim that the now involves the notion of the present. Consider, for example, Ari6totle's claim that we become aware of time whenever the soul pronounces the nows, the one before, the other after (cf. Physics A 11, 219a26-29). This would make no sense as an explanation of our awareness of time if 'the soul pronouncing the nows' consisted merely in it considering (or pro- nouncing) two arbitrary instants t, and t2. For the soul to pronounce a now is for it to be aware of the present moment as present. Such a pronouncement has a certain irrefragible quality: it is not liable to error. For the soul to pronounce the nows is either for it to be aware that at two distinct moments it has pronounced a now or that at a previous moment it has pronounced a now which is distinct from the now it is currently pronouncing. It is only by interpreting the now as involving the notion of the present moment, rather than

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just an arbitrary moment, that one can understand Aristotle's explanation of our aware- ness of time. As another example, consider Aristotle's proof that there must always be time: "Now since time cannot exist and is unthinkable without r6 vvv, but To vvv is a type of mean, being both a beginning and an end, a beginning of future time and an end of the past, it is necessary that there always be time. For the extreme of the completed time will be in some one of TrV viv (since there is nothing to grasp (Xadv) in time apart from To vvv); so since r6 vvv is both a beginning and an end, necessarily there will be time on both sides of it." (Physics 0 1, 25 1 b 19-26). If one translates 'To vvv' as 'a moment' (as e.g. R. P. Hardie and R. K. Gaye do in the Oxford translation) then the argument looks un- necessarily weak. If there was a first moment in time, then it is wrong to say that there was time before that moment. However, to suppose that there was a first now is to make the stronger assumption that there was a first moment which was at some time present. But we can only conceive of a present moment as a division of past and future. On this reading the argument is more compelling and it is, I think, the argument Aristotle offers. 4 Saul Kripke suggested that the present was important for Zeno's Arrow in his lecture "Identity Through Time", delivered at the Eastern Division Meeting, American Philo- sophical Association, 1979. 5Cf. Physics A 11-14. 6 Cf. Physics A 14, 222b30-223a 15. 7Cf. also Physics 237a 14, 239b1, 241a24-26. 8 An atomist need not be bothered by this argument; for it assumes that the motion that occurs in a now is continuous motion, and this the atomist could deny. Treating the now as a temporal atom, he could allow two objects to move at different speeds in the sense that in the next now one object was two spatial atoms removed from where it had been in the previous now, whereas the other object was only one spatial atom removed. One could not divide the now by asking when the faster object was one spatial atom removed; for, according to the atomist, there was no such time. Of course for Aristotle such discontinuous motion was not motion at all. Cf. Physics Zl, 231bl8-232a22 and D. J. Furley, Two Studies in the Greek Atomists, Study 1, Indivisible Magnitudes, Princeton: Princeton University Press, 1967. 9 Cf. Physics 226b 12. 10 There might seem to be a tension between this position and Aristotle's claim that there is a first moment at which a change has occurred (Physics Z5, 235b6-236a7). For suppose that the change that is occurring is that an object is coming to a halt (cf. Physics Z8). Then one might expect that there must be a first moment at which the object is at rest. Indeed under the pressure of such an argument Aristotle does talk about an object resting in a moment (Physics Z5, 236a17-20). But his full reply would, I think, be that there is a first moment of the period that the object is at rest, and if one wishes to say that this is the 'first moment of the object's being at rest' that is all right as long as one is not misled into thinking that the object is resting in an instant. Aristotle does explicitly deny that there can be a first instant of rest, and the reason he gives is that rest, like motion, cannot occur in an instant (Z8, 239a 0- 14). 1 Cf. Physics Z8, 239a23-b3. 12 Cf. Epicurus, fr. 278 Us., and Diodorus Cronos, fr. 121-9 D and Furley, op. cit., pp. 131-5. 13 Owen, "Zeno" op. cit. pp. 157-162. 14 Owen, "Zeno" op. cit. p. 161; cf. p. 157. 15 Penner has attacked Owen for overlooking Aristotle's energeia/kinesis distinction. (T.

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Penner, "Verbs and the identity of actions - a philosophical exercise in the interpretation of Aristotle", in 0. P. Wood and G. Pitcher (eds.) Ryle, London: Macmillan, 1971. Cf. Appendix II). Penner is correct that "Aristotle has no general presumption against saying that if x was 0 -ing throughout a period p, then at any moment t during p he was 0 -ing. It is just that this cannot be said with movement or more generally with kinesis." But though this suggests that Owen's "Zeno" might have been able to use a better example against Aristotle than the case of sleeping (Owen op. cit. p. 157) - which Penner takes to be an energeia and not a kinesis - Owen does explicitly disagree with Aristotle by saying that if x was moving throughout a period p, then x was moving at any moment t during p (p. 160). 16 Owen, "Zeno" op. cit. p. 160. Cf. also Owen's discussion in "Aristotle on Time", op. cit. 17 Owen, "Zeno", op. cit. p. 160-1. 18 Owen, "Zeno", op. cit. p. 161; Vlastos, op. cit. p. 193. 19 For evidence of Zeno's Parmenidean bent see Plato, Parmenides, 127A-128E. For Parmenides' attachment to the present and rejection of the past and future cf. DK 28B8, especially lines 5-6. 20 Vlastos, op. cit. p. 187. Cf. the extended discussion on pp. 184-7. 21 DK 28B8, line 5. 22 J. Barnes, op. cit., p. 283. Thus Barnes takes premiss (I) above to be obviously false. 23 Cf. Barnes op. cit. p. 278: "Aristotle says 'in the now (en tbi nun)'; I say 'at t': some philosophers find a significant difference here; but I think that 'at' is simply the appro- priate English translation of 'en'." I am happy to accept his gloss of'at' for'en', but not his translation (which he gives without comment) of 't' for 'nun'. 24 Cf. Physics 10, 217b33-218a8. 25 This would be the position of one who wishes to deal with Zeno's paradoxes via what Sellars has called the scientific image. See Wilfrid Sellars, Science, Perception and Reality, London: Routledge and Kegan Paul, 1963; chapter 1. Cf. Bertrand Russell, The Principles of Mathematics, London: Allen and Unwin, 1972, pp. 347, 350. 26 This would be the position of someone who wished to deal with Zeno's Arrow via what Sellars has called the manifest image. Cf. Sellars op. cit. See also G. E. L. Owen's remarks on the retrenchability of 'now' in "Aristotle on Time", op. cit. 27 See e.g. 'Zeno's' remarks in Owen, "Zeno", op. cit. p. 159. 28 Cf. e.g. Posterior Analytics A2-3, Metaphysics F4, 1006a5-8, Physics 03, 253a32-b6, b28-30, 254a5-1 1. 29 Cf. e.g. Physics 03, 253a32-b6. 30 Physics 220b32-221 al, 221b7, 22 1b22-23, 22 1b25-26. 31 Physics A 14, 223al6-18. 32 For a fuller discussion of this approach to sceptical arguments see my Aristotle and Logical Theory, Cambridge: Cambridge University Press, 1980; chapter 6. 33 This is a direct consequence of Quine's dictum that no belief is unrevisable. 34 I would like to thank J. N. Butterfield, C. Farrar, M. M. A. Mackenzie, D. Sedley and T. J. Smiley for criticisms of a previous draft.

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