Sorabji Aristotle on Change

Aristotle on the Instant of Change Author(s): Richard Sorabji and Norman Kretzmann Source: Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 50 (1976), pp. 69-89+91-114 Published by: Blackwell Publishing on behalf of The Aristotelian Society Stable URL: http://www.jstor.org/stable/4106825 Accessed: 05/11/2010 14:14 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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ARISTOTLE ON THE INSTANT OF CHANGE Richard Sorabji and Norman Kretzmann I--RichardSorabji The problem "The train leaves at noon", says the announcer. But can it? If so, when is the last instant of rest, and when the first instant of motion? If these are the same instant, or if the first instant of motion precedes the last instant of rest, the train seems to be both in motion and at rest at the same time, and is not this a contradiction? On the other hand, if the last instant of rest precedes the first instant of motion, the train seems to be in neither state, during the intervening period, and how can this be? Finally, to say there is a last instant of rest, but not a first instant of motion, or vice-versa, appears arbitrary. What are we to do? This kind of problem has had a long history. It is to be found already in Plato's Parmenides (156C-157A), and it had a great revival in mediaeval times.' To this day, it remains relevant to the definition of motion at an instant, as we shall see. I want to suggest a solution to it, which turns on treating motion differently from rest. I want then to argue, that though Aristotle has been castigated for attempting to solve the problem by means of a mistaken denial of motion or rest at an instant, he was also attracted by a second solution very close to the one I shall argue for. His treatment of dynamics can to this extent be reassessed. Does it apply to the real world? First we need to consider whether the problem could apply to the real world. It may be doubted whether it could, for the statement of the problem involved a number of assumptions. First, I assumed (what Aristotle argues in the Physics) that time is continuous. This has many implications. It means that time will be infinitely divisible, and there will be no such

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thing as a time-atom, that is, an indivisible period with an indivisible duration. An instant will be not a time-atom, nor any kind of period, but rather the boundary of a period, itself having no duration. Instants, unlike time-atoms, cannot be next to each other (cf. Phys VI 1, 232a 6-11). Rather, between

any two instants, there will be another, indeed, an infinity of others. This is what is involved in time being continuous, and our problem will apply to the real world only if time is so. If there were time-atoms, so that time was not continuous, the train would be in its old postion at one time-atom, and in a new position at the next. The earlier time-atom would be the last time-atom in the period of the train's resting; the later time-atom would be the first time-atom in the period of the train's moving, and our problem would not arise. The problem does arise, however, if Adolf Griinbaum is right that neither quantum theory, nor anything else in modern physics, has given us reason to deny that time is continuous (Modern Science and Zeno's Paradoxes, Middletown, Connecticut 1967).

Another reason why someone might question whether our problem applies to the real world is that a train consists of a mass of moving atoms, and so does the railway track. Can the train have any first instant of motion, or last of rest, if its atoms are moving all the time, and how would these instants be defined? Yet another doubt concerns the fact that a train is not perfectly rigid. When some parts of the train, or of the engine, have started to move, other parts will be lagging behind, so that there is not a single first instant of motion or last of rest for the train as a whole. Both these doubts can be met by raising our problem not about the train as a whole but about some point within the train, such as the centre of mass, and its first instant of motion and last of rest, in relation to some point on the railway track.2In talking of points, rather than of trains, we will be moving beyond the range of observable entities. So far as I can see, then, our problem does apply to the real world, in as much as it applies to unobservable points on a real train. But two further things need to be said. First, the problem would still be of interest, even if it applied only to a world different from ours. Second, we have so far considered only

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one version of the problem, and if this version were inapplicable to the real world, it might still be the case that other versions were applicable. Thus far I have considered only the transition between rest and motion. But our problem can be raised, and was raised by Aristotle, in connexion with other kinds of transition. He discusses the transition from being one colour to being another colour, from being non-existent to being existent, and from being invisible to being visible. (The last case is considered in De Sensu 7, 449a 21-31, and it has interesting connexions with the ancient paradox of the heap or bald man; I regret that space prevents me from discussing it.) In each case, the question can be asked: when is the last instant of the old state, and when the first instant of the new? Or moving from time to space, we may be able to ask where is the last point of the one state, and the first point of the other? In some of these new forms, the question may well apply to our world. Proposed solution for casesof continuous change With the problem now stated and generalized to apply to all kinds of transition, we can start making some suggestions about how to handle it. But first we should be clear how much we need to ask of a solution. The original question was about when the last instant of the old state occurs, and when the first instant of the new. One of the difficulties about answering was that if we said that one of these instants existed, but not the other, we seemed to be being arbitrary. It would be a sufficient solution, if we could show that it would not be arbitrary to prefer one instant to the other. For this purpose, we need only show that there is a reason for preferring one to the other; we need not show that it is mandatory to do so. On this basis, I would suggest that there is a solution available for those cases where the earlier state, or the later state, or both, consists in a continuous change. We can illustrate by considering a transition from rest to motion and back again to rest, provided that the motion is construed as continuous, not jerky. By this I mean that the motion involves passing through an infinity of points, between any two of which there are other points, which are also passed through. Ordinary usage is not precise, but leaves it indeterminate whether we should regard the instant

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of transition between rest and continuous motion as an instant of motion or not. We must therefore make a recommendation about how to regard it, if we want to solve our problem. Fortunately, there are several considerations which would justify the decision to call it an instant of rest, and I shall mention three. First, there is an asymmetry between the series of positions away from the position of rest and the position of rest itself. There can be no first position away from the starting point, or last position away from the finishing point in a continuous motion, or in any other continuous change. Hence there can be no first instant of being away from the starting point or last instant of being away from the finishing point. No such considerations apply to being at the position of rest. This already supplies us with a solution to our paradox, in some of its applications. For if someone were to ask. "when is the last instant of being at the position of rest, and when the first of being away from it?", we could safely reply that the latter instant does not exist. But we can go further. The asymmetry between the position of rest and the positions away from it can provide us with the excuse we want for treating rest differently from motion. It would be perfectly reasonable to mark the asymmetry by saying that just as there is no first or last instant of being away from the position of rest, so equally there is no first or last instant of motion. It would be reasonable, but not mandatory. Reasonableness is all we need in order to escape the charge of arbitrariness. This decision is reinforced by two further considerations. First, let us suppose that not only change of place is continuous, but also change of velocity. In other words, in passing from one velocity to another, an object passes through all the infinitely many intervening velocities. Once again, quantum theory has nothing to say against this assumption. If it is made, then there cannot be a first or last instant of having a velocity greater than zero, for there is no first or last velocity above zero. There is, however, no corresponding objection to there being a first or last instant of having velocity zero. Now it seems more natural, though again it is not mandatory, to connect zero velocity with rest, and velocities above zero with motion. If we do, we get the result that there is no first or last instant of motion, but that there may be first or last instants of rest.

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Further support for this decision is gained, if we consider a special case of coming to a halt. We can imagine the centre of mass of a ball travelling vertically upwards and slowing down until it reverses direction at an instant, without pausing for any period of time at the apex of its journey. We will now have an extra incentive for denying that the centre of mass is in motion at the instant at which it is at the end of its upward journey. For not only will its velocity be zero at that instant, but we could not say that its motion had one direction rather than the opposite direction at that instant. It will be easier, then, to describe the instant of reversing direction as one of rest. And if we do, there can be no last or first instant of motion on either side.3 In arguing for the non-arbitrarinessof the decision to deny first and last instants of continuous motion, we are not saying that the period of motion has no boundary. It will have an instant bounding it on either side, and our only question has been whether that instant should be regarded as one at which there is motion, or rest. We need not worry, incidentally, if a reason is found for saying that rest and motion are not contradictories, and that the instant of transition between rest and motion is neither one of motion nor of rest. We have found no such reason, but if one were found, the resulting view would not be so very different from ours, since at least it would treat the instant of transition as one of non-motion. I do not deny that there are considerations which point to the opposite decision, that the instant of transition is after all one of motion, and it will be as well to bring some of them into the open. First, at the instant of reversing direction, a ball's centre of mass is (as in all cases of coming to a halt) at a different position from that occupied at preceding instants, and (differently from ordinary cases of coming to a halt) it is also at a different position from that occupied at succeeding instants. This difference of position admittedly favours our regarding the instant of reversing direction as one of motion. But I think the consideration is outweighed by the absence of a particular direction and of a positive velocity. A further doubt which might be raised is whether change of acceleration always behaves in the same way as change of velocity. We have supposed that velocity always changes continuously. But does acceleration, the rate of change of velocity,

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always in its turn change continuously, or can it jump discontinuously from zero to, say, one foot per second per second? A similar question could be raised about yet higher derivatives, such as the rate of change of acceleration. Nonetheless, I do not think our solution is seriously threatened. On the one hand, if acceleration does change continuously, there will be no first instant of acceleration above zero, and so we will have yet further incentive to deny a first instant of motion. On the other hand, if acceleration were sometimes to jump discontinuously from zero to something higher, we should admittedly have no obvious general reason for choosing between talk of a last instant when acceleration is zero and a first instant when acceleration is above zero. But even if on some of these occasions we were to talk of a first instant of acceleration above zero, the considerations we rehearsed earlier would still be strong enough to make us hesitate before calling that first instant of positive acceleration an instant of motion. In spite of this defence, our proposal must be understood in a flexible spirit. We should recognize that for everyday purposes, and for many scientific purposes, it simply does not matter which way one talks, One's choice can legitimately be based on the most transient of reasons, or on no reason at all, while the reasons we have given can without penalty be ignored. The point of our reasons is simply that they are available to rebut the charge of arbitrariness in case of need. If a discontinuous jump from zero acceleration were to occur, and if in the context our whole interest were in the acceleration to the exclusion of position and velocity, and in the positive acceleration, rather than in the zero acceleration, then our reasons for denying a first instant of motion might be overridden in that particular context; and this would not matter. The point is that we would still not be forced to be arbitrary. The solution suggested does not in any way preclude physicists' talk of initial velocity. For the initial velocity of a projectile is not a first velocity in its entire motion, but merely the first velocity which it is convenient to consider for the purposes of a given calculation. We should also recognize that the last instant of rest in relation to one point may, of course, be an instant of motion in relation to a different point. Throughout we must be understood as talking of rest or motion in relation to a given point.


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The solution will apply not only to continuous motion, but to all changes which are continuous in the same sort of way. It will apply to changes of size, of temperature, or of colour, if these involve a continuous progress through a series of points between any two of which others are traversed. It will not apply to discontinuous changes, or to other processes or states which lack this kind of continuity. If, for example, we considered the transition from not singing to singing, we should not have the same reasons to deny a first instant of singing. Clearly, arguments based on direction or velocity would be inapplicable. It might be thought that something like our first argument would still be applicable, for the singing lasts through a continuous series of instants, and there can be no first instant after the instant of transition. But our first argument cannot in fact be applied, for it depended on the asymmetry between the single position occupied during a period of rest and the continuous series of positions occupied during a period of motion. In the case of singing, there is no asymmetry, for there is a continuous series of instants traversed during the period of non-singing, just as much as during the period of singing. One might expect the proposed solution to appeal to Aristotle, for in his attack on the atomists he is at great pains to insist that motion, time and space are all alike continuous. He argues hard against the atomists that what has moved must previously have been moving (Phys VI 1, 232a6-18; VI 6 237a17-b22; VI 10o 240b31-241a6); it cannot simply have jerked into its new position (232a6-11; 24ob31-241a6). It was his successorsDiodorus Cronus and Epicurus (or his followers) who were willing to accept jerky motion. (For Diodorus, see Sextus Empiricus, adv. math. 10.48; 85-6; 91-2; 97-o102; 143; cf 12o. For Epicurus or his followers, Themistius, Phys 184.9; and Simplicius, Phys 934-24.) Nonetheless, we shall later see

that Aristotle's solution is somewhat more complicated.

Treatment of other cases Our solution leaves a very large range of cases unsolved. For often neither the earlier state nor the later state is a process of continuous change, as we have remarked. The transition may be from one colour to a different colour, from non-

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existence to existence, or from invisibility to visibility. In these cases, what considerationsare there, to help us to a decision?If we are watchinga recedingaeroplane,or looking for an approachingone, we cannot normallytell at the time whatwill proveto be the last instantof visibility as it recedes, or the last instant of invisibility as it approaches.If we want to registerthis instantas it arrives,we shall normallyhave to

wait until the new state is upon us, before we can do so, and it may then reasonably be held that we are not registering the end of the old state, but, at best, the beginning of the new. This means that, in many contexts, we have a good reason for not talking of the last instant of the old state, but (if it has one) of the first instant of the new. This solution seems to have appealed to Peter of Spain, for certain kinds of case.4 But it needs to be noticed that our interest is not always in registering the instant as it arrives. We may instead want to discuss the instant of transition prospectively or retrospectively. So the present consideration does not provide a solution for all cases, or even for all the cases Peter of Spain applies it to. Aristotle himself may have another consideration relevant to the particular example of visibility. For he classifies seeing as an energeia, and on one interpretation, an energeia has no first instant. This is how J. L. Ackrill interprets Aristotle's idea (e.g., Sens. 446b2) that "he is seeing" entails "he has seen". Ackrill treats the perfect tense "he has seen", like "he has been seeing", as implying an earlier period of seeing ("Aristotle's distinction between Energeia and Kinesis", in New Essays on Plato and Aristotle, ed. R. Bambrough, London 1965, esp. pp. 126-7). This interpretation has been disputed, but if it is correct, it implies that there will not be a first instant of seeing, and therefore not a first instant of seeing the approaching object in the problematic example of De Sensu 449a21-3 1, which was briefly mentioned above, p. 71.

The various considerations we have mentioned do not begin to cover all the cases there are. There may well be unique considerations attaching to particular occasions of discussion. And we must add that there may well be cases in which there are no adequate considerations to guide us. In these last cases, we shall not be able to answer the question, "what is the last


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instant (or point) of the one state and the first of the other?" The most we shall then be able to do is this. If our questioner happens to assume (without reason) that one of the instants (or points) exists, we shall always be able to tell him that in that case the other does not exist. A rival solution An entirely different way of trying to cope with our problem has been advocated by Brian Medlin in his paper "The Origin of Motion" (Mind 1963). Medlin says, in effect, that a thing can be both in motion and at rest at an instant, and equally neither in motion nor at rest at that instant. The first may sound as if it violates the law of contradiction, the second as if it violates the law of excluded middle. But Medlin avoids this, by simply defining motion at an instant, and rest at an instant, in such a way that they are neither contradictories nor contraries of each other.' Given his definitions, all four statements can be true together, namely, that a thing is in motion at an instant, not at rest at that instant, and that it is at rest at that instant, not in motion at it. My objection to this is not so much that it runs the risk of causing confusion, but that it is not sufficient to solve the problem that interests us. Medlin is free to define motion at an instant and rest at an instant in such a way that they are not contradictories or contrariesof each other. But he cannot, and does not, deny that there is a contradictory of the claim that something is in motion at an instant. He himself suggests a way in which we might formulate the contradictory.We could talk of its being the case that something is in motion at a certain instant, and of its not being the case that something is in motion at that instant. Once we have found a formula for picking out the contradictory,we can pose our original problem all over again in terms of the new formula. We shall simply ask what is the last instant when it is not the case that our object is in motion, and what the first instant when it is the case that it is in motion. To this question Medlin himself would agree that we cannot say it is the same instant. When the problem is posed this way, we see that we shall have to fall back on a different solution from Medlin's, such as the one we have advocated, according to which there can be a

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last instant when it is not the case that our object is in motion, but not a first instant when it is the case that it is in motion. Aristotle's treatment. Preliminaries (i): The four main kinds of change and the thesis that they all involve a gradual transition. I shall now turn to Aristotle's solutions of the problem. He is well aware that different kinds of case need different solutions, but, not surprisingly, he looks for solutions of some generality, and does not acknowledge that the matter might ever be decided by the unique interests of a particular context. We shall suggest that in the cases we are going to discuss, Aristotle is attracted by two solutions. Sometimes, like us, he denies that certain continuous changes can have a first or last instant, without however, making it very clear that it is the continuity which precludes this. At other times, he goes further, and denies that there can be any change or stability at an instant. Before expounding his two solutions, we shall have to make some preliminary points clear. A first thing to notice is that Aristotle recognizes only four kinds of change as being changes in the full sense of the word. There is change of quality (as when something changes colour), change of place (in other words, motion), change of size (in growth and diminution), and finally the creation or destruction of substances (Phys III 1, 200oob32-2o0a16; cf V 2; Metaph XI 12).

A second point is that in all four kinds of change, he thinks there is a gradual process of transition (Phys VI 6, 237a17-b3; b9-21). Qualitative change, such as change of colour, is said to take time. Change to a new place or size involves passing through intervening points. The creation of something like a house takes time, and occurs part by part, the foundation before the whole. Aristotle does not by any means think that changes other than the four genuine ones must all involve a gradual process of transition. Indeed, he sees that an infinite regress would be involved, if the gradual process by which something came into being had itself to come into being by a gradual process (Phys V 2, 225b33-226a6). In the very passage where he explains that a house comes into being only part by part, he points out that this cannot be true of things that have no parts (VI 6, 237b 11, b15), points and instants, for example.


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Preliminaries (ii): How can change of colour be gradual? It may be wondered how change of colour can be gradual, given the view stated in the De Sensu (6, 445b21-9g; 446a16-2o), and alluded to in the Physics (VI 5, 236b5-6; b 7-18), that there is only a finite number of discriminable shades. In that case, a change to a new colour would be gradual when it involved passing through a number of intervening shades. But within this process, how could the change from one shade to the next be treated as gradual? Aristotle seems to have two incompatible answers. One answer is implied in Physics V 6 (23ob32-231a1), VI 4 (234bio-2o), VI 9 (24oa19-29), VI 1o (24ob21-31), when he says that certain changes occur part by part.6While a surface is changing from white to the next shade, grey, he says, part of the surface must still be white, and part already grey. The greyness spreads gradually over the surface. This claim, that certain changes occur part by part, is used in combating the view that a partless atom could undergo change or motion. In its turn, the claim has as its ground that, while a thing is actually changing or moving, it cannot yet be in its terminal state, nor can it still be in its initial state. It must therefore be partly in one state, partly in the other, and so must have parts. In the case of motion, it must move part by part into the adjacent area. Diodorus Cronus and certain Epicureans were later to get round this objection to the motion of partless atoms, by denying that an atom actually is moving at any time; rather, at any given time it has moved with a jerk (references above at the end of section on Proposed Solution, p. 75)But Aristotle seems not always to keep in mind the view that these changes occur part by part. For in the De Sensu (6, 447a1-3) he actually denies that qualitative change has to occur part by part, and illustrates how it can happen with the case of a whole pond freezing over at once. In one of the Physics passageswhere he says that qualitative change (en tois enantiois 237b1) is gradual, he speaks as if he must prove this on the basis of time taken, but cannot prove it on the basis of space covered (237a19-28; a29; b2; b21). Why not, if he remembers his view that a surface changes colour part by part? He seems to be forgetting that view, and he probably forgets it again in VIII 8, where he declares that while something is becoming white, it is not yet white (263b27; b3o). This way of

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putting things seems to neglect the idea that there will be a stage of being partly white. How then can the transition from one shade to the next be represented as gradual? A second way of arguing that the transition is gradual serves to refute the idea that part by part changing is indispensible for this purpose. The second way is suggested by what Aristotle says in De Sensu 6. Admittedly, there is only a finite number of discriminable shades, so that discriminable colours form a discontinuous series (445b21-9; 446a16-2o). But nonetheless colours, musical pitches, and other ranges of sensible qualities have a kind of derivative continuity (445b28; b3o, to me kath' hauto suneches). What Aristotle seems to have in mind is that a change to the next discriminable pitch, in the discontinuous series of discriminable pitches, may be produced by a continuous movement of a stopper along a vibrating string. Or in the case of colour, a change to the next discriminable shade, in the discontinuous series of discriminable shades, may be produced by a continuous change in the proportions of earth, air, fire and water in a body. As the stopper moves along the vibrating string, we hear the sound all the time, but we do not hear a change of pitch, until the stopper has moved the distance that corresponds to a quarter tone (446a1-4). Variations of pitch less than a quarter tone are not perceptible except by being part of the whole variation (446a18, hoti en toi holoi), by which Aristotle probably means that they only contribute to the perceptibility of the whole variation. This suggests a way in which Aristotle can maintain that a change to the next discriminable colour or pitch can be continuous. A body is changing to the next discriminable shade all the time that the continuous change in its elemental ingredients is going on, which will eventually lead to its displaying that next discriminable shade. In what follows, we shall only consider Aristotle's treatment of the four genuine kinds of change. This will leave open how he might have treated the many other cases of (non-genuine) change. In connexion with the four genuine kinds, and the transitions involved in them, we find two rather different kinds of treatment.


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Aristotle's first solution The better known one is less satisfactory.It is most fully expressed in connexion with rest and motion in Phys VI 3 (234a24-bg9)and 8 (239alo-b4). Here Aristotle says that there can be neither rest nor motion at an instant. He explicitly cites our problem as one ground for his conclusion (VI 3, 234a34-b5), saying that if something stops moving, the instant of transition between motion and rest is an instant neither of motion nor of rest, since otherwise we could not avoid the contradiction of saying that it is an instant of both motion and rest. Plato had already given the same solution in the Parmenides (156C-157A), when he discussed this very puzzle. At VI 6, 237a14-15, Aristotle extends his treatment of motion to all change, saying that a thing cannot be changing (metaballein) at an instant. An extension is also attempted in VI 8, 239a3-6, where Aristotle denies that there is a first instant of slowing to a halt, by arguing that slowing to a halt implies moving, and that there is no moving at an instant. (To make the argument valid, he ought to show that slowing to a halt at an instant would imply not just moving, but moving at an instant.) Aristotle gives several grounds for denying rest or motion at an instant, besides the need to avoid difficulties about the instant of transition between rest and motion. In VI 3 (234a24-b9), one argument is that to rest is to be in the same state now as then, but an instant does not contain a then. Another argument is that variation of speed would be impossible at an instant, because such variation would imply that the faster body had traversed in less than an instant what the slower body traversedin an instant. Finally, if we cannot speak of motion at an instant, we cannot speak of rest at an instant, since we can only talk of rest where there would have been the possibility of motion. Aristotle allows that, when something stops moving, there is a single instant which is both the last of the period during which the object is moving, and the first of the period during which it is resting (VI 3, 234a34-b5). And something parallel is true when a thing starts moving. But this does not in the least commit him, as he makes very clear, to saying that this is an instant at which the object is moving or resting.

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Since Aristotle thinks his view holds not only for motion and rest, but for change and stability in general, we can apply his remarks, for example, to a change of colour, in which a surface starts off wholly of one shade, and by a gradual process of transition, finishes up wholly of another shade. If we raise problems about the first and last instants of its changing colour, Aristotle will say, for reasons similar to those already quoted from VI 3, that there is no first or last instant, nor indeed any instant, at which it is changing colour, or remaining the same colour. Aristotle sees, however, that this solution is not a complete one. For although he denies that things can change or remain in the same state at an instant, he concedes that there are many other things that can be true of them at an instant. He is quite prepared to allow that what is moving can be at a point (VIII 8, 262a3o; b2o), or level with something (VI 8, 239a35-b3) at an instant. As regards other kinds of change, the object that is changing colour can be white at an instant (VIII 8, 263b2o; 23), or the white have perished and nonwhite have come into being at an instant (263b22). In general, a change can have been completed, and the new state of affairs can have come into being at an instant (VI 5, 235b32-236a7; VI 6, 237a14-15).

In allowing something to be white at an

instant, he is not allowing that it could remain white, or rest in the white state, at an instant.7Since he allows something to be of a certain colour at an instant, he cannot finally dispose of our problem by ruling out of order questions about a first or last instant at which a surface is changing colour. For this still leaves us free to ask about a first instant at which the surface is no longer grey, or wholly grey, and a last instant at which it is not yet white, or wholly white. Aristotle recognizes the need to deal separately with this further question, and this brings us on to the second kind of treatment that we find in his work. Aristotle's second solution In Phys VIII 8, 263b15-264a6, Aristotle discusses a change from not-white to white (or vice versa), and a change from not existing to existing. He thinks of the final state (e.g., white) as being reached by a gradual process of transition, but this is


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one of the passages where he does not construe the process as one of white spreading part by part over the surface. Instead, he implies that throughout the process of transition the surface will remain non-white (263b27; b30). He distinguishes an earlier state, by which he means the state when the surface is still not-white but is changing to white, from a later state, by which he means the final state of being white. Or rather, since he switches his example in mid-discussion, the earlier state is one of being white while changing to not-white, and the later state is one of being not-white, but for simplicity I shall stick to the one example. He then says that there is no last instant of being in the earlier state, but there is a first instant of being in the later. I take it that it is crucial to understanding the passage to notice that the earlier state, of which there is no last instant, is one which involves changing' while the later state does not. His view is generalized in an earlier chapter (VI 5, 235b6-32),' where it is said in connexion with all genuine change that there is a first instant of being in the terminal state after a process of transition. One ground Aristotle gives for his verdict is again the existence of the very problem that interests us. He says that the verdict provides a way (he fails to consider whether it is the only way) of avoiding the contradiction of something being in its old state and in its new state at the same instant (263b11; bl7-21).

He con-

cedes that there is an instant which is equally the end of the period during which white was coming into being and the beginning of the period during which the surface is white, but he insists that at that instant the surface is already in its later state, white (263b9-15; 264a2-3). Aristotle's treatment of the problem here is by and large very much in line with the solution which we have advocated. For we should agree with him that there is no last instant of being not yet white, if changing to white is a continuous process. However, we must to some extent qualify our claim to be in agreement. For when Aristotle gives his reasonsfor denying a last instant of the earlier state, he does not give our reason, the continuity of the process of becoming white. This continuity may well be what influenced him, but if so, he has not managed to identify it explicitly as the reason." There are further passages, besides the pair we have men-

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tioned,whereAristotleis attractedto a view close to our own. We haveso far consideredhis denial of a last instantat which somethingis non-whitewhile becomingwhite. But he alsodiscusseswhetherthere can be a first instantat which something is changing.At VI , 236a7-27,he wantsto show that there is no earliest time at which somethingwas changing (236alS, reading: meteballen),whetherthat earliesttime is construed as a divisible period,or as indivisible.If indivisible,it might be construedeither as an instant, i.e. as a boundarywith no duration,or as an atomof time with an indivisible duration. In 236al7-20 (whatevermay be true of al6-17), he is construing the putative earliest time as a durationlessinstant. And he takesthe view whichwe have advocatedin connexion with continuous changes,that there is not a first instant at which somethingis changing.Moreover,for the first time he actuallycontradictshis othersolutionby sayingthat thereis a last instant at xNhichsomething is resting (contraryto the doctrineof vr 3 and 8, which denies rest at an instant).His ground for denying a first instant at which it is changingis yet again the existence of the kind of problem we are discussing.Once we assumethat thereis a last instantat which it is resting,there cannotbe a firstinstantllat which it is changing. For at such an instant, the object would alreadyhave changedto some extent,l2and it cannothave changedto any extent at the veryinstant at which it is still resting. Aristotle doesnot give as his groundfor denyinga firstinstantat which somethingis changing the view he takes in VI 3 and 8 that thereare no instantsat which somethingis changing.He may insteadbe inSuencedby our kind of consideration;for just as we associatedmovingwith being awayfromthe startingpoint, so he associateschangingwith havingchangedto someextent. And he may be influencedby the fact that in a continuous change there is no first instant of having changed to some extent. But if this is whathasinfluencedhim, he hasagainnot articulatedthe reason. There is one more place where Aristotlecomesclose to our sliew.The themeof PhysicsVI 6 is thatwhathaschangedmust have been changing earlier (237al7-b22), and what was changing earlier must before that have accomplishedsome change (236b32-237al7). so that it has already changed an

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infinite (237al 1; a16) number of times, and you will never get a first in the series of changing and having changed (237b6-7). This implies that there cannot be a first instant of having to some extent changed, not, for example, a first instant of having ceased to be wholly grey, or of having started to be partly white. This implication, which is admittedly not explicitly spelled out by Aristotle in so many words, is precisely what we considered to be true of continuous changes.'3 There are then two strands of thought in Aristotle about the process of transition involved in the four genuine kinds of change. Sometimes he argues that a thing cannot be changing or resting at any instant. But sometimes he argues or implies, in conformity with our view about continuous changes, that there cannot be a first instant at which a thing is changing, nor a first instant at which it has left its initial state, nor a last instant of not having reached its terminal state. Unfortunately, he does not seem to have a firm grasp of the latter point of view. For in Phys. VIII 8, 262a31-b3; -263a3, he appears to contradict it, by assuming that when b2 W a moving object reverses direction, there is a first instant of having left the point of reversal. At least, this is the assumption which he seems to require for his conclusion, which is that the reversing object must spend a period of time at the point of reversal. The assumptions seem to be that there is a first instant of having reached the point of reversal, and a first instant of having left it, and that these cannot be adjacent or identical instants, so must be separated by a pause during which the reversing body rests. Assessment of Aristotle's first solution Let us return briefly to the first strandof thought, according to which a thing cannot be changing or resting at an instant. This view may have appealed to recent philosophers,'"and is not to be lightly dismissed, but it does have severe disadvantages. For it ignores that it is possible, and very useful, to give sense to the idea of changing at an instant. It is possible, so long as we acknowledge that change at an instant is a function of change over a period. It is useful because the velocity of a body in a given direction at an instant is one of several factors from which we can calculate in detail its future

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Much of modern dynamicsdepends on the possibehaviour. of talkingof accelerationat an instant,whereasAristotle bility rob us of this possibility. would of our problem to the definition of motion at an Relevance instant the taskof definingmotionat an instantis by no Nonetheless, easy.And the kind of problemwe have been discussing, means aboutlastand firstinstantsof restor motion,gainsimportance fromthe fact that it needs to be resolved,if we are to obtain asatisfactorydefinitionof motion at an instant. What definiBut tionwill be satisfactorydependsin part on our purposes. our purposes, some ifmotion is continuous,then at least for debe to ought discussionsuggeststhat motion at an instant a For motion. of finedso as to exclude a first or last instant that one be start,we maysuggestthatan instantof motionwill be fallswithin a periodof motion,while an instantof rest will definithis onethat falls within or boundsa periodof rest. But tionmayneed revisionin the light of otherdifficultexamples, suchas that of the ball thrownverticallyupwards,and slowing down until it changes direction at an instant. We found the reasonto regardthis instantas an instantof rest, whereas definitionjust proposedwould make it an instant of motion, But and may need to be revised,for this and other reasons. our formulated, be eventually may however the deSnition discussionsuggeststhat for somepurposesit should be formulated so as to exclude first and last instants of continuous motion,and so as to avoid,if possible,our problemsabout the of relation to first and last instants of rest. The necessity gettingclearaboutthesethingsmaynot alwaysbe appreciated. in BertrandRussellgives a definitionof motion at a moment § 446 of The Principles of Mathematics(London 1903, 2nd edition 1937)and denies,unlike us, that the instant of transidoes tion betweenrestand motioncanbe an instantof rest. He an be can it not or whether not, however,make it so clear ^ *

lnstant

^

ot motlon.

Generalassessmentof Aristotle'stosition I should like to finish with a generalassessmentof Aristotle's his treatmentof our problem. He has earned notoriety for


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refusal to allow motion at an instant. G. E. L. Owen remarks that this refusal not only "spoilt his reply to Zeno" on the paradox of the flying arrow,but also "bedevilled the course of dynamics".15In more detail, Owen explains: "Unable to talk of speed at an instant. Aristotle has no room in his system for any such concept as that of initial velocity or, what is equally important, of the force required to start a body moving. Since he cannot recognize a moment in which the body first moves, his idea of force is restricted to the causing of motions that are completed in a given period of time. And, since he cannot consider any motion as caused by an initial application of force, he does not entertain the Newtonian corollary of this, that if some force F is sufficient to start a motion, the continued application of F must produce not just the continuance of the motion but a constant change in it, namely acceleration. It is the clumsy tools of Aristotelian dynamics, if I am right, that mark Zeno's major influence on the mathematics of science."''16

With the first part of this I entirely agree. Aristotle cannot accommodate the useful concept of initial velocity, by which is meant, of course, not some first velocity in the entire motion, but the first velocity which it is convenient to consider in a given calculation. But what about the second part? A reader might take "a moment in which the body first moves" to be a first instant at which the body is moving. We have argued that it is precisely Aristotle's merit that he denies that there is such an instant; I would not regard this denial as a defect. I would accept, then, some charges against Aristotle, but not others, and at the same time I would draw attention to two merits of his discussion. First, it is a merit that he recognizes that not all cases call for the same treatment. The treatments we have been discussing apply only to those processes of gradual transition which he believes to be involved in the four genuine kinds of change. Second, Aristotle does express the view which we believe to be reasonable for continuous change, namely that there is no first instant at which a thing is changing, or at which it has begun to abandon its original state, and no last instant at which it has not yet achieved its final state."

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NOTES 1 This revival has been discussed by Curtis Wilson in William Heytesbury, Madison Wisconsin, 1956, and now in an illuminating paper by Norman Kretzmann,"Incipit/Desinit" (see n. 18 inf.). This should be read by anyone interested in the continuing history of the subject, although we diverge in our interpretationsof Aristotle. 2 It may be objected that if the atoms of a body are for ever joggling, and if their motions are not equal and opposite so that they cancel each other out, then the centre of mass will also be for ever moving, so that it will have no first instant of motion. We may reply that, even if this is so, we can still ask about the first instant of motion (and last of non-motion) of the centre of massin a given direction, or in responseto a given force. I shall neglect this complication in what follows. 3 This consideration would fall to the ground, if on other occasions the centre of masswere deflected at an angle without the velocity slowing to zero. For then the positive velocity would persuade us to talk of motion at the instant of deflection, and we should then be accustomedto talking of motion without there being one direction rather than another. But if deflection at an angle (as opposed to deflection in a curve) is impossible (and it would involve a discontinuity in the velocity in a given direction), then our considerationcan stand. 4 Accordingto Kretzmannop. cit. It looks as if Peter of Spain failed to see that whether one's interest is in identifying the instant as it arrives is quite independent of whether one is discussing what he calls "permanent" states. or "successive"states, or the beginnings or endings of either. s In effect, he defines, "it was in motion (not at rest) at instant t," by saying something like, "t was either followed, or preceded, or both, by a period throughout which it moved." And he defines, "it was at rest (not in motion) at instant t", roughly as "t waseither followed, or preceded, or both, by a period throughout which it did not move". 6 Aristotle probablyhas this answer in mind also in VI 5, 236b5-8, where he stresses that, even if colour is not in itself divisible, the surface to which it attaches is divisible. ?All he is committed to is the view expressed elsewhere (Phys VIII 8, 264b1), that what is white must remain white over a period. 8 That he thinks the process of change is going on is clear e.g. from 263b21-2, "non-white was coming into being, and white was ceasing to be".

263b26-7, "If what exists now, having been previously non-existent, must have been coming into being, and did not exist while it was coming into being . . .". 264a2, "The time in which it was coming to be". 264a5, "It was coming to be".

9 There is such a thing as the time "when first a thing has changed" (VI 5 235b7-8; b31; b32), i.e. has completed its change. This is an indivisible time (235b32-236a7).And at that instant the thing is already in its new state (235b8; b31-2). 10Aristotle's solution is only appropriate, given his two assumptions that the discriminable shades form a discontinuous series, and that nonetheless the change from one discriminable shade to the next is a continuous one. A quite different treatment of colour changes would be called for, if he took the view (i) that colours form a continuous series. In that case, the possibility would arise of producing a continuous alteration of shade along the spectrum.


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In a continuous change of shade such as this, there could not be a last instant of one shade, nor a first of another. The situation would be different again, if (ii) he took colours as forming a discontinuous series, and also took the change from one colour to the next as being discontinuous. In that case, nothing we have so far said would enable us to decide which colour to speak of at the instant of transition from one colour to the next. As fox (iii) a discontinuous transition through several intervening shades to a distant one, such a transition might be thought of (though it need not be) as having a first instant (namely, the first instant at which a colour other than the original one existed), and a last instant (namely, the last instant at which the penultimate colour was in existence). 11It is confusing that this putative instant is referred to by two letters, A A, and not just one. The reason is that A A stands for the putative earliest time of changing, which is later treated as a period with A and A as its terminal instants. Here, however, it is treated as a durationless instant, so that & is not separate from A. 12 Or "have begun to change". At 236a7-lo "has changed" (metabeblike) is said to be ambiguous between "has completed its change" and "has begun to change". The latter sense is relevant in the present lines (236a19-2o); the former is not. 13 There is another way in which this passage diverges from Aristotle's first solution. Though there is no first instant of having to some extent changed, there is an instant which divides the period of stability from the period of change. This instant is the last instant of the period during which the object is not changing, and Aristotle's view elsewhere (234a34-b5; 263b9-15) suggests that it can also be called the first instant of the period during which the object is changing. But at one point in the present chapter, VI 6, wittingly or not, he casts doubt on the latter description. For he says (237a15) that at any instant of the period during which a thing is changing, it has already changed. This would seem to rule out not only a first instant of having to some extent changed, but also the applicability of the description "first instant of the period during which a thing is changing." Aristotle thereby contradicts for a second time an aspect of his other solution. 14 I am not sure whether this is the intention of Vere Chappell in "Time and Zeno's Arrow", Journal of Philosophy 1962. 15 "The Platonism of Aristotle", Proceedings of the British Academy 1965, p. 148. Similarly "Aristotle" pp. .25-6 in vol. I of The Dictionary of Scientific Biography, ed. C. C. Gillispie, Scribners, N.Y. Owen's papers on the continuum are required reading for students of this subject. 16 "Zeno and the Mathematicians", Proceedings of the Aristotelian Society. 1957-8, pp. 220-2. 17 I have had many helpful discussions on this topic, but I am particularly indebted to Geoffrey Lloyd for a thorough correspondence about some of the texts, to my student Marcus Cohen and to Malcolm Schofield for some very helpful discussions on the philosophical issues, and to Professor Clive Kilmister for patient advice on Newtonian mechanics. 18In Machamer &8Turnbull edd., Motion and Time, Space and Matter (Columbus: Ohio State U.P., 1976), pp. 101-136.

ARISTOTLE ON THE INSTANT OF CHANGE Richard Sorabji and Norman Kretzmann II--NormanKretzmann TIME EXISTS-BUT

HARDLY, OR OBSCURELY

(Physics IV, 10o;217b29-218a33) It is characteristic of Aristotle to preface the explicit presentation of his own view with reviews of difficulties proper to the subject matter and surveys of his predecessors' opinions. He follows that practice in his treatise on time (Physics IV, 10-14), beginning with a clearly delineated investigation of four puzzles (dieporemena) arranged in two pairs.' Regarding the puzzles generally he says that it is good to begin by considering them (217b30), that they are associated with popular or familiar views of time (dia ton exoterikon logon-2 17b30-3 1), and that they, like the opinions of his predecessors,fail to reveal the nature of time (218a31-33). The puzzles have been taken to be unresolved problems which Aristotle afterwards ignores.3 I think, on the contrary, that they are, or involve, arguments designed to show that the most natural and familiar view of the nature of time is incoherent. They fail to reveal the nature of time, but they show what time cannot be. And if they do what I think they do, the four puzzles make a very important contribution to the understanding of Aristotle's own theory of time. The Four Puzzles I. Time consists entirely of two nonexistent parts; and II. The present is not a part of time. The present instant (the now) is either (III) always different or (IV) always the same; but III. If the now is always different, it is so either because IIIa. there are different simultaneous nows-which is impossible-or because

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IIIb. there are differentsuccessivenows-which is impossible-and IV. If the now is alwaysthe same,all events are simultaneous-which is impossible. After a brief introductoryparagraphthe puzzles passage gets under way with the statementsof two counter-intuitive, or at least disconcerting,claimsregardingthe realityof time: Time is altogethernonexistent;Time exists-but hardly,or obscurely.These claims will emergeas forced conclusionsat differentpoints in the developmentof the puzzles;and if I am right about the aim of the puzzles,the conclusionswill constitute embarrassments for the familiar view that time is essentially, really passing. I The development of the first puzzle may be conveniently presented in the following close paraphrase. Time-either infinite time itself or any time one might choose-is composed of two parts. One of those two parts has come to be and is no longer; the other part will be and and is not yet. Therefore neither part of time is. And therefore time is altogether nonexistent, for what is composed of nonexistent parts does not itself exist. There are two obvious objections to this argument. In the first place, it seems plainly false that any time one might choose is divisible into the two parts described in the argument. Suppose we choose the nineteenth century. We might be willing to say that it, like time itself, is composed of a past and a future; pick a date, any nineteenth-century date, and it will mark off a nineteenth-century past and a nineteenthcentury future. We are sometimes willing to extend the use of the words 'past' and 'future' in that way; but Aristotle does not use the Greek words for past and future here. He says of time itself that one part of it has come to be and is no longer and the other part will be and is not yet, and we cannot say that of any time we might choose-of the nineteenth century, for example, or the twenty-first. We can say that only of any time we might choose which, like infinite time itself, includes

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now (but not as either of the limits of the time); suppose we call such a period of time an "N-period". Any and every time is divisible into parts, but only an N-period, no matter how long or how short it may be, is divisible into one part that is no longer and another part that is not yet. Must we, then, consider the first puzzle as applicable only to N-periods? I think we must.' But, of course, any time we might choose which is not an N-period is a time which as a whole either is no longer or is not yet. It is therefore only N-periods that have any chance of counting as existent or real, even if only in part; and so the fact that the first puzzle applies only to them is a mark of economy, not of weakness.5 The second of the two objections is even more obvious than the first. As Fred Miller puts it, "One might be willing to concede the point about the unreality of past and future, and yet reject the conclusion on the grounds that the argument has overlooked a very important part of time: the present".6 We may assimilate this second objection to the first by thinking of it as the claim that an N-period is not exhaustively divisible into a part that is no longer and a part that is not yet. I think Aristotle expects and intends his readers to raise this objection; it sets them up for the second puzzle. The complementary relation of the second puzzle to the first is indicated also by the fact that the second is prefaced by a restatement of the first, as if to drive home the preliminary point. In addition, if any thing with parts is to exist, then, when it exists, all or some of its parts must exist. But, although time is divisible [i.e., although time has parts], some parts of it have been and the others will be, and no part of it exists (218a3-6). In the original statement of the first puzzle Aristotle spoke of two parts of time--one that has been and one that will be. In this restatement of it he speaks, in a way that at first seems unnatural, of parts that have been and other parts that will be. But these plurals are well suited to the considerations raised in connexion with the introduction of the notion of an N-period. All parts, or periods, of time are either N-periods or not. If they are not N-periods, then they are parts that have been or

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parts that will be. If they are N-periods, then they have parts that have been and parts that will be. Before considering the second puzzle directly, it is worth noticing that the language of the first puzzle, both in the original statement and in the restatement, leads naturally to the inference that infinite time itself or any finite N-period is not divisible without remainder into the parts Aristotle describes. In the original statement the one part is no longer7 and the other part "is not yet", and in both the original and the restatement he says of the parts that they have been (or have come to be) and will be. Such expressions cannot apply unless regarding the one part (or each thing in the one part) it has been the case that it is, and regarding the other part (or each thing in the other part) it will be the case that it is. So if time is continuous-and Aristotle of course maintains that it is so (219a12-13; 233alo)-there must be something temporal that is. And so it must not be true that time is exhaustively divisible into what has been and what will be; time must, as the language of the first puzzle itself implies, include what is. But what can that be? It cannot be an N-period, for such a period, like infinite time itself, also includes what has been and what will be. Yes, but even when they are stripped away,8there is a remainder: now, the present instant'-which must be the part of time that is. II That elaboration of the second objection to the first puzzle leads directly to the second puzzle, the heart of which is the flat denial with which the presentation of it begins: "as for the now, it is no part of time" (218a6). What lies behind the denial, of course, is the recognition that the now is only the division between the parts, just as the twenty-four hour day is exhaustively divided into A.M. and P.M., 12: oo M. being only the division between those parts. The now is an instant, the temporal analogue of a geometrical point. But someone who objects to the first puzzle on the grounds that it leaves the present out of account might easily fail to appreciate the force of the second puzzle against his position,


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even if he agrees that the present must be conceived of as a bare instant. Even if he knows that a point has no extension and agrees that an instant is like a point in having no duration, the sophisticated objector may seek further justification for Aristotle's announcement in the succeeding chapter that an instant is no more a part of time than a point is a part of a line (22oa19-20). Why, he will want to know, should he not say of a bisected line segment that it has three parts-the left half, the right half, and the bisector (or point of bisection)? And why, even though an instant has no duration, should he not say of time that it has three parts--one time that is no longer, one time that is not yet, and the instant that is now? Aristotle's answer is contained in the development of the second puzzle, but not, I think, very clearly. And as for the now, it is no part of time; for a part measures, and the whole must be composed of the parts, but it is not thought that time is composed of instants (216a6-8). We seem to have an argument of three premisses: (1) a part measures, (2) the whole must be composed of the parts, (3) time is not composed of instants." We may take it for granted that we are dealing here exclusively with merely quantitative wholes, such as an hour, time, a line segment, space. The strategy of the argument is to lay down criteria for the parts of a merely quantitative whole and to point out that the now (or an instant generally) fails to meet those criteria. If I have understood the argument correctly, Aristotle's intentions in it might be more clearly represented in this way: (i) A part must measure the whole (ii) A part must make a quantitative contribution to the whole instant neither measures nor makes a quantitaAn (iii) tive contribution to time .(iv) An instant generally, or the now in particular, is no part of time.

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The differencesseparating(i) and (ii) from (1) and (2) are slight enough, perhaps,to be consideredonly accidents of diction. (iii) is more substantiallydifferent from (3), but without the addition effected in (iii) I cannot see how the argumenthangs together.With these revisionsthe argument does its job fairlywell, I think, althoughI have somereservations about the firstpremiss. The firstpremissmustmeansomethinglike this: Whatever countsas a part of somewhole is somethingin termsof which that wholemaybe measured.By this criterionan inch maybe countedas a part of a mile or of a journey,and a secondas a part of a day or of a lifetime. And by this criterion,it may be acknowledgedat once, an extensionlesspoint does not count as a part of a line, and a durationlessinstantdoes not count as a part of time. But whatever might be said in general on behalf of the view of the nature of a part on which this criterion is based, it seems peculiarlyinappropriatein this context. Any relevant denial that the now is a part of time must contrastthe now appropriatelywith the past and the future,which have alreadybeen described(in puzzleI) as the partsof time or of any finite N-period;and the past and the future areno bettersuited to serveas measuresof time than is an instant. Past and future are not partsof time in the way in which secondsor centuriesare,and yet thereseemsto be no good reasonto object to their designation,in puzzle I or elsewhere, as partsof time. Parts of the sort that can be used as measures-mensurantparts,we maycall them-must (a)have somefinite quantityand (b) be quantitativelystable.1 As partsof somefinite N-period-to-day,for example-past and future do have finite quantity, but as parts of a finite N-periodtheyare systematicallyunstablequantitatively;their quantities are incessantlychanging relative to each other. Supposesomeoneis now drivingfrom Ithacato New Yorkvia Binghamton.We might measure the length of the journey in miles, of course;we might also measureit in lengths of the carbeing driven-so manyVWs.We might measurethe time of the journey in hours, of course;we might also measureit in times of a particularpart of the journey-so many IthacaBinghamtons.But we could not measurethe distancein terms

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of the distance covered as of now or the time elapsed as of now-and I mean as of now, not as of whatever clock-time it happens to be on the occasion of one of my utterances of 'now'. Thus although every N-period is composed of past and future, past and future are not mensurant parts of a finite N-period. As parts of infinite time itself, past and future are both infinite; they thus lack the first necessary characteristic of mensurant parts--finite quantity. And although they are both (always) infinite, they may be said to be quantitatively unstable as well, since, for instance, today's past cannot be mapped onto yesterday'spast without remainder. The first criterion thus fails to contrast the now with the past and the future in the relevant respect. To observe that a part measures the whole would effect the required contrast only if the now were being compared with some genuinely mensurant part, such as a second. The now, like any other instant, is quantitatively stable-it never varies from zero quantity--but it thus fails to satisfy the other necessary condition of mensurant parts. The now is not a mensurant part of time; but neither are the past and the future. I suspect that Aristotle employed this first criterion casually, realizing that only a positive quantity could serve as a measure but neglecting to take into account the complicating factors I have been pointing out regarding mensurant parts. Even if I am right in thinking that the first criterion in the argument intended to develop the second puzzle is inappropriate, it does, after all, support the conclusion: the now is not a (mensurant) part of time. And in any case the second criterion upholds the conclusion in an appropriate way. If we think of to-day as a salami sausage, then the present instant is a cut dividing the sausage in two. While the salami can be reconstructed out of slices, no matter how thin, it cannot be reconstructed out of cuts, no matter how many. As a salami is not composed of cuts, so a day (or infinite time itself) is not composed of instants. If it is right to say without qualification of any whole that it is the sum of its parts, it must be right to say so of a merely quantitative whole. Whatever counts as a part of a whole which is the sum of its parts is a summand, makes a quantitative contribution to the whole.

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Past and future are summand parts of time; and now is not. Considering the now a part of time is like considering the part in your hair a part of your head. I 8cII Before considering the second pair of puzzles I want to spell out the position on the reality of time which seems to result from the first pair of puzzles together with a few fundamental Aristotelian views on the nature of time. We have been proceeding so far, following Aristotle's lead, on the familiar view that time consists entirely of the past, the now, and the future, which may be characterizedalso as the view that time is essentially passing. On that view the past and the future are the two parts of time, distinguished from each other by the now, which is of course temporal, but not a part of time. The past is not; the future is not. It follows that no part of time is. Ordinarily, if neither of two parts exists, then whatever has been divided exhaustively into those two parts does not exist. One might therefore feel justified, if uncomfortable, at this stage in saying that time does not exist. The uncomfortable feeling should stem not only from the bizarre character of the conclusion but also from the still shadowy status of the now. Although it is not a part of time, it is temporal, and it is. The now is an instant, and "an instant is not time but an attribute" of time (22oa2 1-22). But an instant must be an attribute of time in the essence-delineating sense in which mortal rational animality is an attribute of man, for "it is also evident that neither would an instant exist if time did not exist, nor would time exist if an instant did not exist" (2 19b33-220a1). And since Aristotle certainly does speak as if instants exist, it looks as if it must be in virtue of the existence of an instant (of the now in particular) that he is able to say that "time exists" (222b27). But when he comes to state his own theory of time he says (in the succeeding chapter) that "time does not exist without a motion" (21 8b33). And in leading up to his discussion of Zeno's paradoxes of motion (in Book VI) he says that "a thing can be neither at rest nor in motion at an instant" (239b1-2). So if there is no motion at an instant, and no time without motion, there is no time at an instant. That


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conclusion may seem to be a truism; of course there is no time at an instant-no more than there is space at a point. But if there is no time at an instant, and if an instant is no part of time, and if the two parts of time do not exist, and if "an instant is not time" (22oa1), then even if we acknowledge that

an instant-the now-exists, how does that acknowledgement warrant our concluding that time exists? Well, Aristotle says that the now is an attribute of time, and if an attribute exists, then surely its substrate exists. But where is the time that is to serve as the substrate for that attribute? There simply is none available-not, at any rate, as long as we proceed on the familiar view that time consists entirely in the past, the now, and the future. Moreover,it is, after all, the now which defines

the past-before now-and the future-after now. If the now is to be describedas not time but an attributeof time, it seemsthat the past and the future are to be describedin just that way as well. On the basis of this review of the first pair of puzzlesit seemsquite right to suggestthat theymight lead one to suspectthat "time is either altogethernonexistent,or that it exists,but hardly,or obscurely". III &cIV

The two puzzles constituting the second pair occur as the horns of a dilemma. Either "the now which appears to divide the past and the future [IV] alwaysremains one and the same, or [III] it is always distinct" (218a8-io). It seems clear that the theoretically possible middle ground-that the now sometimes remains one and the same but occasionally changesis ignored as unthinkable. The question Aristotle is considering here resembles the question whether the sun that rises in the morning is always the same or always a new sun, and to that question the middle-ground answer is not unthinkable. But in the case of this question about the now we do not have lots of similar appearancesof some entity or entities, as in the case of the question about the rising sun. Here we are seeking a characterizationof "the now which appearsto divide the past and the future", which is paradigmatically always; it is always now. We seem, therefore, to be offered a forced choice between the only two thinkable accounts of what is always going on if anything at all is going on.

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The choice is an odd one, perhaps odder than it looks at first. For I think we are likely to feel we recognize the first of the two options; it seems to be the account of the now which forms a part of the familiar view of the nature of time-the doctrine of the nunc fluens. "We are inclined to think of the now as surging through history like the crest of a wave along the ocean surface-or 'we think of the present as a spotlight that plays along a line of chorus-girl-like events' ".* These pictures, designed to present the familiar view, certainly seem to be portraying a now that "always remains one and the same". As I shall try to show, however, the association of the first option (puzzle IV) with the nunc fluens doctrine is mistaken. III In his discussion of the dilemma Aristotle takes up the second option first; it thereby becomes the third of the four puzzles to be developed. This third puzzle deals with an account of the now which seems unfamiliar and difficult to characterize precisely; it may be tentatively designated the nunc differens account. Any attempt to characterize it further is best postponed until we have seen how Aristotle handles it. His discussion of it is complicated; perhaps a simple preliminary observation will be helpful. The claim is that the now is always distinct, but there are, one might suppose, two ways in which there might be distinct nows-simultaneously or successively. Thus the development of puzzle III, itself an option in a dilemma, takes the form of a dilemma. The long opening sentence of the discussion (218al1-16) aims at dismissing the possibility of simultaneous distinct nows; the remainder of the paragraph (218a16-21) then attacks the possibility of successive distinct nows. IIIa Most of us are unlikely to feel initially baffled by the notion of successive distinct nows, but what might lead one even to think of simultaneous distinct nows? In the sentence in which he attacks the notion of simultaneous distinct nows Aristotle speaks of "distinct part of time" (218a11-12), and if the parts of time are cut up in a certain (quite natural) way, one might


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be led to think casuallyin terms of simultaneous distinct nows. Take, for example, my lifetime and the lifetime of my cat Suki, who is now dead. The two lifetimes undoubtedly are parts of time (indeed, mensurant parts of time), and they are simultaneous in the respect Aristotle specifies: Suki's lifetime is contained by my lifetime; "..a. no two distinct parts of time exist simultaneously unless one part contains while the other is contained, as the smaller period of time is contained by the greater" (!218al -14). If Suki's lifetime and mine will serve as simultaneous distinct parts of time, it does not seem unnatural to think of them as parallel times, so that for every instant of Suki's lifetime there is a corresponding instant of mine-simultaneous S-instants and NK-instants. The inducement to think in terms of simultaneous distinct nows becomes even stronger if we change the example from me and my cat to you and your absent friend; it may seem entirely natural to think of his or her now as distinct from yours. But the apparent reasonableness that can be built up in the notion of simultaneous distinct nows is, I think, all a consequence of misleadingly specific descriptions of parts of time, such as my or your, his or her, lifetime."3If Aristotle had meant to be considering parts of time distinct in that highly specified way, then he ought to have admitted complete coincidence of two times (or parts of time) as well as containment of one by the other. I think Aristotle's omission of complete coincidence as a case of simultaneity of two distinct parts of time is deliberate. If there is complete coincidence, then there are not two distinct parts of time but only two distinct descriptions of one part of time. If the lifetimes of Robinson and Carpenter are completely coincident, they no more constitute distinct parts of time than do the lifetime of Robinson and the calendar time marked off by the instants of his birth and his death, or the time during which I look at you and the time during which you are looked at by me. According to Aristotle's criterion the cat's lifetime and mine do qualify as two distinct, simultaneous parts of time, but because we are likely to be misled by a consideration of such highly specified parts of time, we will do better to consider merely clock-times. The four-hour period from eight until noon to-day is a part of time, and the three-hour period from

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nine until noon to-day is a part distinct from but simultaneous with the four-hour period. No one is tempted to say that an instant of the shorter period has a corresponding, simultaneous instant in the longer period; the only sensible thing to say is that every instant of the shorter period is an instant of the longer period.'4By the way in which he treats the possibility of distinct and simultaneous parts of time Aristotle shows that he takes time to be linear. On that view of time-so natural to us that it scarcely qualifies as a view-there can no more be distinct but simultaneous instants than there can be distinct but coincident points on a line.'5 But in the complicated conditional sentence we are considering Aristotle does not go directly from the observations about simultaneous parts of time to the consequent "then two instants cannot exist simultaneously" (218a 15).16 He first introduces an additional consideration-to de nun me on proteron de on ananke ephtharthai pote (218a14)-which is open to two interpretations, depending on whether 'nun' is taken as a substantive ('the now' or 'an instant') or as an adverb ('now') contrasting with 'proteron' ('earlier'). The first interpretation may be conveniently designated 'W-A' after Wicksteed and Apostle, who use it in their translations," the second, 'C-R', after Cornford and Ross." W-A: an instant which does not exist but existed earlier must at some time have been destroyed C-R: that which does not exist now but existed earlier must at some time have been destroyed. W-A is supported by the context, which is studded with references to instants or the now, and by the fact that the adverbial use of 'now' seems not to add anything to 'does not exist'. On W-A, however, Aristotle's additional consideration regarding instants seems, although obviously true, irrelevant to his attempt to show that instants cannot be simultaneous. Without this passage the only explicit reason against the possibility of simultaneous instants is the observation that distinct parts of time are simultaneous just in case, and only to the extent to which, one part contains (or overlaps) the other. Even though Aristotle has already said that instants are not parts, it would be helpful, especially at this very early stage of


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his discussion of time, to say expressly that the sort of simultaneity available for distinct parts is unavailable for distinct instants, to point out that there is no possibility that some now of the very recent past might last long enough to contain (or overlap) this now. For instants, of course, differ from parts of time in that respect. A part of time that has been need not be a part of time that is not, that must already have ceased to betake to-day,for example, or any finite N-period. And Aristotle could have made this point by inserting an additional consideration which bears a strong resemblance to W-A: an instant which existed earlier does not exist but must at some time have been destroyed. But the strong resemblance is superficial; the Greek as it stands will not bear this useful interpretation. If we take the natural course of reading 'nun' as a substantive, we are stuck with the pointlessnessof W-A. As for C-R, it has one obvious advantage over W-A (and, for that matter, over the useful but insupportable interpretation) in that it leaves a point to be made in the concluding clause of the sentence-ephtharthai de ananke aei to proteron (218ai 5-16)

("but always the earlier instant

must have

been destroyed")-which on W-A looks like mere repetition. Moreover, on C-R line 14 loses the obvious pointlessness it has on W-A; but it does so, I think, only because its pointlessness has been obscured a little. Instead of being a truism regarding instants in particular it becomes a general truism. Following C-R, the argument against the possibility of simultaneous distinct nows (the development of puzzle IIIa) may be paraphrasedin this way: Suppose that the now is alwaysdistinct; then since no two distinct parts of time exist simultaneously unless one part contains the other, and since that which does not exist now but existed earlier must at some time have been destroyed, two instants cannot exist simultaneously, but always the earlier instant must have been destroyed. But this argument is invalid. The useful interpretation of line 14 (the second premiss) would validate the argument, and it is

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tantalizingly close to the actual text, but I am not bold enough to suggest the emendation that would bring it within reach. Nuances and conflicting interpretations aside, the intended achievement of the complicated conditional sentence is clear enough and may be granted (even if not on the grounds provided in the sentence itself): if the now is always distinct, it cannot be so in virtue of simultaneous distinct nows. If my general outline of the paragraph is correct, it remains to consider the second option of the sub-dilemma (the second half of the third puzzle), the possibility that the now is always distinct in virtue of successive distinct nows. IIIb The consideration of that possibility begins in a way which shows that the final clause of the conditional sentence-"but always the earlier instant must have been destroyed"-is doing duty also as an introduction to the next stage of the discussion; for the development of the second half of the third puzzle may be said to begin (implicitly) in this way: If there are successive distinct nows, then each now that has been must have ceased to be. And the development proceeds in an attempt to show that the notion of an instant's ceasing to be is incoherent. I want to paraphrase that attempt before looking at it more closely. Let t, be some instant that has existed and does not exist. Then tP must, at some instant, have ceased to exist. Let tn be the present instant (now). Then, since tp does not exist, at tp must at some past instant have ceased to exist-either t, itself or at some instant between tp and tn. But tp cannot

have ceased to exist at t,, for that is just exactly when it did exist. So it must be that tp ceased to exist either at the instant immediately after t, or at some subsequent instant (before tn). But there is no instant immediately after tp-"instants, like points on a line, cannot be consecutive to each other" (218a 18-19).19 So t, must have ceased to exist at some instant

between tP and t, (but not immediate to either t, or tn)-

call that instant tin. Let tm be arbitrarily soon after tp; there

will nevertheless be infinitely many instants between t, and tin.

So if tp ceased to exist at

"it would have existed simulti,

ARISTOTLE ON TIlE INSTANT OF CHANGE

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taneously with the infinitely many instants between itself and that other, and that is impossible" (218a 19-21). (The

impossibility of simultaneous instants may be taken to have been shown in the development of puzzle IIIa.) We are faced with the apparently paradoxical conclusion that no instant ever ceases to exist, a situation in which we are bound to review the steps which brought us to such a pass. There is nothing wrong with the rejection of the possibility that t, ceases to exist at some time between to and t, or with the denial of the possibility of consecutive instants. The paradox, if it is a paradox, must stem from the rejection of the possibility that t, ceases to exist at t,.

Mediaeval logicians dealing with problems associated with beginning and ceasing sometimes recognized instantaneous existence-along with permanent and successive existenceto cover primarily just this case of the instant, which begins to exist, exists, and ceases to exist simultaneously.? If instants exist, then surely that is how they exist-instantaneously. As we have seen, Aristotle maintains that instants do exist,2'but in the development of this part of the third puzzle (IIIb) he in effect rejects instantaneous existence as impossible: "the earlier instant cannot have been destroyed at that very instant itself, because it existed then" (218ai6-17). The rejection is certainly plausible; instantaneous existence is at best a problematic notion. But the rejection is, I think, inconsistent with Aristotle's own commitment to the existence of instants. One might suppose that he employs it only for the sake of the development of puzzle IIIb. But if the aim of that development is solely to argue that the now cannot be always distinct in virtue of successive distinct nows, Aristotle has open to him a more powerful line of argument which is consistent with his own position. For he might have argued for that conclusion on the grounds that if "the now which appears to divide the past and the future" were always distinct in virtue of successive distinct nows, there would be one now right after another,2 which is impossible. His neglect of this line is the more surprising in view of the fact that he presents the basis for it in the very development of IIIb: "instants, like points on a line, cannot be consecutive to each other" (218a18-19). As in the development of puzzle IIIa, he seems here to have

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used a bad or at least a suspect argument where a good one lay ready to his hand. I can think of only one explanation for his doing so in this case, and it is not a very good one. The better argument would not yield the conclusion that no instant ever ceases to exist, and that conclusion seems to have a dialectical function in the structure of the second pair of puzzles. If we have to conclude that no instant ever ceases to exist, we are going to feel even more inclined, though for an unexpected reason, to take seriously the claim dealt with in the fourth puzzle-the view that the now "alwaysremains one and the same". When I began considering puzzle III (on page too above), I said that we might be in a better position to characterize the nunc differens account when we had looked at Aristotle's attack on it. The principal clarifying effect of the attack is that it leaves us with only the IIIb version of nunc differens to consider seriously; the IIIa version, of simultaneous distinct nows, is too obviously farfetched. As for successive distinct nows, a theory maintaining that the now "is always distinct" in that respect must maintain that instants cease to be and that one instant immediately succeeds another; puzzle IIIb has raised insuperable difficulties for nunc differens on at least one of those two counts without adding any further details. The absence of detail coupled with the oddness of the original difficulty-Does the now which appears to divide the past and the future always remain one and the same or is it always distinct?--suggests that it is a mistake to think that what Aristotle is dealing with in this second pair of puzzles is two rival accounts of the nature of the now of temporal passage-accounts broadly characterizable as nunc differens and nunc fluens. If we do think in those terms, most of us will feel more at home with the nunc fluens account, the view that time is composed entirely of the past, the now, and the future, and that the passage of time is, more precisely, the passage of the now. But even though the designation "the flowing now" and the familiar images of the rolling wave and the traveling spotlight suggest a now that "always remains one and the same", we cannot take seriously, or imagine Aristotle taking seriously, the notion that there is a persistent instant flowing along the time line, thereby creating the past out of the future.


1o7

Traditional designation and imagery to the contrary notwithstanding, we come closer to depicting accurately the now of temporal passage, "the now which appears to divide the past and the future" if we say that it is "always distinct". Looked at with some awareness of the nature of instants, the so-called nunc fluens seems to disintegrate into the nunc differens. With some hesitation I suggest, then, that what puzzle IIIb really attacks is not some obscure theory regarding the nature of the now but the familiar view that time is essentially, really passing. If time consists, as the first pair of puzzles takes for granted, entirely of the past, the now, and the future, then time is essentially the passing of time. And if time exists but the future and the past do not exist, then the now exists. And if the now exists and time passes, then the now ceases to exist, again and again, infinitely often in any temporal interval no matter how short. But there is an incoherence either in the notion of a now's ceasing to exist or in the notion of one now's succeeding another consecutively. Therefore there is an incoherence either in the notion that the now exists, or in the notion that time exists, or in the notion that time is essentially, really temporal passage. And we know that Aristotle upholds the notions that the now exists and that time exists. III to IV Despite the close connexion between puzzles III and IV as the two horns of the same dilemma, the transition from III to IV shifts the focus of attention from temporal dynamics-the passage of time-to temporal statics-the permanent ordering of events as earlier (or later) than one another." Aristotle presents his own definition of time in terms of temporal statics (as the measurable dimension of motion in respect of earlier and later (at 219b1-2)) and expressly denies that time itself is a motion or change (218bi8-20). This is not the occasion on which to try to expound Aristotle's theory of time, but I cannot in good conscience avoid saying that I think he takes time to be essentially the permanent ordering of events and that he considers the passage of time to be an attribute of, or the appearance of, that linear sequence.2 The stage is set for his theory in the discussion of the puzzles, the first three of which

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seem designed to show that one cannot consistently hold that (A) time is real and (B) the essence of time is temporal passage. Aristotle himself holds (A), but, I maintain, in place of (B) he holds that the essence of time is the fixed temporal order. His discussion of puzzle IV tends to confirm this interpretation since in it the reality of the temporal order is taken for granted and serves as the basis for challenging the view that the now "alwaysremains one and the same".

IV The fourth puzzle is developed in two arguments. The first of them, although it is a simple argument based on elementary facts about the temporal order, may not be quite straightforward in its application to the view under consideration. Moreover, neither is it possible for it to remain always the same; for no finite and divisible thing, whether continuous in one or in more than one dimension, has just one limit, but an instant is a limit, and it is possible to cut off a finite time (218a21-25).

We can pick out a period of time, a period of time must be limited at both ends, instants are the limits of periods of time; therefore there cannot be only one instant. That much of the argument has nothing to do with the passing of time; the conclusion as I have drawn it-there cannot be only one instant -has to be interpreted tenselessly. But the conclusion at which this argument is ultimately aiming is that it is not "possible for it to remain always the same", and the referent of the 'it' is "the now which appears to divide the past and the future". Do those elementary observations about the need for more than one limit for a period of time really weigh against the view that the now which appears to divide the past and the future is always one and the same? I believe they do, and very heavily. Aristotle may be taken to be saying something like this. Every now which appears to divide past and future is an instant, every instant of the past was such a now, and every instant of the future will be such a now. The claim that the now which appears to divide the past from the future is always


10og

one and the same is thus tantamount to the claim that there is only one instant, which is absurd. Would it occur to anyone to make this claim about the now in such a way as to leave himself open to so obvious a refutation? My guess is that someone who is convinced that time is essentially passing and that the past and the future do not exist, and who wants, all the same, to maintain that time exists, might make such a claim. Time, he might say, is one and the same in virtue of being "continuous", and its existence consists in the steady, continuous passage of the now conceived of as the leading edge of becoming. Time exists as the real passage of a single continuously moving now. People certainly have spoken in that way,2 and it might be objected that in this first argument, with its insistence on treating the now simply as a temporal limit, Aristotle fails to address such a view of the now. Perhaps he does, but then at least he requires it to identify itself more precisely. The notion of the now arises as the notion of the division between the past and the future, a temporal limit. A temporal limit may be thought of as an edge, but not as a leading edge; if it's a nunc, it's not fluens. There are infinitely many temporal limits, but if there is a leading edge in time, there is only one. And so it will not do to say-if that is what this claim is out to say-that the leading edge of temporal passage is a now. Even if I have succeeded in showing how this first argument makes contact with the view that the now which appears to divide the past and the future is always one and the same, it may be objected that it attacks it on a technicality, and a technicality which it is particularly easy to stumble over in a language which uses only one word for 'instant' and 'now'. The second argument, I think, remedies this flaw by attacking the view in a way that owes nothing to technicalities. In addition, if to be simultaneous and neither earlier nor later with respect to time is to be at one and the same instant, then if both the earlier and the later were to be at this now [we are discussing], what happened ten thousand years ago would be simultaneous with what is happening to-day, and nothing would be either earlier or later than anything else (2 18a25-30).

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The first argument charged this view with harbouring the absurd notion of a unique temporal limit, but "this now [we are discussing]" may not, perhaps, be fairly and completely described as a temporal limit. This second argument, however, levels a charge which the view cannot evade-namely, that it introduces an enduring now (appropriate only to a timeless mode of existence, eternity). If what we are discussing is always one and the same, and if it is truly now, and if simultaneity is being at one and the same now, then every event is simultaneous with every other event. Although Aristotle introduces this absurdity in terms of temporal dynamics-"ten thousand years ago" and "to-day"-he does not stop with deducing an absurdity in temporal dynamicse.g., and so nothing would be either past or future-but presses on to the fundamental absurdity in temporal statics, the denial of the temporal order-"nothing would be either earlier or later than anything else". The unquestioned basis of both arguments in the development of the fourth puzzle is temporal statics. For any two events, either the one is earlier than the other or the two are simultaneous; not all events are simultaneous; time is infinite and continuous, but it may be divided into finite periods, the limits of which are instants. These propositions of temporal statics are not only relied upon and unchallenged in the discussion of the four puzzles, they also constitute the description of the essence of time as viewed in Aristotle's own theory, as I hope to show another time.m APPENDIX (Physics IV, lo; 217b29-218a33) Next after what has been said comes the discussion b29 of time. Concerning time we would do well, making use of the popular discussions also, to go over the puzzles regarding its existence or nonexistence and then its nature. That time is either altogether nonexistent, or that it b32 exists, but hardly, or obscurely, might be suspected from the following considerations. One part of it has come to be but is not, the other b33 part will be but is not yet; and it is of these two parts that infinite time, or any time one might take, is com-


a3

a8

al

a4

a16

a19

a2 1

a25

I1

posed. But that which is composed of nonexistents might be thought to be incapable of sharing in existence. In addition, if any thing with parts is to exist, then, when it exists, all or some of its parts must exist. But, although time is divisible, some parts of it have been and the others will be, and no part of it exists. And as for the now, it is no part of time; for a part measures, and the whole must be composed of the parts; but it is not thought that time is composed of instants. Again, it is not easy to see whether the now which appears to divide the past and the future alwaysremains one and the same or is always distinct. If it is always distinct, while no two parts of time exist simultaneously unless one part contains while the other is contained, as the smaller period of time is contained by the greater, and if/ /(W-A) an instant which does not exist but existed earlier must at some time have been destroyed/ that which does not exist now but existed ]/(C-R) earlier must at some time have been destroyed/, t en two instants cannot exist simultaneously, but always the earlier instant must have been destroyed. Now the earlier instant cannot have been destroyed at that very instant itself, because it existed then. And it cannot have been destroyed at some later instant. For let us lay it down that instants, like points on a line, cannot be consecutive to each other. Then if indeed it was not destroyed at the succeeding instant but at another, it would have existed simultaneously with the infinitely many instants between itself and that other, and that is impossible. Moreover, neither is it possible for it to remain always the same; for no finite and divisible thing, whether continuous in one or in more than one dimension, has just one limit, but an instant is a limit, and it is possible to cut off a finite time. In addition, if to be simultaneous and neither earlier nor later with respect to time is to be at one and the same instant, then if both the earlier

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and the later were to be at this now [we are discussing], what happened ten thousand years ago would be simultaneous with what has happened to-day, and nothing would be earlier or later than anything else. Let this, then, be the discussion of puzzles faced in a3o connexion with what belongs to time. But what time is and what its nature is is as unclear from the accounts handed down to us as it is from the puzzles just discussed. NOTES 1 Both Hesychius' and Ptolemy's ancient lists of Aristotle's works contain references to a treatise on time, which Ptolemy describes as consisting of one book. Ross believes that those references "probably refer to the essay on time in Phys. iv. 10-14, which may well have been originally a separate treatise" (Aristotle's Physics, p. 6). And the discussion of time begins and ends with formulae of the sort Aristotle uses to open and close treatises. (A translation of the puzzles passage appears as an appendix to this article.) 2 Ross discusses this phrase in his commentary (pp. 595-596) and reviews interpretations of Aristotle's uses of the phrase in his commentary on Metaphysics lo76a28 (pp. 408-410). The logoi might be either claims or arguments, and the designation 'exoterikoi' might mean either of the sort encountered outside the Lyceum or published in one of Aristotle's (lost) dialogues. Ross says of this particular occurrence of the phrase that it seems not that "discussions rather than any special books are meant-discussions peculiar to the Peripatetic school; but in many cases Aristotle had in point of fact developed these in his dialogues. The logoi here referred to are those that are put forward in b33-218a3o" (A's Phys., pp. 595-596). If the logoi are indeed the arguments used to work up the four puzzles, I think they must at least have been "developed" by Aristotle. Since the puzzles constitute difficulties for the view of time held by Plato (among many others), the arguments which support them are very unlikely to have come out of the Academy, and there seems to be no other plausible "exoteric" source for arguments with the aim and level of sophistication these have. Because of these considerations and because the view of time for which the puzzles constitute difficulties is not only Platonic but also intuitive, I am inclined to think that the exoteric logoi are familiar claims rather than arguments with their source outside the Lyceum. 3 See, e.g., G. E. L. Owen in his "Aristotle on Time" (see n. 27 inf.): "Aristotle evidently thought his paradoxes could be evaded . . ." (p. 8); "There is a question here: given that the paradoxes are not systematically and directly answered in the sequel, how and when were they prefaced to the argument? . . ." (n. 27). As I shall try to show, however, the puzzles do not cut against Aristotle's own position but rather prepare the ground for it. The fact that he proceeds without considering them further indicates only that he takes them to have served their purpose. There is no more reason to look for Aristotle's answers to these puzzles than there is to look for Zeno's solutions to his paradoxes.


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4 Richard Sorabji has suggested to me that the phrase 'ho aei lambanomenos chronos' might be read as 'the time that is ever being taken' (rather than as 'any time one might take'). On that reading the phrase appears to refer to refer to any stretch of "present time"--i.e., to any N-period. 5 For purposes of this discussion I am not questioning the legitimacy of describing the past simply as that which is no longer and the future simply as that which is not yet, or of describing the past and the future as therefore nonexistent or unreal. I am grateful to Carl Ginet for a helpful suggestion regarding this first objection. 6 "Aristotle on the Reality of Time", Archiv fiur Geschichte der Philosophie 56 (1974)) 132-155; p. 132. Miller's article is the best discussion I have seen of the topics considered in this article. Although our interpretations differ in many respects, I have learned a great deal from him. ?The description here is actually only "has come to be and is not" (gegone kai ouk estin / 217b34), but what has come to be and is not is no longer. 8 The most famous example of this stripping away is Augustine's in Confessions Book XI, chs. 19-2o, the passage which Miller has christened "the whittling argument" (op. cit., p. 133). - Greek uses one word -'nun'-adverbially in the sense of 'now' and substantivally in the sense of 'instant' or, more particularly, 'the present instant', 'the now'. In the puzzles passage it is frequently important and difficult to decide on the correct interpretation, which is rarely made perfectly unambiguous by the context. 10The force of the "it is . . . thought that" (dokei) cannot be to stigmatize this premiss as mere uninformed opinion; it presents an essential feature of Aristotle's own view of the temporal continuum. Perhaps it can be read instead as an indication that the premiss is beyond serious dispute. 11I am grateful to Richard Boyd for a helpful suggestion regarding these necessary characteristics. 12Miller, op. cit., p. 150; quoting Gale in the latter half of the passage. Such imagery goes back to Broad and McTaggart at least. 13Such descriptions are misleadingly specific because they involve spatially separated sentient beings. They may thus lead us to think of what are supposed to be merely distinct parts of time as distinct events, and there is no relevant difficulty in conceiving of two distinct but completely coincident events. 14What if we choose as the second period the overlapping three-hour period from ten until one to-day? Does Aristotle's description omit that sort of case from consideration? No; the only relevant second period in this case is the two-hour period from ten until noon to-day. The question of distinct and simultaneous parts of time does not arise in connexion with the third hour of the overlapping period, or with the three-hour overlapping period considered only as a whole. 15Taking time to be linear involves taking it to be one-tracked. In this connexion see 218b5, where Aristotle reduces a predecessor's opinion to an absurdity by showing that it allows the possibility that "a plurality of times would exist simultaneously". 16 If it had suited his purposes, Aristotle could have taken a very short way with this thesis. As he points out in the development of puzzle IIIb (218a25ff.), "to be simultaneous . . . is to be at one and the same instant"; and so a "third time" argument could have been generated here. Cf. 218b17: "time is not defined in terms of time".

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17 P. H. Wicksteed (Loeb edn., g929): "the 'now' that is not, but was, must have ceased to be at some time or other"; H. G. Apostle (Indiana U.P., 1969): "a moment which does not exist but existed before must have been destroyed sometime". 1a Cornford's reading appears in his editorial footnote to Wicksteed's translation: "anything that does not exist now, but formerly did exist, must have perished at some moment"; Ross's interpretation appears in his Analysis of this passage: "that which now is not and previously was must have ceased to be" (A's Phys., p. 384). 19As Wicksteed points out in a note to the text here, "This is proved in Book VI, chap. i" (231b6-18). a0See my "Incipit/Desinit" (see n. 28 inf.). Strictly speaking, instantaneous existence is a special case of "permanent" existence; it is therefore not always separately recognized by the logicians who discuss such topics. 2 See p. 98 above. 22 Instants in the temporal continuum, like points on a line, are successive, distinct, and nonconsecutive. But our concern here is not with instants generally but with "the now which appears to divide the past and the future", and if that now is to be analysed into successive distinct nows, each of which begins to exist, exists, and ceases to exist instantaneously, then there must be one such now after another. a Temporal dynamics and temporal statics are associated, respectively, with the relationships McTaggart designated "the A series" and "the B series"; I prefer to call them 'PNF' (for past-now-future) and 'ESL' (for earliersimultaneous-later). All events are permanently ordered in terms of ESL; all and only events within an N-period are transitorily ordered in terms of PNF. 24 I hope to try to substantiate these claims in a later paper. They seem to deserve at least a close look. As Miller points out, "It is a commonplace that Aristotle treats time as something inherently dynamic" (op. cit., p. 147); and in a note to that sentence he says, "For a contrasting view cf. W. Wieland [Die aristotelische Physik (G6ttingen 1962)], p. 327: "Vor allem versteht Aristoteles die Zeit nicht von einem 'flieszenden' Jetzpunkt aus, wie es ihm immer wieder unterstellt wird". 2 Plato is one Aristotle may have heard speak in that way; Donald Williams provides a colourful sampler of more recent remarks of the sort in "The Myth of Passage", The Journal of Philosophy 48(1951). 26 I want to supplement the specific acknowledgements made in earlier notes by expressing special thanks to Fred Miller, who read and carefully criticized the seminar notes from which this paper evolved, to Gail Fine, Terence Irwin, Malcolm Schofield, and Richard Sorabji, for their thoughtful criticisms of an earlier draft, and above all to Eleonore Stump, whose help was unstinting and invaluable. 27 In Machamer & Turnbull edd., Motion and Time, Space and Matter (Columbus: Ohio State U.P., 1976), pp. 3-27. 28 In Machamer & Turnbull, op. cit., pp. 101-136.

Sorabji Aristotle on Change

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