Aristotle reports a paradox given by Zeno designed to cast doubt on movement by showing that an arrow supposedely in flight could never in fact be so, since it actually has to be at rest all the time. In analysing this paradox, we can ask whether Aristotle was accurately reporting Zeno’s original paradox and we can interpret Aristotle’s account of the paradox. We ought to give Aristotle the benefit of the doubt in assuming that he is faithfully reporting Zeno’s actual paradox, unless we have sufficient reasons to think otherwise. I do not think that such sufficient reasons exist. I interpret the paradox as aiming to show that the arrow must be at rest in every ‘now’ and hence it can never be in motion, on the grounds that in any ‘now’ the arrow must be in some one place and that the arrow is at rest when it is in some place. I believe that Aristotle has the philosophical means to successfully counter this paradox.
In Physics VI.9, Aristotle presents a paradox by which Zeno argued against motion fallaciously: “For he says that if everything is always at rest whenever it is [in the place] where it is equal to, and what is in movement is always like that in a now, then a flying arrow is unmoving” (239b1, 5-7). Vlastos suggests that Zeno’s original paradox is different from how Aristotle presents it. There is a nothing Zenonian paradox that has survived in a fragment (Epiphanius, Adv. Haey. 3, 11; H. Diels, Doxogyaphi Graeci, 590) but which does not appear to be reported by Aristotle. This paradox concludes that nothing moves because, if something were to move, it would have to move either in the place in which it is or in the place in which it is not, but it cannot do so in either. The idea behind this paradox seems to be that something cannot move where it is not because it is not there but cannot move where it is because it is already there. I think that this argument should be treated on its own terms as an independent paradox. Vlastos, on the other hand, argues that this is unlikely given that Aristotle does not record this independent paradox. If it were “a self-contained composition” then it would be a fifth paradox separate from the four Aristotle records in Physics VI.9. Vlastos, therefore, asks: “Why then did Aristotle ignore it? Because he thought it a silly puzzle, not worth solving, or too easily solved?” This, together with the claim that it is “a construction trenching so closely on the Arrow” lead him to conclude that it and the Arrow paradox reported by Aristotle really come from an original Arrow Paradox.[1] This original Arrow Paradox is supposedly different from the one Aristotle gives. Vlastos doesn’t think that Zeno gave any premise to do with ‘now’. However, I do not find the argument from the lack of a fifth paradox convincing. There are indeed other Zenonian paradoxes which are not in this precise discussion in Physics VI. For example, there is the paradox about like and unlike in Plato’s Parmenides (Parmenides 127d-e). Therefore, the two arguments need not be fused together.
The other means for doubting Aristotle’s fidelity to reporting the Arrow Paradox is the idea that the premise about ‘now’ seems to be “an Aristotelian plant.”[2] Vlastos argues that the way Aristotle treats νυν is anachronistic when applied to Zeno. But one need not assume that Zeno had anticipated all of Aristotle’s insights into the concept of ‘now’ in order to have some idea of ‘now.’ Everyone has some idea of what ‘now’ means. What we must take into account is that there are two basic uses or meanings of ‘now.’ It can mean the present interval of time (however long you make that interval), such as when I say “it’s raining now” or it can mean the precise present instant more. If I am allowed to speak improperly, we can consider that latter idea as the most nowish now or what is really really now. We are talking about what stands to a period of time as a point stands to a line. Any ‘now’ in this strict sense cannot be a period of time because this strict ‘now’ or strict present would be partly in the past and partly in the future if it were a stretch of time. In an extended interval of time, we can identify an earlier part which stands as past relative to the later part and a later part which stands as future relative to the earlier part. These two senses of ‘now’ it seems to me are exhaustive. When we talk about ‘now’ or the present, we don’t mean anything other than either an interval of time, the loose present or loose sense of now, or an indivisible instant, the strict present or strict sense of now. Therefore, it seems to me that Zeno could very well have talked about ‘now’ and if he did, he would have meant one of these meanings (or confused them together). Soon, I shall utilise Aristotelian principles to argue that whatever meaning we take, Zeno’s paradox will fail. Before that, I will give a brief interpretation of Aristotle’s account of the Arrow Paradox.
Granted that we have taken Aristotle’s account of Zeno’s Arrow Paradox to be faithful to the original, as I believe we ought to, we can now interpret this account. The first premise is a bit obscure: εἰ γὰρ αἰεί...ἠρεμεῖ πᾶν ὅταν ᾖ κατὰ τὸ ἴσον. Many translators have inserted ‘place’ or ‘space’[3] such that Aristotle is saying something like: “everything is always at rest when it is at a place equal to itself.”[4] Alternatively, Zeno wasn’t trying to emphasise that the arrow is in a place equal to itself but that it is in the same one place when it is resting, for however long it is resting.[5] However, whatever the case may be, I believe that the ultimate argument remains the same because the standard interpretation really gets to where this other interpretation gets to more immediately. The arrow is in a place equal to itself and, insofar as it is in fact in the place equal to itself, it is at rest. In short, if something is in a place, it is resting in that place. This does actually seem intuitive. If we consider an arrow in a particular place (call it p), we must surely admit that when the arrow is in p, it remains in p and that when the arrow is in p, it is not moving outside of p. If the arrow was already moving out of p, then part of the arrow would not be in p. But then, p would not be equal to the arrow. Therefore, p would not really be the place where the arrow is after all.
The point of the second premise is to argue that the flying arrow must always be in a given place equal to itself is in any ‘now.’ Again, this seems intuitive. No matter what precisely we mean by ‘now’, we do all have a general idea of what we mean by ‘now.’ Isn’t it uncontroversial to say that the arrow occupies a given place in a given now? Isn’t that intuitive? Surely, the arrow is not nowhere when it is in motion? So, surely it is somewhere. Somewhere seems to entail ‘some place.’
Let us see how the first premise can be compared to Aristotle’s ideas about place and motion. Place, for Aristotle, is an immobile surface. If something is undergoing locomotion, then it is moving from one place to another place. It would not be in any place in the strict sense because to be in a place in a strict sense means to rest in that place. However, it is important for Aristotle that what is in continuous motion from point A to point B does not stop to rest at any point between point A and point B. Any point in between a starting point and a terminus point is only potential, not actual (cf. Physics VIII.8). The points at each end of a line are actual points. If we divide a line into two, then the points which were potential in the middle are made actual and become points at either side of a line. Division gives us actual points. More division gives us more actual points. In a continuous line AC, a mid-point B would be merely potential unless we were to divide that line in half. In a continuous motion AC, a middle moment B would be merely potential unless the thing moving were to stop at B and start again. But then AC would no longer be a continuous motion. We would have two continuous motions: AB and BC. The moment B is then an actual ending moment and starting moment, not merely potential. Therefore, if a moving object like a flying arrow is in continuous motion from A to C, it cannot stop to rest at B. If it stopped to rest at B and then started again, then the motion would not be continuous. If there were no continuous motions, then there would be no motion. If what would be a continuous motion is interrupted in the middle, we have two actually divided motions. Both these motions are still continuous. But, however much we divide, we are still only going to get a definite number of actually divided motions and these will be continuous. We cannot have an infinite amount of actually divided motions. That is impossible (as Zeno would agree) and we could never get from any point to another. Therefore, continuous motion does exist and, in a continuous motion, insofar as it is continuous, there is no rest. If there is no rest, there is no rest in any place. If there is no rest in any place, the moving object is never really ‘in’ a place in the strict sense. It passes through potential place. What it passes through is potentially its place because it could potentially stop there. If ‘to be somewhere’ means ‘to rest in a place’, then a moving object is not ‘somewhere’. Rather, the moving object continuously passes through the potential ‘somewheres’. In any part of the motion, it does not have to rest in some place. Places or ‘somewheres’ are not parts of motion. Smaller motions are parts of motion. These smaller motions do in fact correspond to smaller magnitudes. The parts of motion stand to the whole motion as the parts of the magnitude stand to the whole magnitude traversed in the motion. If we consider a moving object it is always moving through a different part of the whole magnitude when it is moving. It does not rest in any part because not all of its parts are all in one part at any time. If all the parts of the moving object were in one part of the magnitude, then the whole moving object would be in that part of the magnitude. If that was the case, then it would rest there. But the moving object need not rest there if not all its parts are in that part of the magnitude. Because it is constantly moving, it always constantly has parts outside any given part of the magnitude. To conclude, Aristotle would not concede that an arrow rests in any place during its flight.
Nevertheless, he would say that the arrow rests when its flight has ended. The ‘now’ that refers to the point in time when the flight has ended is an actual terminus of that motion. From that ‘now’ onwards, the arrow rests. But any ‘now’ in between the beginning and end of the motion is only a potential ending or starting point. It would be an actual ending point if the arrow were caused to be stopped suddenly at that instant. Any of these intermediate nows could only be the starting points of rest if this happened. However, so long as the motion is continuous, those intermediate nows are not the starting points of rest. If they are not the starting points of rest, they are not periods of time in which rest happens either. Aristotle does not think that these nows are periods of time. Therefore, there is no rest at any now in between the start and end of the flight of the arrow.
As I said, the precise meanings of ‘now’ fall into either a given time interval thought of as present in relation to a past before it and the future after it (eg. ‘it is raining’ now is opposed to the past period of time when it was not raining and the future period of time when it will no longer be raining) or the most precise present, which is an indivisible instant analogous to a point in a line, separate completely from any past or any future (for, the period when it is presently raining encompasses part of the past and part of the future – this is not so with the indivisible now). If Zeno meant that a flying arrow must always be in a given place equal to itself is in any ‘now’, where ‘now’ is taken in the first sense (an interval of time), then Aristotle could refute this. As was seen, Aristotle does not hold that a moving body is ever wholly in one place in a continuous motion. Because it is never wholly in one place, it never rests in one place. Therefore, if ‘now’ refers to an extended interval of time, Zeno’s conclusion is not yielded because the arrow does not rest in that extended interval of time. But Aristotle takes Zeno to be meaning ‘now’ in the second sense (an indivisible instant). If ‘now’ refers to an indivisible instant, Zeno’s conclusion fails because the arrow does not rest at any now before the final terminus. The arrow would only rest in the nows before the final terminus, if those nows were the parts of the time in which the motion takes place. However, the nows are not the parts of time. The only parts of time are times. This is why Aristotle’s simple refutation of the Arrow Paradox is the insistence that time is not composed of indivisible nows. I think that Aristotle is correct about this. No amount of indivisible nows could yield an extent of time. One way to show this is to point out that what is indivisible cannot be contiguous with another indivisible. Since they are not extended, they cannot touch side by side. They have no parts that would enable them to do so. They would either be morphed into one or they would be joined by something divisible. But if they are joined by something divisible, then we have introduced that which does make up divisible things like time.
There are not convincing reasons to suggest that Aristotle misrepresented Zeno’s Arrow Paradox. Given the presentation of the paradox Aristotle provides, I think that his one sentence rebuttal is actually a legitimate rebuttal when seen in the light of Aristotle’s views on place, time, now, indivisibles, divisibles etc. as laid out in other parts of the Physics.
[1] Vlastos, Gregory, ‘A Note on Zeno’s Arrow’, in Furley/Allen (eds.), Studies in Presocratic Philosophy (vol. 2), Routledge 1975, 5
[2] Ibid 7
[3] Somewhat confusing because ‘place’ and ‘space’ really have different meanings and Aristotle would be meaning place rather than space. Place for Aristotle is the innermost immobile surface surrounding a body. This is different from space, which is internal rather than external.
[4] Vlastos 1
[5] κατὰ τὸ ἴσον would then not be referring to the fact that the place it equal to the thing in the place but that the place is the same place as itself. At any moment, the place which the arrow is in is that same one place and no other. I do not have the linguistic ability to verify if that is a viable interpretation of the Greek.
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