The rejection of the reconstruction of Zeno's argument in Physics 8 depends on so much of Aristotle's own furniture -- we are in the last book of the Physics, here, so he has laid out a complete vision of how movement works in nature -- that it would require more than a blog post to explain it. It might do as a beginning to say that Aristotle is wrong about this part. He begins by positing that there might be a kind of infinite motion if it were circular, by which he means the movement of the stars in heaven. We have no reason to believe that now.
Some of his proofs that rectilinear motion cannot be infinite end up applying to the 'circular' motion he intends to consider infinite. He admits this in one case: if two objects are moving in a line, one from A to B and the other from B to A, they will arrest each other. Two trains on the same track, headed in opposited directions, will crash into each other and stop. Yet this would be true of two trains on circular tracks, too, should they meet while headed in opposite directions.
He does not admit that the logic applies to another of his proofs, which is the proof about a thing turning back on its course. If you're moving from A to B, and halfway you decided you prefer A after all, you must come to a stop in the process of reversing your course. Circular motion does not do that because, after all, it's a circle: you can get back to A just by continuing on your course. Yet if you were to reverse course, you would in fact have to stop; Aristotle says you could simply 'turn back' at B without stopping, which implies something other than continuous motion in a circle.
In any case, this proof-by-standstill is important to his last rejection of Zeno. (Note that the points on the line Aristotle is using go A, B, G because "Gamma" is the third letter in Greek.)
We may start as follows: we have three points, starting-point, middle-point, and finishing-point, of which the middle-point in virtue of the relations in which it stands severally to the other two is both a starting-point and a finishing-point, and though numerically one is theoretically two. We have further the distinction between the potential and the actual. So in the straight line in question any one of the points lying between the two extremes is potentially a middle-point: but it is not actually so unless that which is in motion divides the line by coming to a stand at that point and beginning its motion again: thus the middle-point becomes both a starting-point and a goal, the starting-point of the latter part and the finishing-point of the first part of the motion. This is the case e.g. when A in the course of its locomotion comes to a stand at B and starts again towards G: but when its motion is continuous A cannot either have come to be or have ceased to be at the point B: it can only have been there at the moment of passing, its passage not being contained within any period of time except the whole of which the particular moment is a dividing-point.
Here Aristotle is using his potential/actual distinction in a curious way. The 'middle point' on the line is potentially but not actually a destination; we may discuss or think of it as the finishing of the first half of a continuous motion from A to G, but it isn't actually so. The proof is that the motion doesn't actually stop at B, the midpoint, and then resume. Rather, the motion from A to G, being continuous, just happens to pass over B.
If you recall that Zeno's Stadium argument was built on the need to pass an infinity of midpoints to complete a finite motion, you can probably see where Aristotle is going with this. The time it takes to cross B isn't an actual period of time spent passing B, just a potential division of the actual time it took to go from A to G. Thus, just as we can speak of B as 'numerically one but theoretically two,' i.e., both the ending of the first half of the motion to G and the beginning of the second half, we don't have to speak of B at all. It's not actually there; it's just conceptually so.
The way I said this the last time was that the points on the line don't exist in our three dimensional world, being one dimensional, in quite the same way that the three dimensional ground does. You may cross endless infinities of one dimensional objects in any simple motion through the third (fourth?) dimension.
So, in any case, Aristotle returns to Zeno.
The same method should also be adopted in replying to those who ask, in the terms of Zeno’s argument, whether we admit that before any distance can be traversed half the distance must be traversed, that these half-distances are infinite in number, and that it is impossible to traverse distances infinite in number-or some on the lines of this same argument put the questions in another form, and would have us grant that in the time during which a motion is in progress it should be possible to reckon a half-motion before the whole for every half-distance that we get, so that we have the result that when the whole distance is traversed we have reckoned an infinite number, which is admittedly impossible. Now when we first discussed the question of motion we put forward a solution of this difficulty turning on the fact that the period of time occupied in traversing the distance contains within itself an infinite number of units: there is no absurdity, we said, in supposing the traversing of infinite distances in infinite time, and the element of infinity is present in the time no less than in the distance.
You will recall that I described this as the infinities of time and space being 'geared together,' so that there was always enough time to cross the space because whatever divisions exist fit each other like the teeth of two gears (one space, and one time). There can be more divisions or fewer, but however many there are, there are exactly as many on both sides. Since motion in space is always motion in time, turning the one gear turns the other in the same way and to the same degree.
Aristotle is satisfied, but wants a theoretical answer and not just a practical one.
But, although this solution is adequate as a reply to the questioner (the question asked being whether it is possible in a finite time to traverse or reckon an infinite number of units), nevertheless as an account of the fact and explanation of its true nature it is inadequate. For suppose the distance to be left out of account and the question asked to be no longer whether it is possible in a finite time to traverse an infinite number of distances, and suppose that the inquiry is made to refer to the time taken by itself (for the time contains an infinite number of divisions): then this solution will no longer be adequate, and we must apply the truth that we enunciated in our recent discussion, stating it in the following way.
By abandoning distance and focusing on time alone, the 'gearing' solution is no longer viable. So what then?
In the act of dividing the continuous distance into two halves one point is treated as two, since we make it a starting-point and a finishing-point: and this same result is also produced by the act of reckoning halves as well as by the act of dividing into halves. But if divisions are made in this way, neither the distance nor the motion will be continuous: for motion if it is to be continuous must relate to what is continuous: and though what is continuous contains an infinite number of halves, they are not actual but potential halves. If the halves are made actual, we shall get not a continuous but an intermittent motion. In the case of reckoning the halves, it is clear that this result follows: for then one point must be reckoned as two: it will be the finishing-point of the one half and the starting-point of the other, if we reckon not the one continuous whole but the two halves. Therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible: if they are potential, it is possible.
It looks like Zeno wins a point here: Aristotle admits that it is impossible to travel through an actual set of infinities. Now Aristotle thinks that, since continuous motion can be observed to exist, this proves that the divisions aren't actual, but merely potential. Yet Aristotle himself has maintained, especially in the Physics, that potentiality is first actuality; for example, that lumber is potentially a house because it is actually the kind of thing that could become a house. To say that there are potential divisions is thus to say that there are, in a way, actual divisions. And if so, an apparently-observed continuous motion is impossible -- which is exactly what Zeno wanted to prove. Q.E.D., Aristotle.
We can rescue Aristotle here by suggesting that he means 'potential' in a different way here; perhaps one that is closer to the way he said 'theoretically' when he was talking about one point being spoken of as both a start and an end. Let's see how that move works with the rest of what he has to say.
For in the course of a continuous motion the traveller has traversed an infinite number of units in an accidental sense but not in an unqualified sense: for though it is an accidental characteristic of the distance to be an infinite number of half-distances, this is not its real and essential character.
Aristotle is now attempting to discuss this in terms of another distinction that was important to his physics and metaphysics, which is the distinction between essence and accidents. This doesn't look like it works to me for the same reason that the potential/actual distinction does: accidents are indeed accidental, in the sense that they happen-to-be but might-be-otherwise, but they nevertheless are are. Indeed, they are actual: a blue table might be blue accidentally, but it is actually blue. Stating that the divisions are accidental but not essential will not save Aristotle from Zeno.
It is also plain that unless we hold that the point of time that divides earlier from later always belongs only to the later so far as the thing is concerned, we shall be involved in the consequence that the same thing is at the same moment existent and not existent, and that a thing is not existent at the moment when it has become. It is true that the point is common to both times, the earlier as well as the later, and that, while numerically one and the same, it is theoretically not so, being the finishing-point of the one and the starting-point of the other: but so far as the thing is concerned it belongs to the later stage of what happens to it.
This argument is closer to the 'theoretical, not real' move. If we make all these theoretical divisions, Aristotle says, we fall into logical contradictions. For example, if every moment that is numerically one is treated as 'really' two, both a start and an end, then the moment at which a thing finishes coming to be is also a moment at which it isn't, quite yet.
But this is no answer to Zeno! His whole point was that our account of motion (including any sort of coming-to-be, which can be discussed as a kind of motion) leads to logical contradictions. Aristotle argues that the fact that the contradictions pop up is a reason to dismiss the idea that these divisions as fully real; Zeno's point is that the contradictions come up whenever we try to discuss the ways motion could occur. They're only differing over whether to dismiss the motion as a consequence of the logical contradictions, or to dismiss the reality of our theoretical framework.
This does not get better in the longer explication of it, in which Aristotle briefly introduces 'time atoms' of the sort he rejected in the Physics 6 argument I treated the last time.
Let us suppose a time ABG and a thing D [i.e. "Delta"; and note that for some reason Gamma has to be between Alpha and Beta for this argument to work as a line --Grim], D being white in the time A and not-white in the time B. Then D is at the moment G white and not-white: for if we were right in saying that it is white during the whole time A, it is true to call it white at any moment of A, and not-white in B, and G is in both A and B. We must not allow, therefore, that it is white in the whole of A, but must say that it is so in all of it except the last moment G. G belongs already to the later period, and if in the whole of A not-white was in process of becoming and white of perishing, at G the process is complete. And so G is the first moment at which it is true to call the thing white or not white respectively. Otherwise a thing may be non-existent at the moment when it has become and existent at the moment when it has perished: or else it must be possible for a thing at the same time to be white and not white and in fact to be existent and non-existent. Further, if anything that exists after having been previously non-existent must become existent and does not exist when it is becoming, time cannot be divisible into time-atoms. For suppose that D was becoming white in the time A and that at another time B, a time-atom consecutive with the last atom of A, D has already become white and so is white at that moment: then, inasmuch as in the time A it was becoming white and so was not white and at the moment B it is white, there must have been a becoming between A and B and therefore also a time in which the becoming took place. On the other hand, those who deny atoms of time (as we do) are not affected by this argument: according to them D has become and so is white at the last point of the actual time in which it was becoming white: and this point has no other point consecutive with or in succession to it, whereas time-atoms are conceived as successive. Moreover it is clear that if D was becoming white in the whole time A, the time occupied by it in having become white in addition to having been in process of becoming white is no more than all that it occupied in the mere process of becoming white.
It turns out that Aristotle's final answer to Zeno is much weaker than his earlier one. Yes, the contradictions he discusses arise, and they arise whether or not time can be divided into indivisibles, i.e., time atoms. But that was Zeno's point all along.
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