Plato's Parminedes Preface: Zeno II & Aristotle I

Sticking with the same two sources as yesterday, the Stanford article (by one John Palmer) and Aristotle's Physics 6, I'll now walk through how Aristotle treats Zeno's arguments. 

Note that the Stanford article doesn't seem to think Aristotle was fair to Zeno. He objects to Aristotle's "incomplete presentation," which doesn't offer any "indication of how these four arguments might have functioned within the kind of dialectical scheme indicated by Plato’s Parmenides." This is part of a general concern he raises about how these arguments are "reconstructed." A point I think is worth raising is that the "reconstruction" seems to have started immediately: 

Furthermore, Aristotle implies that people were reworking Zeno’s arguments soon after they were first propounded. In Physics 8.8, after giving a basic reconstruction of the so-called Stadium paradox (see below, sect. 2.2.1) recalling its presentation in Physics 6.9, Aristotle then notes that some propound the same argument in a different way; the alternative reconstruction he then describes (Arist. Ph. 8.8, 263a7–11) is in effect a new version of the original argument.

Now, plausibly the reason for this rapid "reconstruction" was the lack of reliable accounts of exactly what Zeno said, given the mostly oral and somewhat limited writing culture of ancient Greece. I reject this as likely, however; the best exploration of the oral culture of ancient Greece I know is Albert Lord's The Singer of Tales, which demonstrates inter alia that these oral approaches worked very well at preserving important details. They could widely alter stories in length, judging the importance of audience attention and interest, but even the abbreviated versions would be accurate to the heart of the story. 

Rather, I think it is likely that the original forms of Zeno's paradoxes were rapidly disposed of by the brilliant thinkers of Socrates' and Plato's generation. What most likely happened, and what I suspect Aristotle is noting, is that other thinkers were finding more plausible ways of arguing for the point that Zeno had made. "He who strives for the stars may stumble on a straw," and perhaps Zeno's striving at his highly original arguments missed a few things; but people who weren't satisfied with the easy out constructed sounder proofs of the same point. 

In any case, take it as read that we only have the one thing (from yesterday) that we think is what Zeno really said; but also that these arguments are interesting enough that even if you find a way to 'resolve' them you shouldn't set them aside. Maybe someone could find a way to resolve your resolution, too; maybe there's another approach that makes the argument better. It seems to me as if that was probably a big part of the program in what was one of the most interesting times and places for debate in human history. 

So, on to the first problem:

Aristotle begins this part of his Physics with a more basic approach to explaining how things function. He is going to need this furniture to reject some of Zeno's arguments, so it makes sense to lay it out. He begins the book with a discussion of the nature of a contiuum.

Now if the terms 'continuous', 'in contact', and 'in succession' are understood as defined above things being 'continuous' if their extremities are one, 'in contact' if their extremities are together, and 'in succession' if there is nothing of their own kind intermediate between them-nothing that is continuous can be composed 'of indivisibles': e.g. a line cannot be composed of points, the line being continuous and the point indivisible. For the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct).

The number line is a standard contemporary example of a continuum, but again it can be conceptually distracting because it is different from the physical objects under discussion. For example, a number line has not got extremities; it is infinitely extensive in both directions. For Aristotle, the open air might constitute a continuum; a stretch of ground might be thought of that way (as indeed he shall use it in a moment). The stretch begins here and finishes there, but we can talk and think about it as one thing that stretches for however long it does, rather than a bunch of pieces of ground next to one another.

Nevertheless, Aristotle is definitely doing the thing I'm trying to be careful not to do, which is mixing mathematical and physical concepts A line cannot be composed of points, and a line drawn across the ground is a continuum that is composed of ground, not of the points on the line drawn across it. 

So the first paradox is the paradox of motion. I won't block-quote the Stanford discussion of this paradox because it is easily linked, but it may be helpful to read it first because it's a good summary of the problem. Here is what Aristotle says about it.

Hence Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.

The passage over the infinite, then, cannot occupy a finite time, and the passage over the finite cannot occupy an infinite time: if the time is infinite the magnitude must be infinite also, and if the magnitude is infinite, so also is the time. This may be shown as follows. Let AB be a finite magnitude, and let us suppose that it is traversed in infinite time G, and let a finite period GD of the time be taken. Now in this period the thing in motion will pass over a certain segment of the magnitude: let BE be the segment that it has thus passed over. (This will be either an exact measure of AB or less or greater than an exact measure: it makes no difference which it is.) Then, since a magnitude equal to BE will always be passed over in an equal time, and BE measures the whole magnitude, the whole time occupied in passing over AB will be finite: for it will be divisible into periods equal in number to the segments into which the magnitude is divisible. Moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to ass over some magnitude, say BE, in a finite time, and if this BE measures the whole of which it is a part, and if an equal magnitude is passed over in an equal time, then it follows that the time like the magnitude is finite. That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given. The same reasoning will also show the falsity of the assumption that infinite length can be traversed in a finite time. It is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible.

This conclusion follows not only from the present argument but from the consideration that the opposite assumption implies the divisibility of the indivisible. For since the distinction of quicker and slower may apply to motions occupying any period of time and in an equal time the quicker passes over a greater length, it may happen that it will pass over a length twice, or one and a half times, as great as that passed over by the slower: for their respective velocities may stand to one another in this proportion. Suppose, then, that the quicker has in the same time been carried over a length one and a half times as great as that traversed by the slower, and that the respective magnitudes are divided, that of the quicker, the magnitude ABGD, into three indivisibles, and that of the slower into the two indivisibles EZ, ZH. Then the time may also be divided into three indivisibles, for an equal magnitude will be passed over in an equal time. Suppose then that it is thus divided into KL, Lm, MN. Again, since in the same time the slower has been carried over Ez, ZH, the time may also be similarly divided into two. Thus the indivisible will be divisible, and that which has no parts will be passed over not in an indivisible but in a greater time. It is evident, therefore, that nothing continuous is without parts.

The basic point that Aristotle is making here is that time and space are both divisible magnitudes, and that they are what I would call "geared together." That is, because motion in space also entails motion in time, you don't get a paradox of the sort Zeno is trying to set up. However long it takes to travel across the infinite divisions occupies enough of the equally infinitely divisible magnitude of time to allow for it. 

(Contemporary physics offers us "spacetime," which makes this point that time and space are geared together even more emphatically.)

The other point that Aristotle wants to clarify is that both of these 'infinitely divisible' magnitudes are not made up of indivisibles: "the line is not made up of points," and time is not made up of indivisible moments of 'now.' Properly a point doesn't belong to the same dimension as physical reality; it exists here only conceptually, as a one dimensional point on a two-dimensional line in what is actually three dimensional space (or four dimensional spacetime, perhaps). The error of assuming that the points are fundamental to the line drawn across the space is what gives rise to the error that Zeno is propounding. 

This turns out to be Aristotle's resolution of another of Zeno's paradoxes, which he disposes of very rapidly with the same furniture. 

Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.

That paradox is 2.2.3 in the Stanford piece, which treats it more seriously than Aristotle does. His account of how the argument works is that, at any given moment of time, the arrow must occupy a space exactly equal to its length. Yet this means the arrow is resting, because it neither extends into space it does not occupy in this moment, nor does it leave space it does not occupy. If it is resting at any random point of time, given that all points of time are the same, at every point it is resting; and thus it cannot move, because there is no extension at any point in time that we could call motion. 

A more natural way of saying this might be that a flying arrow, at a frozen moment in time, is motionless; and since every length of time is composed of an infinite number of frozen moments, the arrow cannot be flying at all. Motion is impossible because at each of the divisions (a 'point in time' rather than a physical point) has no ability to sustain motion because the points are not extended objects. 

Aristotle's rejection is a rejection of the whole frame, as above. There are no unextended points, not actually in our three dimensional world (or four, etc). Zeno is wrong not merely mathematically, but metaphysically: he is wrong about the nature of reality, which cannot actually be divided into indivisible points. Neither space nor time can be, so says Aristotle.

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