Zeno
1.
Alexander says that the second argument, from the dichotomy, is Zeno's and that he claims that if what exists has magnitude and is divided, then it will be many and no longer one, thus proving that the one does not exist... Alexander seems to have taken his opinion that Zeno does away with the one from Eudemus's writings. For in his Physics Eudemus says:
Then does this not exist although some one thing does exist? That was the puzzle. They report that Zeno said that he would be able to talk about what exists if only someone would explain to him what on earth the one was. He was puzzled, it seems, because each perceptible item is called many things both by way of predication and by being divisible into parts, whereas points are nothing at all (for he thought that what neither increases when added nor decreases when subtracted was not an existent thing).
Now it is indeed likely that Zeno argued on both sides, by way of intellectual exercise (that is why he is called 'two-tongued') and that he actually published arguments of this sort to raise puzzles about the one. But in his treatise, which contains many arguments, he shows in each case that anyone who says that several things exist falls into inconsistencies.
There is one argument in which he shows that if several things exist they are both large and small - so large as to to be infinite in magnitude, so small as to have no magnitude at all. Here he shows that what has no magnitude, no mass, and no bulk, does not even exist. For, he says,
if it were added to anything else, it would not make it larger. For if it is of no magnitude but is added, [the other thing] cannot increase at all in magnitude. Thus what is added will therefore be nothing. And if when it is subtracted the other thing is no smaller - and will not increase when it is added again - then clearly what was added and subtracted was nothing. [29 B 2]
Zeno says this not to do away with the one but in order to show that the several things each possess a magnitude - a magnitude which is actually infinite by virtue of the fact that, because of infinite divisibility, there is always something in front of whatever is taken. And he shows this having first shown that they possess no magnitude from the fact that each of the several things is the same as itself and one. (Themistius actually says that Zeno's argument established that what exists is one from the fact that it is continuous and indivisible; 'for if it were divided,' he says, 'it would not strictly speaking be one beacuse of the infinite divisibility of bodies.' But Zeno seems rather to say that there do not exist several things.)
Porphyry holds that the argument from dichotomy belonged to Parmenides who attempted to show by it that what exists is one. He writes as follows:
Parmenides had another argument, the one based on dichotomy, which purports to show that what exists is one thing only and, moreover, partless and indivisible. For were it divisible, he says, let it have been cut in two - and then each of its parts in two. Since this goes on for ever, it is clear, he says, that either some final magnitudes will remain which are minimal and atomic and infinite in number, so that the whole thing will be constituted from infinitely many minima; or else it will disappear and be dissolved into nothing, and so be constituted from nothing. But these consequences are absurd. Therefore it will not be divided but will remain one. Again, since it is everywhere alike, if it is really divisible it will be divisible everywhere alike, and not dibisible in one place and not another. Then let it have been divided everywhere. It is clear, again, that nothing will remain but that it will disappear; and if it is constituted at all, it will again be constituted from nothing. For if anything remains, it will not yet have been divided everywhere. Thus from these considerations too it is evident, he says, that what exists will be indivisible and partless and one...
Porphyry is right here to refer to the argument from dichotomy as introducing the indivisible one by way of the absurdity consequent upon division; but it is worth asking whether te argument is really Parmenides; rather than Zeno's, as Alexander thinks. For nothing of the sort is stated in the Parmenidian writings, and most scholars ascribe the argument from dichotomy to Zeno - indeed it is mentioned as Zeno's in Aristotle's work On Motion [i.e. Physics 239b9]. And why say more when it is actually found in Zeno's own treatise? For, howing that if several things exist the same things are finite and infinite, Zeno writes in the following words:
If several things exist, it is necessary for them to be as many as they are, and neither more nor fewer. But if they are as many as they are, they will be finite. If several things exist, the things that exist are infinite. For there are always others between the things that exist, and again others between them. And in this way the things that exist are infinite. [B 3]
And in this way he has proved infinity in quantity from the dichotomy. As for infinity in magnitude, he proved that earlier in the same argument. For having first proved that if what exists had no magnitude it would not even exist, he continues:
But if it exists, it is necessary for each thing to have some bulk and magnitude, and for one part of it to be at a distance from the other. And the same argument applies to the protriding part. For that too will have a magnitude, and a part of it will protrude. Now it is all one to say this once and to say it for ever. For it will have no last part of such a sort that there is no longer one part in front of another. IN this way if there exist several things it is necessary for them to be both small and large - so small as not to have a magnitude, so large as to be infinite. [B 1]
Perhaps, then, the argument from dichotomy is Zeno's, as Alexander holds, but he is not doing away with the one but rather with the many (by showing that those who hypothesize them are committed to inconsistencies) and is thus confirming Parmenides' argument that what exists is one.
(Simplicius, Commentary on the Physics 138.3-6,
138.29-140.6, 140.18-141.11)
2.
Zeno argues fallaciously. For if, he says, everything is always at rest when it is in a space equal to itself, and if what is travelling is always in such a space at any instant, then the travelling arrow is motionless. That is false; for time is not composed of indivisible instants - nor is any other magnitude.
Zeno's arguments about motion which provide trouble for those who try to resolve them are four in number.
The first maintains that nothing moves because what is travelling must first reach the half-way point before it reaches the end. We have discussed this earlier.
The second is the so-called Achilles. This maintains that the slower thing will never be caught when running by the fastest. For the pursuer must first reach the point from which the pursued set out, so that the slower must always be ahead of it. This is the same argument as the dichotomy, but it differs in that the additional magnitudes are not divided in half. Now it follows from the argument that the slower is not caught, and the same error is committed as in the dichotomy (in both arguments it follows that you do not reach the end if the magnitude is divided in a certain way - but here there is the additional point that not even the fastest runner in fiction will reach his goal when he pursues the slowest); hence the solution must also be the same. And it is false to claim that the one ahead is not caught: it is not caught while it is ahead, but nonetheless it is caught (provided you grant that they can cover a finite distance).
Those, then, are two of the arguments. The third is the one we have just stated, to the effect that the travelling arrow stands still. It depends on the assumption that time is composed of instants; for it that is not granted the inference will not go through.
The fourth is the argument about the bodies moving in the stadium from opposite directions, an equal number past an equal number; the one group starts from the end of the stadium, the other from the middle; and they move at equal speed. He thinks it follows that half the time is equal to its double. The fallacy consists in claiming that equal magnitudes moving at equal speeds, the one past a moving object and the other past a stationary object, travel for an equal length of time. But this is false.
For example, let the stationary equal bodies be AA; let BB be those beginning from the middle, equal in number and in magnitude to them; and let CC be those beginning from the end, equal in number and in magnitude to them and equal in speed to the Bs. It follows that, as they move past one another, the first B and the first C are at the end at the same time. And it follows that the C has travelled past all of them but the B past half of them. Hence the time is half for each of the two is alongside each for an equal time. At the same time it follows that the first B has travelled past all the Cs; for the first C and the first B will be at opposite ends at the same time (being, as he says, alongside each of the Bs for a time equal to that for which it is alongside each of the As) - because both are alongside the As for an equal time. That is the argument, and it rests upon the falsity we have mentioned.
(Aristotle, Physics 239b5-240a18)
3.
Zeno's argument assumes that it is impossible to traverse an infinite number of things, or to touch an infinite number of things individually, in a finite time. But this is false. For both lengths and times - and indeed all continua - are said to be infinite in two ways: either by division or in respect of their extremities. Now it is not possible to touch a quantitatively infinite number of things in a finite time, but it is possible so to touch things infinite by division. For time itself is infinite in this way. Hence it follows that what is infinite is traversed in an infinite and not in a finite time, and that the infinite things are touched at infinitely not at finitely many instants.
Aristotle, Physics 233a21-31
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