Commentary Aristotle Physics
Lecture 2
It is impossible for magnitude and motion to be composed of indivisible parts
The same reasoning applies equally to magnitude, to time, and to motion: either all of these are composed of indivisibles and are divisible into indivisibles, or none.
This may be made clear as follows. If a magnitude is composed of indivisibles, the motion over that magnitude must be composed of corresponding indivisible motions.
For instance, if the magnitude ABC is composed of the indivisibles A, B, and C, each corresponding part of the motion DEZ of O over ABC is indivisible.
Therefore, since where there is motion there must be something that is in motion and, where there is something in motion, there must be motion, therefore the being-moved will also be composed of indivisibles.
So O traversed A when its motion was D, B when its motion was E, and C similarly when its motion was Z.
Now, if a thing that is in motion must move from one place to another, it cannot, at the moment when it was in motion, both be in motion and, at the same time, have completed its motion at the place to which it was in motion.
For instance, if a man is walking to Thebes, he cannot be walking to Thebes and at the same time have completed his walk to Thebes.
And, as we saw, O traverses the partless section A in virtue of the presence of the motion D. Consequently, if O actually moved through A after being in motion through it, the motion must be divisible, for at the time when O was in motion, it neither was at rest nor had completed its passage, but was in an intermediate state.
But, if that which is walking will, when walking, have completed its walk, then it will be where it is walking and have completed its motion where it is in motion.
And if a thing is in motion over the whole ABC and its motion is the three D, E, and Z, and if it is not in motion at all over the partless section A but has completed its motion over it, then the motion will consist not of motions, but of instants.
And the motion will take place by a thing’s having completed a motion without having been in motion, for on this assumption, it has gone over A without going.
So something will have gone over, although not going over at any time: for it has gone over this without going over it.
Since, then, everything must be either at rest or in motion, and O is therefore at rest in each of the sections A, B, and C, it follows that a thing can be continuously at rest and at the same time in motion, for as we saw, O is in motion over the whole ABC and at rest in any part (and consequently in the whole) of it.
Moreover, if the indivisibles composing DEZ are motions, it would be possible for a thing, in spite of the presence in it of motion, to be not in motion, but at rest. But, if they are not motions, it would be possible for motion to be composed of something other than motions.
758. Because the arguments presented in the previous lecture clearly apply to lines and other continuous quantities having position, in which continuous contact is properly found, the Philosopher now wishes to show that the same reasoning applies to magnitudes and time and motion.
And it is divided into two parts:
first, he proposes his intention;
second, he proves his proposition, at “this may be made” (231b20; [759]).
He says therefore first (231b18) that any argument that shows that a magnitude is composed or not composed of indivisibles, and divided or not divided into indivisibles, applies also to time and motion, since whatever is granted in regard to any of them would necessarily be true of the others.
759. Then, at “this may be made” (231b20), he proves this proposition:
first, in regard to magnitude and motion;
second, in regard to time and magnitude, at “and, if length and motion” (232a18; [766]).
About the first, he does three things:
first, he presents his proposition;
second, he gives an example, at “for instance, if the magnitude” (231b22; [760]);
third, he proves his proposition, at “therefore, since where there is” (231b25; [761]).
The proposition is this. If a magnitude is composed of indivisibles, then the motion that traverses it will likewise be composed of indivisible motions equal in number to the indivisibles of which the magnitude is composed.
760. Of this, he gives the following example (231b22). Let the line ABC be composed of the three indivisibles A, B and C, and let O be an object in motion over the distance of the line ABC, so that DEZ is its motion. Now, if the parts of the distance or of the line are indivisibles, then the parts of the motion are indivisibles.
Then, at “therefore, since where there is” (231b25; [761]), he proves his proposition.
About this, he does three things:
first, he lays down some premises necessary for his proof;
second, he proves that, if a magnitude is composed of points, then the motion is composed not of motions, but of instants, at “and, as we saw” (232a1; [762]);
third, he shows that it is impossible for motion to be composed of instants, at “and the motion will take place” (232a9; [763]).
761. Therefore, he first lays down two presuppositions.
The first (231b25) is that, according to each part of the motion under consideration, something must be in motion, and conversely, if something is in motion, a motion must be in it. Now, if this is true, then the mobile O is being moved through A, which is part of the entire magnitude, by means of that part of the motion that is D, and through B (another part of the magnitude) by that part of the motion that is E, and through C (the third part of the magnitude) by that part of the motion that is Z. In other words, single parts of motion correspond to single parts of the magnitude.
He sets out the second presupposition at “now, if a thing that is” (231b28), and he says that what is being moved from one terminus to another is not, at the same time, both being moved and finished moving, anymore than a man going to Thebes is, at the time while he is going, already there.
He presupposes these two statements as per se evident. As to the statement that, when motion is present, something must be in the state of being moved, a like situation is apparent in all accidents and forms, for in order that something be white, it must have whiteness, and conversely, if whiteness exists, something is white.
As to the statement that being moved and having been moved are not simultaneous, we appeal to the very successive nature of motion, since it is impossible that any two elements of time coexist, as we explained in book 4. Hence, it is impossible that having been moved, which is the terminus of motion, be simultaneous with being in motion.
762. Then, when he says, and, as we saw (232a1), he uses these presuppositions to prove his proposition. For if it is true that, whenever a part of motion is present, something has to be in motion, and if it is in motion, there must be motion present, then, if the mobile O is in motion with respect to an indivisible part of the magnitude—namely, A—there is in O that part of the motion we called D. Accordingly, O is being moved through A and has completed its motion either at the same time or not at the same time.
If not at the same time but after being in motion, it follows that A is divisible, for while O was in motion, it neither was resting at A (with the rest preceding motion) nor had passed through the entire distance A, since it then would not still be in motion through A, as nothing is in motion through a distance it has already traversed. Consequently, it must be midway. Therefore, when it is in motion through A, it has already passed through part of A and is now in another part of A. Consequently, A is divisible, contrary to our supposition.
But if it is in motion through A and has completed its motion at the same time, it follows that it arrived while it was coming, and it will have completed its motion while it was being moved, which is against the second presupposition.
From this, it is clear that no motion is possible when the magnitude is indivisible, for there are only two choices: either things can be in motion at the same time that their motion is over, or the magnitude must be divisible.
Therefore, assuming that nothing can be in motion through the indivisible A, if someone should say that a mobile is in motion through the entire magnitude ABC and that the whole motion by which it is in motion is DEZ, and moreover that nothing can be in motion, but only in the state of completed motion through the indivisible A, it follows that the motion consists not of motions, but of instants.
Now, the reason we say that it follows that the motion is not composed of motions is that, since the part of the motion that is D corresponds to the part of the magnitude that is A, if D were a motion, the mobile should be in motion through A, for when motion is present, the mobile is being moved. But it was proved that the mobile is not in motion through A as indivisible, but in the state of having completed its motion when it had traversed this indivisible.
Consequently, what remains is that D is not a motion, but a moment. (The state of completed motion is called a “moment,” just as being moved is called “motion”; moreover, the moment is related to motion as point is related to line.) And the same holds for the other parts of the motion and of the magnitude. Consequently, it follows necessarily that, if a magnitude is composed of indivisibles, then a motion is composed of indivisibles, that is, of instants; and this is what he intended to show.
763. But, since it is not possible for a motion to be composed of instants any more than for a line be composed of points, at “and the motion will take place” (232a9), he exposes this impossibility by concluding to three absurdities.
The first of these is that, if motion is composed of instants and a magnitude of indivisibles in such a way that, through an indivisible part of a magnitude, things are not in motion, but in the state of completed motion, it will follow that something has completed a motion without having been in motion, because it was assumed that, in regard to the indivisible, something arrived without going, because it was not able to be in motion in that indivisible. Hence, it follows that something has finished a motion without previously being in motion. But this is no more possible than for an event to be past without having been present.
764. But, because a person who claimed that motion is composed of instants might grant this strange state of affairs, Aristotle gives another impossibility in the following argument, at “since, then, everything must be” (232a12). Anything capable of being in motion and at rest must be either in motion or at rest.
But, in our original example, while the mobile is in A, it is not being moved (and likewise when it is at B, and when it is at C); therefore, it must be at rest while at A and while at B and while at C. Therefore, it follows that a thing is at the same time continually at rest and continually in motion.
He now proves that this follows. We have agreed that it is in motion throughout the entire length ABC, and again that it is at rest in relation to each part. But what is at rest in relation to each and every part is at rest throughout the whole. Consequently, it is at rest throughout the entire length. Thus, it follows that, throughout the entire length, it is continually in motion and continually at rest—which is wholly impossible.
765. He gives the third absurdity at “moreover, if the indivisibles” (232a15). It has been shown that, if a magnitude is composed of indivisibles, so also the motion. Now, those indivisibles of motion—namely, D and E and Z—are such that each of them is either a motion or not. If each is a motion, then, since each of them corresponds to an indivisible part of the magnitude (in which something is not in motion, but in the state of completed motion), it will follow that a mobile is not in motion, but at rest, even though a motion exists—which is against the first presupposition. If each is not a motion, it follows that motion is composed of non-motions, which is no more possible than that a line be composed of non-lines.