Commentary Aristotle Physics
Lecture 21
Preliminaries for proving that the first mover must be without parts and without magnitude
We have now to assert that the first mover must be without parts and without magnitude, beginning with the establishment of the premises on which this conclusion depends.
One of these premises is that nothing finite can cause motion during an infinite time. We have three things: the mover, the moved, and, third, that in which the motion takes place, namely the time.
These are either all infinite or all finite or partly—that is to say two of them or one of them —finite and partly infinite.
Let A be the motion, B the moved, and C the infinite time.
Now, let us suppose that D moves E, a part of B. Then the time occupied by this motion cannot be equal to C, for the greater the amount moved, the longer the time occupied.
It follows that the time Z is not infinite. Now we see that, by continuing to add to D, I shall use up A, and by continuing to add to E, I shall use up B, but I shall not use up the time by continually subtracting a corresponding amount from it, because it is infinite.
Consequently, the duration of the part of C that is occupied by all A in moving the whole of B will be finite.
Therefore, a finite thing cannot impart to anything an infinite motion. It is clear, then, that it is impossible for the finite to cause motion during an infinite time.
It has now to be shown that, in no case, is it possible for an infinite force to reside in a finite magnitude. This can be shown as follows.
We take it for granted that the greater force is always that which in less time than another does an equal amount of work when engaged in any activity, such as in heating or sweetening or throwing—in fact, in causing any kind of motion.
Then, that on which the forces act must be affected to some extent by our supposed finite magnitude possessing an infinite force as well as by anything else, in fact, to a greater extent than by anything else, since the infinite force is greater than any other.
But, then, there cannot be any time in which its action could take place. Suppose that A is the time occupied by the infinite power in the performance of an act of heating or pushing, and that AB is the time occupied by a finite power in the performance of the same act.
Then, by adding to the latter another finite power and continually increasing the magnitude of the power so added, I shall at some time or other reach a point at which the finite power has completed the motive act in the time A.
For by continual addition to a finite magnitude, I must arrive at a magnitude that exceeds any assigned limit, and in the same way, by continual subtraction, I must arrive at one that falls short of any assigned limit.
So, we get the result that the finite force will occupy the same amount of time in performing the motive act as does the infinite force. But this is impossible.
Therefore, nothing finite can possess an infinite force. So, it is also impossible for a finite force to reside in an infinite magnitude.
It is true that a greater force can reside in a lesser magnitude, but the superiority of any such greater force can be still greater if the magnitude in which it resides is greater.
Now let AB be an infinite magnitude. Then BC possesses a certain force that occupies a certain time, let us say the time EZ, in moving D. Now, if I take a magnitude twice as great as BC, the time occupied by this magnitude in moving D will be half of EZ (assuming this to be the proportion); so, we may call this time ZT.
That being so, by continually taking a greater magnitude in this way, I shall never arrive at the full AB, but I shall always be getting a lesser fraction of the time given. Therefore, the force must be infinite, since it exceeds any finite force. Moreover, the time occupied by the action of any finite force must also be finite, for if a given force moves something in a certain time, a greater force will do so in a lesser time, but still a definite time, in inverse proportion. But a force must always be infinite—just as a number or a magnitude is—if it exceeds all definite limits.
This point may also be proved in another way—by taking a finite magnitude in which there resides a force the same in kind as that which resides in the infinite magnitude, such that this force will be a measure of the finite force residing in the infinite magnitude.
It is plain, then, from the foregoing arguments, that it is impossible for an infinite force to reside in a finite magnitude or for a finite force to reside in an infinite magnitude.
1141. After showing of what sort the first motion is, the Philosopher here shows of what sort is the first mover.
And it is divided into two parts:
first, he mentions his intention;
second, he carries out his proposal, at “one of these premises” (266a12; [1142]).
He says first (266a10), then, that, since it was said above that the first mover is immobile, now we must assert that the first mover is indivisible and has no magnitude, as being wholly incorporeal. But, before we show this, certain things necessary for this proof must be settled in advance.
1142. Then, at one of these premises (266a12), he carries out his proposal:
first, he sets out things required for proving the main proposition;
second, he proves the main proposition, at “now that these points” (267b17; [1172]).
About the first, he does three things:
first, he shows that an infinite motion supposes an infinite power;
second, he shows that an infinite power cannot exist in a magnitude, at “it has now to be shown” (266a24; [1146]);
third, he shows that the first mover must be one that causes a continuous and undying motion, at “but, before proceeding” (266b27; [1160]).
He says first (266a12) that, among the things to be established before the main proposition, one is that it is impossible for anything of finite power to cause motion for an infinite time. This he now proves. There are three things in every motion: one of which is what is moved, another of which is the mover, and the third of which is the time in which the motion occurs. But all three must be infinite, or all three finite, or some finite and some infinite, that is, either two only or one only. Suppose, therefore, that A is the mover, B the mobile, and C the infinite time. Then let D, a part of A, move E, a part of B. Under these conditions, it could be concluded that D moves E in a time not equal to time C (in which A moved B), but in less time.
For it has been proved in book 6 that the entire mobile requires more time to pass a certain point than it takes for a part of it. Therefore, since the time C is infinite, it follows that the time in which D moves E will not be infinite, but finite. So, let that time be Z, such that, just as A moves B in the infinite time C, D moves E in the finite time Z. But, since D is part of A, then, if we add to D by subtracting from A, A will eventually be entirely taken away or used up, since it is finite, and every finite is used up by subtraction, if the same quantity is continually taken away, as said in book 3. And likewise, B will be used up, if continual subtractions are made from it and added to E, because B is also finite. But no matter how much is taken from the time C—even if the same amount is continually taken away—all of C will not be used up, because it is infinite.
From this, he concludes that the entire A moves the entire B in a finite time, which is part of C. And this does indeed follow from the premises, because additions are made to the time of the motion in the same ratio as they are made to the mobile and to the mover. Since, therefore, by subtracting from the entire mobile and mover and by adding to their parts, the whole mobile and the whole mover are at length used up, such that all that was in the whole is added to the part, it will follow that, by proportional additions being made to the time, there will result a finite time in which the whole mover will move the whole mobile. Thus, if the mover is finite and the mobile also finite, the time too must be finite.
According to this, therefore, it is not possible that anything be moved by a finite mover with an infinite motion according to an infinite time. And, so, what was first proposed is now plain: it does not happen that a finite mover should cause motion for an infinite time.
1143. But Avicenna raises a difficulty about this demonstration of Aristotle. For it seems not to be universal, since there exists a finite mover and mobile from which nothing can be subtracted or taken away, such as a heavenly body, which nevertheless was not excluded from Aristotle’s proof. Hence, it seems that the proof is either particular or it proceeds from a false assumption.
To this objection, Averroes answers in his commentary that although nothing can be subtracted from the heavenly body, yet the conditional is true that, if a part be taken away from the body, that part will move or be moved in less time than the whole body.
For there is nothing to prevent a conditional from being true, even if its antecedent be impossible, as is patent from this conditional: “If a man flies, he has wings.” But whatever takes away the truth of a true conditional is false, even though the antecedent of the conditional be false. Now, the truth of the above conditional cannot stand with the statement that the finite moves for an infinite time, as is evident through Aristotle’s deduction. Thus, from the truth of the foregoing conditional, Aristotle concludes that it is impossible for a finite thing to cause motion for an infinite time.
However, it may be said more briefly that, when Aristotle speaks in his demonstrations of removing or subtracting, it does not always have to be understood in the sense of destroying a thing’s continuity, which is impossible in a heavenly body; rather, subtraction can be understood in the sense of designating. For example, I can, without disturbing the continuity of a piece of wood, designate by touch or thought a certain point as though dividing the whole, and in this way, I can remove a part from the whole and say that there is less whiteness in that part than in the whole. In like manner, it can be said that there is less power to move in a part of a heavenly body—a part removed by designating it—than in the whole.