Commentary Aristotle Physics
Lecture 4. No continuum is indivisible
And, if either is infinite, so is the other, and the one is so in the same way as the other—that is, if time is infinite in respect of its extremities, length is also infinite in respect of its extremities; if time is infinite in respect of divisibility, length is also infinite in respect of divisibility; and if time is infinite in both respects, magnitude is also infinite in both respects.
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 time, but an infinite one, and the contact with the infinites is made by means of moments not finite in number, but infinite.
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 pass 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 from what has been said, then, that neither a line nor a plane, nor in fact anything continuous, can be atomic. This conclusion follows not only from the present argument but also 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 in any period of time, but in an equal time, the quicker passes over a greater length, it may happen that it will pass over a length twice as great, or in the ratio of three to two, since their respective velocities may be 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 ABCD, into three indivisibles, and that of the slower into the two indivisibles: EZ and 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, and MN.
Again, since the slower has been carried over EZ and ZH in the same time, 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 time, but in a greater one. It is evident, therefore, that nothing continuous is without parts.
777. After showing that magnitude and time are subject to similar divisions, the Philosopher now shows that, if either is finite or infinite, so is the other.
About this, he does three things:
first, he states the proposition;
second, from this, he settles a doubt, at “hence Zeno’s argument” (233a21; [779]);
third, he proves the proposition, at “the passage over” (233a31; [780]).
778. He says first (233a17) that, if either of these two (time and magnitude) is infinite, so is the other; likewise, both will be infinite in the same manner.
He explains this by distinguishing two ways of being infinite, saying that, if time is infinite in respect of its extremities, then magnitude, too, is infinite in that way. Now, time and magnitude are said to be infinite in their extremities, because they lack extremities. It is as though we imagined that a line is not terminated at any points, or that time is not terminated at a first or final instant.
Moreover, if time is infinite through division, so also is a length. And this is the second way in which something is infinite. But something is said to be infinite through division because it can be divided to infinity; which, of course, pertains to the definition of a continuum, as was said. Consequently, if time is infinite in both ways, so, too, is length.
It is fitting that these two ways of being infinite be set in contrast:
For the first way is taken from the viewpoint of indivisible extremities that are absent;
the second is taken from the viewpoint of the indivisibles that are intermediate, for a line is divided according to points within the line.
779. Then, at “hence, Zeno’s argument” (233a21), he uses these facts to refute Zeno, who tried to prove that nothing is moved from one place to another—for example, from A to B.
For it is clear that there is an infinitude of intermediate points between A and B, since a continuum is divisible to infinity. Therefore, if something were to be moved from A to B, it would have to bridge the infinite and touch each of the infinites, and this cannot be done in finite time. Therefore, nothing can be moved through even the smallest distance during a period of finite time, however great.
The Philosopher, therefore, says that this argument is based on a false opinion, for length and time and any magnitude are said to be infinite in two ways, as we have said: according to division and according to their extremities. Accordingly, if there were things (a mobile and a distance) infinite in regard to quantity, which is to be infinite at the extremities, they could not touch one another in finite time. But, if they are infinite in respect of division, they will touch, because time also, which is finite in respect of quantity, is infinite in respect of division.
Hence two things follow: the infinite can be traversed not in finite time, but infinite time; and the infinite points of a magnitude are traversed in the infinite “nows” of time, but not in the finite “nows.”
But it should be noted that this solution is ad hominem and not ad veritatem, as Aristotle will explain in book 8.
780. Then, at “the passage over” (233a31), he proves what he stated above as a proposition.
First, he restates the proposition;
second, he proves it, at “this may be shown” (233a34).
He says first (233a31) that no mobile can traverse an infinite distance over finite time, nor a finite distance over infinite time; rather, if the time is infinite, then the magnitude must be infinite, and vice versa.
Then, at “this may be shown” (233a34), he proves the proposition:
first, that the time cannot be infinite if the magnitude is finite;
second, that, if the length is infinite, the time cannot be finite, at “the same reasoning” (233b14; [784]).
781. He proves the first part of the proposition with two arguments.
The first (233a34) is this. Let AB be a finite magnitude, and let G be an infinite time. Take GD as a finite part of this infinite time. Now, since the mobile traverses the entire magnitude AB in the entire time G, then, in the part of this time that is GD, it will traverse some part, BE, of the magnitude. But, since the magnitude AB is finite and greater than BE, which is finite and less, then either BE is an exact measure of AB or it will be less or greater.
These are the only relationships that a lesser finite quantity can bear to a greater finite quantity, as is evident in numbers. For three, which is less than six, measures it twice, but three taken twice does not measure five, which is greater than three, but exceeds it; nor does it measure seven, but is less than seven. But, if three were taken thrice, that product would exceed even seven.
Now, it makes no difference in which of these three ways BE is related to AB, for the same mobile will always traverse a magnitude equal to BE in a time equal to GD. But BE is either an exact measure of AB or exceeds it, if taken a sufficient number of times. Therefore, GD should also exactly measure the entire time G or exceed it, if GD is repeated frequently enough. Consequently, the whole time G (in which the entire finite magnitude was traversed) must be finite, since there was a corresponding segment of time for every segment of magnitude.
782. The second reason is given at “moreover, if it is” (233b7): although it be granted that a mobile traverse the finite magnitude AB in infinite time, it cannot be granted that it will traverse any magnitude at random in infinite time, because we see finite magnitudes being traversed in finite times.
So, let BE be the finite magnitude that is traversed in a finite time. But BE, since it is finite, will measure AB, which is also finite. Now, the same mobile will traverse a magnitude equal to BE in a finite time equal to that in which it traversed BE. Thus, the number of magnitudes equal to BE that will form AB corresponds to the number of equal times required to form the entire time consumed. Hence the entire time was finite.
783. This second reason is different from the first: in the first, BE was taken to be part of the magnitude AB, but here it is taken as a separate magnitude.
Then, he shows the necessity of this second reason when he adds that infinite time (233b11). For someone could cavil by saying that, just as the whole magnitude AB is traversed in infinite time, so would every part of it be, and thus the part BE would not be traversed in finite time. But, because it cannot be granted that any magnitude at random is traversed in infinite time, it was necessary to present the second reason, in which BE is a different magnitude that is traversed in finite time.
And he adds that it is clear that the mobile body does not cross the magnitude BE in an infinite time if the time be taken as limited in one direction—that is, if we take some magnitude other than the first one. This other magnitude is called BE and is crossed in a finite time.
For if the body crosses part of the magnitude in less time than the whole, then this magnitude BE must be finite, the limit in one direction being given, that is, AB.
It is as if he said that, if the time in which BE is traversed is finite and less than the infinite time in which AB is traversed, then BE is necessarily less than AB and must be finite, since AB is finite.
784. Then, at “the same reasoning” (233b14), he posits that the same proof leads to an impossibility if the length is said to be infinite and the time finite, because a part of the infinite length will be taken as finite, just as a finite part of infinite time was taken.
785. Then, at “it is evident” (233b15), he proves that no continuum is indivisible.
First, he says that an inconsistency would otherwise follow;
second, he gives the demonstrations that lead to that inconsistency, at “for since the distinction” (233b19; [786]).
He says, therefore (233b15), that it is clear from what has been said that no line or plane, that is, surface, or any continuum is atomic, that is, indivisible:
first of all, on account of the foregoing, namely, that it is impossible for any continuum to be composed of indivisibles, although a continuum can be composed of continua;
second, because it would follow that an indivisible can be divided.
786. Then, at “for since the distinction” (233b19), he gives the proof that leads to this inconsistency. In this proof, he makes use of certain facts already established.
One of these is that, in any finite time, the faster and the slower can be in motion.
The second is that the faster will traverse more distance in equal time.
The third is that there can be excess of speed over speed and of length traversed over length traversed, according to varying proportions, such as according to a double proportion, which is the proportion of two to one; and according to one and a half times (which, by another name, is also called a “sesquialter” proportion), where the one is to the other as a whole plus a half, as the proportion of three to two; or according to any other proportion.
With these presuppositions, he proceeds thus. Let this be the ratio of the faster to the slower, that the one is faster in the ratio of three to two; and let the faster traverse one magnitude ABCD composed of three indivisible magnitudes AB, BC, and CD. During the same time according to the given ratio, the slower will traverse a magnitude of two indivisible magnitudes, which form the magnitude EZI. And, because the time is divided as the magnitude, the time in which the faster traverses the three indivisible magnitudes must be divided into three indivisibles, because the equal magnitude must be traversed in equal time. So, let the time KLMN be divided into three indivisibles. But, because during that time the slower traverses EZI, which are two indivisible magnitudes, the time can be divided into two halves.
Consequently, it follows that an indivisible has been divided. For the slower had to traverse one indivisible magnitude in one and a half indivisibles of time, since it cannot be said that it traverses one indivisible magnitude in one indivisible time, for then the faster would not have been moved ahead of the slower. Therefore, what remains is that the slower traverses an indivisible magnitude in more than one indivisible and less than two indivisibles of time. Thus the indivisible time will have had to be divided.
In like manner, it follows that an indivisible magnitude is divided if the slower manages to move through three indivisible magnitudes in three indivisible times. For the faster will in one indivisible time be moved through more than one indivisible magnitude and less than two.
Therefore, it is clear that no continuum can be indivisible.