Commentary Aristotle Physics
But, so long as continuity endures in a continuum, there is an infinity of intermediates not in act, but in potency, for if someone should make some middle actual, it will be due to division, as has been said, insofar as it is taken as the beginning of one and the end of the other. In that case, the continuum will not remain, but it will stop; that is, the intermediates that are now in act will not be infinite, but there will be a stop in them.
This shows up especially in the case of one who wishes to count the intermediates, because he will have to count one point as two, inasmuch as it is the end of one half and the beginning of the other. And this, I say, takes place when the whole continuum is not counted as one, but two halves are counted in it. For if the whole continuum is taken as one, it has already been stated that, then, an intermediate point will not be taken as an actual end and beginning, but potentially only.
With these facts in mind, the answer to be given to one who asks whether infinites in time or in a magnitude may be traversed is that, in one sense, it does happen and, in another, it does not happen. For when one has infinites in act, it is impossible that they be traversed, but when they are potentially infinite, they can be traversed.
And so, since the intermediates in a continuum are infinite only in potency, it does happen that infinites are traversed, because what is in continuous motion traverses per accidens what is infinite, namely, what is infinite in potency. But, per se, it has traversed a finite line that happens to have an infinitude of intermediates in potency. The line itself, however, in its substance and definition, is distinct from those infinite intermediates. For a line is not a composite of points, but points may be designated in a line insofar as it is divided.
1119. Then, at “it is also plain” (263b9), he resolves the third doubt.
About this, he does three things:
first, he mentions the doubt and its solution;
second, he explains each with an example, at “let us suppose” (263b15; [1120]);
third, he draws a corollary from the foregoing, at “further, if anything” (263b26; [1122]).
First, therefore (263b9), he states the doubt that is wont to arise with respect to generation and ceasing to be. For what is generated ceases not to be and begins to be. But the time assigned for the existence of a thing that is generated or has ceased to be must be different from the one assigned to its non-existence. For example, if fire is generated from air, then, in the whole time AB, there was not fire, but air, but in the entire time BC, there is fire. Since, therefore, this point of time, B, is common to both times, it seems that, in that common instant, the fire both exists and does not exist.
The Philosopher therefore solves this doubt, saying that it is plain that, unless someone holds that the point of time that divides a prior time from a later one always belongs only to the later—that, in that instant, the thing is in the state that it subsequently has—it follows that the same is simultaneously being and non-being and that, when something has been produced, it is non-being. For it is then produced when generation terminates, namely, in that now that divides the prior time and the later. If, therefore, in the entire prior time, it was non-being, in that now also, when it has already been generated, it is also non-being, since this now is the end of the prior time.
How these impossibilities do not follow he explains by adding that one and the same sign as to number (that is, the now) is common to both times, namely, to the prior and to the subsequent. But, although it be one as to subject, it is not one in account, but two, for it is the end of the prior time and the beginning of the subsequent. But, if we take the now as it is a thing, that is, if it be taken as it is one in reality, it always belongs with the subsequent passion.
Or, in other words, although the now is the end of the prior time and the beginning of the subsequent, and is thus common to both, insofar as it belongs to the thing—that is, insofar as it is compared to the thing that is being moved—it always belongs to the subsequent passion, because the thing being moved is in that instant being subject to the passion of the subsequent time.
1120. Having given the objection and its solution, he explains both with examples.
And first he clarifies the objection, at “let us suppose” (263b15).
He says, therefore, to let ACB be the time and D the thing that is being moved, such that, in time A, D is white and, in B, it is non-white. It seems therefore to follow that, in C, it is both white and non-white. How this follows he now explains. If it is white in the entire time A, then, at any time taken in A, it is white; and likewise, if it is non-white in the entire time B, it follows that, at any time taken in B, it is non-white. Since, therefore, C is taken in both—being both the end of the former and the beginning of the latter—it seems to follow that, in C, it is white and non-white.
1121. Second, at “we must not allow” (263b20), he illustrates the solution given above. And he says that we must not concede that it is white at any point of time in A, for the ultimate now, which is C, must be excepted, for it is already later, that is, it is the ultimate terminus of the change. For example, if the white was coming to be or ceasing to be in the entire time A, then in C, it is not ceasing to be or becoming white, but already become or ceased to be.
But what has already been made exists, and what has already ceased to be does not exist. Hence, it is clear that, in C, it is first true to say “this is white” if the generation of white has terminated there, or to say “this is not white” if the ceasing to be of white has terminated there. Or, if that is not stated, the above-mentioned inconsistencies follow: when something has been already generated, it is still non-existent, and when it has ceased to be, it is still a being. Or it also follows that something is at once white and non-white, and universally, being and non-being.
1122. Then, at “further, if anything” (263b26), he draws a certain corollary from the foregoing: time is not divided into indivisible times, for, should one suppose this, it would be impossible to solve the doubt previously mentioned.
He says therefore that it is necessary that whatever is first a non-being and later is a being comes to be at some time; and again, it is necessary that, when something is coming to be, it is not.
Now, if these two assumptions are true, it is impossible for time to be divided into times that are indivisible. For let a time be divided into indivisible times, and then let A be the first indivisible time and B the second and subsequent time.
Now, D, which was previously not white and later is white, was becoming white in time A, and at that time was not white. But one must suppose that it has been made white in some indivisible time that is next, that is, subsequent, to A, namely, in time B in which it is now white.
Now, if it was becoming white in A, it follows that it was not white in A; in B, however, it is white. Since, therefore, between non-existence and existence an instance of generation occurs (because nothing passes from non-existence to existence but by generation), it follows that an act of generation occurs between time A and time B.
Therefore, between A and B there will be an intermediate time in which it was becoming white (since, in time B, D is already generated).
And similarly, since it is becoming white in that intermediate indivisible time, it is not white, and hence, for the same reason, it will be necessary to posit still another intermediate time, and so on to infinity, because we cannot assume that it is becoming white and is white in the same period of time.
But the argument is not the same if one states that the times are not divided into indivisible times. For according to this, we will say that it is one and the same time in which it was coming to be and was produced. But it was coming to be, and was non-being, in the entire preceding time, and it was produced and a being in the final now of the time, which instant is not related to the preceding time as contiguous or subsequent, but as its terminus. But, if one assumes indivisible times, they are necessarily consecutive.
But it is plain according to the foregoing that, if we do not assume indivisible times, then, if something comes to be white in the entire time A, the time in which it was coming to be and was completely made is no greater than the time in which it was coming to be alone.
For it is coming to be in the entire time, but in the final terminus of that time, it was completely made. But time plus its terminus is not something greater than the time by itself, any more than a point adds any magnitude to a line. But, if indivisible times are assumed, then it is clear from the foregoing that there must be more time in coming to be and completely being than in coming to be alone.
Finally, in summary, he argues for his main intention, saying that the foregoing arguments and ones like them are the appropriate ones to convince us that a reflected motion is not continuous.