Lecture 17 — Certain difficulties are answered

So, this is how we must meet the difficulty that then arises, which is as follows. Suppose that the line E is equal to the line Z, that A proceeds in continuous locomotion from the extreme point of E to C, and that, at the moment when A is approaching B, D is receding in uniform locomotion and with the same velocity as A from the extremity of Z to I.

Then, says the argument, D will have reached I before A has reached C, for that which makes an earlier start and departure must make an earlier arrival.

The reason for the late arrival of A, then, is that it has not simultaneously come to be and ceased to be at B; otherwise, it will not arrive later. For this to happen, it will be necessary that it should come to rest there.

Therefore, we must not hold that there was a moment when A arrived to be at B and that D was in motion from the extremity of Z at the same moment.

For the fact that A was at B will involve the fact of its also having departed from there, and the two events will not be simultaneous, whereas the truth is that A is at B in a division of time and does not occupy time there.

In this case, therefore, where the motion of a thing is continuous, it is impossible to use this form of expression.

On the other hand, in the case of a thing that turns back in its course, we must do so. For suppose that, in the course of its locomotion, I proceeds to D and then turns back and proceeds downward again; then the extreme point D has served as finishing point and as starting point for it, one point thus serving as two.

Therefore, I must have come to rest there, as it cannot have come to be at D and departed from D simultaneously, for in that case, it would simultaneously be there and not be there at the same moment.

And here we cannot apply the argument used to solve the difficulty stated above: we cannot argue that H is at D at a sectional point of time and has not come to be or ceased to be there. For here the goal that is reached is necessarily one that is actually existent, not potentially.

Now, the point in the middle is potential, but this one is actual, and regarded from below, it is a finishing point, while regarded from above, it is a starting point, such that it stands in these same two respective relations to the two motions.

Therefore, that which turns back in traversing a rectilinear course must, in so doing, come to rest. Consequently, there cannot be a continuous rectilinear motion that is eternal.

The same method should be adopted in replying to those who ask, in the terms of Zeno’s argument, whether we admit that (1) before any distance can be traversed, half the distance must be traversed; (2) but these are infinite; and (3) it is impossible to traverse the infinite.

Some, on the lines of this same argument, put the questions in another form and would have us grant that, in the time during which a motion is in progress, it should be possible to reckon a half motion before the whole for every half distance that we get, such that we have the result that, when the whole distance is traversed, we have reckoned an infinite number, which is admittedly impossible.

Now, when we first discussed the question of motion, we put forward a solution of this difficulty turning on the fact that the period of time occupied in traversing the distance contains within itself an infinite number of units.

There is no absurdity, we said, in supposing the traversing of infinite distances in infinite time, and the element of infinity is present in the time no less than in the distance.

But, although this solution is adequate as a reply to the questioner (the question asked being whether it is possible in a finite time to traverse or reckon an infinite number of units), nevertheless as an account of the fact and explanation of its true nature, it is inadequate.

For suppose the distance to be left out of account and the question asked to be no longer whether it is possible in a finite time to traverse an infinite number of distances, and suppose that the inquiry is made to refer to the time taken by itself (for the time contains an infinite number of divisions): then this solution will no longer be adequate.

And we must then apply the truth that we enunciated in our recent discussion, stating it in the following way.

In the act of dividing the continuous distance into two halves, one point is treated as two, since we make it a starting point and a finishing point; this same result is also produced by the act of reckoning halves as well as by the act of dividing into halves.

But, if divisions are made in this way, neither the distance nor the motion will be continuous. For if it is to be continuous, motion must relate to what is continuous, and though what is continuous contains an infinite number of halves, they are not actual halves, but potential.

If the halves are made actual, we shall get not a continuous motion, but it will stop. In the case of reckoning the halves, it is clear that this result follows, for then one point must be reckoned as two: it will be the finishing point of the one half and the starting point of the other, if we reckon not the one continuous whole but the two halves.

Therefore, to the question whether it is possible to pass through an infinite number of units either of time or of distance, we must reply that, in one sense, it is and, in another, it is not. If the units are actual, it is not possible; if they are potential, it is possible.

For in the course of a continuous motion, the traveler has traversed an infinite number of units in an accidental sense, but not in an unqualified sense. Though it is an accidental characteristic of the distance to be an infinite number of half distances, this is not its real and essential character.

It is also plain that, unless we hold that the point of time that divides earlier from later always belongs only to the later so far as the thing is concerned, then the same thing is at the same time both existent and not existent, and a thing is not existent at the moment it has become.

It is true that the point is common to both times, the earlier as well as the later, and that, while numerically one and the same, it is theoretically not so, being the finishing point of the one and the starting point of the other. But, so far as the thing is concerned, it belongs to the later stage of what happens to it.

Let us suppose a time ACB and a thing D, the latter being white in the time A and not white in the time B. Then D is at the moment C white and not white, for if we were right in saying that it is white during the whole time A, it is true to call it white at any moment of A and not white in B, and C is in both A and B.

We must not allow, therefore, that it is white in the whole of A, but must say that it is so in all of it except now, C. C belongs already to the later period, and if, in the whole of A, not-white was in process of becoming and white of corruption, then the process is complete at C.

And, so, C is the first moment at which it is true to call the thing white or not white, respectively.

Otherwise, a thing may be non-existent when it has become and existent when it has perished; or else it must be possible for a thing at the same time to be white and not white, and in fact to be and to not be.

Further, if anything that exists after having been previously non-existent must become existent and does not exist when it is generation, time cannot be divisible into time atoms.

For suppose that D was generating white in the time A and that, at another time B, which is a individual moment of time concurrent with the last of A, D had already become white, and so is white at that moment.

Then, inasmuch as it was generating white in the time A, and so was not white, and it is white at the moment B, there must have been a generation between A and B, and therefore also a time in which the generation took place.

On the other hand, those who deny atoms of time (as we do) are not affected by this argument: D has become (and so is) white at the last point of the actual time in which it was generating white, and this point has no other point contiguous with or consecutive to it, whereas time atoms are conceived as successive.

Moreover, it is clear that, if D was generating white in the whole time A, then the time occupied by it in having become white in addition to having been in process of generating white is no more than all that it occupied in the mere process of generating white.

These and such like, then, are the arguments for our conclusion that derive cogency from the fact that they have a special bearing on the point at issue.

1112. After showing that a reflected motion is neither continuous nor one, the Philosopher now settles some doubts on the basis of what has gone before.

And it is divided into three parts, according to the three doubts he resolves from the foregoing:

the second part begins at the same method (263a4; [1115]);

the third begins at it is also plain (263b9; [1119]).

About the first, he does two things:

first, he sets forth the doubt;

second, he solves it, at therefore, we must not hold (262b15; [1114]).

1113. He therefore says first (262b8) that what was said in order to prove that a reflected motion is not continuous (Lect. 16, n. 1109.) may be applied to solving a certain doubt, which is this. Assume two equal magnitudes, one called E and the other called Z. Let A and D be two equally swift mobiles, such that A is continuously moved from the beginning of the magnitude E to C, and D along Z to I. And let us assume that, in the magnitude E, there is an intermediate point B, which is as far from C as a like point Z on Z is distant from I. Let us further assume that, at the same time that A, in its continuous motion, is approaching B, D is, in its continuous motion, receding from Z and going to I. Now, since these motions are regular and equally swift, it will follow that D will arrive at I before A arrives at C, because the one that starts first will first arrive to the end of an equal distance. But D left Z before A left B, because D left Z when A was arriving at B. Therefore, according to this, A did not simultaneously arrive at B and leave B, and it consequently follows that it departed after it arrived, for if it arrives and departs at the same time, it will not have begun to move later. And, so, it is necessary that A, while being carried along, rests in B. Therefore, a continuous motion will be composed of periods of rest, as Zeno claimed in book 6. (Lect. 11, n. 866.)

1114. Then, at therefore, we must not hold (262b15), he resolves this doubt in the light of the foregoing. For the objection supposed that A, in its continuous motion, arrives at a point, B, in the magnitude and that, at the same time that A arrived at B, D left the point Z—which is against what was had above. For it was said above (Lect. 16, n. 1109.) that, when something is being moved continuously, it neither arrived at nor departed from any intermediate point. Therefore, what the objection assumes must not be assumed, namely, that, when A was at (that is, approached) B, D was departing from Z. For if it be granted that A arrived at B, then, for the same reason, it should be granted that it left B, and that this did not occur simultaneously, but in two instants, such that it was at rest in the intermediate time between the two instants.

But, as was said previously, when something was being continuously moved, it was neither departing from nor approaching a given point, but was simply there—and this not for a time, because then it would have been resting, but in a division of time, that is, in some now, which divides time.

Therefore, what the objection assumed—namely, that A arrived at and that D left some intermediate point—is impossible to state in a continuous motion.

But, in a reflected motion, this must be stated. For if a mobile, I, is moved to the point D and is then rebounded, it is plain that the mobile uses the ultimate, which is D, as a beginning and as an end; that is, the point is used for two things. Hence, it had to be at rest there.

Nor can it be said that it simultaneously arrived at and left D, for then it would have been there and not have been there in the same instant. For whatever has been moved exists in the terminus to which it was being moved, and whatever begins to be moved is not in the terminus from which it begins to be moved.

But, when we use the expression “to be at” or “to approach,” we mean that a motion is being terminated at that point, and when we say “to be away from” or “to depart,” we mean that the motion is beginning. Hence, it is necessary that whatever arrives at (or is at) a point be in it, while what is leaving it or is departing from it be not in it. Since, therefore, it is impossible to be and not to be in a given point at the same time, it is consequently impossible to be at once at and away from the same, as the objection more than once assumed.

It should be noted that he here uses different letters from those used above. Here I is the mobile and D the terminus; above, it was the opposite.

But the solution given for continuous motion is not to be used with respect to a reflected motion. For it cannot be said that the mobile I is in the terminus D, from which it began to be reflected, only in the division of time—that is, only during the now—and that the mobile neither arrived at nor departed from the same, as was said with respect to a continuous motion.

For in a reflected motion, an end must be reached that is an actual end, and not merely a potential one, as the intermediate point in a continuous motion was only potentially a beginning and an end. What is an intermediate point of a continuous motion is only potentially a beginning and an end, but the point from which a reflected motion begins is actually a beginning and an end. For example, in the case of a stone falling to earth and bouncing upward, the same thing is the end of the downward motion of a stone and the beginning of its upward motion.

Therefore, just as, in the magnitude in which a motion is occurring, a point from which the motion is reflected is actually both a beginning and end, so also in the motions themselves, there is actually an end of one and a beginning of the other. And this would not be so unless an interval of rest occurred.

Therefore, it is necessary that what is reflected in a straight line be at rest. And, so, it follows that, on a straight magnitude, there cannot be a continuous and perpetual motion, because no straight magnitude is infinite. And, so, there could not be perpetual, continuous, rectilinear motion unless reflection is involved.

1115. Then, at the same method (263a4), he presents the second doubt.

About this, he does three things:

first, he mentions the doubt;

second, he rejects a solution given in book 6, at now, when we first (263a11; [1116]);

third, he gives the true solution, at and we must then apply (263a22; [1118]).

He says first (263a3) that, by the same method, using the things shown above, (Lect. 16, n. 1109.) one can block those who give the objection of Zeno and wish to argue in the following manner: whatever is being moved must first cross what is intermediate before arriving at the end; but, between any two termini, there are infinite intermediates on account of a magnitude’s infinite divisibility; and so it is impossible to traverse the intermediates, because infinites cannot be traversed; therefore, nothing can arrive at any terminus by motion.

Again, the same difficulty can be presented under another form, as some do in fact propose it. Whatever traverses a whole must previously traverse the half; and, since the half is again divided in half, half of the half must be first traversed. And, thus, whatever is being moved counts off every half as it reaches it. But such halves can be multiplied to infinity. Therefore, it follows that, if anything traverses an entire magnitude, it has counted off an infinite number, which is plainly impossible.

1116. Then, at now, when we first (263a11), he rejects the solution he had presented above in book 6. (Lect. 4, n. 779; cf. bk. 6, lect. 11, n. 863.)

First, he says what it is;

second, he sets it aside, at but, although this solution (263a15; [1117]).

He says therefore that the foregoing objection was answered in book 6 when motion in general was being discussed, on the ground that, just as a magnitude is divided infinitely, so also is time. Consequently, time possesses infinities in itself in the same way as a magnitude. And, so, it is not unfitting if the infinites in a magnitude be traversed in the infinites that are in time. For it is not inconsistent for an infinite magnitude to be traversed in an infinite time. But, as shown in book 6, the infinite is found in magnitude and in time in the same way. (Lect. 4, n. 778ff.)

1117. Then, at but, although this solution (263a15), he sets aside this solution. And he says that this solution is sufficient to answer the questioner who asked whether it was possible in a finite time to traverse and count off infinites. This question was dismissed by saying that a finite time possesses infinities in which the magnitudinal infinites can be traversed. But that solution does not reach the truth of the matter, because, if someone should omit to ask about the magnitude and fails to ask whether it is possible to traverse infinities in finite time, but asked rather this same question about time, namely, whether the infinites that are in time can be traversed—since time is divided to infinity—then the previous solution would not answer this question. Consequently, another solution must be sought.

1118. Then, at and we must then apply (263a22), he gives the true solution in the light of his premises above. (Lect. 16, nn. 1109–10.) And he says that the true solution of the present doubt requires us to repeat what was stated in the immediately foregoing arguments: when someone divides a continuum into two halves, he uses the one point at which the continuum is divided as two, because he is making it serve both as the beginning of one part and as the end of the other. He does this by numbering and by dividing into two halves.

But when a continuum has been divided in this manner, it is no longer a continuum, whether it be a magnitude (such as a line) or a motion that is divided, for a motion cannot be continuous unless of something continuous, namely, the motion of the subject, the time, and the magnitude traversed. Therefore, the divided counts, and by counting, it breaks the continuity.

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 had, 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.