LECTURE 17 (262 b 10-264 a 7)

FROM THE FOREGOING CERTAIN DIFFICULTIES ARE ANSWERED

1112. After the Philosopher has shown that reflex motion cannot be continuous and one, he here uses the foregoing to answer certain difficulties.

             This discussion is divided into three parts corresponding to the three difficulties which he answers from the foregoing. The second part begins where he says, 'The same method . . .' (263 a 4). The third part begins where he says, 'It is also plain . . .' (263 b 10).

             Concerning the first part he makes two points. First he states the difficulty. Secondly, where he says, 'Therefore we must not hold . . .' (262 b 18), he answers the difficulty.

             1113. He says, therefore, first that the same argument which he used to prove that reflex motion is not continuous may also be used to answer a certain difficulty.

             For the first difficulty is as follows. Let there be two equal magnitudes, one of which is E and the other F. Assume also two mobile objects of equal velocity, one of which is A and the other D. Let A be moved continuously from the extremity, that is, from the beginning of the magnitude, toward C, and let D be moved toward G. And in the magnitude E let us designate an intermediate point B which is the same distance from C as F is from G in the magnitude F. And let us grant that while A by its own continuous motion approaches point B, the mobile object D by its own continuous motion withdraws from F and approaches G. Since the motions of each mobile object are regular and of equal velocity, it follows that D reaches G before A reaches C. For that which departs first arrives first at the end of an equal magnitude. D, however, departed from F before A departed from B, for D departed when A reached B. Therefore, since A does not arrive at B and depart from it at the same time, it follows that it departs later than it arrives. For if it were to arrive and depart at the same time, it would not begin to be moved afterwards. And so it must be that although A is moved continuously, it rests at B. And thus a continuous motion will be composed of states of rest, just as Zeno held, as was explained above in Book VI.

             1114. Next where he says, 'Therefore we must not hold . . .' (262 b 18), he answers this difficulty by using the foregoing. The above objection supposed that while A is being moved continuously, it arrives at the point B in the middle of the magnitude, and at the same time that A arrives at B, D withdraws from some other point, namely, from F. But this is contrary to what has gone before. For it was said above that when a thing is moved continuously, it can neither be present nor absent, that is, it cannot arrive at or depart from some mid-point.

             Therefore we cannot hold what this objection proposes, namely, that when A arrives at B, D simultaneously departs from F. For if it is granted that A arrives at B, for the same reason it is granted that it departs from B. And this will not occur at the same time, but in two instants, so that it will be at rest in the intermediate time.

             But as was said earlier, when a thing is moved continuously, it is not present or absent at some mid-point. Rather it was there simply, not indeed through some time, for then it would be at rest, but in a division of time, that is, in some 'now', which divides time.

             Hence, what the objection proposes, namely, that A would be present at and D absent from some mid-point, cannot be said of continuous motion.

             But this must be said of reflex motion.

             For if the mobile object G is moved to point D and then is reflected, it is clear that the mobile object uses the terminus D as a beginning and an end, that is, one point as two. Hence it must be said that it rests there.

             Nor can it be said that it arrives at and departs from D simultaneously. For then it would follow that simultaneously in the same instant it would be and not be there. For whatever has been moved is in the terminus to which it was moved. And whatever begins to be moved is not in the terminus from which it begins to be moved. For when we say 'present at' or 'arrives at', we mean that the motion is terminated at that point. When, however, we say 'absent from' or 'departs from', we mean that motion begins. Hence whatever arrives at or is present at some point must be in that point. And whatever departs from or is absent from some point is not in it. Therefore, since it is impossible for a thing to be and not to be in some point at the same time, consequently it is impossible for it to be simultaneously present in and absent from the same thing, as was proposed many times above.

             It should be noted that here he is using the letters in a different way than he did before, using G for the mobile object and D for the terminus--just the opposite of the above.

             Moreover, the explanation given above for continuous motion must not be applied to reflex motion. For it cannot be said that the mobile object G is in the terminus D, from which its reflex motion begins, only in a division of time, that is, in a 'now', and that the mobile object does not arrive at and depart from the same thing, as was said of continuous motion. For a reflex motion must come to an end which is an end in act and not just in potency, while in continuous motion a mid-point was only potentially a beginning and an end. For a mid-point in a continuous motion is only potentially a beginning and an end, but that from which a reflex motion begins is actually a beginning and an end. The end of a downward motion, for example, of a stone, is the beginning in act of a reflex upward motion, when the stone falling to the earth rebounds upward.

             Hence in a magnitude in which motion occurs, the point of reflection is actually a beginning and an end. Thus in these motions there is an actual end of one motion and a beginning of the other. This could not occur unless an intermediate state of rest intervenes. Therefore that which is reflected in a straight line must be at rest. And so it follows that motion in a straight magnitude cannot be continuous and eternal. For a straight magnitude is not infinite, and thus a straight continuous motion cannot be eternal unless it is reflected.

             1115. Next where he says, 'The same method . . .' (263 a 4), he states the second difficulty.

             Concerning this he makes three points. First he raises the difficulty. Secondly, where he says, 'Now when we first . . .' (263 a 11), he rejects a certain answer given above in Book VI. Thirdly, where he says, '. . . and we must apply . . .' (263 a 22), he gives the true answer.

             He says, therefore, first, that using the same method employed above, we can reply to those who advance Zeno's objection and who wish to argue as follows.

             Before a thing which is moved reaches its terminus, it must cross half the distance to it. But between any two termini there is an infinity of half-distances, because magnitude is divisible to infinity. And so it is impossible to cross these half-distances, since an infinity cannot be crossed. Therefore, nothing can reach any terminus by moving.

             The same objection may be formulated differently, as some have proposed.

             Whatever passes through a whole first passes through half of it. And since the half-distance is itself divided in half, the mobile object must first cross half of the half-distance. Hence, whatever is moved numbers each half-distance by arriving at it. But these half-distances can be taken to infinity. And so it follows that if a thing traverses a whole magnitude, it has numbered an infinite number, which is obviously impossible.

             1116. Next where he says, 'Now when we first . . .' (263 a 11), he rejects an answer to this objection which he gave above in Book VI.

             First he repeats this answer, and secondly he rejects it, where he says, 'But, although this solution . . .' (263 a 15).

             He says, therefore, first that the above objection was answered in Book VI when, speaking of motion in general, he said that time is infinitely divisible, just as is magnitude. Hence in the same way both time and magnitude have infinities in themselves. And so there is no inconsistency if the infinities which are in magnitude are crossed in the infinities which are in time. For it is not inconsistent for an infinite magnitude to be crossed in an infinite time. And as was shown in Book VI, the infinite is found in time and in magnitude in the same way.

             1117. Next where he says, 'But, although this solution . . .' (263 a 15), he rejects this explanation.

             He says that this solution provides a satisfactory answer to one who asks the question in this way: Is it possible to number and pass through the infinite in a finite time? This question was answered by the statement that a finite time has infinities in which the infinities which are in magnitude can be crossed. But this answer is not sufficient for the truth of the thing. For if one neglects to make an inquiry about the magnitude, and if one neglects to ask if infinities can be crossed in a finite time, one might raise the same question about time itself, asking if the infinities which are in time could be passed through since time is infinitely divisible. The above solution is not a satisfactory answer to this question, and so another answer should be sought.

             1118. Next where he says, '. . . and we must apply . . .' (263 a 22), he gives the true solution according to what he has established above.

             He says that in order to answer this difficulty according to the truth we must say what we proposed in the arguments given above; namely, if one divides a continuum into two halves and then uses the one point at which the continuum is divided as if it were two points, then one makes that point both the beginning of one part and the end of the other. Moreover one does this by numbering and by dividing into two halves.

             When a continuum has been so divided, however, it will not be a continuum, whether a magnitude, for example, a line, is divided or whether motion is divided. For no motion can be continuous unless there be continuity in the subject and in the time and in the magnitude which the motion crosses. Therefore, by dividing, one numbers, and by numbering, one breaks up the continuum.

             But as long as the continuity of a continuum endures, there is an infinity of halves, not actually, but potentially. For if one causes a half to exist in act, this occurs by division, as was said, insofar as the division is taken as the beginning of one part and the end of the other. Thus the continuum will not remain, but it will cease, that is, the actual halves will not be infinite but will come to an end. This is especially clear if one wishes to number the halves. For he must number one point as two, insofar as it is the end of one half and the beginning of the other. This occurs when the whole continuum is not numbered as one, but rather the two halves are numbered in themselves. For if the whole continuum is taken as one, then, as was said, the mid-point is not taken as a beginning and an end in act but only in potency.

             As a result, he who asks whether it is possible to pass through an infinity of time or of magnitude must receive the answer that in a certain sense this is possible and in a certain sense this is not possible. For it is not possible to pass through an actual infinity; but it is possible to pass through a potential infinity. And so, since in a continuum the mid-points are only potentially infinite, it is possible to pass through this infinity. For that which is moved continuously passes accidentally through a potential infinity. Per se it crosses a finite line which happens to have in it an infinity of mid-points in potency. But the line itself, with respect to its substance and nature [ratio], is different from these infinite mid-points. For a line is not composed of points. Rather points can be designated in a line insofar as it is divided.

             1119. Next where he says, 'It is also plain . . .' (263 b 10), he answers the third difficulty.

             Concerning this he makes three points. First he sets forth the problem and its solution. Secondly, where he says, 'Let us suppose . . .' (263 b 16), he clarifies each with an example. Thirdly, where he says, 'Further, if anything . . .' (263 b 26), he introduces a corollary from what he has said.

             First he sets forth a problem which is usually raised in regard to generation and corruption. That which is generated ceases to not-be and begins to be. Some time must be assigned, therefore, to the existence of that which is generated or corrupted, and a different time to its nonexistence. For example, if fire is generated from air, in the whole time AB it was not fire but air. But in the whole time BC it is fire. Since, then, this point of time B is common to both times, it seems that in that common instant the same thing simultaneously is and is not fire.

             In solving this problem, the Philosopher says that it is clear that unless the point of time which divides the earlier and the later always refers to the later thing, that is, at this moment the thing exists in the same way as it does in the following time, it follows that the same thing is a being and is not a being at the same time, and it also follows that when a thing has been made it is non-being. For it was made when the generation ended, namely, in that 'now' which divides the earlier time from the later time. And so, if it did not exist in the whole earlier time, in this 'now' when it has been generated, it still does not exist. For this 'now' is the end of the earlier time.

             He shows how this inconsistency does not follow, however, by adding that the point which is one and the same in number, namely, the 'now', is common to each time, namely, the earlier and the later. But although it is one in subject, in reason [ratio] it is not one, but two. For it is the end of the earlier time and the beginning of the later time. But if the 'now' which is the thing is taken in itself, that is, if it is taken insofar as it is one in the thing, it is always referred to the later passion.

             Or to put it another way: Although this 'now' is the end of the earlier time and the beginning of the later time, and thus is common to both, nevertheless, with respect to the thing, that is, insofar as it is compared to the thing which is moved, it always pertains to the later passion. For the thing which is moved is in that 'now' the subject of the passion of the later time.

             1120. Having set forth the problem and its solution, he clarifies each with examples. First he clarifies the problem where he says, 'Let us suppose . . .' (263 b 16).

             He asks us to assume a time ACB and an object D which is moved. In the time A, D is white, and in B it is not white. It seems to follow that in C it is both white and not white. He explains how this follows by adding that if it is white in the whole time A, it follows that it is white in any part of A. Similarly, if it is not white in the whole of time B, it follows that it is not white in any part of B. Since, then, C is found in both times, because it is the end of one and the beginning of the other, it seems to follow that in C the thing is white and not white.

             1121. Secondly, where he says, 'We must not allow . . .' (263 b 20), he explains the answer given above.

             He says that it must not be conceded that the object is white in every part of A. Rather we must exclude the ultimate 'now' C, which is the ultimate terminus of the mutation. For example, if a white thing either was coming to be or was being corrupted in the whole of A, in C it is neither being corrupted nor is it becoming white. Rather it has already been made or corrupted. Moreover, that which has been made, is; and that which has been corrupted, is not. And so it is clear that it is primarily true to say that it is white in C, if the generation of whiteness ends there; or that it is not white, if the corruption of whiteness ends there. And if this is not said, the inconsistencies set forth above will follow, namely, when a thing has been generated, at that point it is non-being, and when it has been corrupted, at that point it is being. And it will also follow that a thing is white and not white at the same time, and universally it is being and non-being.

             1122. Next where he says, 'Further, if anything . . .' (263 b 26), he introduces a corollary from the foregoing, namely, time is not divided into indivisible times. For the previous problem could not be answered if this were not granted.

             He says, then, that whatever first is not a being and afterwards is a being must at some time become a being. Furthermore, it is necessary that when a thing is coming to be, it is not. If, however, these two suppositions are true, it is impossible for time to be divided into indivisible times.

             For let time be divided into indivisible times, and let the first indivisible time be A. Let the second one, which is consecutive to A, be B. And let D, which was not white earlier, and afterwards is white, become white in A, when it was not white. We must grant that this is done in some indivisible and consecutive time, namely, B, in which it already is. If, however, it was becoming white in A, it follows that it was not white in A. In B, however, it is white. Therefore, since between being and nonbeing there is an intervening generation because nothing passes from non-being to being except by a generation, it follows that a generation intervened between A and B. Hence there will be some time intervening between A and B in which it became white. (For it was granted that in the time B, D has been generated.)

             And similarly, when it becomes white in that intermediate, indivisible time, it is not white; and so, for the same reason we ought to posit another intermediate time, and so on to infinity. And so it cannot be held that it comes to be and has come to be in the same time.

             But the same reasoning does not apply if it is denied that time is divisible into indivisible times. For in this case we say that the time in which it was coming to be and is made is one and the same. It was coming to be and was non-being in the whole preceding time. It has been made, however, and is being in the ultimate 'now' of time which is not related to the preceding time as consecutive to it, but as its terminus. But if indivisible times are posited, they must be related as consecutive.

             It is clear, moreover, from the foregoing that if indivisible times are not posited, and if a thing becomes white in the whole time A, the time in which it was made and came to be is not greater than that in which it only came to be. For it came to be in the whole time, but it was made in the ultimate terminus of the time. The time, however, and the terminus of the time are not greater than the time alone, just as a point adds nothing to the magnitude of a line. But if indivisible times are posited, it is clear from the foregoing that the time in which it comes to be and is made is greater than the time in which it only comes to be.

             Finally he summarizes and concludes to his main point. He says that the above and similar arguments serve as proper proofs that reflex motion is not continuous.