Commentary Aristotle Physics
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 generated, 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 an 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: