Does Turning Back Involve a Stop?

Suppose the line E is equal to the line Z. A proceeds in continuous locomotion from the extreme point of E to G. When A is at B, D is proceeding in uniform locomotion and with the same velocity as A from the extremity of Z to H.

Then D will have reached H before A has reached G because that which makes an earlier start and departure must make an earlier arrival.

A will arrive late because 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 a stand there.

Therefore we must not hold that there was a moment when A came to be at B and that at the same moment D was in motion from the extremity of Z: for the fact of A’s having come to be at B will involve the fact of its also ceasing to be there, and the two events will not be simultaneous, whereas the truth is that A is at B at a sectional point 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.

Suppose H in the course of its locomotion proceeds to D and then turns back and proceeds downwards again: then the extreme point D has served as finishing-point and as starting-point for it, one point thus serving as two:

Therefore H must have come to a stand there: 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.

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, not potentially, existent. 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, so 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 a stand. Consequently there cannot be a continuous rectilinear motion that is eternal.

The same method should also be used to reply to Zeno.

Zeno argues that:

• before any distance can be traversed, half the distance must be traversed.
• these half-distances are infinite in number
• it is impossible to traverse distances infinite in number

He would have us grant that in the time during which a motion is in progress it is possible to reckon a half-motion before the whole for every half-distance that we get. In this way, we have the result that when the whole distance is traversed we have reckoned an infinite number, which is impossible.

I solved this difficulty turning on the fact that the period of time occupied in traversing the distance contains within itself an infinite number of units.

It is not absurd to traverse infinite distances in infinite time. The element of infinity is present in the time no less than in the distance.

This solves the question: Is it possible in a finite time to traverse or reckon an infinite number of units?

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 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 2 halves, 1 point is treated as 2, since we make it a startingpoint 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 motion if it is to be continuous must relate to what is continuous:

Though what is continuous contains an infinite number of halves, they are not actual but potential halves. If the halves are made actual, we shall get not a continuous but an intermittent motion.

Counting the halves gives this result. For then one point must be counted as 2.

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 2 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 a sense it is and in a sense 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 traveller has traversed an infinite number of units in an accidental sense but not in an unqualified sense: for 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, we shall be involved in the consequence that the same thing is at the same moment existent and not existent, and that a thing is not existent at the moment when it has become.

The point is common to both times, the earlier as well as the later.

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.