Commentary Aristotle Physics
Lecture 11
The arguments of Zeno, who tried to deny all motion, are answered
Zeno’s reasoning, however, is fallacious when he says that, if everything is at rest when it occupies an equal space and if what is in locomotion is always occupying such a space at any moment, then the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
Zeno’s arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number.
The first asserts the non-existence of motion on the ground that what is in locomotion must arrive at the half-way stage before it arrives at the goal. This we have discussed above.
The second is the so-called “Achilles,” and it amounts to this: in a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, such that the slower must always hold a lead.
This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves.
The result of the argument is that the slower is not overtaken, but it proceeds along the same lines as the bisection-argument (for in both, a division of the space in a certain way leads to the result that the goal is not reached, though the “Achilles” goes further, in that it affirms that even the quickest runner in tragic tradition must fail in his pursuit of the slowest), such that the solution must be the same.
And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead, but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed. These, then, are two of his arguments.
The third is that already given above, to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments; but if this assumption is not granted, the conclusion will not follow.
The fourth argument is that concerning bodies moved in an equal manner on a race course with equal velocity in opposite directions—the one originally between the goal and the midpoint, and the other between the midpoint and the start.
This, he thinks, involves the conclusion that half a given time is equal to double that time.
The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest, which is false.
For instance (so runs the argument), let AAA be the bodies of equal size which are standing still, and let BBB the bodies equal in number and in size to AAA and originally occupying the half of the course from the starting post to the middle of AAA, and let CCC be those originally occupying the other half from the goal to the middle of AAA, equal in number, size, and velocity to BBB.
Then three consequences follow: first, as BBB and CCC pass one another, the first B reaches the last C at the same moment as the first C reaches the last B.
Second, at this moment, the first C has passed all of AAA, whereas the first B has passed only half of AAA, and has consequently occupied only half the time occupied by the first C, since each of the two occupies an equal time in passing each A.
Third, at the same moment, all of BBB has passed all of CCC, for the first C and the first B will simultaneously reach the opposite ends of the course, since (so says Zeno) the time occupied by the first C in passing each B is equal to that occupied by it in passing each A, because an equal time is occupied by both the first B and the first C in passing all of AAA.
This is the argument, but it presupposed the previously stated fallacious assumption.
Nor, in reference to contradictory change, shall we find anything unanswerable in the argument that, if a thing is changing, say, from not-white to white and is in neither condition, then it will be neither white nor not-white. For the fact that it is not wholly in either condition will not preclude us from calling it white or not-white.
We call a thing white or not-white not necessarily because it is wholly either in one or the other, but because most of its parts or the most essential parts of it are so: not being in a certain condition is different from not being wholly in that condition.
It is the same in the case of being and non-being and all other conditions that stand in a contradictory relation: while the changing thing must of necessity be in one of the two opposites, it is never wholly in either.
Again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest on the ground that such things in motion—themselves and their parts—will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion.
For in the first place, the parts do not occupy the same position for any period of time; in the second place, the whole also is always changing to a different position;
for if we take the orbit as described from a point A on a circumference, it will not be the same as the orbit as described from B or C or any other point on the same circumference, except in an accidental sense—the sense, that is to say, in which a musical man is the same as a man.
Thus, one orbit is always changing into another, and the thing will never be at rest. And it is the same with the sphere and all other things that are moved within themselves.
860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion. About this, he does three things: first, he answers the arguments of Zeno, who denies that motion exists at all; second, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at “our next point is that what is” (240b8; [872]). third, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at “our next point is that no process” (241a26; [879]).
About the first, he does two things: first, he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion;
second, he explains all his arguments in order, at “Zeno’s arguments” (239b9; [862]).
861. He says, therefore (239b5), that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved—even what seems to be in rapid motion, as an arrow in flight. And this was his argument. Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant, it is either in motion or at rest. But it is not in motion; therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then it is at rest throughout the entire time and not in motion.
Now, this argument could be answered by appealing to something already proved: in an instant, there is neither motion nor rest. But such a solution would not cripple Zeno’s intention, for he is satisfied to show that, through the entire time, there is no motion. This seems to follow if there is no motion at any instant of the time. Therefore, Aristotle answers in a different manner and says that the conclusion both is false and does not follow from the premises.
For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence, it does not follow that a thing is not in motion in a given time just because it is not in motion in any instant of that time.
862. Then, at Zeno’s arguments (239b9), he lists in order all the arguments that Zeno used for destroying motion. About this, he does three things: first, he shows how he destroyed local motion with his arguments; second, he shows how he destroyed the other types of change, at “nor, in reference to” (240a19; [870]). third, how in particular he destroyed circular motion, at “again, in the case” (240a29; [871]).