We have seen how Empedocles and Anaxagoras attempted to restore the world of appearances which had been so nearly destroyed by the logic of Parmenides. At the same time the disciples of Pythagoras were continuing to pursue their method of analyzing reality into a system of numbers — numbers expressing points, lines or geometrical shapes and proportions. No one, in fact, had really faced the full force of Parmenides' assertion "As it was in the beginning is now and ever shall be." One cannot help feeling that in the systems of his successors the idea of motion, whether under the guise of condensation and rarefaction, "mingling" or Nous, has somehow slipped in through the back door.
Now, in the middle of the fifth century and in the youth of Socrates, the doctrine of Parmenides is once more vigorously asserted by Zeno, a pupil and fellow-townsman of Parmenides himself, and by Melissos, a native of Samos.
Zeno, according to Plato, was born about 490 B.C., and visited Athens, in company with Parmenides, when he was about forty years old. Aristotle describes him as the inventor of dialectics and it is certainly true that, though the principles of dialectics are evident in Parmenides' own writing, Zeno was the first to give to what may be called either "logic-chopping" or "rational thought" a dramatic and a particularly incisive expression. His method was to take up the hypothesis of his philosophical opponent and from it to deduce contradictory conclusions. The effect of the meth-od is to suggest that those who disagree with the doctrine of the Parmenidean "One" and try to "save appearances" will find themselves involved in difficulties much greater than those which they have been attempting to avoid.
His three best-known paradoxes may be summarized as follows:
1. You cannot get to the other side of a racecourse. To do so, you must first get halfway across, and to do this, you must first get halfway to the halfway point, and so on ad infinitum. Therefore you can never start at all.
2. Achilles can never catch up with the tortoise. To do so he must first reach the point from which the tortoise started, and by that time the tortoise will have got a little further on. By the time he makes up this further distance, the tortoise will have moved a little more, and so on ad infinitum.
3. The arrow in flight is at rest. At any given moment it must occupy a space equal to itself. Therefore it cannot move.
Of these paradoxes the first two are directed against the hypothesis that matter or being is infinitely divisible, or, to be more precise, that a line is made up of an infinite number of points. The third paradox (and a fourth which, for reasons of space, I have omitted) is directed at the hypothesis that matter or being is made up of a finite number of indivisibles. We are left to conclude that if being is neither infinitely divisible nor composed of a finite number of divisibles, it must be, as Parmenides had concluded, a continuum. It may be thought that there is something obviously absurd about Zeno's arguments, and in a sense this is true. However, anyone who cares to investigate the enormous and still growing literature devoted to their discussion will soon realize that his logical knots are very skillfully tied.
The system of Parmenides is explained with at least as great clarity as that of Parmenides himself by Melissos of Samos, who also, in one important respect, makes an addition to the theory. Of Melissos himself we know nothing except that he commanded the Samian fleet against Athens in 442-440 B.C., defeated Pericles and, probably, was later defeated by him. He differs from Parmenides in concluding that "what exists" is infinite in space as well as in time. I must admit that I cannot understand why Parmenides did not come to this conclusion himself, since if his sphere had anything outside it, that "anything" must be "nothing," and there is no "nothing." We should note too an interesting line of thought in fragment 8. Here Melissos asserts that if one is going to believe in "a many" (and of course he thinks one should not) then each one of the many must have all the characteristics of the Parmenidean "One." It seems something of an exaggeration to say, as Burnet does of this statement, "In other words, the only consistent pluralism is the atomic theory"; but there is no doubt that the atomic theorists were influenced by this line of thought.
It may be interesting to read the following fragments in conjunction with those of Parmenides. The fragments are from Burnet
1a. If nothing is, what can be said of it as of something real?
1. What was was ever, and ever shall be. For, if it had come into being, it needs must have been nothing before it came into being. Now, if it were nothing, in no wise could anything have arisen out of nothing.
2. Since, then, it has not come into being, and since it is, was ever, and ever shall be, it has no beginning or end, but is with-out limit. For, if it had come into being, it would have had a beginning (for it would have begun to come into being at some time or other) and an end (for it would have ceased to come into being at some time or other); but, if it neither began nor ended, and ever was and ever shall be, it has no beginning or end; for it is not possible for anything to be ever without all being.
3. Further, just as it ever is, so it must ever be infinite in magnitude.
4. But nothing which has a beginning or end is either eternal or infinite.
5. If it were not one, it would be bounded by something else.
6. For if it is (infinite), it must be one; for if it were two, it could not be infinite; for then they would be bounded by one another.
7. So then it is eternal and infinite and one and all alike. And it cannot perish nor become greater, nor does it suffer pain or grief. For, if any of these things happened to it, it would no longer be one. For if it is altered, then the real must needs not be all alike, but what was before must pass away, and what was not must come into being. Now, if it changed by so much as a single hair in ten thousand years, it would all perish in the whole of time.
Further, it is not possible either that its order should be changed; for the order which it had before does not perish, nor does that which was not come into being. But, since nothing is either added to it or passes away or is altered, how can any real thing have had its order changed? For if anything became different, that would amount to a change in its order.
Nor does it suffer pain; for a thing in pain could not all be. For a thing in pain could not be ever, nor has it the same power as what is whole. Nor would it be alike, if it were in pain; for it is only from the addition or subtraction of something that it could feel pain, and then it would no longer be alike. Nor could what is whole feel pain; for then what was whole and what was real would pass away, and what was not would come into being. And the same argument applies to grief as to pain.
Nor is anything empty. For what is empty is nothing. What is nothing cannot be.
Nor does it move; for it has nowhere to betake itself to, but is full. For if there were aught empty, it would betake itself to the empty. But, since there is naught empty, it has nowhere to betake itself to.
And it cannot be dense and rare; for it is not possible for what is rare to be as full as what is dense, but what is rare is at once emptier than what is dense.
This is the way in which we must distinguish between what is full and what is not full. If a thing has room for anything else, and takes it in, it is not full; but if it has no room for anything and does not take it in, it is full.
Now, it must needs be full if there is naught empty, and if it is full, it does not move.
8. This argument, then, is the greatest proof that it is one alone; but the following are proofs of it also. If there were a many, these would have to be of the same kind as I say that the one is. For if there is earth and water, and air and iron, and gold and fire, and if one thing is living and another dead, and if things are black and white and all that men say they really are, — if that is so, and if we see and hear aright, each one of these must be such as we first decided, and they cannot be changed or altered, but each must be just as it is. But, as it is, we say that we see and hear and understand aright, and yet we believe that what is warm becomes cold, and what is cold warm; that what is hard turns soft, and what is soft hard; that what is living dies, and that things are born from what lives not; and that all those things are changed, and that what they were and what they are now are in no way alike. We think that iron, which is hard, is rubbed away by contact with the finger; and so with gold and stone and everything which we fancy to be strong, and that earth and stone are made out of water; so that it turns out that we neither see nor know realities. Now these things do not agree with one another. We said that there were many things that were eternal and had forms and strength of their own, and yet we fancy that they all suffer alteration, and that they change from what we see each time. It is clear, then, that we did not see aright after all, nor are we right in believing that all these things are many. They would not change if they were real, but each thing would be just what we believed it to be; for nothing is stronger than true reality. But if it has changed, what was has passed away, and what was not is come into being. So then, if there were many things, they would have to be just of the same nature as the one.
9. Now, if it were to exist, it must needs be one; but if it is one, it cannot have body; for, if it had body it would have parts, and would no longer be one.
10. If what is real is divided, it moves; but if it moves, it cannot be.