Plato and the Other Companions of Sokrates, 3rd ed. Volume 1

Again — If existing things be many and discontinuous, each of these must exist in a place of its own. Nothing can exist except in some place. But the place is itself an existing something: each place must therefore have a place of its own to exist in: the second place must have a third place to exist in and so forth ad infinitum.20 We have here a farther reductio ad impossibile of the original hypothesis: for that hypothesis denies the continuity of space, and represents space as a multitude of discontinuous portions or places.

Another argument of Zeno is to the following effect:—“Does a grain of millet, when dropped upon the floor, make sound? No. — Does a bushel of millet make sound under the same circumstances? Yes. — Is there not a determinate proportion between the bushel and the grain? There is. — There must therefore be the same proportion between the sonorousness of the two. If one grain be not sonorous, neither can ten thousand grains be so.”21

To appreciate the contradiction brought out by Zeno, we must recollect that he is not here reasoning about facts of sense, phenomenal and relative — but about things in themselves, absolute and ultra-phenomenal realities. He did not deny the fact of sense: to appeal to that fact in reply, would have been to concede his point. The adversaries against whom he reasoned (Protagoras is mentioned, but he can hardly have been among them, if we have regard to his memorable dogma, of which more will be said presently) were those who maintained the plurality of absolute substances, each for itself, with absolute attributes, apart from the fact of sense, and independent of any sentient subject. One grain of millet (Zeno argues) has no absolute sonorousness, neither can ten thousand such grains taken together have any. Upon the hypothesis of absolute reality as a discontinuous multitude, you are here driven to a contradiction which Zeno intends as an argument against the hypothesis. There is no absolute sonorousness in the ten thousand grains: the sound which they make is a phenomenal fact, relative to us as sentients of sound, and having no reality except in correlation with a hearer.22

Other memorable arguments of Zeno against the same hypothesis were those by which he proved that if it were admitted, motion would be impossible. Upon the theory of absolute plurality and discontinuousness, every line or portion of distance was divisible into an infinite number of parts: before a moving body could get from the beginning to the end of this line, it must pass in succession over every one of these parts: but to do this in a finite time was impossible: therefore motion was impossible.23

A second argument of the same tendency was advanced in the form of comparison between Achilles and the tortoise — the swiftest and slowest movers. The two run a race, a certain start being given to the tortoise. Zeno contends that Achilles can never overtake the tortoise. It is plain indeed, according to the preceding argument, that motion both for the one and for the other is an impossibility. Neither one nor the other can advance from the beginning to the end of any line, except by passing successively through all the parts of that line: but those parts are infinite in number, and cannot therefore be passed through in any finite time. But suppose such impossibility to be got over: still Achilles will not overtake the tortoise. For while Achilles advances one hundred yards, the tortoise has advanced ten: while Achilles passes over these additional ten yards, the tortoise will have passed over one more yard: while Achilles is passing over this remaining one yard, the tortoise will have got over one-tenth of another yard: and so on ad infinitum: the tortoise will always be in advance of him by a certain distance, which, though ever diminishing, will never vanish into nothing.

The third Zenonian argument derived its name from the flight of an arrow shot from a bow. The arrow while thus carried forward (says Zeno) is nevertheless at rest.24 For the time from the beginning to the end of its course consists of a multitude of successive instants. During each of these instants the arrow is in a given place of equal dimension with itself. But that which is during any instant in a given place, is at rest. Accordingly during each successive instant of its flight, the arrow is at rest. Throughout its whole flight it is both in motion and at rest. This argument is a deduction from the doctrine of discontinuous time, as the preceding is a deduction from that of discontinuous space.

A fourth argument25 was derived from the case of two equal bodies moved with equal velocity in opposite directions, and passing each other. If the body A B were at rest, the other body C D would move along the whole length of C D in two minutes. But if C D be itself moving with equal velocity in the opposite direction, A B will pass along the whole length of C D in half that time, or one minute. Hence Zeno infers that the motion of A B is nothing absolute, or belonging to the thing in itself — for if that were so, it would not be varied according to the movement of C D. It is no more than a phenomenal fact, relative to us and our comparison.

These Zenonian reasonings are memorable as the earliest known manifestations of Grecian dialectic, and are probably equal in acuteness and ingenuity to anything which it ever produced. Their bearing is not always accurately conceived. Most of them are argumenta ad hominem: consequences contradictory and inadmissible, but shown to follow legitimately from a given hypothesis, and therefore serving to disprove the hypothesis itself.26 The hypothesis was one relating to the real, absolute, or ultra-phenomenal, which Parmenides maintained to be Ens Unum Continuum, while his opponents affirmed it to be essentially multiple and discontinuous. Upon the hypothesis of Parmenides, the Real and Absolute, being a continuous One, was obviously inconsistent with the movement and variety of the phenomenal world: Parmenides himself recognised the contradiction of the two, and his opponents made it a ground for deriding his doctrine.27 The counter-hypothesis, of the discontinuous many, appeared at first sight not to be open to the same objection: it seemed to be more in harmony with the facts of the phenomenal and relative world, and to afford an absolute basis for them to rest upon. Against this delusive appearance the dialectic of Zeno was directed. He retorted upon the opponents, and showed that if the hypothesis of the Unum Continuum led to absurd consequences, that of the discontinuous many was pregnant with deductions yet more absurd and contradictory.

He exhibits in detail several of these contradictory deductions, with a view to refute the hypothesis from whence they flow; and to prove that, far from performing what it promises, it is worse than useless, as entangling us in contradictory conclusions. The result of his reasoning, implied rather than announced, is — That neither of the two hypotheses are of any avail to supply a real and absolute basis for the phenomenal and relative world: That the latter must rest upon its own evidence, and must be interpreted, in so far as it can be interpreted at all, by its own analogies.

But the purport of Zeno’s reasoning is mistaken, when he is conceived as one who wishes to delude his hearers by proving both sides of a contradictory proposition. Zeno’s contradictory conclusions are elicited with the express purpose of disproving the premisses from which they are derived. For these premisses Zeno himself is not to be held responsible, since he borrows them from his opponents: a circumstance which Aristotle forgets, when he censures the Zenonian arguments as paralogisms, because they assume the Continua, Space, and Time, to be discontinuous or divided into many distinct parts.28 Now this absolute discontinuousness of matter, space, and time, was not advanced by Zeno as a doctrine of his own, but is the very doctrine of his opponents, taken up by him for the purpose of showing that it led to contradictory consequences, and thus of indirectly refuting it. The sentence of Aristotle is thus really in Zeno’s favour, though apparently adverse to him. In respect to motion, a similar result followed from the Zenonian reasonings; namely, to show That motion, as an attribute of the Real and Absolute, was no less inconsistent with the hypothesis of those who opposed Parmenides, than with the hypothesis of Parmenides himself:—That absolute motion could no more be reconciled with the doctrine of the discontinuous Many, than with that of the Continuous One:—That motion therefore was only a phenomenal fact, relative to our sensations, conceptions, and comparisons; and having no application to the absolute.

In this phenomenal point of view, neither Zeno nor Parmenides nor Melissus disputed the fact of motion. They recognised it as a portion of the world of sensation and experience; which world they tried to explain, well or ill, by analogies and conjectures derived from itself.

Though we have not the advantage of seeing the Zenonian dialectics as they were put forth by their author, yet if we compare the substance of them as handed down to us, with those dialectics which form the latter half of the Platonic dialogue called Parmenides, we shall find them not inferior in ingenuity, and certainly more intelligible in their purpose. Zeno furnishes no positive support to the Parmenidean doctrine, but he makes out a good negative case against the counter-doctrine.

Zeller and other able modern critics, while admitting the reasoning of Zeno to be good against this counter-doctrine, complain that he takes it up too exclusively; that One and Many did not exclude each other, and that the doctrines of Parmenides and his opponents were both true together, but neither of them true to the exclusion of the other. But when we reflect that the subject of predication on both sides was the Real (Ens per se) it was not likely that either Parmenides or his opponents would affirm it to be both absolutely One and Continuous, and absolutely Many and Discontinuous.29 If the opponents of Parmenides had taken this ground, Zeno need not have imagined deductions for the purpose of showing that their hypothesis led to contradictory conclusions; for the contradictions would have stood avowedly registered in the hypothesis itself. If a man affirms both at once, he divests the predication of its absolute character, as belonging unconditionally to Ens per se; and he restricts it to the phenomenal, the relative, the conditioned — dependent upon our sensations and our fluctuating point of view. This was not intended either by Parmenides or by his opponents.

If, indeed, we judge the question, not from their standing-point, but from our own, we shall solve the difficulty by adopting the last-mentioned answer. We shall admit that One and Many are predicates which do not necessarily exclude each other; but we shall refrain from affirming or denying either of them respecting the Real, the Absolute, the Unconditioned. Of an object absolutely one and continuous — or of objects absolutely many and discontinuous, apart from the facts of our own sense and consciousness, and independent of any sentient subject — we neither know nor can affirm anything. Both these predicates (One — Many) are relative and phenomenal, grounded on the facts and comparisons of our own senses and consciousness, and serving only to describe, to record, and to classify, those facts. Discrete quantity or number, or succession of distinct unities — continuous quantity, or motion and extension — are two conceptions derived from comparison, abstracted and generalised from separate particular phenomena of our consciousness; the continuous, from our movements and the consciousness of persistent energy involved therein — the discontinuous, from our movements, intermitted and renewed, as well as from our impressions of sense.

We compare one discrete quantity with another, or one continual quantity with another, and we thus ascertain many important truths: but we select our unit, or our standard of motion and extension, as we please, or according to convenience, subject only to the necessity of adapting our ulterior calculations consistently to this unit, when once selected. The same object may thus be considered sometimes as one, sometimes as many; both being relative, and depending upon our point of view. Motion, Space, Time, may be considered either as continuous or as discontinuous: we may reason upon them either as one or the other, but we must not confound the two points of view with each other. When, however, we are called upon to travel out of the Relative, and to decide between Parmenides and his opponents — whether the Absolute be One or Multitudinous — we have only to abstain from affirming either, or (in other words) to confess our ignorance. We know nothing of an absolute, continuous, self-existent One, or of an absolute, discontinuous Many.

Some critics understand Zeno to have denied motion as a fact — opposing sophistical reasoning to certain and familiar experience. Upon this view is founded the well-known anecdote, that Diogenes the Cynic refuted the argument by getting up and walking. But I do not so construe the scope of his argument. He did not deny motion as a fact. It rested with him on the evidence of sense, acknowledged by every one. It was therefore only a phenomenal fact relative to our consciousness, sensation, movements, and comparisons. As such, but as such only, did Zeno acknowledge it. What he denied was, motion as a fact belonging to the Absolute, or as deducible from the Absolute. He did not deny the Absolute or Thing in itself, as an existing object, but he struck out variety, divisibility, and motion, from the list of its predicates. He admitted only the Parmenidean Ens, one, continuous, unchanged, and immovable, with none but negative predicates, and severed from the relative world of experience and sensation.