Aristotle (Phys. VI. 9) explains this dialectic further; Zeno’s treatment of motion was above all objectively dialectical. But the particulars which we find in the Parmenides of Plato are not his. For Zeno’s consciousness we see simple unmoved thought disappear, but become thinking movement; in that he combats sensuous movement, he concedes it. The reason that dialectic first fell on movement is that the dialectic is itself this movement, or movement itself the dialectic of all that is. The thing, as self-moving, has its dialectic in itself, and movement is the becoming another, self-abrogation. If Aristotle says that Zeno denied movement because it contains an inner contradiction, it is not to be understood to mean that movement did not exist at all. The point is not that there is movement and that this phenomenon exists; the fact that there is movement is as sensuously certain as that there are elephants; it is not in this sense that Zeno meant to deny movement. The point in question concerns its truth. Movement, however, is held to be untrue, because the conception of it involves a contradiction; by that he meant to say that no true Being can be predicated of it.
Zeno’s utterances are to be looked at from this point of view, not as being directed against the reality of motion, as would at first appear, but as pointing out how movement must necessarily be determined, and showing the course which must be taken. Zeno now brings forward four different arguments against motion; the proofs rest on the infinite divisibility of space and time.
(a) This is his first form of argument:—“Movement has no truth, because what is in motion must first reach the middle of the space before arriving at the end.” Aristotle expresses this thus shortly, because he had earlier treated of and worked out the subject at length. This is to be taken as indicating generally that the continuity of space is presupposed. What moves itself must reach a certain end, this way is a whole. In order to traverse the whole, what is in motion must first pass over the half, and now the end of this half is considered as being the end; but this half of space is again a whole, that which also has a half, and the half of this half must first have been reached, and so on into infinity. Zeno here arrives at the infinite divisibility of space; because space and time are absolutely continuous, there is no point at which the division can stop. Every dimension (and every time and space always have a dimension) is again divisible into two halves, which must be measured off; and however small a space we have, the same conditions reappear. Movement would be the act of passing through these infinite moments, and would therefore never end; thus what is in motion cannot reach its end. It is known how Diogenes of Sinope, the Cynic, quite simply refuted these arguments against movement; without speaking he rose and walked about, contradicting them by action.[60] But when reasons are disputed, the only valid refutation is one derived from reasons; men have not merely to satisfy themselves by sensuous assurance, but also to understand. To refute objections is to prove their non-existence, as when they are made to fall away and can hence be adduced no longer; but it is necessary to think of motion as Zeno thought of it, and yet to carry this theory of motion further still.
We have here the spurious infinite or pure appearance, whose simple principle Philosophy demonstrates as universal Notion, for the first time making its appearance as developed in its contradiction; in the history of Philosophy a consciousness of this contradiction is also attained. Movement, this pure phenomenon, appears as something thought and shown forth in its real being—that is, in its distinction of pure self-identity and pure negativity, the point as distinguished from continuity. To us there is no contradiction in the idea that the here of space and the now of time are considered as a continuity and length; but their Notion is self-contradictory. Self-identity or continuity is absolute cohesion, the destruction of all difference, of all negation, of being for self; the point, on the contrary, is pure being-for-self, absolute self-distinction and the destruction of all identity and all connection with what is different. Both of these, however, are, in space and time, placed in one; space and time are thus the contradiction; it is necessary, first of all, to show the contradiction in movement, for in movement that which is opposed is, to ordinary conceptions, inevitably manifested. Movement is just the reality of time and space, and because this appears and is made manifest, the apparent contradiction is demonstrated, and it is this contradiction that Zeno notices. The limitation of bisection which is involved in the continuity of space, is not absolute limitation, for that which is limited is again continuity; however, this continuity is again not absolute, for the opposite has to be exhibited in it, the limitation of bisection; but the limitation of continuity is still not thereby established, the half is still continuous, and so on into infinity. In that we say “into infinity,” we place before ourselves a beyond, outside of the ordinary conception, which cannot reach so far. It is certainly an endless going forth, but in the Notion it is present, it is a progression from one opposed determination to others, from continuity to negativity, from negativity to continuity; but both of these are before us. Of these moments one in the process may be called the true one; Zeno first asserts continuous progression in such a way that no limited space can be arrived at as ultimate, or Zeno upholds progression in this limitation.
The general explanation which Aristotle gives to this contradiction, is that space and time are not infinitely divided, but are only divisible. But it now appears that, because they are divisible—that is, in potentiality—they must actually be infinitely divided, for else they could not be divided into infinity. That is the general answer of the ordinary man in endeavouring to refute the explanation of Aristotle. Bayle (Tom. IV. art. Zénon, not. E.) hence says of Aristotle’s answer that it is “pitoyable: C’est se moquer du monde que de se servir de cette doctrine; car si la matière est divisible à l’infini, elle contient un nombre infini de parties. Ce n’est donc point un infini en puissance, c’est un infini, qui existe réellement, actuellement. Mais quand-même on accorderait cet infini en puissance, qui deviendrait un infini par la division actuelle de ses parties, on ne perdrait pas ses avantages; car le mouvement est une chose, qui a la même vertu, que la division. Il touche une partie de l’espace sans toucher l’autre, et il les touche toutes les unes après les autres. N’est-ce pas les distinguer actuellement? N’est-ce pas faire ce que ferait un géomètre sur une table en tirant des lignes, qui désignassent tous les demi-pouces? Il ne brise pas la table en demi-pouces, mais il y fait néanmoins une division, qui marque la distinction actuelle des parties; et je ne crois pas qu’Aristote eut voulu nier, que _si_ l’on tirait une infinité de lignes sur un pouce de matière, on n’y introduisît une division, qui réduirait en infini actuel ce qui n’était selon lui qu’un infini virtual.” This _si_ is good! Divisibility is, as potentiality, the universal; there is continuity as well as negativity or the point posited in it—but posited as moment, and not as existent in and for itself. I can divide matter into infinitude, but I only can do so; I do not really divide it into infinitude. This is the infinite, that no one of its moments has reality. It never does happen that, in itself, one or other—that absolute limitation or absolute continuity—actually comes into existence in such a way that the other moment disappears. There are two absolute opposites, but they are moments, i.e. in the simple Notion or in the universal, in thought, if you will; for in thought, in ordinary conception, what is set forth both is and is not at the same time. What is represented either as such, or as an image of the conception, is not a thing; it has no Being, and yet it is not nothing.
Space and time furthermore, as _quantum_, form a limited extension, and thus can be measured off; just as I do not actually divide space, neither does the body which is in motion. The partition of space as divided, is not absolute discontinuity [Punktualität], nor is pure continuity the undivided and indivisible; likewise time is not pure negativity or discontinuity, but also continuity. Both are manifested in motion, in which the Notions have their reality for ordinary conception—pure negativity as time, continuity as space. Motion itself is just this actual unity in the opposition, and the sequence of both moments in this unity. To comprehend motion is to express its essence in the form of Notion, _i.e._, as unity of negativity and continuity; but in them neither continuity nor discreteness can be exhibited as the true existence. If we represent space or time to ourselves as infinitely divided, we have an infinitude of points, but continuity is present therein as a space which comprehends them: as Notion, however, continuity is the fact that all these are alike, and thus in reality they do not appear one out of the other like points. But both these moments make their appearance as existent; if they are manifested indifferently, their Notion is no longer posited, but their existence. In them as existent, negativity is a limited size, and they exist as limited space and time; actual motion is progression through a limited space and a limited time and not through infinite space and infinite time.
That what is in motion must reach the half is the assertion of continuity, i.e. the possibility of division as mere possibility; it is thus always possible in every space, however small. It is said that it is plain that the half must be reached, but in so saying, everything is allowed, including the fact that it never will be reached; for to say so in one case, is the same as saying it an infinite number of times. We mean, on the contrary, that in a larger space the half can be allowed, but we conceive that we must somewhere attain to a space so small that no halving is possible, or an indivisible, non-continuous space which is no space. This, however, is false, for continuity is a necessary determination; there is undoubtedly a smallest in space, i.e. a negation of continuity, but the negation is something quite abstract. Abstract adherence to the subdivision indicated, that is, to continuous bisection into infinitude, is likewise false, for in the conception of a half, the interruption of continuity is involved. We must say that there is no half of space, for space is continuous; a piece of wood may be broken into two halves, but not space, and space only exists in movement. It might equally be said that space consists of an endless number of points, i.e. of infinitely many limits and thus cannot be traversed. Men think themselves able to go from one indivisible point to another, but they do not thereby get any further, for of these there is an unlimited number. Continuity is split up into its opposite, a number which is indefinite; that is to say, if continuity is not admitted, there is no motion. It is false to assert that it is possible when one is reached, or that which is not continuous; for motion is connection. Thus when it was said that continuity is the presupposed possibility of infinite division, continuity is only the hypothesis; but what is exhibited in this continuity is the being of infinitely many, abstractly absolute limits.
(b) The second proof, which is also the presupposition of continuity and the manifestation of division, is called “Achilles, the Swift.” The ancients loved to clothe difficulties in sensuous representations. Of two bodies moving in one direction, one of which is in front and the other following at a fixed distance and moving quicker than the first, we know that the second will overtake the first. But Zeno says, “The slower can never be overtaken by the quicker.” And he proves it thus: “The second one requires a certain space of time to reach the place from which the one pursued started at the beginning of the given period.” Thus during the time in which the second reached the point where the first was, the latter went over a new space which the second has again to pass through in a part of this period; and in this way it goes into infinity.
c d e f g B A c d e f g B A B, for instance, traverses two miles (c d) in an hour, A in the same time, one mile (d e); if they are two miles (c d) removed from one another, B has in one hour come to where A was at the beginning of the hour. While B, in the next half hour, goes over the distance crossed by A of one mile (d e), A has got half a mile (e f) further, and so on into infinity. Quicker motion does not help the second body at all in passing over the interval of space by which he is behind: the time which he requires, the slower body always has at its avail in order to accomplish some, although an ever shorter advance; and this, because of the continual division, never quite disappears.
Aristotle, in speaking of this, puts it shortly thus. “This proof asserts the same endless divisibility, but it is untrue, for the quick will overtake the slow body if the limits to be traversed be granted to it.” This answer is correct and contains all that can be said; that is, there are in this representation two periods of time and two distances, which are separated from one another, i.e. they are limited in relation to one another; when, on the contrary, we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical. In ordinary language we solve the matter in the easiest way, for we say: “Because the second is quicker, it covers a greater distance in the same time as the slow; it can therefore come to the place from which the first started and get further still.” After B, at the end of the first hour, arrives at d and A at e, A in one and the same period, that is, in the second hour, goes over the distance e g, and B the distance d g. But this period of time which should be one, is divisible into that in which B accomplishes d e and that in which B passes through e g. A has a start of the first, by which it gets over the distance e f, so that A is at f at the same period as B is at e. The limitation which, according to Aristotle, is to be overcome, which must be penetrated, is thus that of time; since it is continuous, it must, for the solution of the difficulty, be said that what is divisible into two spaces of time is to be conceived of as one, in which B gets from d to e and from e to g, while A passes over the distance e g. In motion two periods, as well as two points in space, are indeed one.
If we wish to make motion clear to ourselves, we say that the body is in one place and then it goes to another; because it moves, it is no longer in the first, but yet not in the second; were it in either it would be at rest. Where then is it? If we say that it is between both, this is to convey nothing at all, for were it between both, it would be in a place, and this presents the same difficulty. But movement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first makes motion possible. Zeno, in the deduction made by him, brought both these points into forcible opposition. The discretion of space and time we also uphold, but there must also be granted to them the overstepping of limits, i.e. the exhibition of limits as not being, or as being divided periods of time, which are also not divided. In our ordinary ideas we find the same determinations as those on which the dialectic of Zeno rests; we arrive at saying, though unwillingly, that in one period two distances of space are traversed, but we do not say that the quicker comprehends two moments of time in one; for that we fix a definite space. But in order that the slower may lose its precedence, it must be said that it loses its advantage of a moment of time, and indirectly the moment of space.
Zeno makes limit, division, the moment of discretion in space and time, the only element which is enforced in the whole of his conclusions, and hence results the contradiction. The difficulty is to overcome thought, for what makes the difficulty is always thought alone, since it keeps apart the moments of an object which in their separation are really united. It brought about the Fall, for man ate of the tree of the knowledge of good and evil; but it also remedies these evils.
(c) The third form, according to Aristotle, is as follows:—Zeno says: “The flying arrow rests, and for the reason that what is in motion is always in the self-same Now and the self-same Here, in the indistinguishable;” it is here and here and here. It can be said of the arrow that it is always the same, for it is always in the same space and the same time; it does not get beyond its space, does not take in another, that is, a greater or smaller space. That, however, is what we call rest and not motion. In the Here and Now, the becoming “other” is abrogated, limitation indeed being established, but only as moment; since in the Here and Now as such, there is no difference, continuity is here made to prevail against the mere belief in diversity. Each place is a different place, and thus the same; true, objective difference does not come forth in these sensuous relations, but in the spiritual.
This is also apparent in mechanics; of two bodies the question as to which moves presents itself before us. It requires more than two places—three at least—to determine which of them moves. But it is correct to say this, that motion is plainly relative; whether in absolute space the eye, for instance, rests, or whether it moves, is all the same. Or, according to a proposition brought forward by Newton, if two bodies move round one another in a circle, it may be asked whether the one rests or both move. Newton tries to decide this by means of an external circumstance, the strain on the string. When I walk on a ship in a direction opposed to the motion of the ship, this is in relation to the ship, motion, and in relation to all else, rest.
In both the first proofs, continuity in progression has the predominance; there is no absolute limit, but an overstepping of all limits. Here the opposite is established; absolute limitation, the interruption of continuity, without however passing into something else; while discretion is presupposed, continuity is maintained. Aristotle says of this proof: “It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow.”
(d) “The fourth proof,” Aristotle continues, “is derived from similar bodies which move in opposite directions in the space beside a similar body, and with equal velocity, one from one end of the space, the other from the middle. It necessarily results from this that half the time is equal to the double of it. The fallacy rests in this, that Zeno supposes that what is beside the moving body, and what is beside the body at rest, move through an equal distance in equal time with equal velocity, which, however, is untrue.”
In a definite space such as a table (A B) let us suppose two bodies of equal length with it and with one another, one of which (C D) lies with one end (C) on the middle (g) of the table, and the other (E F), being in the same direction, has the point (E) only touching the end of the table (h); and supposing they move in opposite directions, and the former (C D) reaches in an hour the end (h) of the table; we have the result ensuing that the one (E F) passes in the half of the time through the same space (i k) which the other does in the double (g h); hence the half is equal to the double. That is to say, this second passes (let us say, in the point l) by the whole of the first C D. In the first half-hour l goes from m to i, while k only goes from g to n.
In the second half-hour l goes past o to k, and altogether passes from m to k, or the double of the distance.
This fourth form deals with the contradiction presented in opposite motion; that which is common is given entirely to one body, while it only does part for itself. Here the distance travelled by one body is the sum of the distance travelled by both, just as when I go two feet east, and from the same point another goes two feet west, we are four feet removed from one another; in the distance moved both are positive, and hence have to be added together. Or if I have gone two feet forwards and two feet backwards, although I have walked four feet, I have not moved from the spot; the motion is then nil, for by going forwards and backwards an opposition ensues which annuls itself.
This is the dialectic of Zeno; he had a knowledge of the determinations which our ideas of space and time contain, and showed in them their contradiction; Kant’s antinomies do no more than Zeno did here. The general result of the Eleatic dialectic has thus become, “the truth is the one, all else is untrue,” just as the Kantian philosophy resulted in “we know appearances only.” On the whole the principle is the same; “the content of knowledge is only an appearance and not truth,” but there is also a great difference present. That is to say, Zeno and the Eleatics in their proposition signified “that the sensuous world, with its multitudinous forms, is in itself appearance only, and has no truth.” But Kant does not mean this, for he asserts: “Because we apply the activity of our thought to the outer world, we constitute it appearance; what is without, first becomes an untruth by the fact that we put therein a mass of determinations. Only our knowledge, the spiritual, is thus appearance; the world is in itself absolute truth; it is our action alone that ruins it, our work is good for nothing.” It shows excessive humility of mind to believe that knowledge has no value; but Christ says, “Are ye not better than the sparrows?” and we are so inasmuch as we are thinking; as sensuous we are as good or as bad as sparrows. Zeno’s dialectic has greater objectivity than this modern dialectic.
Zeno’s dialectic is limited to Metaphysics; later, with the Sophists, it became general. We here leave the Eleatic school, which perpetuates itself in Leucippus and, on the other side, in the Sophists, in such a way that these last extended the Eleatic conceptions to all reality, and gave to it the relation of consciousness; the former, however, as one who later on worked out the Notion in its abstraction, makes a physical application of it, and one which is opposed to consciousness. There are several other Eleatics mentioned, to Tennemann’s surprise, who, however, cannot interest us. “It is so unexpected,” he says (Vol. I., p. 190), “that the Eleatic system should find disciples; and yet Sextus mentions a certain Xeniades.”
D. HERACLITUS.
If we put aside the Ionics, who did not understand the Absolute as Thought, and the Pythagoreans likewise, we have the pure Being of the Eleatics, and the dialectic which denies all finite relationships. Thought to the latter is the process of such manifestations; the world in itself is the apparent, and pure Being alone the true. The dialectic of Zeno thus lays hold of the determinations which rest in the content itself, but it may, in so far, also be called subjective dialectic, inasmuch as it rests in the contemplative subject, and the one, without this movement of the dialectic, is abstract identity. The next step from the existence of the dialectic as movement in the subject, is that it must necessarily itself become objective. If Aristotle blames Thales for doing away with motion, because change cannot be understood from Being, and likewise misses the actual in the Pythagorean numbers and Platonic Ideas, taken as the substances of the things which participate in them, Heraclitus at least understands the absolute as just this process of the dialectic. The dialectic is thus thre-fold: (_α_) the external dialectic, a reasoning which goes over and over again without ever reaching the soul of the thing; (_β_) immanent dialectic of the object, but falling within the contemplation of the subject; (_γ_) the objectivity of Heraclitus which takes the dialectic itself as principle. The advance requisite and made by Heraclitus is the progression from Being as the first immediate thought, to the category of Becoming as the second. This is the first concrete, the Absolute, as in it the unity of opposites. Thus with Heraclitus the philosophic Idea is to be met with in its speculative form; the reasoning of Parmenides and Zeno is abstract understanding. Heraclitus was thus universally esteemed a deep philosopher and even was decried as such. Here we see land; there is no proposition of Heraclitus which I have not adopted in my Logic.
Diogenes Laertius says (IX. 1) that Heraclitus flourished about the 69th Olympiad (500 B.C.), and that he was of Ephesus and in part contemporaneous with Parmenides: he began the separation and withdrawal of philosophers from public affairs and the interests of the country, and devoted himself in his isolation entirely to Philosophy. We have thus three stages: (_α_) the seven sages as statesmen, regents and law-givers; (_β_) the Pythagorean aristocratic league; (_γ_) an interest in science for its own sake. Little more is known of Heraclitus’ life than his relations to his countrymen the Ephesians, and according to Diogenes Laertius (IX. 15, 3), these were for the most part found in the fact that they despised him and were yet more profoundly despised by him—a relationship such as we have now-a-days, when each man exists for himself, and despises everyone else. In the case of this noble character, the disdain and sense of separation from the crowd emanates from the deep sense of the perversity of the ordinary ideas and life of his people: in reference to this, isolated expressions used on various occasions are still preserved. Cicero (Tusc. Quæst. V. 36) and Diogenes Laertius (IX. 2) relate that Heraclitus said: “The Ephesians all deserve to have their necks broken as they grow up, so that the town should be left to minors” (people now say that only youth knows how to govern), “because they drove away his friend Hermodorus, the best of them all, and gave as their reason for so doing that amongst them none should be more excellent than the rest; and if any one were so, it should be elsewhere and amongst others.” It was for the same reason that in the Athenian Democracy great men were banished. Diogenes adds: “His fellow-citizens asked him to take part in the administration of public affairs, but he declined, because he did not like their constitution, laws and administration.” Proclus (T. III. pp. 115, 116, ed. Cousin) says: “The noble Heraclitus blamed the people for being devoid of understanding or thought. ‘What is,’ he says, ‘their understanding or their prudence? Most of them are bad, and few are good.’” Diogenes Laertius (IX. 6) furthermore says: “Antisthenes cites, as a proof of Heraclitus’ greatness, that he left his kingdom to his brother.” He expresses in the strongest manner his contempt for what is esteemed to be truth and right, in the letter preserved to us by Diogenes (IX. 13, 14), in which, to the invitation of Darius Hystaspes, “to make him acquainted with Greek wisdom—for his work on Nature contains a very forcible theory of the world, but it is in many passages obscure—to come to him and explain to him what required explanation” (this is certainly not very probable if Heraclitus’ turn of mind was also Oriental), he is said to have replied: “All mortal men depart from truth and justice and are given over to excess and vain opinions according to their evil understandings. But I, since I have attained to an oblivion of all evil, and shun the overpowering envy that follows me, and the vanity of high position, shall not come to Persia. I am content with little and live in my own way.”
The only work that he wrote, and the title of which, Diogenes tells us, was by some stated to be “The Muses” and by others “On Nature,” he deposited in the temple of Diana at Ephesus. It seems to have been preserved until modern times; the fragments which have come down to us are collected together in Stephanus’ _Poësis philosophica_ (p. 129, seq.). Schleiermacher also collected them and arranged them in a characteristic way. The title is “Heraclitus, the Dark, of Ephesus, as represented in fragments of his work and by the testimony of the ancients,” and it is to be found in Wolf and Buttmann’s “Museum of ancient Learning,” vol. I. (Berlin, 1807) pp. 315-533. Seventy-three passages are given. Kreuzer made one hope that he would work at Heraclitus more critically and with a knowledge of the language. He made a more complete collection, particularly from grammarians; however, as, for lack of time, he left it to be worked up by a younger scholar, and as the latter died, it never came before the public. Compilations of the kind are as a rule too copious: they contain a mass of learning and are more easily written than read. Heraclitus has been considered obscure, and is indeed celebrated for this; it also drew upon him the name of _σκοτεινός_. Cicero (De Nat. Deor. I. 26; III. 14; De Finib. II. 5) takes up a wrong idea, as often happens to him; he thinks that Heraclitus purposely wrote obscurely. Any such design would, however, be a very shallow one, and it is really nothing but the shallowness of Cicero himself ascribed by him to Heraclitus. Heraclitus’ obscurity is rather a result of neglecting proper composition and of imperfect language; this is what was thought by Aristotle (Rhet. III. 5), who, from a grammatical point of view, ascribed it to a want of punctuation: “We do not know whether a word belongs to what precedes or what succeeds.” Demetrius is of the same opinion (De Elocutione, § 192, p. 78, ed. Schneider). Socrates, as Diogenes Laertius relates (II. 22; IX. 11-12), said of this book: “What he understood of it was excellent, and what he did not understand he believed to be as good, but it requires a vigorous (_Δηλίου_) swimmer to make his way through it.” The obscurity of this philosophy, however, chiefly consists in there being profound speculative thought contained in it; the Notion, the Idea, is foreign to the understanding and cannot be grasped by it, though it may find mathematics quite simple.