Our Knowledge of the External World as a Field for Scientific Method in Philosophy Part 8

"I see, Parmenides, said Socrates, that Zeno is your second self in his writings too; he puts what you say in another way, and would fain deceive us into believing that he is telling us what is new. For you, in your poems, say All is one, and of this you adduce excellent proofs; and he on the other hand says There is no Many; and on behalf of this he offers overwhelming evidence. To deceive the world, as you have done, by saying the same thing in different ways, one of you affirming the one, and the other denying the many, is a strain of art beyond the reach of most of us.

"Yes, Socrates, said Zeno. But although you are as keen as a Spartan hound in pursuing the track, you do not quite apprehend the true motive of the composition, which is not really such an ambitious work as you imagine; for what you speak of was an accident; I had no serious intention of deceiving the world. The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who scoff at him and show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is an address to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of the many if carried out appears in a still more ridiculous light than the hypothesis of the being of the one."

[31] _Parmenides_, 128 A-D.

Zeno's four arguments against motion were intended to exhibit the contradictions that result from supposing that there is such a thing as change, and thus to support the Parmenidean doctrine that reality is unchanging.[32] Unfortunately, we only know his arguments through Aristotle,[33] who stated them in order to refute them. Those philosophers in the present day who have had their doctrines stated by opponents will realise that a just or adequate presentation of Zeno's position is hardly to be expected from Aristotle; but by some care in interpretation it seems possible to reconstruct the so-called "sophisms"

which have been "refuted" by every tyro from that day to this.

[32] This interpretation is combated by Milhaud, _Les philosophes-geometres de la Grece_, p. 140 n., but his reasons do not seem to me convincing. All the interpretations in what follows are open to question, but all have the support of reputable authorities.

[33] _Physics_, vi. 9. 2396 (R.P. 136-139).

Zeno's arguments would seem to be "ad hominem"; that is to say, they seem to a.s.sume premisses granted by his opponents, and to show that, granting these premisses, it is possible to deduce consequences which his opponents must deny. In order to decide whether they are valid arguments or "sophisms," it is necessary to guess at the tacit premisses, and to decide who was the "h.o.m.o" at whom they were aimed.

Some maintain that they were aimed at the Pythagoreans,[34] while others have held that they were intended to refute the atomists.[35] M.

Evellin, on the contrary, holds that they const.i.tute a refutation of infinite divisibility,[36] while M. G. Noel, in the interests of Hegel, maintains that the first two arguments refute infinite divisibility, while the next two refute indivisibles.[37] Amid such a bewildering variety of interpretations, we can at least not complain of any restrictions on our liberty of choice.

[34] _Cf._ Gaston Milhaud, _Les philosophes-geometres de la Grece_, p. 140 n.; Paul Tannery, _Pour l'histoire de la science h.e.l.lene_, p. 249; Burnet, _op. cit._, p. 362.

[35] _Cf._ R. K. Gaye, "On Aristotle, _Physics_, Z ix." _Journal of Philology_, vol. x.x.xi., esp. p. 111. Also Moritz Cantor, _Vorlesungen uber Geschichte der Mathematik_, 1st ed., vol. i., 1880, p. 168, who, however, subsequently adopted Paul Tannery's opinion, _Vorlesungen_, 3rd ed. (vol. i. p. 200).

[36] "Le mouvement et les partisans des indivisibles," _Revue de Metaphysique et de Morale_, vol. i. pp. 382-395.

[37] "Le mouvement et les arguments de Zenon d'elee," _Revue de Metaphysique et de Morale_, vol. i. pp. 107-125.

The historical questions raised by the above-mentioned discussions are no doubt largely insoluble, owing to the very scanty material from which our evidence is derived. The points which seem fairly clear are the following: (1) That, in spite of MM. Milhaud and Paul Tannery, Zeno is anxious to prove that motion is really impossible, and that he desires to prove this because he follows Parmenides in denying plurality;[38]

(2) that the third and fourth arguments proceed on the hypothesis of indivisibles, a hypothesis which, whether adopted by the Pythagoreans or not, was certainly much advocated, as may be seen from the treatise _On Indivisible Lines_ attributed to Aristotle. As regards the first two arguments, they would seem to be valid on the hypothesis of indivisibles, and also, without this hypothesis, to be such as would be valid if the traditional contradictions in infinite numbers were insoluble, which they are not.

[38] _Cf._ M. Brochard, "Les pretendus sophismes de Zenon d'elee,"

_Revue de Metaphysique et de Morale_, vol. i. pp. 209-215.

We may conclude, therefore, that Zeno's polemic is directed against the view that s.p.a.ce and time consist of points and instants; and that as against the view that a finite stretch of s.p.a.ce or time consists of a finite number of points and instants, his arguments are not sophisms, but perfectly valid.

The conclusion which Zeno wishes us to draw is that plurality is a delusion, and s.p.a.ces and times are really indivisible. The other conclusion which is possible, namely, that the number of points and instants is infinite, was not tenable so long as the infinite was infected with contradictions. In a fragment which is not one of the four famous arguments against motion, Zeno says:

"If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number.

"If things are a many, they will be infinite in number; for there will always be other things between them, and others again between these. And so things are infinite in number."[39]

[39] Simplicius, _Phys._, 140, 28 D (R.P. 133); Burnet, _op. cit._, pp. 364-365.

This argument attempts to prove that, if there are many things, the number of them must be both finite and infinite, which is impossible; hence we are to conclude that there is only one thing. But the weak point in the argument is the phrase: "If they are just as many as they are, they will be finite in number." This phrase is not very clear, but it is plain that it a.s.sumes the impossibility of definite infinite numbers. Without this a.s.sumption, which is now known to be false, the arguments of Zeno, though they suffice (on certain very reasonable a.s.sumptions) to dispel the hypothesis of finite indivisibles, do not suffice to prove that motion and change and plurality are impossible.

They are not, however, on any view, mere foolish quibbles: they are serious arguments, raising difficulties which it has taken two thousand years to answer, and which even now are fatal to the teachings of most philosophers.

The first of Zeno's arguments is the argument of the race-course, which is paraphrased by Burnet as follows:[40]

"You cannot get to the end of a race-course. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on _ad infinitum_, so that there are an infinite number of points in any given s.p.a.ce, and you cannot touch an infinite number one by one in a finite time."[41]

[40] _Op. cit._, p. 367.

[41] Aristotle's words are: "The first is the one on the non-existence of motion on the ground that what is moved must always attain the middle point sooner than the end-point, on which we gave our opinion in the earlier part of our discourse." _Phys._, vi. 9. 939B (R.P.

136). Aristotle seems to refer to _Phys._, vi. 2. 223AB [R.P. 136A]: "All s.p.a.ce is continuous, for time and s.p.a.ce are divided into the same and equal divisions.... Wherefore also Zeno's argument is fallacious, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time. For there are two senses in which the term 'infinite' is applied both to length and to time, and in fact to all continuous things, either in regard to divisibility, or in regard to the ends. Now it is not possible to touch things infinite in regard to number in a finite time, but it is possible to touch things infinite in regard to divisibility: for time itself also is infinite in this sense. So that in fact we go through an infinite, [s.p.a.ce] in an infinite [time] and not in a finite [time], and we touch infinite things with infinite things, not with finite things." Philoponus, a sixth-century commentator (R.P. 136A, _Exc.

Paris Philop. in Arist. Phys._, 803, 2. Vit.), gives the following ill.u.s.tration: "For if a thing were moved the s.p.a.ce of a cubit in one hour, since in every s.p.a.ce there are an infinite number of points, the thing moved must needs touch all the points of the s.p.a.ce: it will then go through an infinite collection in a finite time, which is impossible."

Zeno appeals here, in the first place, to the fact that any distance, however small, can be halved. From this it follows, of course, that there must be an infinite number of points in a line. But, Aristotle represents him as arguing, you cannot touch an infinite number of points _one by one_ in a finite time. The words "one by one" are important. (1) If _all_ the points touched are concerned, then, though you pa.s.s through them continuously, you do not touch them "one by one." That is to say, after touching one, there is not another which you touch next: no two points are next each other, but between any two there are always an infinite number of others, which cannot be enumerated one by one. (2) If, on the other hand, only the successive middle points are concerned, obtained by always halving what remains of the course, then the points are reached one by one, and, though they are infinite in number, they are in fact all reached in a finite time. His argument to the contrary may be supposed to appeal to the view that a finite time must consist of a finite number of instants, in which case what he says would be perfectly true on the a.s.sumption that the possibility of continued dichotomy is undeniable. If, on the other hand, we suppose the argument directed against the partisans of infinite divisibility, we must suppose it to proceed as follows:[42] "The points given by successive halving of the distances still to be traversed are infinite in number, and are reached in succession, each being reached a finite time later than its predecessor; but the sum of an infinite number of finite times must be infinite, and therefore the process will never be completed." It is very possible that this is historically the right interpretation, but in this form the argument is invalid. If half the course takes half a minute, and the next quarter takes a quarter of a minute, and so on, the whole course will take a minute. The apparent force of the argument, on this interpretation, lies solely in the mistaken supposition that there cannot be anything beyond the whole of an infinite series, which can be seen to be false by observing that 1 is beyond the whole of the infinite series 1/2, 3/4, 7/8, 15/16, ...

[42] _Cf._ Mr C. D. Broad, "Note on Achilles and the Tortoise,"

_Mind_, N.S., vol. xxii. pp. 318-9.

The second of Zeno's arguments is the one concerning Achilles and the tortoise, which has achieved more notoriety than the others. It is paraphrased by Burnet as follows:[43]

"Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it."[44]

[43] _Op. cit._

[44] Aristotle's words are: "The second is the so-called Achilles. It consists in this, that the slower will never be overtaken in its course by the quickest, for the pursuer must always come first to the point from which the pursued has just departed, so that the slower must necessarily be always still more or less in advance." _Phys._, vi. 9. 239B (R.P. 137).

This argument is essentially the same as the previous one. It shows that, if Achilles ever overtakes the tortoise, it must be after an infinite number of instants have elapsed since he started. This is in fact true; but the view that an infinite number of instants make up an infinitely long time is not true, and therefore the conclusion that Achilles will never overtake the tortoise does not follow.

The third argument,[45] that of the arrow, is very interesting. The text has been questioned. Burnet accepts the alterations of Zeller, and paraphrases thus:

"The arrow in flight is at rest. For, if everything is at rest when it occupies a s.p.a.ce equal to itself, and what is in flight at any given moment always occupies a s.p.a.ce equal to itself, it cannot move."

[45] _Phys._, vi. 9. 239B (R.P. 138).

But according to Prantl, the literal translation of the unemended text of Aristotle's statement of the argument is as follows: "If everything, when it is behaving in a uniform manner, is continually either moving or at rest, but what is moving is always in the _now_, then the moving arrow is motionless." This form of the argument brings out its force more clearly than Burnet's paraphrase.

Here, if not in the first two arguments, the view that a finite part of time consists of a finite series of successive instants seems to be a.s.sumed; at any rate the plausibility of the argument seems to depend upon supposing that there are consecutive instants. Throughout an instant, it is said, a moving body is where it is: it cannot move during the instant, for that would require that the instant should have parts.

Thus, suppose we consider a period consisting of a thousand instants, and suppose the arrow is in flight throughout this period. At each of the thousand instants, the arrow is where it is, though at the next instant it is somewhere else. It is never moving, but in some miraculous way the change of position has to occur _between_ the instants, that is to say, not at any time whatever. This is what M. Bergson calls the cinematographic representation of reality. The more the difficulty is meditated, the more real it becomes. The solution lies in the theory of continuous series: we find it hard to avoid supposing that, when the arrow is in flight, there is a _next_ position occupied at the _next_ moment; but in fact there is no next position and no next moment, and when once this is imaginatively realised, the difficulty is seen to disappear.

The fourth and last of Zeno's arguments is[46] the argument of the stadium.

[46] _Phys._, vi. 9. 239B (R.P. 139).

The argument as stated by Burnet is as follows:

First Position. Second Position.

A .... A ....

B .... B ....

C .... C ....

"Half the time may be equal to double the time. Let us suppose three rows of bodies, one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions. By the time they are all in the same part of the course, B will have pa.s.sed twice as many of the bodies in C as in A. Therefore the time which it takes to pa.s.s C is twice as long as the time it takes to pa.s.s A. But the time which B and C take to reach the position of A is the same. Therefore double the time is equal to the half."

Gaye[47] devoted an interesting article to the interpretation of this argument. His translation of Aristotle's statement is as follows:

"The fourth argument is that concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, pa.s.sing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the s.p.a.ce between the goal and the middle point of the course, and the other that between the middle point and the starting-post. This, he thinks, involves the conclusion that half a given time is equal to double the time. The fallacy of the reasoning lies in the a.s.sumption that a body occupies an equal time in pa.s.sing with equal velocity a body that is in motion and a body of equal size that is at rest, an a.s.sumption which is false. For instance (so runs the argument), let A A ... be the stationary bodies of equal size, B B ... the bodies, equal in number and in size to A A ..., originally occupying the half of the course from the starting-post to the middle of the A's, and C C ... those originally occupying the other half from the goal to the middle of the A's, equal in number, size, and velocity, to B B ... Then three consequences follow. First, as the B's and C's pa.s.s one another, the first B reaches the last C at the same moment at which the first C reaches the last B. Secondly, at this moment the first C has pa.s.sed all the A's, whereas the first B has pa.s.sed only half the A's and has consequently occupied only half the time occupied by the first C, since each of the two occupies an equal time in pa.s.sing each A. Thirdly, at the same moment all the B's have pa.s.sed all the C's: 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 pa.s.sing each of the B's is equal to that occupied by it in pa.s.sing each of the A's, because an equal time is occupied by both the first B and the first C in pa.s.sing all the A's. This is the argument: but it presupposes the aforesaid fallacious a.s.sumption."

[47] _Loc. cit._

First Position. Second Position.