1 Arist. Phys. Z, 9. 239 b 9 sqq.
2 Cf. Diog. ix. 25 (R. P. 130).
3 Plato, Parm. 128 c (R. P. 130 d).
4 The technical terms used in Plato’s Parmenides seem to be as ol Zeno himself. The ὑπόθεσις is the provisional assumption of the truth a certain statement, and takes the form εἰ πολλά ἐστι or the like. The word does not mean the assumption of something as a foundation, but the setting before one’s self of a statement asa problem to be solved (Ionic ὑποθέσθαι, Attic προθέσθαι). If the conclusions which necessarily follow : from the ὑπόθεσις (τὰ συμβαίνοντα) are impossible, the ὑπόθεσις is ** destroyed” (cf. Plato, Rep. 533 c 8, τὰς ὑποθέσεις ἀναιροῦσα). The , author of the Περὶ ἀρχαίης ἰατρικῆς (c 1) knows the word ὑπόθεσις in a similar sense.
362 Early Greek Philosophy
theory of Parmenides had led to conclusions which
contradicted the evidence of the senses, and Zeno’s
object was not to bring fresh proofs of the theory
itself, but simply to show that his opponents’ view
led to contradictions of a precisely similar nature.
Zeno and 158. That Zeno’s dialectic was mainly directed
Pythagorean- : . '
τς against the Pythagoreans is certainly suggested by
Plato’s statement, that it was addressed to the adversaries of Parmenides, who held that things were ‘a many.”' Zeller holds, indeed, that it was merely the popular form of the belief that things are many that Zeno set himself to confute ; 2 but it is surely not true that ordinary people believe things to be “a many” in the sense required. Plato tells us that the premisses of Zeno’s arguments were the beliefs of the adversaries of Parmenides, and the postulate from which all his contradictions are derived is the view that space, and therefore body, is made up of a number of discrete units, which is just the Pythagorean doctrine. Nor is it at all probable that Anaxagoras is “aimed αἱ. We know from Plato that Zeno’s book was the work of his youth.* Suppose even that it was written when he was thirty, that is to say, about 459 B.c., Anaxagoras had just taken up his abode at Athens at that time,” and it is very unlikely that Zeno had ever heard of him. There is, on the other hand, a great deal to be said for the view that Anaxagoras had read the work of Zeno, and that his emphatic adhesion to the doctrine
1 The view that Zeno’s arguments were directed against Pythagoreanism has been maintained in recent times by Tannery (Sczence helléne, pp. 249 sqq.), and Baiumker (Das Problem der Materie, pp. 60 sqq.).
2 Zeller, p. 589 (Eng. trans. p. 612).
3 This is the view of Stallbaum in his edition of the Parmenzdes (pp. 25 544.)
4 Parm., loc, cit. 5 Chap. VI. § 120.
The Younger Eleatics 363
of infinite divisibility was due to the criticism of his younger contemporary.’
It will be noted how much clearer the historical
position of Zeno becomes if we follow Plato in assign-
ing him to a somewhat later date than is usual. We
have first Parmenides, then the pluralists, and then the
criticism of Zeno. This, at any rate, seems to have
been the view which Aristotle took of the historical
development.”
159. The polemic of Zeno is clearly directed in
the first instance against a certain view of the unit.
Eudemos, in his Physics,> quoted from him the saying
that “if any one could tell him what the one was, he
would be able to say what things are.” The com-
mentary of Alexander on this, preserved by Simplicius,*
is quite satisfactory. “As Eudemos relates,” he says,
“Zeno the disciple of Parmenides tried to show that
it was impossible that things could be a many, seeing
that there was no unit in things, whereas ‘many’
means a number of units.” Here we have a clear refer-
ence to the Pythagorean view that everything may be
reduced to a sum of units, which is what Zeno denied.”
1 Cf. for instance Anaxagoras, fr. 3, with Zeno, fr. 2; and Anaxagoras, fr. 5, with Zeno, fr. 3.
® Arist. Phys. A, 3. 187 a 1 (R. P. 134 b). See below, § 173.
8. Simpl. Phys. p. 138, 32 (R. P. 134 a).
* Simpl. Phys. p. 99, 13, ὡς γὰρ ἱστορεῖ, φησίν (᾿Αλέξανδρος), Eddnuos, Ζήνων ὁ Παρμενίδου γνώριμος ἐπειρᾶτο δεικνύναι ὅτι μὴ οἷόν re τὰ ὄντα πολλὰ εἶναι τῷ μηδὲν εἶναι ἐν τοῖς οὖσιν ἕν, τὰ δὲ πολλὰ πλῆθος εἶναι ἑνάδων. This is the meaning of the statement that Zeno ἀνήρει τὸ ἕν, which is not Alexander’s (as implied in R. P. 134 a), but goes back to no less an authority than Eudemos. It is perfectly correct when read in
connexion with the words τὴν yap στιγμὴν ὡς τὸ ὃν λέγει (Simpl. PAys. p- 99, 11).
5 It is quite in order that Mr. Bertrand Russell, from the standpoint of pluralism, should accept Zeno’s arguments as ‘‘ immeasurably subtle and profound” (Principles of Mathematics, p. 347). We know from Pilato, however, that Zeno meant them as a veductio ad absurdum of pluralism.
What is the unit,?
The Fragments.
364 Early Greek Philosophy
160. The fragments of Zeno himself also show that
this was his line of argument. I give them according
to the arrangement of Diels.
If the one had no magnitude, it would not even be. . . . But, if it is, each one must have a certain magnitude and a certain thickness, and must be at a certain distance from another, and the same may be said of what is in front of it; for it, too, will have magnitude, and something will be in front of it! It is all the same to say this once and to say it always ; for no such part of it will be the last, nor will one thing not be-compared with another.? So, if things are a many, they must be both small and great, so small as not to have any magnitude at all, and so great as to be infinite. R. P. 134.
For if it were added to any other thing it would not make it any larger; for nothing can gain in magnitude by the addition of what has no magnitude, and thus it follows at once that what was added was nothing.® But if, when this is taken away from another thing, that thing is no less; and again, if, when it is added to another thing, that does not increase, it is plain that what was added was nothing, and what was taken away was nothing. R. P. 132.
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.
1 I formerly rendered ‘‘the same may be said of what surpasses it in smallness ; for it too will have magnitude, and something will surpass it in smallness.”” This is Tannery’s rendering, but I now agree with Diels in thinking that ἀπέχειν refers to μέγεθος and προέχειν to πάχος. Zeno is showing that the Pythagorean point has really three dimensions.
2 Reading, with Diels and the MSS., οὔτε ἕτερον πρὸς ἕτερον οὐκ ἔσται. Gomperz’s conjecture (adopted in R. P.) seems to me arbitrary.
8. Zeller marks a lacuna here. Zeno must certainly have shown that the subtraction of a point does not make a thing less; but he may have done so before the beginning of our present fragment.
Ἐπ υ le ie.
eG eee ee eee eee eee ee eee eel eee iid
The Younger Eleatics 365
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. moe, 133.7
161. If we hold that the unit has no magnitude—
and this is required by what Aristotle calls the argu-
ment from dichotomy,’—then everything must be in-
finitely small. Nothing made up of units without
magnitude can itself have any magnitude. On the
other hand, if we insist that the units of which things
are ΕΝ are something and not nothing, we must
hold that everything is infinitely great. The line is
infinitely divisible ; and, according to this view, it will
be made up of an infinite number of units, each of
which has some magnitude. |
That this argument refers to points is proved by an
instructive passage from Aristotle’s Metaphysics? We
read there— :
If the unit is indivisible, it will, according to the pro- position of Zeno, be nothing. That which neither makes anything larger by its addition ‘to it, nor smaller by its sub-
_ traction from it, is not, he says, a real thing at all; for clearly
what is real must be a magnitude. And, if it is a magnitude, it is corporeal ; for that is corporeal which is in every dimen- sion. ‘The other things, z.e. the plane and the line, if added in one way will make things larger, added in another they will produce no effect ; but the point and the unit cannot make things larger in any way.
From all this it seems impossible to draw any other
1 This is what Aristotle calls ‘‘the argument from dichotomy” (PAys. A, 3. 187 a1; R. P. 134 b). Ifa line is made up of points, we ought to be able to answer the question, ‘‘ How many points are there in a given line?” On the other hand, you can always divide a line or any part of it into two halves; so that, if a line is made up of points, there will always be more of them than any number you assign.
2 See last note: 3 Arist. AZet. B, 4. 1001 Ὁ 7.
The unit.
Space.
Motion.
366 Early Greek Philosophy
conclusion than that the “one” against which Zeno
argued was the “one” of which a number constitute a
“many,” and that is just the Pythagorean unit.
162. Aristotle refers to an argument which seems
to be directed against the Pythagorean doctrine of
space,’ and Simplicius quotes it in this form :”
If there is space, it will be in something ; for all that is is in something, and what is in something is in space. So space will be in space, and this goes on ad infinitum, therefore there is no space. R. P. 135.
What Zeno is really arguing against here is the
attempt to distinguish space from the body that.
occupies it. If we insist that body must be 222 space,
then we must go on to ask what space itself is in.
This is a “reinforcement” of the Parmenidean denial
of the void. Possibly the argument that everything
must be “in” something, or must have something
beyond it, had been used against the Parmenidean
theory of a finite sphere with nothing outside it.
163. Zeno’s arguments on the subject of motion
have been preserved by Aristotle himself. The system
of Parmenides made all motion impossible, and his
successors had been driven to abandon the monistic
hypothesis in order to avoid this very consequence.
Zeno does not bring any fresh proofs of the im-
possibility of motion; all he does is to show that a
pluralist theory, such as the Pythagorean, is just as
unable to explain it as was that of Parmenides.
Looked at in this way, Zeno’s arguments are no mere
1 Arist. Phys. A, 1. 209 a 23; 3. 210 b 22 (R. P. 135 a).
‘2 Simpl. Phys. p. 562, 3 (R. P. 135). The version of Eudemos is given in Simpl. Phys. p . 563, 26, ἀξιοῖ yap πᾶν τὸ ὃν ποῦ εἶναι " εἰ δὲ ὁ τόπος τῶν ὄντων, ποῦ ἂν εἴη ; οὐκοῦν ἐ ἄλλῳ τόπῳ κἀκεῖνος δὴ ἐν ἄλλῳ καὶ οὕτως εἰς τὸ πρόσω.
The Younger Eleatics 367
quibbles, but mark a great advance in the conception of quantity. They are as follows :-—
(1) 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 space, and you cannot touch an infinite number one by one in a finite time.” |
(2) 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 [1.3
The “hypothesis” of the second argument is the
same as that in the first, namely, that the line is a
_ series of points; but the reasoning is complicated by
the introduction of another moving object. The
difference, accordingly, is not a half every time, but
diminishes in a constant ratio. Again, the first
argument shows that no moving object can ever
traverse any distance at all, however fast it may move ;
the second emphasises the fact that, however slowly
it moves, it will traverse an infinite distance.