The Cambridge Companion to Early Greek Philosophy
Page 22
Aristotle solves the paradox by denying that time is made up of “nows.” Since time is continuous, there are no adjacent instants. For any two instants t1 and t2, there is another instant, t3, between them. Since motion is continuous too, there is a one-to-one correspondence between the instants during the arrow’s flight and the positions it occupies during its flight. The arrow moves from d(t1) to d(t2) during the interval between t1 and t2; for any position d(t3) on the arrow’s path between d(t1) and d(t2) there is a time t3 between t1 and t2 at which the arrow is at d(t3), and for any time t3 between t1 and t2, there is a position d(t3) on the arrow’s path between d(t1) and d(t2) which the arrow occupies at t3. So motion does not involve instantaneous jumps from place to place or from time to time.
This answer is helpful but only to a point. If space, time, and motion are continuous, they are not composed of minimal units of some definite size, as B1 (p. 138) demonstrates. They are not composed of units of zero size, either, according to B2 (p. 138). According to Aristotle, they are composed of intervals, not points; as we have seen, for Aristotle motion involves moving some interval of distance in some interval of time. But the solution of the previous problem, that in a sense there can be motion at an instant, implies that in a sense time, space, and motion are composed of (a continuum of) points. Not only can we speak coherently of velocity at instant t1, we can also speak coherently of motion over the time interval from t1 to t2 as the sum of motions at all the instants from t1 to t2 If we know what the velocity is at each point, we can determine the motion over the entire interval by taking the definite integral of the velocity over the interval from t1 to t2.
But now something like the original problem recurs. Since motion involves being at different places at different times, there still remains the problem of how the arrow gets from one place to another, or, for that matter, from one time to another. It is not a matter of jumping from one place or time to the next, for in a continuous stretch no point is “next” to another. But it is still a matter of getting from d(t1) to d(t2) through all the intervening positions and from t1 to t2 through all the intervening times.
The answer is that it does the former by being in all the intervening positions and it does the latter by being at all the intervening times. For the arrow to move continuously from d(t1) to d(t2) over the time interval from d(t1) to d(t2) is a matter of its occupying different positions from d(t1) to d(t2) at all the different times during t1–t2 continuously, without any periods of rest and without changing the direction of motion. Likewise, moving throughout the time interval from t1 to t2 is a matter of moving during all the different times during the interval from t1 to t2 Since motion takes place, strictly speaking, over intervals of space and time and only derivatively at points and instants, it follows that if the arrow moves continuously over the spatial interval from d(t1) to d(t2), it gets through all the intervening positions. Likewise, if it moves continuously during the time interval from t1 to t2, it gets through all the intervening times.
At the level of the individual points and instants, the answer is that for the arrow to be moving (to have a velocity unequal to zero) when it is occupying a given position d(t) is for it to be moving over some interval of space which includes d(t). Also, for the arrow to be moving at instant t is for it to be moving in some interval of time which includes t. Again, the problem of how the arrow gets from one position to another or from one time to another during its flight is solved by pointing out that that is precisely what it means for the arrow to be flying.
THE PARADOXES: CONCLUSION
I have selected some of these paradoxes for their historical and philosophical importance, and others because they seem to display characteristic features of Zeno’s way of thinking. Other paradoxes survive, each with its own peculiar twists. But by now enough material is at hand to bring this sketch to a conclusion.
The arguments considered here attack plurality and motion. Another argument attacks the reliability of the sense of hearing and another attacks the belief that things have locations. Since ordinary people (as opposed, perhaps, to some philosophers) believe in plurality and motion, rely (at least to some extent) on the senses, and think that some things have locations, there is some reason to think that Zeno directed his paradoxes against ordinary views about the world, and I have presented the paradoxes along these lines. Plato’s statement that Zeno’s book “is a defence of Parmenides’ argument against those who make fun of it, saying that if there is One, the argument has many ridiculous consequences that contradict it” (Parm. 128c-d) does not conflict with this view. Ordinary unphilosophical people who heard Parmenides’ poem would very likely ridicule it for these very reasons.
At one time it was commonly believed that Zeno composed his paradoxes with particular philosophers and mathematicians in mind, primarily the Pythagoreans. Others have argued that Zeno was out to disprove all possible theories of space, time, and motion; he set out not just to refute the ideas of ordinary people or those of particular philosophers, but· to construct stumbling-blocks for all possible theories of the nature of space, time, and motion. For example, the Dichotomy and the Achilles attack theories that space and time are infinitely divisible and the Flying Arrow attacks theories that space and time are finitely divisible.22 However, the evidence for these views is slim to nonexistent and they have gone out of fashion. What remains is the fact that Zeno attacks common sense, which is not the exclusive prerogative of philosophers.
Plato’s testimony about Zeno is left standing as a viable interpretation. We have found no good reason to doubt that Zeno’s purpose was to support Parmenides in the ways discussed above (pp. 134–36, 143 and n.12). And even though Plato says that all of Zeno’s arguments attack plurality, this need not mean that Plato was unaware of some of the surviving paradoxes, including those directed against motion. I find it plausible that Plato used the statement “all is one” as emblematic of Eleaticism as a whole. Saying that Zeno argued against advocates of plurality then becomes simply a way of expressing what is true – Zeno argued against views that contradict any of the tenets of Eleaticism. If so, then all the surviving paradoxes could come from the book Plato mentions.23
NOTES
1 For a sceptical line on these chronological indications, see Mansfeld [32] 64–68.
2 But we cannot be certain that Zeno ever visited Athens. Diogenes Laertius (IX.28) says he never left Elea. However, Plutarch reports that Zeno instructed Pericles (Pericles 4.3), and Plato (Alcibiades I 119a) says that he earned a good deal of money teaching in Athens, which suggests at least one lengthy visit.
3 How many books did Zeno write? Plato mentions only one, and in a way that discourages us from thinking that there were others. The Suda lists four titles but inspires little confidence; it is even unclear which of the four is the one Plato describes. See Lee [324] 8.
4 For this reading of Zeno, see Barnes [14] 234–35.
5 Proclus, In Parm. 694.23–25; Elias, In Cat. 109.17–30.
6 This count includes the two arguments (not in DK) that Proclus attributes to Zeno in his Commentary on Plato’s Parmenides, 769.22ff. and 862.25ff., translated and discussed in Dillon [327] and Dillon [326] respectively.
7 Barnes [14] 207.
8 See McKirahan [10] 169.
9 Barnes [14] 235.
10 Barnes [14] 236.
11 This is essentially the reconstruction of Cornford [285] 68. For other reconstructions, all of them containing fallacies, see Barnes [14] 237–38, McKirahan [10] 182–83.
12 I follow Owen [338] 46 in this interpretation of Eudemus’ testimony. Others have seen in it a rejection of Parmenidean monism.
13 Simplicius reports that Zeno argued that they are “unlimited in size,” that is, infinitely large (In Phys. 140.34). But the direct quotation from Zeno does not say “in size”; if we stick to the quoted text, we can reconstruct the argument so as to avoid the fallacy.
14 This argument has been seen as fundamental in the origins of fifthcent
ury atomism. On this, see Furley [400] ch. 6 and Taylor, this volume p. 182. For Epicurus’ claim that atoms have theoretically indivisible minimum parts, see Epicurus, Letter to Herodotus 56–59 and Furley [400] chs. 1, 8.
15 Some hold that this argument tells against Parmenides’ monism too, proving that the One can be divided and so is not really one. This conclusion obtains only if Parmenides conceived of the One as spatially extended, which is a reason for adopting an interpretation of Parmenides on which the One is not extended in space. See McKirahan [10] 172–73, but Sedley, this volume p. 121, opts for the other interpretation.
16 The bibliography is immense. For a sample of philosophers’ reactions to Zeno, see especially Russell [339] and [340]; Ryle [341]; Grünbaum [334]; and Salmon [328], as well as the flurry of notes on the paradoxes of motion that appeared in Analysis between 1951 and 1954 (see bibliography [348–55]). Barnes [14] sets the standard for careful logical dissection of the arguments.
17 I shall not discuss the fourth paradox, which is known as “The Stadium” and as “The Moving Rows” and which has been subject to widely differing interpretations. On one interpretation, the paradox is a valid argument against an atomistic conception of time (see Tannery [131]; Lee [324]; Kirk and Raven [4] (1st edn); and Owen [338]); on another it has nothing to do with such a view of time, and commits a gross logical blunder (see Furley [400], KRS [4], and Barnes [14]).
18 There is no evidence that the Dichotomy was half of an antinomy – not that this is decisive; Aristotle’s particular interest in the paradox would have led him to disregard the other limb.
19 The fact that in relativistic mechanics an object in motion shrinks is obviously irrelevant to a historical interpretation of Zeno’s argument.
20 For another reconstruction of the argument, which does not use these premises, see Vlastos [344] 3–18.
21 Although the usual mathematical definition requires t to be one of the endpoints of the interval, this formula is equivalent.
22 For the history of these interpretations of Zeno, see especially Tannery [131]; Cornford [285]; Raven [226]; and Owen [338]; and the criticisms contained in works mentioned in Barnes [14] 617 n.5 and 618 n.6.
23 I presented versions of this chapter at California State University at San Francisco and at the University of Texas at Austin. The final version has benefited from the lively discussion that on both occasions followed the talk, as well as from the comments of Jim Bogen and Sandy Grabiner, the latter of whom was an invaluable aid in mathematical matters.
DANIEL W. GRAHAM
8 Empedocles and Anaxagoras: Responses to Parmenides
There is no question that Parmenides’ poem was a watershed in the history of early Greek philosophy. No serious thinker could ignore his work. And yet it seems to pose insuperable problems for cosmology and scientific inquiry. The first generation to follow Parmenides includes thinkers who wished to continue the tradition of Ionian speculation. But how would they confront Parmenides? What would they make of him and what effect would his arguments have on their work? The first neo-Ionians1, as they have been called, were Empedocles and Anaxagoras.2 Despite some salient differences, the two philosophers have much in common in their approach. They are near contemporaries,3 and as we shall see, they make similar moves in their approach to scientific speculation. Let us first examine the systems of Empedocles and Anaxagoras, and then discuss their responses to Parmenides.
I. EMPEDOCLES AND ANAXAGORAS
After warning us to seek a balance in our evaluation of sensory evidence (DK 31 B3), Empedocles goes on to identify the basic constituents of the universe and to develop a cosmology based on those constituents. There are four “roots,” (rizômata): earth, water, air, and fire (B6), which combine in whole-number ratios to form compounds. For instance, bone consists of two parts earth, four parts fire, and two parts water (B96) and blood of equal portions of the four roots (B98). The roots always exist in their own right, but they do not always appear to us because they are sometimes mixed with each other. In effect the four roots are unchanging stuffs that came to be known in antiquity as the four elements. In a striking simile, Empedocles compares nature to painters:
As when painters decorate offerings,
men well trained by wisdom in their craft,
who when they grasp colourful chemicals with their hands,
mixing them in combination, some more, some less,
from them provide forms like to all things,
creating trees and men and women,
beasts and birds and water-nourished fish,
and longlived gods mightiest in honours. (B23.1-8)
As a painter can, with a few colours, represent diverse forms of very different things, so can nature, with a few elements, create all natural substances.
In a discussion of these realities, Empedocles also introduces two personified forces, Love and Strife, with Love uniting the elements and Strife separating them (B17. 19ff.). Empedocles describes Love and Strife as spatially extended but invisible. There is some controversy over how they act, but evidently Love joins unlike elements together while Strife separates them. No force seems necessary to combine for example, earth with earth, but some power is needed to make earth combine with water or air or fire. Love and Strife interact to shape the world. Love brings elements together into a harmonious arrangement, finally uniting all things into a perfectly homogeneous mixture in a cosmic Sphere (sphairos). Eventually, however, Strife enters into the Sphere from outside, shattering its unity and precipitating a separation of the elements. From the separated parts of the Sphere comes a cosmos in which the different masses of earth, water, air, and fire appear, and plants and animals arise. At this point there is controversy over what happens. On one account, Strife continues to separate the elements until earth, water, air, and fire are completely dissociated from each other and stratified in their concentric layers, permitting no compounds and no living things; at this point Love begins to expand from the centre of the cosmic Sphere and again forms compounds including living things.4 Fragment B35 seems to suggest this view:
…when Strife had reached the innermost depth
of the vortex, and Love comes to be in the middle of the circle,
there all these things come together to be one only,
not suddenly, but willingly joining together, this from here and that from there.
And as they were being mixed, ten thousand races of beasts were poured out
but many things unmixed stood apart from those mixed,
all those that Strife still held back in midair; for not yet blamelessly
did it completely stand apart at the final limits of the circle,
but some members remained in the limbs, some moved outside them.
As far as it kept fleeing forth, so far did ever
the gentleminded blameless immortal rush of Love go on. (B35.3-13)
Love causes compounding to take place as it occupies the battlefield and Strife retreats to the periphery of the cosmos. On another account, there is never a complete separation of the elements but only an ongoing struggle between Strife and Love, which Love eventually wins as it again forms the Sphere in an unending cyclical process.5
On the former view, there are two separate creations of plants and animals, one during the stage when Strife is increasing, and another during the stage when Love is increasing. During the increase of Strife “whole-natured forms” emerge from earth as the cosmic separation of elements is taking place. These are gradually differentiated, at least in some cases, into viable living creatures. Later, this generation will perish as Strife completely separates every element into its own stratum. As Love begins to assert itself, first detached limbs are formed from the elements; these limbs join together in chance combinations to form monsters such as “manfaced oxkind” and “oxheaded mankind.” Unable to survive, these monsters perish. But when limbs come together in viable combinations, the resulting beasts survive and reproduce. In his account of genera
tion from limbs, Empedocles provides a kind of precursor to modern biological theories. Although he does not enunciate a theory of stepwise evolution, his theory does presuppose a principle of natural selection to account for existing species. Aristotle (Phys. II.8) criticizes Empedocles for assigning too great a role to chance in the production of natural kinds, but in this Empedocles is closer to modern science than is Aristotle.
Many details of Empedocles’ cosmic cycle remain unclear, but it is clear that his main theme is the ceaseless alternation between the processes of union and division that produce one out of many and many out of one:
and these things never cease continually alternating,
at one time all coming together into one by Love,
at another time each being borne apart by the enmity of Strife.
Thus, inasmuch as they are wont to grow into one from more
and in turn with the one growing apart they become more
they are born and do not enjoy a steadfast life;
but inasmuch as they never cease continually alternating,
they are ever immobile in the cycle. (B17.6-13)
Empedocles recognizes the symmetry of the contrary processes of unification and division by balancing antithetical lines; he recognizes the continuity of the process by reiterations of his descriptions. In his cycle both one and many have a place. And there is a kind of changelessness manifest in the repetitions of the cycle itself, as line 13 makes explicit. Thus Empedocles posits a one and a many, motion and rest, and indeed rest in motion, as features of his dynamic world view.