Zeno's Paradoxes
I. Introduction
Parmenides is a pivotal figure among the pre-Socratics. While the cosmologies of the early pre-Socratics seemed dogmatic and arbitrary, Parmenides developed a position based on reasons and argumentation. The position he developed, an extreme monism, was also at odds with the early cosmologies. In defending the view that what is real is permanent, the Parmenidean position denied the reality of change. Such a view was greeted with scepticism and ridicule. But Parmenides was not without his defenders. The most famous of these was Zeno of Elea (f. 460 B.C.), who is best known for a series of paradoxes formulated for the purpose of criticizing the critics of Parmenides.
Four of these paradoxes, on motion, seem designed to challenge the intelligibility of spatio-temporal change. The bulk of this chapter is devoted to an examination of these paradoxes and their implications for our understanding of space and time. A fifth, the paradox of plurality, can be construed as raising questions about the intelligibility of spatial (or temporal) extendedness. If the paradoxes hold up, then, they would constitute a refutation of the possibility of change and extension. In effect, they would be a defense of the Parmenidean thesis that what is real is One, Unchanging and Indivisible.
We know very little about Zeno's life. What information we do have is based primarily on a story in one of Plato's dialogues (Plato, Parmenides, 127A-128E). Zeno, while a youth, had written a series of arguments designed "to protect the arguments of Parmenides against . . . the partisans of the many, whose attack I return with interest by retorting on them that their hypotheses of the existence of many, if carried out, appears to be still more ridiculous than the hypothesis of the existence of one" (cf. Barnes, 1979 I, 233).
Unfortunately, Zeno's work has been lost. All we possess are paraphrases by commentators and critics. The paradoxes of motion are known only from their formulation by Aristotle, whose purpose was to criticize and refute them. This presents a number of problems of interpretation, not the least of which are that (1) we do not know against whom the paradoxes were directed, and (2) we do not know exactly how Zeno originally formulated the paradoxes. The result is that the paradoxes admit of a wide variety and range of interpre- tations and a correspondingly wide spectrum of opinion as to their worth and profundity (Barnes, 1979 I, 231).
Plato, apart from his few remarks on Zeno's purposes in the Parmenides, does not discuss Zeno's paradoxes, and he does not mention the paradoxes of motion at all. Aristotle discusses them but does not mention them seriously (see below). In general, the view of the ancients was that Zeno was a clever "paradoxer," but neither he nor his arguments were to be taken seriously. Recently, however, Zeno has been rehabilitated. Bertrand Russell, for example, thinks that the paradoxes are "immeasurably subtle and profound" (Salmon, 1970, 7). "They are not . . . mere foolish quibbles: they are serious arguments, raising difficulties which it has taken two thousand years to answer . . .Zeno's arguments, in some form, have afforded grounds for almost all the theories of space, time and infinity which have been constructed from his day to our own" (Salmon, 1970, 47, 54). Other modern interpreters agree (Salmon, 1970; Grünbaum, 1967). It is our conviction that the modern commentators are right which has led to the inclusion of a chapter on Zeno in this book.
There is, however, a problem of presentation. In what follows we do not pretend to give a faithful portrait of the arguments or the intentions of the historical Zeno. Indeed, given the paucity of the evidence and wide diversity of views regarding Zeno, such a portrait is probably impossible to draw. What we offer instead is a plausible reconstruction of Zeno based on the information we possess from Aristotle and other ancient commentators. Our aim is to present the paradoxes in such a way that their significance for our understanding of space and time is made clear, without, at the same time, doing too much violence to the argument of the historical Zeno, insofar as we know it.
II. The Paradoxes of Motion.
Aristotle reports on four paradoxes of motion which have come down to us as
(1) the Dichotomy
(2) the Achilles
(3) the Arrow
and
(4) the Stadium.
Our procedure will be as follows. First, we present each paradox as it is reported in Aristotle. Then, we present a reconstruction of the argument which makes explicit the underlying assumptions. Next, we discuss Aristotle's critique of the paradox in question and assess the cogency of his remarks. Finally, we try to indicate the significance of the paradox for our understanding of space, time and motion.
In following this line of attack, we are departing somewhat from our usual procedure in this section of the book, which is to focus on the historical development of the concepts of space and time. This is necessitated by the difficulties in recapturing the historical Zeno, on the one hand, and the importance of the paradoxes for our subject matter, on the other hand. We proceed to a discussion of the paradoxes of motion.
1. The Dichotomy
Aristotle reports the first paradox, known as the Dichotomy in one brief line:
[It is] . . . the one about a thing's not moving because what is traveling must arrive at the half-way point before the end (Physics 239 b 9-14).
The paradox is reconstructed as follows.
Figure 3-1: The Dichotomized Line
Suppose an object, A, is required to move from S to F. In order to get to F, A must first arrive at the halfway point M. Having reached M, A has now to travel from M to F. Before it can get to F from M, it must first arrive at M', which is halfway between M and F. This continues indefinitely, since the distance between S and F is infinitely divisible. In order to reach F from S, A must pass through the infinite sequence of points M, M', M", . . . etc. Passing through an infinite sequence of points, however, involves completing an infinite number of tasks (i.e., first A arrives at M, then A arrives at M', then
. . .). But, an infinite number of tasks cannot be completed. Therefore, A cannot move from S to F.
Zeno's basic argument can be formalized as follows:
(1) If anything moves from one place to another, then it
performs infinitely many tasks.
(2) Nothing can perform infinitely many tasks.
Therefore,
(3) Nothing moves. (adapted from Barnes, 1979 I, 263)
The argument, as it stands, is valid, i.e., the conclusion must be true, if both premises are true. Given that we are unpersuaded by Parmenidean monism, we reject the conclusion as false. It follows that at least one of the two premises must be false.
One might want to reject the first premise on the grounds that moving from S to F over a distance L is one task, not many, let alone infinitely many. There is some justice to this. However, the fact is that the interval can be divided as many times as one wishes. A could, in fact, stop anywhere along the route. Thus, it seems arbitrary to deny that infinitely many tasks are performed (cf. Barnes, 1979 I, 263f.).
Given that we are prepared to accept (1), we must reject (2), if we are to deny the conclusion. Aristotle's initial remarks on the Dichotomy are, in effect, an attack on the second premise. Aristotle says
Zeno's argument assumes a falsehood--that one cannot pass through an unlimited number of things or touch an unlimited number of things individually in a finite time. For both length and time--and, in general, whatever is continuous--are called unlimited in two ways: either by division or as to their extremes. Now one cannot touch things unlimited in respect of quantity in a limited time, but one can touch things unlimited by division--for the time itself is unlimited in this way. (Physics 233a, 21-31; from Barnes 261-2)
Aristotle's remarks can be construed as an argument against (2) in the following way. One might ask, what leads Zeno to think that (2) is true? He thinks that it is true, says Aristotle, on the basis of the following argument:
(4) To complete an infinite number of tasks would require an infinite amount of time.
(5) Nothing can do anything which requires an infinite amount of time.
Therefore,
(2) Nothing can perform infinitely many tasks.
(5) seems true. Aristotle attacks (4) by distinguishing two senses in which something may be said to be infinite. Something may be infinite by division or infinite by extremes. To say something is infinite by division is just to say that it is infinitely divisible, but not that it is infinitely large or infinitely long. To say that something is infinite by extremes is just to say that it is infinite in extent, i.e. infinitely large, infinitely long, etc.
Aristotle's suggested solution to the Dichotomy is that Zeno has failed to realize that something can be both finite (in extent) and infinite (by division). The finite length from S to F is infinitely divisible. Let T be the time it takes A to move from S to F. T is also infinitely divisible, but finite as well. What this means is that (4) is false. But, if (4) is false, then the argument for (2) is undermined. We haven't, thereby, shown (2) to be false, but, on the other hand, we no longer have a reason to think that (2) is true.
At first sight, this may seem to be a satisfactory resolution of the problem. But, a clever defender of Zeno would not be satisfied. Aristotle's solution, he might argue, merely pushes the problem back one step. The problem has not to do with the fact that the time it takes to perform an infinite number of tasks is infinite, but just with the fact that an infinite number of tasks cannot be performed. Thus, a similar problem arises with respect to T. Just as L was infinitely divisible, so T is, as Aristotle himself argues. If the original problem was to--reconstruct L from an infinite number of parts, the new problem suggested by Aristotle's solution is to reconstruct T from the infinite number of parts which result from its being divided. Aristotle, himself, recognized that his first solution was inadequate
. . . for if someone were to forget about the length and
about asking whether one can traverse unlimited things in a limited time, and were to make this inquiry of the time itself (for the time has unlimited divisions), this solution is no longer adequate. (Physics,263a, 15-22; from Barnes, 1979 I, 266)
Aristotle's suggestion at this point is to make another distinction, this time between actual and potential infinities (Physics, 263b, 3-9). The point is that if we subdivide the length L or the time T, we may wind up with either an actual infinity of parts or a potential infinity of parts. As long as we only have a potential infinity of parts, then Aristotle seems to think that there is no problem in moving a distance L or enduring through a time T, since the parts don't actually exist.
This solution is not adequate either. Even if Zeno were to accept the distinction between actual and potential infinite divisibility, Aristotle offers no reason for us to think that the same problem does not arise for potential infinities as rises for actual infinities. Between Zeno and Aristotle, the argument seems to be a standoff. But, there is another problem with Aristotle's solution. The solution assumes that L and T are only potentially infinitely divisible. But, regardless of whether anyone could actually ever complete an infinite subdivision of an interval, the fact of the matter seems to be that the interval is actually composed of an infinite sequence of segments each of which is one half the size of its immediate predecessor. Thus, the segment L is composed of the elements
L , L , L , L , . . .
2 4 8 16
This sequence has an infinite number of terms. The length L is somehow the "sum" of all these segments. One might reconstruct a Zenoan argument for a version of (2) as follows:
(6) Every interval can be subdivided into an infinite number of non-zero parts.
(7) To reconstruct the interval we would have to
"stitch together" ("sum" up) all the parts.
(8) The sum of an infinite number of non-zero parts is infinite, i.e. can never be completed.
Therefore,
(9) No intervals can be constructed.
(9) is a cousin of (2), once it is realized that reconstructing the interval involves performing an infinite number of additions. The weak link in this argument is (8). The Greeks did not know how to sum a series consisting of an infinite number of terms. (8) is a piece of bad arithmetic. On this view, which is shared by many modern commentators, Zeno failed to realized that the series
1 + 1 + 1 . . .
2 4 8
sums up to 1 (cf. Barnes, 1979 II, 265: Salmon, 1970, 26).
The mathematical solution seems to offer an easy resolution of the Dichotomy. However, even if it does, one must realize that the appropriate mathematical tools were 2000 years in the making, with a proper foundation only beginning to emerge in the 19th century. Furthermore, the mathematical solution only undermines a particular argument for (2); it does not show that (2) is false. Many commentators, familiar with the mathematical facts, are persuaded that there is more to the paradox, even granting that the sum of an infinite series can be finite. (See, e.g. Barnes 1979 I, ch. 13; Salmon 1970.) The fact is that undermining arguments for (2) does not prove that (2) is false. In order to lay the Dichotomy to rest once and for all, one must produce a cogent argument establishing the falsity of (2). We leave this (nontrivial) task to the interested reader.
2. The Achilles
Second is the one called Achilles. This says that the slow will never be caught in running by the fastest. For the pursuer must first get to where the pursued started from so that it is necessary that the slower should always be some distance ahead. (Physics, 239b, 14-18; from Barnes, 1979 I, 273)
Achilles, the fastest of the Greek warriors, is in a race with a slow opponent, which tradition has taken to be a tortoise. Since the tortoise is such a sluggard, it is only fair to give the tortoise a head start. Under these circumstances, Zeno argues, Achilles can never catch up with the tortoise, no matter how fast he runs.
Figure 3-2: Achilles/tortoise diagram
Let A represent Achilles and T the tortoise. Achilles starts at A0, the tortoise starts at T0. In order to catch the tortoise, Achilles must, at least, get to where the tortoise started from. But, by then, the tortoise has moved to T1. In the time it takes Achilles to get to T1, the tortoise has moved to T2. This procedure continues indefinitely, with the tortoise always just ahead of Achilles. Therefore, Achilles never catches up with the tortoise.
The Achilles seems to be a variation on the Dichotomy, the only difference being that the end of the race course is moving away from Achilles at a rate slower than Achilles is running toward the end. The argument can be formalized
as follows:
(1) For every i, if Achilles is at Ti, then the tortoise
is at Ti+1 .
Therefore,
(2) For every point P on the race course, if Achilles is at P, the tortoise is at some point P' such that P' is further from A0 than P. (Adapted from Barnes, 1979 I, 274)
(1) is true, as Zeno's argument shows. (2), however, does not follow from (1) alone. In addition, Zeno needs a premise of the form
(3) For all i , the distance from T0 to T is shorter than the distance from To to Ti+1 .
But (3) is false. In order to see that it is false, consider the distance that Achilles must run in order to catch the tortoise. Let the distance Ti+1 - Ti = [Delta]i. Then, in order to catch the tortoise, Achilles must run the
distance
D = [Delta]0 + [Delta]1 + [Delta]2 + . . .
Since Achilles is running faster than the tortoise, each [Delta]i+1 is smaller than the preceding [Delta]i. In order for (3) to be true, the sum D would have to be the infinite. But, the series D is analogous to the series generated by the problem in the Dichotomy. The only difference is that each successive term is not one half the size of its predecessor. Aristotle recognized this, and argued that the solution in both cases must be the same (239b, 11-26). Given that any finite distance can be traversed, which is what is at issue in the Dichotomy, the Achilles adds nothing conceptually new.
3. The Arrow
The third paradox of motion reported by Aristotle is known as the Arrow. The conclusion is that "a flying arrow is stationary" (Physics, 239b, 65-7; Apostle, 1969, 122).
The basic idea seems to be as follows. At every instant during its flight a moving arrow is at some point in its trajectory and occupies a portion of space equal to itself. Hence, it does not move during this instant. But, then, it does not move during any instant of its flight. Therefore, the arrow never moves.
Aristotle responds by rejecting what he takes to be an underlying assumption of the argument, namely, that temporal intervals are made up of indivisible moments of time. This suggests the following reconstruction of the argument:
(1) At every moment during its flight, the arrow occupies a space equal to itself.
(2) When the arrow occupies a space equal to itself, it is not moving.
(3) If the arrow is not moving when it occupies a space
equal to itself, then it must be at rest in that space.
Therefore,
(4) At every moment during its flight, the arrow is at rest.
Therefore,
(5) The arrow never moves.
Aristotle makes two comments about the arrow and their connection is somewhat obscure. First, he argues that, properly speaking, an object cannot be said to be either moving or at rest at a moment. In order for us to properly say that "A rests" or "A moves," we must consider a finite interval of time over which A is either resting or moving (Physics, 238a4-239bS). In effect, Aristotle is challenging premise (3). It is certainly true that the arrow does not move when it occupies a space equal to itself. However, it is not the case that the arrow must, then, be at rest. That would only be true if the arrow did not move for some finite time interval. If we are considering a durationless moment, at which the arrow occupies a space equal to itself, then it is not at rest at that moment. Thus, the antecedent of (3) is true, but the consequent of (3) is false. Therefore, (3) is false.
Given this view, Aristotle's second comment is puzzling. He suggests that the error in Zeno's argument is the false presumption that time is composed of indivisible moments (Physics, 239b, 7-9, 31-34). The implication seems to be that if Zeno's implicit assumption that time is composed of moments were correct, then the conclusion would follow. What is puzzling about this, is that Aristotle has just argued that neither motion nor rest can properly be predicated of something at a moment. (3) and (4) are still false, for the reasons given above, and the fact that time is not composed of indivisible moments (which Aristotle argues for elsewhere, cf. Chapter 5 below) is irrelevant (cf. Vlastos, 1975, 197 fr. 21).
Aristotle suggests that were time to be composed of moments the Arrow argument would be valid. But, even if we were to grant that (4) were true, the conclusion (5) would not follow. The argument as reconstructed is invalid. It commits what logicians call "the fallacy of composition." From 'Every part of X is P' one cannot conclude that 'X is P.' For example, from 'Every part of the water molecule is an atom' it does not follow that 'The water molecule is an atom.'
It is possible to reconstruct the arrow paradox in a way which makes it valid. Barnes suggests the following reconstruction:
(6) If the arrow is moving at t, then (at t) the arrow
occupies a space equal to its own volume.
(7) If the arrow occupies (at t) a space equal to its
own volume, then it is not moving at t.
From (6) and (7), we may conclude
(8) If the arrow is moving at t, then the arrow is not
moving at t.
Therefore,
(9) The arrow is not moving at t. (Barnes, 1979 I, 276- 285)
The inference from (8) to (9) is valid and licensed by the rule "'If not ~P, then P' implies 'P'." Barnes argues that (6) is trivially true, since an object, at an instant, is where it is, but that (7) is false (Barnes, 1979 I, 278, 283). One reason for thinking (7) is true is that, at an instant, there does not seem to be time enough for the arrow to move. But Barnes argues that we must distinguish 'A moves' from 'A is moving.' The former is only true if there are two points P1 and P2 such that A moves from P1 to P2. There is no time, at an instant, for A to do that. But, A can be moving at an instant nonetheless, since to be moving from P1 to P4 at an instant does not entail that there are points P3 and P4 between which A moves at that instant. Barnes illustrates with the following analogy.
Similarly if I smoke my pipe, there is a plug of tobacco which I consume in the process; and if I am smoking a pipe at t then there is a plug of tobacco which I am consuming at t. But it does not follow--and it is not true--that if I am smoking a pipe at t, then there is a plug of tobacco which I consume at t. (Barnes 1979 I, 283)
For those unpersuaded by this analogy, essentially the same reconstruction, which does not rely on it, can be found in Salmon (1970, 38-39).
(10) At every moment t of its flight, the arrow is at some point in its trajectory.
(11) If at every moment t of its flight, the arrow is at
some point in its trajectory (i.e., occupies a space equal to its own volume), then the arrow is not moving at t.
From (10) and (11), we can conclude, via modus ponens that
(12) The arrow is not moving at t.
Salmon argues that (11) is false. His argument is based on the fact that modern mathematics allows us to speak meaningfully of 'motion at an instant' or 'rest at an instant.' Using the differential calculus, we can define a concept of instantaneous velocity and also instantaneous rest. We can then say that an object is moving at an instant t if its instantaneous velocity at t is non-zero. Similarly, we can say that an object is resting at an instant if its instantaneous velocity at t equals O. On this construction, (11) is false. The antecedent of (11) is true: at every moment t of its flight, the arrow is at some point in its trajectory. But, the consequent of (11) is false: it need not be the case that the instantaneous velocity of the arrow at t = 0. In fact, consider the simplest case of an arrow moving uniformly (i.e. with constant non-zero velocity v) over an interval I. Then, at each moment during its flight, the instantaneous velocity of the arrow is equal to v. The conclusion (12) is also, of course, false, and, thus, the arrow paradox is undermined.
Salmon's solution has several virtues. First, it indicates the relevance of modern mathematics for the resolution of the paradoxes. It does this by showing how, on one construction, at least, the modern mathematical framework, which, incidentally, underlies our modern conception of space and time, can be brought into play to resolve the paradox. Second, Salmon's reconstruction bypasses the need to resort to Aristotle's claims that the concepts of rest and motion at an instant are nonsensical. In fact, the modern mathematical solution shows that Aristotle's claims are false. This is a good thing, since modern mathematical analysis also rejects Aristotle's other claim, namely, that time is not composed of moments (see Grünbaum, 1967).
4. The Stadium
The fourth Zenoian paradox of motion is known as the Stadium. Aristotle reports it as follows:
The fourth is an argument concerning two rows with an equal number of bodies all of equal length, the rows extending from the opposite ends of the stadium to the midpoint and moving in opposite directions with the same speed; and the conclusion in this argument, so Zeno thinks, is that the half of an interval of time is equal to its double . . . For example, let AlA2A3A4 be a set of stationary bodies all of equal length, BlB2B3B4 another equal set of moving bodies starting on the right from the middle of the A's and having lengths equal to the A's, and ClC2C3C4 a third equal set with speed equal to and contrary to that of the B's, also of lengths equal to those of the A's and ending on the right with the end of the stadium [Figure 3-4a].
[Insert Figure 3-4a]
Now as the B's and the C's pass one another, Bl will be over C4 at the same time that Cl will be over B4 [Figure 3-4b].
[Insert Figure 3-4b]
Thus, (1) Cl will have passed all the B's but only half the A's, and, as Cl takes an equal time to go through each B as through each A, its time in covering half of the A's will be half that in covering all the B's. Also, (2) during this same time the B's will have passed all since Cl takes an equal time to pass each A as each B, Cl and Bl will reach the contrary ends of the course at the same time because each of them takes an equal time to pass each A [Figure 3-4c]. (Aristotle, Physics, 240a, 5-15; from Apostle, 1969)
[Insert Figure 3-4c]
The argument can be more formally set out as follows:
(1) At tl, the A's, B's, and C's, each one unit long,
are arranged as in figure 3-4a.
(2) The B's move to the right at some fixed speed which we may take, for convenience, to be one unit per unit time.
(3) The C's move to the left at the same speed of one unit per unit time.
(4) The A's are stationary.
(5) Let t2 be the time at which the front of C1 has passed every A.
(6) The front of Cl has passed 2 A's in the interval
T = t2 - t1.
(7) The front of Cl has passed 4 B's in the interval
T = t2 - t1.
(8) The time it takes the front of Cl to pass each A is equal to the time it takes the front of Cl to pass each B.
From (6), (7) and (8), we conclude
(9) T = 2T.
But, this is absurd. Therefore,
(10) Motion is impossible.
Aristotle's rejoinder is short and swift. He rejects (8) as false. The fallacy, he says, lies in Zeno's assumption that "an object with an equal speed takes an equal time to pass a moving body as to pass a stationary body of an equal length . . ." (Physics, 240a, 3-4). Aristotle's remark is quite correct and, at first glance, the paradox does not seem to be very interesting or exciting. However, two points contribute to its importance.
First, Aristotle does not pick up on the implications of this remark. What he says is quite true, but he does not stop to explain why it is true. The obvious reason why it is true is that motion (or speed) is not an intrinsic property of an object in itself, but rather is a relational property that the object bears to something else. In other words, it is improper to speak simply of the 'speed of C,' without specifying with respect to what the speed of C is being measured. The speed of C1 with respect to the A's is not the same of the speed of Cl with respect to the B's. C1 is moving faster with respect to the B's than it is with respect to the A's. It is for this reason that the time it takes the front of C1 to pass each A is not equal to the time it takes the front of C1 to pass each B.
Zeno's paradox, in effect, points out the (important) fact that motion is always motion with respect to some frame of reference. Whether Zeno actually intended this to be the point is not clear from the textual evidence. What is clear, and significant, is that neither Aristotle nor any of the other early commentators on Zeno picked up on the fact that what the paradox shows is that 'A moves' is actually a disguised relational statement of the form 'A moves with respect to __________' (cf. Barnes, 1979 I, 292f.). The full significance of the relational character of motion was not made fully clear until the 18th century critics of Newton (cf. Chapter 8 below).
Second, some commentators, unhappy with the presumed triviality of the stadium, have opted instead for the belief that Aristotle must have misconstrued Zeno's argument (Barnes, 1979 I, 291). On this interpretation, Zeno is taken to be assuming that space and time are composed of atoms (i.e. indivisible units) of time and length. The A's, B's, and C's are supposed to represent individual atoms of spatial link. The argument can then be constructed in a way which avoids the false premise (8). The result is supposedly an argument against the view that spatial and temporal magnitudes can be decomposed into atoms. This version of the argument picks up after (7), assuming the necessary emendation, as follows:
(8') In the time it takes C1 to pass one A [A1], i.e. one atomic unit of time T, C1 passes 2 B's. [This follows from a consideration of premises (6) and (7)].
Therefore,
(9') In that time T, C1 must have first passed B1 and then passed B2.
(10') The time interval T can be subdivided into TB1 (the time during which C1 is passing B1) and TB2 (the time during which C1 is passing B2) such that T = TB1 + TB2, and TB1 comes before TB2.
But, (10) contradicts the assumption that T is an atomic (i.e. indivisible) unit of time. Therefore,
(11') Motion in opposite directions is impossible.
This version, unlike the original version, does not seem to harbor any false assumptions. Since the conclusion is false, and the argument is valid, there must be an implicit assumption which is false. The implicit assumption which is singled out as false is the assumption that space and time are composed of atoms (cf. the articles by Owens and Grünbaum in Salmon, 1970).
Can we attribute this argument to Zeno? Some have tried to, but two points count against it.
(1) Aristotle, who himself rejected the view that space and time are made up of atomic parts, does not construe the argument as directed against that view. He must have missed the point of it. This is conceivable but unlikely.
(2) One is hard pressed to say against whom Zeno would have been directing the argument. There is no textual evidence that any pre-Socratics held the atomistic view of time. Zeno might have been just systematically excluding all the possible ways in which a pluralist might have argued for the reality of space, time and motion, but this is to credit Zeno with more cleverness than some commentators are willing to allow.
Even if we cannot attribute the argument against atomistic spaces and times to the historical Zeno, the Stadium paradox still can be so construed, and, as such has implications for modern theories of space and time which suggest to some that there might be indivisible units of time (chronons) or indivisible units of length (chorons) (see Finkelstein, 1969).
This concludes our discussion of Zeno's four paradoxes of motion. Some commentators (e.g., Owens in Salmon, 1970), have argued that the paradoxes should be understood as a systematic attack on the intelligibility of change. Either change occurs continuously or discretely (i.e. via atomic "jumps"). The Dichotomy and the Achilles are supposed to rule out the former alternative. The Arrow and the Stadium are supposed to rule out the latter alternative. The textual basis for attributing such a master plan to Zeno ranges from slim to none, and the thesis that Zeno had such a master plan in mind is extremely controversial (Barnes, 1979 I, 233f.). Nevertheless, if we forget about what Zeno himself might have had in mind and only look at the paradoxes in terms of their implications for the structure of space, time and motion, then such an architectonic seems more plausible. If one accepts that the Stadium rules out an atomistic conception of space and time, then the only alternative is that space and time are continua. The Dichotomy and Achilles, designed to block this option, are overcome by the recognition that an infinite-series of terms can add up to a finite sum. Zeno appears defeated. But, the game is not over so quickly. There is another Zenoian paradox, the paradox of plurality, which raises serious questions about the concept of a continuum. Unlike the Dichotomy and the Achilles, it does not appear to be soluble using the mathematical techniques of infinite series (cf. Salmon, 1970, 15, and the Grünbaum article in Salmon, 1970).
III. Zeno's Paradox of Plurality
Zeno's paradox of plurality is not a paradox of motion. Rather, it is a paradox which can be construed as an attack on the intelligibility of extension. In this sense, it is a defense of Parmenidean monism--that the One is Absolute and Indivisible.
Our source for the paradox is Simplicius, a 6th century A.D. commentator on Aristotle. Neither Aristotle nor Plato refers to this particular paradox. As Simplicius states it, it goes like this:
. . . what is, is one only, and accordingly without parts and indivisible. For . . . if it were divisible, then suppose the process of dichotomy to have taken place: then either there will be left certain ultimate magnitudes, which are minima and indivisible, but infinite in number, and so the whole will be made up of minima but of an infinite number of them; or else it will vanish and be divided away into nothing. Both of which conclusions are absurd . . . (Simplicius, 139, 27 from H.D.P. Lee, 1967, p. 13).
The underlying argument can be made explicit in the following way. Consider any spatially extended physical object, a. (The argument goes through equally well for a temporally extended object, i.e. one which endures throughout a time period T.)
(1) The magnitude of the interval occupied by a, Ia, is
greater than 0.
(2) Ia can be divided in half, and the remaining parts divided in half, ad infinitum.
(3) The ultimate parts produced by the infinite division in (2) will either
(a) be an infinite number of small atomic parts of some positive magnitude, or
(b) be an infinite number of parts of zero magnitude.
(4) If the ultimate parts are an infinite number of elements with some positive magnitude, then the sum of the parts (= the interval Ia) will be an object of infinite magnitude.
(5) If the ultimate parts are an infinite number of elements of zero magnitude, then the sum of these parts (= the interval Ia) will be an object of zero magnitude.
Therefore,
(6) Either the interval Ia has zero magnitude or it is
infinitely large.
But, the argument was couched in terms of any extended object a. Hence, any extended object either has no size or is infinitely large.
The logic seems impeccable. Thus, one of the premises must be false. Premise (2) assumes that the magnitude of Ia is continuous. To reject (2) would commit one to an atomistic view of space, and that has been refuted by the second version of the Stadium paradox.
Premise (3) seems to exhaust all the alternatives. One might think that an appeal to infinitesimals, i.e., non-zero magnitudes smaller than any positive magnitude is a viable third alternative. But, even if we accept this, the sum of any two infinitesimals is an infinitesimal, and a premise analogous to (5) would insure that the sum of all the infinitesimals would produce an object no bigger than an infinitesimal.
Premise (4) is unimpeachable. At first glance, one might think that it could be impeached using the same technique that solved the Dichotomy. There, recall, we were faced with an infinite sum of positive magnitudes which added up to a finite, not an infinite, number. Alas, there is a crucial difference between this case and that. In the Dichotomy, the series of terms to be summed was such that each term was smaller than the previous term. As one considers terms far out in the series, they become as close to O as one likes. In the present case, on the other hand, all the terms are presumed to be of the same finite (albeit small) size. When such a series of terms is added together, the series diverges (i.e. approaches infinity as more and more terms are added on).
The only remaining premise is (5). It certainly seems unimpeachable as well. It is axiomatic that if one starts from nothing and adds nothing to it, the result is nothing, no matter how many times one does the trick. Nevertheless, Adolf Grünbaum has argued, in effect, that this premise is false (Grünbaum, 1967, Ch. 3).
Grünbaum argues that the trick to getting something from nothing is the recognition, by 19th century mathematicians, that some infinities are "larger" than others. This result is part of one of the most powerful theories of mathematics developed by Georg Cantor in the 19th century. This theory, which has wide ranging implications for and applications in modern mathematics and science is called "set theory." According to Cantor's set theory, in fact, there are an infinite number of different sized infinities. Luckily, according to Grünbaum, the resolution of the paradox of plurality rests on a consideration of only two of the smallest infinities.
The details of Grünbaums's argument are too complicated for us to reproduce here. But, the basic idea is fairly simple. Consider the infinite divisibility of a continuous magnitude. (Even Aristotle was capable of recognizing this possibility, at least, in principle.) Take a length L and begin to divide it in half. Call the first cut (i.e. the midpoint) cut 1 (!)
[Insert Figure 3-5]
Call the second cut, cut 2, and so on It is clear that we arranged it so that each cut in the process of infinitely dividing L is uniquely associated with one of the natural numbers 1, 2, 3, . . . Any set or group of objects (in this case, the "cuts" of L) which can be so associated with the natural numbers (itself a set, usually represented as N), is said to be countably infinite or denumerable. The reason for the name is obvious. Since we can associate each member of the set we are interested in with a different natural number, in some sense we are "counting" the members of the set. Of course, we can't really count such sets because we can never get through all the natural numbers by starting with 1 and counting steadily on. A finite set will then be one which can be so associated with some finite sequence of natural numbers. The set of planets of the solar system, e.g., is finite because we only need the numbers from 1 through 9 to count all the planets. The size of a set, finite or infinite, is called the cardinal number (or cardinality) of the set. The cardinality of the set of planets is 9. The cardinality of the set of cuts in the infinite division of the length L is equal to the cardinality of the natural numbers, which Cantor called aleph-null [insert symbol]. Other sets whose cardinality is [aleph]0 include the set of odd numbers, the set of even numbers, the set of positive integers, the set of negative integers, the set of positive plus negative integers, the set of positive and negative integers plus 0, the set of rational numbers and a host of other sets. If it seems puzzling that all these sets could be the same size, this is just an indication that our intuitions based on finite arithmetic do not all hold up in the realm of infinite arithmetic.
The next fact to realize is that although every infinite set is at least as large as the set of natural numbers, some infinite sets ("most," in fact) are larger than [aleph]0. These sets are called uncountable or nondenumerable, again for obvious reasons. The elements of such sets cannot be counted, that is, they cannot all be made to correspond, one for one, with the elements of N. One such set (and here we approach the paradox of plurality) is the set of points that make up the length of segment L (or any other "continuous" magnitude, for that matter). Suppose L is laid out on the X-axis of some coordinate system from 0 to 1.
[Insert Figure 3-6]
Aristotle's procedure of infinite division will, it is clear, pick out all the rational points (i.e. all the fractions of the form a) lying between 0 and 1. An ingenious argument by
b
Cantor shows, however, that the number of points in the interval from 0 to 1 includes many other irrational points, and that the total set of rational and irrational points in the interval from 0 to 1 (indeed, in any interval) is uncountable. (For details the reader may consult Salmon,
[1970, 251-268] or any standard mathematical text on set theory.)
The gist of Grünbaum's resolution of the paradox of plurality rests on this last point. The truth or falsity of premise (5) ("If the ultimate parts are an infinite number of elements of zero magnitude, then the sum of these parts will be an object of zero magnitude") depends on whether one is considering a countably infinite number of elements of zero magnitude or an uncountably infinite number. If one "sums" up a countable infinite number of "points" (i.e., elements of zero magnitude), one does wind up with an interval whose total length is 0. Aristotle, and the rest of the ancients, were, thus, justified in denying that a continuum is made up of points, if one imagines as he (and they) did, that all the points on a continuum can be reached by infinite subdivision. On the other hand, one can show, using 20th century mathematical techniques, that by "summing" up an uncountable number of "points," or elements of length 0, one can wind up with a finite, non-zero, length. The argument depends upon generalizing the idea of "length" and is far too complicated for us to consider here (again see Grünbaum, 1967, Ch. 3).
The import of all of this for our consideration of Zeno's paradox of plurality can be construed as an insightful dilemma about the constitution of spatial and temporal continua. Of course, as was the case with some of our earlier reconstructions of the paradoxes of motions, it would be rash, in the light of the textual evidence or lack of evidence, to attribute too much to Zeno himself. But, this only goes to show that Zeno's paradoxes can be construed in such a way as to transcend the historical Zeno. In so doing, they raise a number of important points about modern views of space and time and, thus, justify our consideration of them.
IV. The Significance of Zeno
Zeno's paradoxes, as the preceding discussion should have made clear, are subject to widely divergent interpretations. The task of trying to figure out what Zeno himself was up to is a job for classical scholars. What makes Zeno important for our story is the light that his paradoxes shed on the concepts of space and time. In so using the paradoxes, we transcend the historical Zeno, and follow the modern critics, such as Grünbaum and Salmon, who see the paradoxes as going "to the very heart of space, time and motion" and requiring, for their resolution, "the subtlety of modern physics, mathematics, and philosophy" (Salmon, 1970, 44, 5). In this final section, we examine some of the subtleties raised by the paradoxes.
The first point, which one might think trivial, is that the paradoxes of motion emphasize the intimate connection between motion, on the one hand, and space and time, on the other. Well, of course, space, time and motion are related What could be more obvious? Nothing, perhaps, but what is not obvious is that consideration of motion, and, in particular, what it means to move, can clarify the structure of space and time. The Stadium, construed as an argument for a relational analysis of motion (x moves only insofar as x moves with respect to y) suggests questions about the nature of space and time as well. If motion is to be understood relationally, how about space and time? What makes these questions interesting is that they are still open questions today. We can and will have more to say about the relational or non-relational character of space and time in the latter half of this book. Construed as an argument against an atomistic theory of time or space, its structural implications are clear.
The success of modern quantum theory in showing that many properties of nature come in quantized chunks has led to speculation that space and time come in discrete chunks as well (see, e.g., Finkelstein, 1969). [It is an open] question whether this is so or whether it would be useful to think of space and time in such a way. If so, the Stadium paradox would have to be dealt with.
The second point is non-trivial and concerns the implications of the paradoxes for our understanding of the continuum. We find it natural to think of space and time as being somehow continuous, i.e., not made up of space or time "atoms." But the paradox of plurality and to a somewhat lesser extent, the Dichotomy and Achilles present challenges to our understanding of the nature of continua. Even if one grants that the Dichotomy and the Achilles can be handled by a proper appreciation of infinite series, one must recognize that the "proper appreciation" requires the use of fairly sophisticated mathematical methods only fully understood within the past 200 years. The insubstantiality of space and time stand in marked contrast to the substantial mathematics which must be employed to ferret out all their mysteries.
The paradox of plurality, on the other had, serves to emphasize how different our modern conception of a continuum is from that of the Greeks. For Aristotle and the Greeks in general, continuity meant infinite divisibility. An interval cannot be partitioned into a set of points by an infinite sequence of divisions. But a continuum, in the modern sense, is pointlike. The resolution of the paradox of plurality requires the use of 20th century mathematical techniques which, in the process, illuminates the structure of spatial lengths and temporal intervals.
The third point raised by the paradoxes is the significance of the concept of infinity for our understanding the "local" structure of space and time. Of course, if one conceives of space as extending indefinitely and of time as having no beginning or no end, it is clear that space and time would be infinite. What the paradoxes show is that our understanding of "local" stretches of space and time also requires us to employ the idea of infinity. The Dichotomy and Achilles illustrate the role of infinite divisibility in our understanding of continua, as Aristotle clearly realized. The resolution of the Arrow paradox, insofar as it rests on the concept of instantaneous velocity, also involves a tacit appeal to the concept of infinity. This comes about because the instantaneous velocity is defined as the "limit" of an infinite sequence of "average" velocities, and thus a proper appreciation of infinite sequences is required. That is, suppose we want to find the instantaneous velocity of an object at a point P (Vp) in an interval L. We determine Vp in the following way. The average velocity of the object over the distance = distance/time = L/T. Now consider an infinite sequence of intervals [Delta] Xi around P, each smaller than its predecessor. The average velocity of the object over [Delta] Xi is [Delta] Vi = [Delta] Xi / [Delta] ti. Let L = [Delta] X0, T = [Delta] t0. The net result is a sequence of average velocities L, [Delta] X1, [Delta] X2,
T [Delta] tl [Delta] t2 , . . .
which are such that as i -> infinity, [Delta] X gets smaller and smaller around P. In the limit, as i -> infinity,
[Delta] Xi -> Vp, the velocity at point P.
[Delta] ti
Finally, the resolution of the paradox of plurality requires the fact that the cardinality of the continuum is an uncountable, rather than countable finite number. The solution of this paradox, thus, rests on our being able to extend our concept of number to include infinite as well as finite numbers. This is so, no matter how "small" the interval of space or time one is considering, and is related to the (paradoxical, perhaps) point that all intervals contain the same number of points (cf. Salmon, 1970, 251-268).
Finally, as Salmon points out, the paradoxes are not puzzles that can be solved by merely invoking high powered mathematics (Salmon, 1970, 16). More is at stake than that. It must be recalled that the paradoxes were initially moves in the struggle between reason and our senses. This struggle continues to the modern day and is central to the working of science. One of the aims of science is to reduce the sensory experiences of human beings, and thereby the world, to some intelligible order. The standard procedure is to construct a model of the available data that provides insight into the phenomena under investigation.
In physics, in general, and theories of space and time, in particular, these models tend to be mathematical. The idea, then, is to produce a mathematical model which makes sense of our experience. Zeno's paradoxes can be seen as early arguments against the adequacy of certain models to account for the familiar facts of spatial, temporal and motional experiences. Our experiences tell us things move, but my paradoxes (we might imagine Zeno saying) imply that our understanding of motion and its underlying spatio-temporal arena is incomplete.
The important point is to realize that producing scientific models involves two distinct, though related, tasks. First, one must produce a coherent model with enough structure to ensure that one can draw implications concerning the parameters of the model. Theoretical mathematics is of great help here. The theoretical model of the continuum, shaped with the help of modern mathematics, provides us with a model of space and time with a great deal of structure. Second, one must ask: is the model of space and time we have (theoretically) constructed a model of the space and time of the physical world which we daily experience? The models, in other words, must be tested. In order to be tested we must identify the parameters of the model with observable or measurable features of our environment. This is a non-trivial task.
The point of Salmon's remark should now be clear. Even if the latest mathematical techniques give us a handle on "mathematical spaces" and "mathematical times" which are not subject to Zeno's paradoxes, the paradoxes will not be completely laid to rest until such models can be shown to be adequate to represent what we take to be the essential features of experienced time and experienced space. How to work out this problem is what requires the subtleties of modern philosophy as well as modern physics and mathematics.
As an illustration of the problem consider Grünbaum's analyslis of the paradox of plurality. His solution relies on treating spatial (temporal) intervals as continua. With this mathematical model the problem disappears--at the theoretical level. But, what about at the experiential level? There is a fundamental limit to our capacity to make measurements and our observational evidence is always discrete. First, our physiological and psychological reaction times are finite and, hence, our sensory experience is ultimately atomic and particulate in structure. We can, however, construct measuring instruments with faster reaction times and greater spatial resolution. However, physicists recently have raised questions about the meaningfulness of applying the classical motions of space and time to very small intervals. There seem to be theoretical limits to the smallness of spatial and temporal intervals that can be measured. On this view, it is not appropriate to speak of "space" at intervals shorter than 10-33 cm. or "time" at intervals shorter than 10-43 sec. The implication is that even our measuring instruments cannot be made infinitely precise. Thus, there appears to be a lower limit to the applicability of continuous models to experiential data. Can we then reconcile the discreteness of our observational experience with our use of continuous theoretical models? This question leads to vexing issues in the theory of measurement. It is a variation on a Parmenidean theme: How to reconcile the voice of reason with the voice of experience.
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