## 1. Life and Writings




The dramatic occasion of Plato's dialogue, *Parmenides*, is a 
visit to Athens by the eminent philosopher Parmenides and Zeno, his 
younger associate, to attend the festival of the Great Panathenaea. 
Plato describes Parmenides as about sixty-five years old, Zeno as 
nearly forty, and Socrates, with whom they converse, as "quite 
young then," which is normally taken to mean about twenty. 
Given that Socrates was a little past seventy when executed by the 
Athenians in 399 B.C.E., this description suggests that Zeno was born 
about 490 B.C.E. He would appear to have been active in Magna Graecia, 
that is, the Greek-speaking regions of southern Italy, during the 
mid-fifth century B.C.E. There is otherwise little credible information
about the circumstances of his life. Diogenes Laertius's brief 
"Life of Zeno" (D.L. 9.25-9) is largely taken up with 
stitching together conflicting reports of his involvement in a brave 
plot to overthrow one of the local tyrants, but how much truth these 
reports contain cannot be determined. Although Diogenes also says 
that Zeno so loved his native Elea that he had no interest in 
immigrating to Athens, this report is not inconsistent with his 
having spent some time there; and Plutarch's report that Pericles 
heard Zeno expounding on the nature of things in the manner of 
Parmenides (Plu. *Pericles* 4.5) suggests that Zeno may indeed
have visited Athens and read his famous book, as Plato's 
*Parmenides* implies, to a group of intellectually keen 
Athenians. Vivid evidence of the cultural impact of Zeno's arguments 
is to be found in the interior of a red-figure drinking cup (Rome, 
Mus. Villa Giulia, inv. 3591) discovered in the 
Etrurian city of Falerii and dated to the mid-fifth century B.C.E. It 
depicts a heroic figure racing nimbly ahead of a large tortoise and 
has every appearance of being the first known "response" 
to the Achilles paradox.




Plato's *Parmenides* depicts Socrates going as a young man to 
hear Zeno reading from the famous book he has brought to Athens for 
the first time. Parmenides himself and some others, including 
Pythodorus (the dramatic source of Plato's report) are portrayed as 
entering toward the end of the reading so that they hear only a 
little of Zeno's recitation. Plato then presents an exchange between 
Socrates and Zeno, the first part of which is as follows:



> 
> 
> Once Socrates had heard it, he asked Zeno to read the first hypothesis
> of the first argument again, and, after it was read, he said:
> "What do you mean by this, Zeno? 'If the things that are
> are many, that then they must be both like and unlike, but this is
> impossible. For neither can unlike things be like, nor like things
> unlike'? Is this not what you say?" "Yes,"
> said Zeno. "Then if it is impossible both for things unlike to
> be like and for like things to be unlike, then it's also impossible
> for there to be many things? For if there were many things, they would
> incur impossibilities. So is this what your arguments intend, nothing
> other than to maintain forcibly, contrary to everything normally said,
> that there are not many things? And do you think that each of your
> arguments is a proof of this very point, so that you consider yourself
> to be furnishing just as many proofs that there are not many things as
> the arguments you have written? Is this what you say, or do I not
> understand correctly?" "Not at all," said Zeno,
> "but you have understood perfectly well what the treatise as a
> whole intends" (Pl. *Prm*. 127d6-128a3).




While the dialogue's scenario, and thus this exchange, are clearly 
fictional, this passage is nonetheless normally taken as indicating 
that Zeno composed a single treatise comprising numerous arguments, 
cast in the form of antinomies, all purporting to demonstrate the 
untenability of the commonsense presumption that there are many 
things. 




While the later tradition unreliably ascribes other works to Zeno, 
there is some interesting evidence in the commentary on the 
*Parmenides* by the Athenian Neoplatonist Proclus (5th c. 
C.E.) that he was familiar with a work transmitted under Zeno's name 
containing forty arguments or *logoi* (Procl. *in Prm*.
694, 17-18 Steel). Much of what Proclus says about Zeno in his 
commentary simply recasts what is already present in the above 
exchange, but this comment that this work of Zeno's contained forty 
arguments, taken with certain other things he says, suggests that 
Proclus had access to a work with some sort of Zenonian pedigree, a 
work known to earlier commentators as well (as evidenced by Procl. 
*in Prm*. 630.26ff., especially 631.25-632.3). If there was a 
work available in later antiquity entitled *The Forty Arguments of
Zeno*, it is however unlikely to have been a fair replica of any 
original treatise of Zeno's. In the first place, some of Proclus' 
apparent references to this work suggest that it fathered upon Zeno 
arguments akin to some of those in Parmenides' own elaborate 
dialectical exercise later in the *Parmenides*. Furthermore, 
Aristotle implies that people were reworking Zeno's arguments soon 
after they were first propounded. In *Physics* 8.8, after 
giving a basic reconstruction of the so-called Stadium paradox (see 
below, sect. 2.2.1) recalling its presentation in *Physics* 
6.9, Aristotle then notes that some propound the same argument in a 
different way; the alternative reconstruction he then describes 
(Arist. *Ph*. 8.8, 263a7-11) is in effect a new 
version of the original argument.




Returning to the *Parmenides* passage, it should also be noted
that Socrates' description of Zeno's book, which Plato has Zeno 
endorse, indicates that its arguments had a certain structure and 
purpose. Specifically, the passage indicates that all Zeno's 
arguments opposed the common-sense assumption that there are many 
things. It might also suggest that these arguments took the form of 
antinomies like the one Socrates specifically cites, so that the 
general pattern of Zeno's argumentation would have been: if there are
many things, these must be both *F* and not-*F*; but 
things cannot be both *F* and not-*F*; therefore, it 
cannot be the case that there are many things. Although this 
description has inspired some to attempt to accommodate the extant 
paradoxes (of motion, plurality, and place) within a unified 
architecture that would have provided the plan for Zeno's original 
book, if in fact he wrote only one, none of these attempts have 
proved convincing. Since Plato's description is in a number of 
respects difficult to square with what we know from other sources of 
Zeno's actual arguments, one should be wary of making it the basis 
for hypotheses regarding the book's plan of organization. For one 
thing, the paradoxes of motion reported by Aristotle do not evidently
target the assumption that there are many things, nor do they take 
the form of antinomies. Moreover, only one of the arguments against 
plurality elsewhere reported, the antinomy of limited and unlimited, 
conforms to the pattern of argumentation exemplified in the antinomy 
of like and unlike described by Plato's Socrates (see below, 2.1.1). 
The remaining argument, the antinomy of large and small (see 2.1.2), 
purports to show not only that the assumption that there are many 
things leads to an apparent contradiction, but, rather more 
ambitiously, it purports to reduce each of the contradictory 
consequences to absurdity. Plato does not actually state, of course, 
that all Zeno's arguments took the form of antinomies. In the end, if
the characterization of Zeno's treatise by Plato's Socrates in the 
passage above is not quite accurate, there remains no more plausible 
view from antiquity regarding the general thrust of his arguments, to
the extent that there may have been a single one. One can, moreover, 
easily broaden Socrates' specification of the target to encompass the
arguments against motion and place by changing it to the slightly 
more complex thesis that there are many things that move from place 
to place. Socrates might easily have been taking it for granted that,
for Zeno, such motion goes along automatically with plurality. What 
we know of Zeno's arguments certainly accords with the notion that 
they were meant to challenge ordinary assumptions about plurality and
motion. His arguments are quite literally 
"para-doxes"--from the Greek *para* 
("contrary to" or "against") and 
*doxa* ("belief" or 
"opinion")--arguments for conclusions contrary to 
what people ordinarily believe. What more there might be to say about
Zeno's purposes will be discussed below, after presentation of what 
we know of his actual arguments.


## 2. The Extant Paradoxes




The task of reconstructing Zeno's arguments is sometimes 
insufficiently distinguished from the task of developing responses to
them. How one reconstructs Zeno's reasoning certainly determines to 
some extent what will constitute an effective response. The danger is
that one's idea of how to formulate an effective response may affect 
one's reconstruction of Zeno's actual reasoning, particularly if one 
imports into his arguments concepts more developed or precise than 
the ones with which he was actually operating. In some cases, as with
the one called the Achilles, the paradox's power derives to a 
significant extent from the very simplicity of the notions it 
deploys. The reconstructions provided here therefore aim to preserve 
something of the manner of Zeno's own argumentation as we know it 
from verbatim quotation of at least portions of some of the preserved
paradoxes. More formal reconstructions are possible and available. As
already noted, at least one effort at improving Zeno's argumentation 
was already known to Aristotle. But such efforts can come at the cost
of historical accuracy, which is the primary goal of this article. 
How it might be possible to improve Zeno's arguments will be left to 
others. Since it is also essential to appreciate just how much (or 
how little) we know of Zeno's arguments, the primary evidence for 
each major argument is presented along with a reconstruction.


### 2.1 The Arguments Against Plurality


#### 2.1.1 The Antinomy of Limited and Unlimited




In his commentary on book 1 of Aristotle's *Physics*, the
Alexandrian Neoplatonist Simplicius (6th c. C.E.) quotes verbatim
Zeno's argument that if there are many things, they are limited and
unlimited, as follows: "If there are many things, it is
necessary that they be just so many as they are and neither greater
than themselves nor fewer. But if they are just as many as they are,
they will be limited. If there are many things, the things that are
are unlimited; for there are always others between these entities, and
again others between those. And thus the things that are are
unlimited" (Zeno fr. 3 DK, i.e., Simp. *in
Ph*. 140.29-33 Diels). This is the only Zenonian antinomy
that has the appearance of being preserved in its entirety.




The argument here may be reconstructed as follows. Its overall 
structure is: If there are many things, then there must be finitely 
many things; and if there are many things, then there must be 
infinitely many things. The assumption that there are many things is 
thus supposed to have been shown to lead to contradiction, namely, 
that things are both finitely and infinitely many. The particular 
argument for the first arm of the antinomy seems to be simply: If 
there are many things, then they must be just so many as they are. If
the many things are just so many as they are, they must be finitely 
many. Therefore, if there are many things, then there must be 
finitely many things. Simplicius somewhat loosely describes the 
antinomy's second arm as demonstrating numerical infinity through 
dichotomy (Simp. *in Ph*. 140.33-4). In fact, the argument 
depends on a postulate specifying a necessary condition upon two 
things being distinct, rather than on division *per se*, and 
it may be reconstructed as follows: If there are many things, they 
must be distinct, that is, separate from one another. Postulate: Any 
two things will be distinct or separate from one another only if 
there is some other thing between them. Two representative things, 
*x*1 and *x*2, will be distinct 
only if there is some other thing, *x*3, between 
them. In turn, *x*1 and *x*3 will
be distinct only if there is some other thing, 
*x*4, between them. Since the postulate can be 
repeatedly applied in this manner unlimited times, between any two 
distinct things there will be limitlessly many other things. 
Therefore, if there are many things, then there must be limitlessly 
many things.


#### 2.1.2 The Antinomy of Large and Small




In the same stretch of his commentary on Aristotle's 
*Physics*, Simplicius reports at length one of Zeno's numerous
arguments designed to show how the claim that there are many things 
leads to contradiction. "One of these," Simplicius says, 
"is the argument in which he demonstrates that if there are 
many things, they are both large and small: so large as to be 
unlimited in magnitude, and so small as to have no magnitude. Indeed,
in this argument he shows that what has neither magnitude nor 
thickness nor bulk would not even exist. 'For if', he 
says, 'it were added to another entity, it would not make it 
any larger; for since it is of no magnitude, when it is added, there 
cannot be any increase in magnitude. And so what was added would just
be nothing. But if when it is taken away the other thing will be no 
smaller, and again when it is added the other thing will not 
increase, it is clear that what was added and what was taken away was
nothing'" (Zeno fr. 2 DK = Simp. *in Ph*. 
139.7-15). After thus quoting this portion of the argument, 
Simplicius continues: "Zeno says this because each of the many 
things has magnitude and is infinite [reading *apeiron* 
instead of ms. *apeiron*], given that something is 
always in front of whatever is taken, in virtue of infinite division;
this he shows after first demonstrating that none have magnitude on 
the grounds that each of the many is the same as itself and 
one" (Simp. *in Ph*. 139.16-19). Soon after this, 
Simplicius records the argument for unlimited magnitude he has 
alluded to in the first part of the passage just quoted, as follows: 
"Infinity in respect of magnitude he earlier proves in the same
way. For having first shown that, if what is does not have magnitude,
it would not even exist, he continues: 'But if it is, each must
have some magnitude and thickness, and one part of it must extend 
away from another. And the same account applies to the part out 
ahead. For that part too will have magnitude and will have part of it
out ahead. Indeed, it is the same to say this once as always to keep 
saying it; for no such part of it will be last, nor will one part not
be related to another. Thus if there are many things, they must be 
both small and large, so small as to have no magnitude, and so large 
as to be unlimited'" (Zeno fr. 1 DK, = Simp. *in 
Ph*. 140.34-141.8). 




Simplicius only alludes to Zeno's argument for smallness, without 
setting it out: he says that Zeno derived the conclusion that 
"none have magnitude on the grounds that each of the many is 
the same as itself and one." Although this is not much to go 
on, the argument may plausibly be reconstructed as follows. Each of 
the many is the same as itself and one. Whatever has magnitude can be
divided into distinguishable parts; whatever has distinguishable 
parts is not everywhere the same as itself; thus, whatever has 
magnitude is not everywhere, and so is not genuinely, the same as 
itself. Whatever is not the same as itself is not genuinely one. 
Thus, whatever has magnitude is not genuinely one. Therefore, each of
the many has no magnitude. The basic assumption here is that to be 
"the same as itself" is what it means for something to be
"one" in the strict sense Zeno envisages, whereas any 
magnitude, which will have distinguishable parts in virtue of being 
spatially extended, will fail to be strictly one and self-identical. 




The evidence in Simplicius indicates that Zeno then transitioned to 
the antinomy's other arm, the unlimited largeness of things, via the 
following lemma: since what has no magnitude would be nothing, each 
of the many must have some magnitude. Simplicius's report of how Zeno
specifically argued for the second arm's conclusion, that each of the
many is of unlimited magnitude, pertains primarily to its apparent 
sub-argument for the interim conclusion that each thing has 
limitlessly many parts, which ran as follows. Each of the many has 
some magnitude and thickness (from the lemma). Whatever has some 
magnitude and thickness will have (distinguishable) parts, so that 
each of the many will have parts. If *x* is one of the many, 
then *x* will have parts. Since each of these parts of 
*x* has some magnitude and thickness, each of these parts will
have its own parts, and these parts will in turn have parts of their 
own, and so on, and so on, without limit. Thus each of the many will 
have a limitless number of parts. Whether or not Zeno then made 
explicit how the antinomy's final conclusion followed from this, here
is a plausible reconstruction of the rest of the reasoning was 
presumably supposed to go: Every part of each thing has some 
magnitude; the magnitude of any object is equal to the sum of the 
magnitudes of its parts; and the sum of limitlessly many parts of 
some magnitude is a limitless magnitude. Therefore, the magnitude of 
each of the many is limitless. 




Taken as a whole, then, this elaborate *tour de force* of an 
argument purports to have shown that, if there are many things, each 
of them must have simultaneously no magnitude and unlimited 
magnitude.


### 2.2 The Paradoxes of Motion




Aristotle is most concerned with Zeno in *Physics* 6, the book
devoted to the theory of the continuum. In *Physics* 6.9, 
Aristotle states that Zeno had four arguments concerning motion that 
are difficult to resolve, gives a summary paraphrase of each, and 
offers his own analysis. The ancient commentators on this chapter 
provide little additional information. Thus reconstruction of these 
famous arguments rests almost exclusively on Aristotle's incomplete 
presentation. Note that Aristotle's remarks leave open the 
possibility that there were other Zenonian arguments against motion 
that he deemed less difficult to resolve. More importantly, 
Aristotle's presentation gives no indication of how these four 
arguments might have functioned within the kind of dialectical scheme
indicated by Plato's *Parmenides*.


#### 2.2.1 The Stadium, or The Dichotomy




"First," Aristotle says, "there is the argument 
about its being impossible to move because what moves must reach the 
half-way point earlier than the end" (*Ph*. 
6.9, 239b11-13). He says no more about this argument here 
but alludes to his earlier discussion of it in *Physics* 6.2, 
where, after arguing that both time and magnitude are continuous, he 
asserts: "Therefore the argument of Zeno falsely presumes that 
it is not possible to traverse or make contact with unlimited things 
individually in a limited time" (233a21-3). 
Subsequently, in *Physics* 8.8, he again raises the question 
of how to respond "to those posing the question of Zeno's 
argument, if one must always pass through the half-way point, and 
these are unlimited, and it is impossible to traverse things 
unlimited" (263a4-6), and he proceeds to offer what 
he claims is a more adequate solution than the one presented in 
*Physics* 6.2. The argument Aristotle is alluding to in these 
passages gets its name from his mention in *Topics* 8.8 of 
"Zeno's argument that it is not possible to move or to traverse
the stadium" as a prime example of an argument opposed to 
common belief yet difficult to resolve (160b7-9).




The following reconstruction attempts to remain true to this evidence
and thus to capture something of how Zeno may originally have argued.
For anyone (*S*) to traverse the finite distance across a 
stadium from *p*0 to *p*1 within 
a limited amount of time, *S* must first reach the point half 
way between *p*0 and *p*1, namely
*p*2. 



![S is on a line at p0 on a line that extends to p1. p2 is halfway between p0 and p1 and p3 is halfway between p0 and p2. p4 is halfway between p0 and p3.](graphic1.jpg)



Before *S* reaches *p*2, *S* must 
first reach the point half way between *p*0 and 
*p*2, namely *p*3. Again, before 
*S* reaches *p*3, *S* must first 
reach the point half way between *p*0 and 
*p*3, namely *p*4. There is a 
half way point again to be reached between *p*0 and
*p*4. In fact, there is always another half way 
point that must be reached before reaching any given half way point, 
so that the number of half way points that must be reached between 
any *p*n and any *p*n-1 is 
unlimited. But it is impossible for *S* to reach an unlimited 
number of half way points within a limited amount of time. Therefore,
it is impossible for *S* to traverse the stadium or, indeed, 
for *S* to move at all; in general, it is impossible to move 
from one place to another.


#### 2.2.2 The Achilles




Immediately after his brief presentation of the Stadium, Aristotle 
introduces the most famous of Zeno's paradoxes of motion, that of 
Achilles and the Tortoise: "Second is the one called 
'Achilles': this is that the slowest runner never will be
overtaken by the fastest; for it is necessary for the one chasing to 
come first to where the one fleeing started from, so that it is 
necessary for the slower runner always to be ahead some" 
(*Ph*. 6.9, 239b14-18). Simplicius adds the 
identification of the slowest runner as the tortoise (*in Ph*.
1014, 5). Aristotle remarks that this argument is merely a variation 
on the Dichotomy, with the difference that it does not depend on 
dividing in half the distance taken (*Ph*. 6.9, 
239b18-20), and his analysis, such as it is, emphasizes 
that this paradox is to be resolved in the same way as the first 
paradox of motion. Whether this is actually the case is debatable.



If a tortoise starts ahead of Achilles in a race, the tortoise will
never be overtaken by Achilles. Let the start of the race be
represented as follows: 


![At position 0, A is at a0, the beginning of the line, and T is at t0, the midpoint of the line](graphic2.jpg)



During the time it takes Achilles to reach the point from which the 
tortoise started (*t*0), the tortoise will have 
progressed some distance (*d*1) beyond that point, 
namely to *t*1, as follows:



![At Postion 1, A is now at a1 = t0, T is at t1 = t0 + d1](graphic3.jpg)



Likewise, during the time it then takes Achilles to reach the new 
point the tortoise has reached (*t*1), the tortoise
will have progressed some new distance (*d*2) 
beyond the tortoise's new starting point, namely to 
*t*2, as follows:



![At Postion 2, A is now at a2 = t1, T is at t2 = t1 + d2](graphic4.jpg)



The tortoise will again have progressed some further distance 
(*d*3) beyond *t*2, namely to 
*t*3, in the time it takes Achilles to move from 
*a*2(=*t*1) to 
*a*3(=*t*2). In fact, during the 
time it takes Achilles to reach the tortoise's location at the 
beginning of that time, the tortoise will always have moved some 
distance ahead, so that every time Achilles reaches the tortoise's 
new starting point, the tortoise will be ahead some. Therefore, the 
slowest runner in the race, the tortoise, will never be overtaken by 
the fastest runner, Achilles.


#### 2.2.3 The Arrow




Aristotle's discussion of the relation of motion and time in 
*Physics* 6.8 prepares the way for his objection to the 
Zenonian paradox of motion he mentions at the very beginning of 
*Physics* 6.9: "Zeno reasons fallaciously; for he says 
that if every thing always is resting whenever it is against what is 
equal, and what moves is always in the now, then the moving arrow is 
motionless" (*Ph*. 6.9, 239b5-7). Aristotle 
remarks that Zeno relies on the false supposition that time is 
composed of indivisible "nows" or instants 
(b8-9), a point he soon repeats in identifying the 
argument purporting to show that "the moving arrow is standing 
still" as the third of Zeno's paradoxes of motion 
(b30-3). In his *Life of Pyrrho*, Diogenes Laertius
reports, "Zeno abolishes motion, saying, 'What moves 
moves neither in the place it is nor in a place it is not" 
(D.L. 9.72 = Zeno B4 DK; cf. Epiphanius, *Against the 
Heretics* 3.11). This report, which Diels and Kranz took to 
preserve a genuine fragment of Zeno's book, appears to suggest how 
the argument that the moving arrow is at rest may have figured as 
part of a broader argument against motion. Diogenes, however, is not 
a particularly good source for Zeno's arguments: his *Life of 
Zeno* notes only that he was the first to propound the 
"Achilles" argument, along with many others (D.L. 9.29). 
It is just as likely, therefore, that Diogenes' report depends 
on an intervening attempt to couch the paradoxes of motion reported 
by Aristotle in the dilemmatic form Plato indicates was typical of 
Zenonian argumentation.




Even if Diogenes' report happens to be reliable, we still must rely 
on Aristotle in trying to reconstruct the argument that, as in 
Diogenes' report, what moves does not move in the place where it is. 
(We get no indication from him of any argument of Zeno's to show that
what moves does not move where it is not; perhaps that was thought 
self-evident.) And Aristotle's evidence in this instance is an even 
more meager basis for reconstruction than usual. Thus, according to 
Aristotle, the moving arrow (*A*) is actually standing still. 
The argument for this conclusion seems to be as follows: What moves 
is always, throughout the duration of its motion, in the now, that is
to say, in one instant of time after another. So, throughout its 
flight, *A* is in one instant of time after another. At any 
particular instant during its flight (*t*), *A* 
occupies a place exactly equivalent to its length, that is, 
*A* is "against what is equal." But whatever is 
against what is equal is resting. So *A* is resting at 
*t*. But *t* is no different than the other instants 
during *A*'s flight, so that what is the case with 
*A* at *t* is the case with *A* at every other 
instant of its flight. Thus *A* is resting at every instant of
its flight, and this amounts to the moving arrow always being 
motionless or standing still.


## 2.2.4. The Moving Rows




"Fourth," Aristotle says, "is the one about the
things in the stadium moving from opposite directions, being of equal
bulk, alongside things of equal size, with some moving from the end of
the stadium and some from the middle, at equal speed, in which case he
supposes it turns out that half the time is equal to its double"
(*Ph*. 6.9, 239b33-240a1).




Aristotle's ensuing discussion of what he takes to be Zeno's mistakes
is based on an *exempli gratia* scenario normally taken as a
basis for reconstruction, as it is here. "For example,"
Aristotle says, "let the resting equal masses be those marked
AA, let those
marked BB be beginning from the middle, being
equal in number and size to these, and let those marked
CC be beginning from the end, being equal in
number and size to these, and moving at the same speed as
the *B*s" (Arist. *Ph*. 6.9,
240a4-8). 





|  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
| A A A A |
| B B B B | - |
| - | C C C C |

Diagram 1 |  | 

|  |
| --- |
| A A A A |
| B B B B |
| C C C C |

Diagram 2 |





Diagram 1 represents a plausible way of understanding what Aristotle
envisages as the starting position in Zeno's paradox, even though his
description of this position is somewhat underdetermined. Aristotle
continues: "It follows that the
leading B and the
leading C are at the end at the same time,
once they move past one another"
(240a9-10). This description suggests a final
position as represented in Diagram 2. Since we have no other
indication of how Zeno himself thought he could derive the conclusion
Aristotle reports, that "half the time is equal to its
double," from the description of this situation, we have to rely
on Aristotle for this as well: "It follows," Aristotle
says, "that the [leading] C has gone
past all [the Bs], while the
[leading] B has gone past half
[the As], so that the time is half; for each
of the two is alongside each other for an equal amount of time. But it
also follows that the leading B has gone past
all the
Cs; for the leading C
and the leading B will be at the opposite ends
at the same time, because both are alongside
the As for an equal time"
(204a10-17). Apparently, Zeno somehow meant to infer
from the fact that the leading B moves past
two As in the same time it moves past all
four Cs that half the time is equal to its
double. The challenge is to develop from this less than startling fact
anything more than a facile appearance of paradox. Since it is
stressed that all the bodies are of the same size and that the moving
bodies move at the same speed, Zeno would appear to have relied on
some such postulate as that a body in motion proceeding at constant
speed will move past bodies of the same size in the same amount of
time. He could have argued that in the time it takes all
the Cs to move past all
the Bs, the leading B
moves past two As or goes two lengths, and the
leading B also moves past
four Cs or goes four lengths. According to the
postulate, then, the time the leading B
travels must be the same as half the time it travels. Unfortunately,
the evidence for this particular paradox does not enable us to
determine just how Zeno may in fact have argued. Aristotle thinks the
argument depends upon a transparent falsehood, and one must therefore
keep in mind, if it seems he was right, that Aristotle's presentation
and reconstruction may itself be colored by his desire to bear out his
accusation.


### 2.3 Other Paradoxes




Aristotle also gestures toward two additional ingenious arguments by 
Zeno, versions of which were also known to Simplicius.


#### 2.3.1 The Millet Seed




"Zeno's argument is not correct, that any portion of millet 
seed whatsoever makes a sound" (Arist. *Ph*. 
7.5, 250a20-1). The version of this argument known to 
Simplicius represents Zeno as engaged in a fictional argument with 
Protagoras, wherein he makes the point that if a large number of 
millet seeds makes a sound (for example, when poured out in a heap), 
then one seed or even one ten-thousandth of a seed should also make 
its own sound (for example, in that process) (Simp. *in Ph*. 
1108.18-28). Aristotle's report is too slight a basis for 
reconstructing how Zeno may in fact have argued, and Simplicius is 
evidently reporting some later reworking. The evidence nonetheless 
suggests that Zeno anticipated reasoning related to that of the 
sorites paradox, apparently invented more than a century later.


#### 2.3.2 A Paradox of Place




Toward the end of the introduction to his analysis of place, 
Aristotle notes that "Zeno's difficulty requires some 
explanation; for if every thing that is is in a place, it is clear 
that there will also be a place of the place, and so on to 
infinity" (Arist. *Ph*. 4.1, 209a23-5). His 
subsequent statement of the problem is even briefer but adds one key 
point: "Zeno raises the problem that, if place is something, it
will be in something" (Arist. *Ph*. 
4.3, 210b22-3; cf. Eudemus fr. 78 Wehrli, [Arist.] *De 
Melisso Xenophane Gorgia* 979b23-7, Simp. *in 
Ph*. 562, 3-6). Zeno would appear to have argued as follows. 
Everything that is is in something, namely a place. If a place is 
something, then it too must be in something, namely some further 
place. If this second place is something, it must be in yet another 
place; and the same reasoning applies to this and each successive 
place *ad infinitum*. Thus, if there is such a thing as place,
there must be limitless places everywhere, which is absurd. 
Therefore, there is no such thing as place. This argument could well 
have formed part of a more elaborate argument against the view that 
there are many things, such as that if there are many things, they 
must be somewhere, i.e. in some place; but there is no such thing as 
place and thus no place for the many to be; therefore, there are not 
many things. This is, however, only speculation.


## 3. Zeno's Purposes




The commonly found claim that Zeno aimed to defend the paradoxical 
monism of his Eleatic mentor, Parmenides, is based upon the 
speculations by the young Socrates of Plato's *Parmenides* on 
Zeno's ulterior motives. After the portion of the exchange between 
Socrates and Zeno quoted above (sect. 1), Socrates turns to 
Parmenides and says:



> 
> 
> In a way, he has written the same thing as you, but he's changed it
> around to try to fool us into thinking that he's saying something
> different. For you say in the verses you've composed that the all is
> one, and you do a fine and good job of providing proofs of this. He,
> on the other hand, says there are not many things, and he too provides
> numerous and powerful proofs. Given that one says "one"
> and the other "not many," and that each speaks in this way
> so as to appear to have said none of the same things, when you are in
> fact saying virtually the same thing, what you've said seems said in a
> way that's beyond the powers of the rest of
> us. (Pl. *Prm*. 128a6-b6)




Socrates virtually accuses Zeno of having plotted with Parmenides to 
conceal the fundamental identity of their conclusions. With so many 
readers of Plato accustomed to taking Socrates as his mouthpiece in 
the dialogues, it is not surprising that this passage has served as 
the foundation for the common view of Zeno as Parmenidean legatee and
defender, by his own special means, of Eleatic orthodoxy. 
Unfortunately, this use of the Platonic evidence is unjustifiably 
selective, even prejudicial, in the weight it accords the words put 
in Socrates' mouth. Plato immediately has Zeno disabuse Socrates of 
his suspicions about the book's ulterior purpose. 




Zeno this time replies that Socrates has not altogether grasped the 
truth about his book. First, he says, the book had nothing like the 
pretensions Socrates has ascribed to it (*Prm*. 128c2-5). Zeno
is made to explain his actual motivation as follows:



> 
> 
> The treatise is in truth a sort of support for Parmenides' doctrine
> against those attempting to ridicule it on the ground that, if one is,
> the doctrine suffers many ridiculous consequences that contradict it.
> This treatise, therefore, argues against those who say the many are,
> and it pays them back with the same results and worse, intending to
> demonstrate that their hypothesis "if many are" suffers
> even more ridiculous consequences than the hypothesis of there being
> one, if one pursues the issue sufficiently. (Pl. *Prm*.
> 128c6-d6)




Zeno's account of how he defended Parmenides against those who 
ridiculed him is designed to correct Socrates' mistaken impression 
that Zeno was basically just arguing for the same thing as 
Parmenides, that the all is one. Zeno is portrayed as trying to 
correct this mistaken view of his purposes as born of a superficial 
understanding of Parmenides' doctrine. Zeno's arguments against 
plurality will seem to entail Parmenides' doctrine only if his 
thesis, "one is" (*hen esti*), is taken to mean 
that only one thing exists. However, the elaborate examination of 
this very thesis, "one is" (*hen esti*), by 
Parmenides himself in the latter part of the dialogue shows that 
Plato thinks it is not to be understood in any such trivial sense. 
For not only does Parmenides end up examining the relation of his One
to other things, which would have been impossible if his doctrine 
entailed their non-existence, but the relation other things have to 
the One actually proves responsible in a way for their existence. 
Zeno cannot be supposing that his arguments against plurality 
entailed the doctrine of Parmenides when that doctrine is represented
in this same dialogue by Parmenides himself as something altogether 
more involved than the simple thesis that only one thing exists. 
Nevertheless, Zeno's description of the persons who attempted to 
ridicule Parmenides is perfectly compatible with *their* 
having understood the thesis, "one is" (*hen 
esti*), as an assertion that only one thing exists. Zeno's 
arguments constitute an *indirect* defense of 
Parmenides--"a sort of support" (*boetheia 
tis*, 128c6)--because they do nothing to disabuse his 
detractors of their superficial understanding of his doctrine. 
Instead, as Zeno says, he tried to show that the assumption that 
there are many things has consequences every bit as unpalatable as 
those Parmenides' critics suppose his position has (cf. Procl. *in
Prm*. 619.15-21). Thus, while Zeno accepts Socrates' point that 
his own arguments aim to show that there are not many things, he 
corrects Socrates' impression that, in arguing this point, he was 
just saying the same thing as Parmenides in a different form. 




The evidence of Plato's *Parmenides*, then, does not license 
the conventional view that Zeno's arguments against plurality and 
motion were intended to support the strict monism of Parmenides. 
Claims to the contrary have rested upon selective and prejudicial use
of this evidence due to the tendency to privilege Socrates' remarks 
on Zeno's purposes over Zeno's own qualifications and corrections of 
that analysis. What Plato actually suggests is that Zeno aimed to 
show those whose superficial understanding of Parmenides had led them
to charge him with flying in the face of common sense, that common 
sense views concerning unity and plurality are themselves riddled 
with latent contradictions. Such is, essentially, the judgment of 
Jonathan Barnes: "Zeno was not a systematic Eleatic solemnly 
defending Parmenides against philosophical attack by a profound and 
interconnected set of reductive argumentations. Many men had mocked 
Parmenides: Zeno mocked the mockers. His *logoi* were designed
to reveal the inanities and ineptitudes inherent in the ordinary 
belief in a plural world; he wanted to startle, to amaze, to 
disconcert. He did not have the serious metaphysical purpose of 
supporting an Eleatic monism" (Barnes 1982, 236). However, 
whether the historical Zeno was actually involved in anything like 
the dialectical context Plato envisages for him must remain 
uncertain. Even if there were already in Zeno's day individuals who 
mistook Parmenides' position for the thesis that only one thing 
exists, the idea that Zeno's arguments were motivated by a desire to 
respond to such individuals in kind is as historically unverifiable 
as the claim Plato puts in his mouth that his book was stolen and 
circulated before he could decide for himself whether to make his 
arguments public (Pl. *Prm*. 128d7-e1). 




Nevertheless, just as Socrates' initial remark that Zeno's 
arguments were all designed to show that there are not in fact many 
things remains basically plausible, so there are elements in Zeno's 
account of his own purposes that have the ring of historical truth 
and that square well with other evidence. Plato has Zeno continue his
second response to Socrates (quoted above) by saying, "It was 
written by me in such a contentious spirit when I was still young. 
... You are mistaken in this regard, then, Socrates, that you suppose
it was written, not under the influence of youthful contentiousness, 
but under that of a more mature ambition" (Pl. *Prm*. 
128d6-e3). The point is repeatedly made that Zeno's book was written 
in a spirit of youthful contentiousness or "love of 
victory" (*philonikia*, *Prm*. 128d7, e2). The 
more mature Zeno seems a little embarrassed by the combative manner 
evident in the arguments of his younger days, as well he might since 
that spirit would have come to be seen as typical of the eristic 
controversialists who sprang up in the sophistic era. Plato gives yet
another nod to the idea that Zeno was a forerunner of eristic 
contentiousness when he has him say that his book "contradicts 
(*antilegei*) those who say the many are" (*Prm*.
128d2-3). This suggestion that Zeno was a practitioner of what came 
to be known as "antilogic," or the art of contradiction, 
is consistent with Plato's representation of him in other dialogues 
as something of a sophist. In the *Alcibiades*, Socrates 
reports that Pythodorus and Callias each paid Zeno a hundred minae to
become clever and skilled in argument (*Alc*. 119a3-6; cf. 
*Prm*. 126b-c). Teaching for payment is of course one hallmark
of the professional educators who styled themselves experts in 
wisdom. That Plato saw Zeno as a practitioner of the specific brand 
of argument known as antilogic is evidenced by the 
*Phaedrus*'s famous description of him as the 
"Eleatic Palamedes" for his ability to make the same 
things appear to his audience both like and unlike, one and many, 
moving and at rest (*Phdr*. 261d6-8). Again, at the beginning 
of the *Sophist*, when Theodorus introduces the Eleatic 
Visitor as an associate of Parmenides and Zeno and their followers, 
Socrates expresses concern that the Visitor may be "some god of
refutation" until Theodorus reassures him that the Visitor is 
more moderate than those who spend their time in eristic and 
competitive disputation (*Sph*. 216a-b). Plato's references 
thus consistently connect Zeno with the rise of eristic disputation, 
and it is perfectly plausible that his arguments against plurality 
and motion would have been well-known examples of making the weaker 
case seem the stronger. 




The portrait of Zeno and his tactics that emerges from Plato's 
references makes it seem natural that Aristotle, in one of his lost 
dialogues, entitled *Sophist*, spoke of Zeno as the inventor 
of dialectic (D.L. 8.57; cf. 9.25; S.E. *M*. 7.7). Precisely 
what Aristotle meant by this remains a matter of speculation, given 
that Aristotle also attributes the invention of dialectic to Socrates
(Arist. *Metaph*. M.4, 1078b25-30) and to Plato 
(*Metaph*. A.6, 987b31-3); he says he himself 
invented the *theory* of it (*SE* 
34, 183b34-184b8). There is also the question of
whether Aristotle viewed Zeno's arguments as more eristic than 
properly dialectical. The difference, according to Aristotle, is that
dialectical arguments proceed from *endoxa* or "views 
held by everyone or by most people or by the wise, that is, by all, 
most, or the especially famous and respected of the wise," 
whereas eristic arguments proceed from what only seem to be, or what 
seems to follow from, *endoxa* (*Top*. 
1.1, 100a29-30, b22-5). Aristotle clearly 
believes that some of Zeno's assumptions have only a specious 
plausibility (see *Top*. 8.8, 160b7-9, *SE* 
24, 279b17-21, *Ph*. 1.2, 233a21-31, 
*Metaph*. B.4.1001b13-16), so that they would
by Aristotle's own criteria be examples of eristic rather than
properly dialectical arguments. For Aristotle, then, Zeno was a
controversialist and paradox-monger, whose arguments were nevertheless
both sophisticated enough to qualify him as the inventor of dialectic
and were important for forcing clarification of concepts fundamental
to natural science. Aristotle's view of Zeno thus seems largely in
accordance with Plato's portrayal of him as a master of the art of
contradiction. 




Should we then think of Zeno as a sophist? Certainly Isocrates, the 
rhetorician and contemporary of Plato, did not hesitate to lump 
Gorgias, Zeno, and Melissus together as among the other 
"sophists" flourishing in the era of Protagoras and all 
producing tedious treatises advocating the most outrageous claims 
(Isoc. *Hel*. [*Orat*. 10] 2-3). While there are 
difficulties in giving precise definition to the term 
"sophist," one feature common to those normally classed 
as such that Zeno lacks is an interest in the interrogation of 
cultural norms and values. Zeno's influence, however, on the great 
sophists who were his contemporaries and, more generally, on the 
techniques of argumentation promulgated among the sophists seems 
undeniable. Protagoras' development of the techniques of antilogic, 
rooted in his claim that there are two opposed arguments on every 
matter (D.L. 9.51), seems likely to have been inspired by Zeno's 
novel forms of argumentation as well as by his advocacy of the most 
counter-intuitive of theses. Zeno's influence is especially clear, 
moreover, in Gorgias' treatise, "On Nature, or On What Is 
Not," both in its penchant for argumentation via antithesis and
*reductio* and in its use of premises drawn straight from Zeno
himself (see [Arist.] *MXG* 979a23, b25,
b37). It is even possible that the famous circle of 
contemporary intellectuals the great Athenian statesman Pericles 
gathered around himself provided a major conduit for Zeno's impact on
the first generation of sophists. Plutarch, at any rate, records that
"Pericles heard Zeno of Elea discoursing on nature in the 
manner of Parmenides, and practicing a kind of skill in 
cross-examination and in driving one's opponent into a corner by 
means of contradictory argument" (Plu. *Per*. 4.5). The 
skill Plutarch attributes to Zeno, still evident in the fragmentary 
remains of his arguments, is just the kind of skill in argument 
manifested in a great deal of sophistic practice. Although doubts 
have been raised about the reliability of Plutarch's report that 
Zeno, like Damon and Anaxagoras, was one of the many contemporary 
intellectuals whose company was avidly pursued by Pericles, there is 
little that seriously tells against it. Thus George Kerferd has 
argued both that the patronage of Pericles and his keen interest in 
the intellectual developments of his day must have been critically 
important to the sophistic movement and that Zeno's paradoxes were a 
profound influence on the development of the sophistic method of 
antilogic, which he sees as "perhaps the most characteristic 
feature of the thought of the whole period" (Kerferd 1981, 
18-23, 59ff., 85).




The evidence surveyed here suggests that Zeno's paradoxes were 
designed as provocative challenges to the common-sense view that our 
world is populated by numerous things that move from place to place. 
His apparent demonstrations of how the common-sense view is fraught 
with contradiction made him an influential precursor of sophistic 
antilogic and eristic disputation. It is not surprising that someone 
like Isocrates should have viewed Zeno as a sophist to be classed 
with Protagoras and Gorgias. To ask whether Zeno was in fact a 
sophist, a practitioner of antilogic, an eristic controversialist, or
a proper dialectician is to some extent inappropriate, for these 
designations all acquired their normal meaning and range of 
application only after Zeno's time. While he perhaps does not fit 
exactly into any of these categories, still his development of 
sophisticated methods of argumentation to produce apparent proofs of 
the evidently false conclusions that motion is impossible and that 
there are not in fact many things made it quite natural for Plato, 
Aristotle, Isocrates, and others to refer to him under all these 
labels. 




It is remarkable that, while many of the responses to Zeno's 
paradoxes, and even some modern formulations of the paradoxes 
themselves, depend on advanced mathematical techniques, Zeno's 
original arguments do not themselves appear to have involved any 
particularly complicated mathematics. Several of the paradoxes 
involve no specifically mathematical notions at all. The Achilles is 
perhaps the best example since it employs only very ordinary notions,
such as getting to where another has started from. The other extant 
arguments for the most part deploy similarly prosaic notions: being 
somewhere or being in a place, being in motion, moving past something
else, getting halfway there, being of some size, having parts, being 
one, being like, being the same, and so on. Where Zeno seems to have 
leapt ahead of earlier thinkers is in deploying specifically 
quantitative concepts, most notably quantitative concepts of limit 
(*peras*) and the lack of limit (*to apeiron*). Earlier
Greek thinkers had tended to speak of limitedness and unlimitedness 
in ways suggesting a qualitative rather than a quantitative notion. 
While one might suppose that Zeno's turn to a more strictly 
quantitative conception of limit and limitlessness could have been 
inspired by his familiarity with Pythagorean philosophers and 
mathematicians in Magna Graecia, we can in fact trace the philosophy 
of limiters and unlimiteds only back as far as Philolaus, a 
Pythagorean roughly contemporary with Socrates and thus a good deal 
younger than Zeno.




Whatever may have spurred Zeno's development of his collection of 
paradoxes, his arguments quickly achieved a remarkable level of 
notoriety. They had an immediate impact on Greek physical theory. 
Zeno's powerful principle that any spatially extended entity must be 
limitlessly divisible would profoundly impact the development of the 
subtle and powerful physical theories of both Anaxagoras, who accepts
the principle, and the early atomists, Leucippus and Democritus, who 
reject it. Zeno's arguments also had a formative influence on 
Aristotle's own theory of the continuum and of continuous motion. 
More generally, Zeno's arguments made it necessary for Greek natural 
philosophers to develop something more than an everyday conception of
the composition of material bodies. His arguments, perhaps more than 
anything else, forced the Greek natural philosophers to develop 
properly *physical* theories of composition as opposed to the 
essentially chemical theories of earlier thinkers such as Empedocles.
That mathematicians and physicists have worked ever since to develop 
responses to the more ingenious of his paradoxes is remarkable, 
though perhaps not surprising, for immunity to his paradoxes might be
taken as a condition upon the adequacy of our most basic physical 
concepts. He may even have offered his collection of paradoxes to 
provoke deeper consideration of the adequacy of theretofore 
unexamined notions. If so, it is likewise remarkable that he 
simultaneously developed forms of argument--most notably, 
*reductio ad absurdum* by means of antinomical and/or regress 
arguments--that have ever since been fundamental to 
philosophical probing of conceptual adequacy.