The works of the three "Eleatic" philosophers have much in common, but that is not to say that all three were trying to do the exact same thing, or that all three can be found to be in agreement on all points.
In Parmenides' poem, a goddess declares that it is necessary to say and to conceive (think, take it) that what is is (fr. 6). (Most likely she means that this is necessary in or for inquiry, given what she says in fr. 8, and given the reference to roads of inquiry at 6.3.) She also says that on the only road of inquiry that she considers to be at all viable or possible, we must say and take it that what is is one, whole, ungenerated, indestructible, complete, unmoving, unique, continuous, and limited.
Zeno tried to show that the assumption that multiple things are or exist has contradictory implications. That is, he tried to show that on the analysis of things that was (and to some extent still is) offered, there cannot coherently or consistently be taken to be multiple things. He also tried to show that contradictions result from assuming that there is motion and accepting the standard analysis of motion; and that contradictions result from assuming that there are things that have extension in space and accepting the standard analysis of that.
Melissus wrote in support of the claim that what
is is one. His understanding of what this claim involved and of
what it implied led him to argue for some conclusions that were
very different from key statements made by the goddess in
Parmenides' poem. Perhaps most notably, Melissus argues that what
is must be spatially and temporally unlimited, whereas Parmenides'
goddess holds that what is must (in or for inquiry) be said and
taken to be limited - and she is not clear, or not explicit, about
its relationships to existence in space or time.
It is important to note the form in which each of these philosophers writes. In Parmenides, the narrator of the poem recounts a journey to the realm of a goddess, and this goddess discusses roads of inquiry and the opinions of mortals. In her discussion of roads of inquiry, she talks of how that which is must be said and taken to be, in or for inquiry at very least. All of the remarks in Parmenides' fragments that concern the way what is must be are in fact remarks about the way what is must be (or the way it must be said and taken to be) on or for roads of inquiry; and they all seem to be part of a speech by a goddess character.
Zeno, in contrast, discusses how what is cannot be, or how it cannot coherently be taken to be. He does not make use of characters or narrative, as far as we know. He also does not in the surviving fragments talk of what that which is could be like, or characteristics it could or must have (or be taken to have); he only mentions what that which is could not be like (or characteristics, e.g. multiplicity, that it could not coherently be taken to have). That is, Zeno only says that what is could not be (or could not coherently be taken to be) multiple, and that there could not be (or could not coherently be taken to be) multiple things. There is no evidence that he said that what is is one. Perhaps he thought that our customary ways of looking at that which we say exists (or at that which we say is, or at that which we say is the case) are so flawed that we will run into contradictions if we say that what is is one, and that we will run into contradictions if we say that multiple things exist.
Melissus, on the other hand, does say that what is is one. He also says that what is must be spatially and temporally unlimited. He says this in his own voice, i.e. not through a character in a poem.
These differences in the forms of their remarks have significant ramifications for the interpretation of these philosophers' works, as will be seen below.
(Note: The abbreviation 'frr.' means 'fragments.')
I. Parmenides. Parmenides wrote a poem in which a young man, the narrator of the poem, is transported to the realm of a thea, a female divinity. This goddess promises to have him learn "everything, both the...heart of...[T]ruth; and the opinions of mortals, in which there is no true assurance" (fragment 1/McKirahan 11.1, lines 28-30).
However, there is no evidence that this goddess presents the
young man with a direct statement or description of the "heart of
[T]ruth". Instead, she discusses "roads of inquiry" that she takes
to be conceivable (first mentioned in fr. 2/McKirahan 11.2). Then,
after she has talked about the last one (8.1-49/McKirahan 11.8,
lines 1-49), the one that she seems to take to be the most viable
(see e.g. 8.34-38) and the one that has something to do with truth
(8.50-52), she abruptly changes the subject and talks about the
opinions of mortals. Thus it may be helpful to look at the
surviving fragments as falling into two main sections: the story
of the journey (1.1-23) and the speech of the goddess (the rest of
the fragments, with the possible exception of frr. 3,5, and
"Cornford's Fragment" [=McKirahan 11.3, 11.5, 11.19 respectively],
for which we have no clearly decisive indications as to
placement). The goddess's speech itself can also be seen to have
two parts (though perhaps there once were more): a section on the
roads of inquiry that can be conceived, and a section on the
opinions of mortals.
A. Fragment 1: The journey to the goddess.
Ancient sources report that Parmenides' poem began with these
lines. In fragment 1, the poem's narrator (who may or may not be
intended to be Parmenides himself) recounts an unusual chariot
ride. Drawn by mares and led by the Heliades (daughters
of the Sun, Sun-maidens), the narrator's chariot goes beyond the
beaten paths of humans to the gates of the roads of Night and Day.
The gates are guarded by Dike (Justice, Right), but the
Sun-maidens persuade Dike to open the doors, and the
chariot is able to proceed to the home of an unnamed goddess. This
goddess welcomes the narrator and makes the speech that takes up
the rest of the fragments (again, frr. 3,5, and "Cornford's
Fragment" may be exceptions).
1. The wording of the first line suggests that the mares continue
to conduct the narrator in the chariot, even after the goddess's
speech is done. That is, his quest or journey may not be complete;
it may be an ongoing process or way of living.
2. We know that the narrator is male, because the goddess refers
to him as "young man" (kouros). All but one of the other
figures in the fragments are female: the mares, the daughters of
the Sun, Dike, Ananke (Necessity or
Constraint), Moira (Fate or Portion), the goddess who
speaks, Truth, Persuasion, Themis (Custom, Right), and
the unnamed divinity in frr. 12 and 13. The only other male figure
in the fragments is Eros, god of love or desire, in fr.
3. A number of the figures and items mentioned in fr. 1 appear in
other contexts in other fragments; in interpreting their meanings
it is important to be aware of the differences and the parallels
among contexts. For example, Dike, Themis, and
Moira show up again in fr. 8. Night and Day in fr. 1
would seem to have some connection with night and light (or fiery
stuff) at 8.56-61, and frr. 9 and 12. Ananke appears in
frr. 8 and 10. The sun is mentioned in frr. 1,10,11, and 15. Roads
appear in frr. 1,2,6,7, and 8.
4. Some authors have suggested that the fact that Parmenides has a goddess make pronouncements is a sign that he intends a sort of "argument from authority": a sign that he wants his readers to accept these pronouncements unquestioningly, and that he himself accepts them unquestioningly.
But there are reasons to suspect that such an interpretation may not be correct. For one thing, the goddess argues for the things she says; she provides support and explanation for her claims. She does not insist that the young man accept her statements as true just because she is a goddess (or because she could kill him if he refused to accept them); rather, she gives reasons for her assertions, and asks him to think about them.
It is noteworthy that the goddess argues at all. The poem of
Parmenides provides the first known example of written deductive
arguments in Greek philosophy. The level of abstraction (e.g. the
fact that the goddess talks about being as a whole or in general,
or about that-which-is as a whole or in general; and the fact that
she takes as her subject what it is we'd need to say and to think
about what is, in order for inquiry to be possible) seems to be an
innovation as well.
5. In fragment 2 the goddess announces that she will discuss "the only roads of inquiry conceivable". What might be meant by 'road of inquiry'? It may be helpful to note here that the expressions 'avenue of inquiry' and 'avenue of exploration' are sometimes used today, and these could well be the sort of thing the goddess intends. A road of inquiry would seem to be a path that one follows, or makes, in seeking something. (The word Parmenides uses, 'dizesis,' can be translated as 'inquiry' or as 'seeking'.) It would be a path in though or conceptions; the steps one takes on the road would be the questions one asks, the ways one looks for answers, and so on. What defines the path or road would then be the ways of looking at the world that are involved in the steps one takes. For example, a person who believes that everything that exists in the world is matter and energy (and that everything that does not immediately appear to be matter or energy must be explainable in terms of the effects of matter and energy) would look at the world quite differently, might ask different questions, and would accept different kinds of answers, from a person who believed that matter and energy do not exist; and from a person who believed that matter, energy, and some other kinds of entities exist.
In fragments 6 and 7 she seems to rule out, or to bar the human from following, certain roads; from the way she describes them, she seems to see them as unviable (presumably, unviable for inquiry, since they were supposed to be roads of inquiry). There and in fr. 8 she tries to get the young man to reason away from assumptions based on ordinary experience to some indications about the character of to eon, "that which is" or "what is" or "being" (or "Being", in some translations; Parmenides did not use capital letters, so we don't know if he would want us to capitalize the initial 'b')(1). It's important to note that she is talking about how "what is" would have to be (or have to be said and taken to be) on an apparently viable road of inquiry. That is, she does not say anything about how "what is" must be (or how we must say and take it to be) independent of contexts of inquiry. Perhaps even outside contexts of inquiry we must take it that "what is" is as the goddess says it is (or must be) in fr. 8; or perhaps not. To ask whether outside of contexts of inquiry "what is" must be a certain way, would itself be an inquiry, and thus subject to the restrictions the goddess mentions.
On the road of inquiry discussed at the beginning of fr. 8, "signs" indicate how "what is" must be one, continuous, ungenerated, indestructible, etc. Why must it be that way? I would suggest that we must take it to be that way in order for inquiry to be possible and coherent; and that it must be that way if inquiry is to be able to get valid results (whether it does is anotherquestion). "What is" would have to be taken to be continuous, for example, because if it were not, what would separate its "parts" or "members"? The separator could not be something nonexistent, or "non-being", because then it could not be said to be what separates the parts or members of "being" or "what is". The separator could not be said to be something existent other than "what is", because "what is" is all that is. So "what is" would seem to have to be continuous, or to have to be taken to be so.
"What is" would have to be taken to be ungenerated, if inquiry is
to be possible, for the following reasons. To say that it was
generated, that it came to be, would seem to imply first that
there was a time at which it did not exist. Then we would
have to say either that "what is" came from nothing, from
non-existence; or that it came from something else. If we suppose
that it came into being from nothing, then it would seem that
things can pop into being at random, or without an existent
source. Then we could not have any way of seeking anything, since
we would not be able to tell when and where it could be found.
Nothing would be known to be stable enough to allow the "process
of elimination" to be used as a search tool. But perhaps it was
only "what is" as a whole that came into being from nothing.
Still, we would not be able to take steps that would be
efficacious (or known to be efficacious) in seeking anything. Why
not? The problem would be that if "what is" came to be from
nothing, then we could not trace anything to its source; we could
not know whether the relationships we might think things have with
each other really do obtain. The same problem would arise if we
said that "what is" came to be from something else: if that
something else was not "what is", or was utterly unlike "what is"
(because if there was any continuity or commonality between the
"two" then that would be the ultimate nature of "what
is", and we'd have to ask whether that came into being),
then the relationships that obtain within "what is" would not have
any analogue in the "source" of "what is". And we'd have to ask
where that source came from, etc. If we could not answer these
questions, inquiry could not be known to be viable.
6. According to the goddess, "what is" must be this way, in or
for inquiry, because it is bound by Dike (Justice), Ananke
(Necessity), and Moira (Fate or Portion). The standard
Greek understandings of these forces would give them powers or
implications that can be seen to correspond (with respect to to
eon) to the effects the goddess ascribes to them. The
standard Greek understandings of these forces, and indeed the
language of the whole poem, invoke multiplicity, change,
non-existence, and lots of other things that the goddess says are
incompatible with the way "what is" must be on the road of inquiry
of fr. 8. We could not reach the conclusion that "what is" must be
continuous, on that road, without invoking distinctions
(discontinuities), etc. In other words, the goddess shows that
there are contradictions and incoherences in the basic conceptions
that make inquiry possible and that we use in everyday life (and
that she uses to point these out to us). It's paradoxical, but
that's why we're mortals: we're mortals insofar as we have lacks
that prevent us from being able to resolve these.
At 8.50, the goddess abruptly changes the subject, and describes
what she says are human opinions about cosmology, biology, and the
naming or classification of things in general. She says that
mortals lay down two forms or opinions to specify (name)
appearances; these two are light (a high-up, lightweight, fiery
thing) and night (a dark, heavy thing). While these two are
fundamental aspects of many cultures' ways of looking at what
exists, no person of culture in Parmenides' area and time is known
to have held that light and night are the only
fundamental components or aspects of all things. Thus
the goddess may be saying that mortals' opinions about what exists
imply that mortals believe that light and night (or
something equivalent) are at the foundation of all things. This is
not to say that the goddess or Parmenides thinks that mortals are
justified or accurate in holding these opinions; perhaps the
goddess or Parmenides thinks that this is about as good as mortals
can do when it comes to describing their world. (Perhaps P. thinks
that this is the best that mortals can do if they are using the
kind of language and conceptions that enable them to understand
his poem.) If light and night name the most basic distinctions
that mortals make, it is perhaps fitting that the young man in the
chariot passes beyond the gates from which they are supposed to
issue -- especially since this is the last landmark
mentioned before he enters the realm of the goddess.
Zeno was concerned to show that our understandings of time,
motion, change, position, magnitude, unity, singleness, and
plurality are incoherent; or that they result in contradictions.
Many writers have supposed that Zeno tried to prove (or thought he did prove) that there is no motion, that there are not multiple things, and so on. In fact, that is how Aristotle presents Zeno's work, and Aristotle and his school are our main sources on Zeno. Some of these writers then criticize Zeno, saying that he failed to prove those things.
But a careful look at what Aristotle presents of Zeno suggests
that Zeno may not have intended to prove that there is no motion,
etc. Instead, Zeno may have been trying to show that what people
in his time (and perhaps even now) mean by "motion" does not make
sense, i.e. that whatever is going on when we say that a thing is
moving could not possibly be what we think we mean when we say
that a thing is moving, etc. That is, Zeno may not have been
trying to show that your eyes are not doing anything when you read
this, nor that the keys of the keyboard on which this was typed
were not doing anything different during the typing from what they
were doing when no typing was going on. Instead, he may have been
trying to show that the account of movement that was (and perhaps
still is) given is incoherent; that the claims that were (or are)
usually made about what movement involves lead to contradictions.
Similarly, he may have been trying to show not that only one thing
exists or is, but that the assumptions involved in the claim that
multiple things exist or are lead to contradictions. (This does
not rule out that he might have thought that the assumptions
behind the claim that there is only one [thing] result in
contradictions as well.)
If in fact Zeno meant only to show that our accounts of motion, multiplicity, and so on are incoherent and inconsistent (result in contradictions), then his arguments can be seen to make more sense, and to be more effective, than they would if he meant to show that there is no motion or multiplicity, etc. (An interesting discussion of whether or to what extent anyone has ever refuted Zeno's arguments can be found in Thomas Heath, A History of Greek Mathematics, Chapter VIII.)
At very least, then, Zeno felt that our analysis or understanding of what is going on when we say there is motion (or multiple things, or different positions in space, etc.) leads to inconsistencies no matter which way we turn.
Zeno's method was to try to show that accepting motion or
multiplicity etc. as a quality of what exists leads to two
possible alternatives. These alternatives would each lead to a
contradiction or impossible consequence, and would also
contradict one another.
Let us take the first of Zeno's paradoxes(2) quoted in your text (McKirahan p. 184, 12.3 and 12.4) as an example. It can be applied to time or change as well as to space or magnitude, as we will see, but let us start with magnitude.(3)
Consider the things that are supposed to exist in space. We, like people in Zeno's time, generally say that there are spatial things, things that "take up space" or "exist in space". Maybe not all things that we say exist "take up space", but we generally say that at least some things do. We suppose that these things have sizes, or spatial magnitudes, or dimensions; each such thing is supposed to take up a certain amount of space, i.e. to have a certain size or magnitude.
An unspoken assumption that the Greeks held (and that we generally do too(4)) is that each thing that exists in space has a definite size or magnitude, a definite beginning and end. After all, if we are going to claim that multiple things exist in space, we must have some idea of where one thing "A" ends and where the next thing "B" starts - otherwise, we would not be justified in claiming that there are these two distinct things A and B. (And if they're not distinct from each other, how are they two?) Zeno will argue that if we can't tell where any one thing A ends and the next thing B begins, we would seem to lack justification (certainly we would lack the traditional justification) for asserting that there are multiple things at all.
If we consider the things that we suppose to have spatial magnitude, we note that each such thing can be divided into parts, at least conceptually. Even when we can't actually break a particular thing, we may speak of the right and left sides of it, or the top five inches of it, etc. We tend to agree, that is, that each thing has parts or units: a six-inch-long piece of wood is longer than a five-inch piece by a "unit" of one inch.
Consider, then, the parts or units of the things that we say have size or spatial magnitude.
Either the smallest units or parts of which those things are
composed have no (that is, zero) magnitude, or they have some
(positive) magnitude. Or, so goes the common Greek and
modern conception of these matters. Let us see what the
consequences of each of these possibilities are.
(a) Suppose that the smallest units or parts of spatial things have zero magnitude. Then if you add one of these units to a thing "C", then that thing would not get any larger - that is, the sum of the two would be no larger than C alone. Then it would be impossible for there to be anything that had positive magnitude, since all magnitudes would be multiples of the smallest magnitude. Thus all magnitudes would be constituted of many units of the smallest magnitude.
But we just saw that on supposition (a), adding the smallest
magnitude to something will not make that thing larger. So adding
up lots of smallest-magnitude units gives you a sum or aggregate
that is no larger than one smallest-magnitude unit - which is zero
magnitude. Put another way, if all magnitudes are multiples of
that smallest unit, they are all zero. Yet we say that there are
things with positive magnitude, things that have size greater than
zero. We even claim to see such things. So the smallest parts
of things should have some positive magnitude -- which
contradicts premise (a).
(b) So, let's suppose instead the other possibility given above: Suppose that the smallest units or parts of spatial things have some positive magnitude. In that case, says Zeno, let's look harder at those things that we say have definite (limited, finite) sizes. If you examined a very small end piece of one of those things, you'd notice that that end piece had an end piece with a magnitude. Then you'd notice that that end piece had an outermost piece (or area or section) with a magnitude. Every time you would come to what you thought was the end of what you thought was the smallest piece, you would notice that this end itself was an even smaller piece with magnitude.
Or, to look at it another way, imagine that you are trying to get an exact measurement of something that is supposed to exist in space. You hold the thing up against a ruler, and you notice that the end of the thing falls in between two of the smallest markings on the ruler. So you get a ruler with even smaller divisions, and it happens again: the end of the object falls in between two of the smallest markings on the new ruler. You now have a somewhat more accurate measurement than you originally had, but it's still not an exact measurement. So you get rulers with smaller and smaller divisions, and finally the end of the object does come out right on a ruler marking. But then you look at the object and the ruler under a magnifier, and you notice that the ruler marking line has some thickness. You can see that the end of the object falls somewhere in the middle of the thickness of one of the ruler marking lines. Where exactly does it fall? You need even a more finely-divided ruler in order to find out, it seems. Since all ruler markings must have a thickness and since any two ruler markings must have a distance between them, the end of the object will always fall in something that can be divided further.(5) No smallest positive magnitude will be found. Further, there will not be determinable endpoints to things. These conclusions undermine hypothesis (b).
So now we are stuck: either we can't find the smallest piece of a thing or the smallest magnitude with which we can measure; or there must be things or units that have zero magnitude. If we choose the first, we can't find exactly where a thing ends, so things are unlimited or indefinite(6) in size; there are always more divisions of the measuring device or of the thing. Each thing could be seen as indefinitely large (if we tried to measure it outward from its center or other internal point); and each thing could be seen as indefinitely small (if we tried to measure the outermost extents of the things that were supposed to surround it). The exact number of spatial objects might be indeterminate too, for we might not be able to tell where one supposed spatial object left off and the next began, so we could not with support the claim that there were in fact two distinct spatial objects. Moreover, if we cannot find the smallest piece of a spatial thing, we cannot express spatial magnitudes as multiples of some smallest unit. That conflict with our usual belief that there are things that have definite magnitudes. (Again, to be sure that there are multiple spatial things, it seems we would have to be able to distinguish them clearly from one another, i.e. find clear distinctions between them -- exactly what we have seen we cannot do if we assume that there is a smallest positive magnitude.)
So it would seem that we would have to choose the second alternative from the preceding paragraph, that is, choose to say that there is a smallest magnitude, and that this smallest magnitude is equal to zero. But we have seen in our examination of hypothesis (a) above that we cannot coherently (i.e. without contradiction) suppose that there is a unit of zero magnitude.
Thus it seems that if we suppose that there are distinct things with finite magnitudes then we must suppose that there is a smallest magnitude; and we must also suppose that there cannot be a smallest magnitude. If we suppose that there are distinct things of finite magnitude, we must also hold either that there are no things of positive magnitude, or that there are no things of finite or determinate positive magnitude. Thus it would seem to be impossible to hold (without contradicting oneself) that there are multiple distinct things in space. (Again, in order for us to say with justification that there are multiple things, we must be able to find clear distinctions between those things; but if the limits of spatial things are unclear we could not with certainty tell one from another, nor tell with certainty that there were indeed multiple different things.)
Similarly, according to Zeno we cannot identify exactly when a
process starts because we could not find a smallest interval of
time, an interval that is not made up of smaller (non-zero) ones
-- for if an interval is made up of smaller ones, during which
smaller one did the process start; and during which division of
the smaller one, etc.?--. So we cannot according to Zeno have a
coherent account of change.
We have seen that it is often said that all three "Eleatics" claimed that what is (to eon in Parmenides or to on in Zeno and Melissus: a difference of dialect) is one; or that all three wrote in support of the claim that what is is one. We have also seen that Parmenides and Zeno may not have held or supported the claim that what is is one.
But the third "Eleatic", Melissus, did write in support of the claim that what is is one. His understanding of what was involved in or implied by this claim led him to argue for some conclusions that are very different from key statements made by the goddess in Parmenides' poem. Most notably, Parmenides' goddess holds that on or for a certain road of inquiry at least, what is (to eon) must be taken to be limited; whereas Melissus argues that what is must be spatially and temporally unlimited (apeiron). Parmenides' goddess is discussing a certain road of inquiry, and is not necessarily at this point saying anything about space and time; Melissus is not discussing roads of inquiry (i.e. he seems to be making unconditional claims about what exists), and he does specifically mention space and time.
One thing that might account in part for this difference is that Melissus seems to have a different focus from those of Parmenides or Zeno. Parmenides' goddess is concerned about what must be said and conceived, in the most general way, regarding what is. Zeno examines the consequences of supposing that there are multiple things (especially in space and time, but conceivably also in general). He draws conclusions about what cannot exist or take place or be thought without contradiction, but does not as far as we know make any claims about what kind of entity he might think does exist or the kind of situation he thinks does obtain. Melissus, however, thinks that at least some terms of the kind we use to describe the things we usually (if perhaps erroneously) think exist or are, can in fact be used to describe the unity of "being" ("what is") that Melissus thinks exists or obtains (see e.g. his use of 'arrangement', 'healthy', and perhaps 'full', in fr. 7).
Melissus suggests that the ways in which we describe the multiple things we say are lead to unavoidable contradictions; what is can't be as we say it is, he thinks. That is, Melissus assumes that if the account or description of a thing leads to a contradiction, then the thing can't really exist, or can't really be as we had said it was. Zeno and Parmenides did not take this step; they found contradictions in our acounts of things, but did not conclude that this necessarily meant that those things did not exist. Rather, Parmenides and Zeno call into question our basic ways of speaking and conceiving of things, or at least investigate whether our ways of speaking and conceiving of things are adequate means of finding truth through inquiry.
Moreover, Melissus argued that if the claim that there are multiple things leads to a contradiction, there cannot be multiple things - and concludes from this that there is only one thing. Zeno seems to have thought that there were also problems with the claim that there is only one thing, but Melissus did not.
Once Melissus has rejected the things he felt led to contradictions, he went on to look at the implications of the notion of existence, or of the notion that anything at all exists. When he does this, he concludes that what is must be one, infinite, eternal, continuous, unchanging, and so on; and that change, multiplicity, division, creation, destruction, etc. must be illusion. The question may arise as to what in M.'s unity could account for this change and multiplicity even if the change and multiplicity are illusory. How could what must be one and undivided produce the illusion of division? For even if it is an illusion, we really have the illusion, and there's supposed to be a distinction or division between the illusion and the truth (or the truth behind the illusion), so would that not mean that some division or distinction is real?
We do not know whether M. ever responded to this sort of
question. But he was not alone in trying to take P.'s and Z.'s
observations into account when trying to formulate an
understanding of what exists (including appearances). Empedocles,
Anaxagoras, and Philolaus, among others, tried to meet the same
challenge, but did so in very different ways.
1. It is not entirely clear what is meant by "what is" or "being" (to eon) in Parmenides. The term can be translated as 'what is', 'being', 'what is real', 'what is the case', 'what occurs', 'what obtains', 'being actual', and so on. If you think that a thing exists or that a situation obtains or that something is the case, you are considering it as belonging to "to eon". If you think that something can exist, be the case, etc., you would seem to be considering it to belong at least potentially to "to eon". Thoughts, ideas, situations, material objects, colors, sounds -- anything you say exists or obtains, you are considering as belonging to "what is", to to eon.
It does seem as though the goddess is talking about "what is" as a whole or in general; or about what it is that all things said to exist or obtain are supposed to have in common. She does not state directly what she thinks this is, nor does she say anything about what specifically should count as "what is", what qualifies as "being", etc. She does suggest that it is incoherent and/or inconsistent for us to suppose, in or for inquiry at very least, that to eon is not one, that it came to be or could perish, that it changes, and so on.
Some commentators (e.g. Coxon, Owen) claim that Parmenides or the goddess thinks that only what is thought (ideas, perhaps) exists or can exist. But they base this claim on a combination of a mistranslation of fr. 6, a dubious translation of 8.34-36, and one of many possible translations of fr. 3. Fragment 3 is half a hexameter line in length and is not a complete sentence, however, so its reference is not at all clear. Without further evidence, then, these commentators' interpretation cannot be found to be valid.
2. The word 'paradox' comes from a Greek word meaning 'contrary to opinion'. In the sense in which it is used to describe Zeno's work, it refers to an argument that derives self-contradictory or mutually contradictory conclusions by valid deduction from accepted or apparently acceptable premises.
3. The fragments are not numbered according to their order in Zeno's book; we don't know in what order they occurred in Zeno's book. They are numbered in your text in an order that the editor thinks makes sense, and they are numbered in the best-known or standard (Diels-Kranz) collection according to the order in which they are quoted in later writers who discuss Zeno. The earliest source in which we find quotations (or something close to quotations) of the passages from Zeno that are now designated as fragments 1 and 2 is Simplicius' commentary on Aristotle's Physics. (Simplicius lived about one thousand years after Zeno.) Simplicius says that what he is presenting first is the second part of a certain section of Zeno's work, and that what he is presenting second is the first part.
4. In modern physics, we may try not to hold this assumption, but problems arise then when it comes to specifying what an "object" is, and what the relationships are between "middle-sized objects" - roughly, anything bigger than a molecule and smaller than a galaxy - and the not-always-particle-like submolecular entities that are supposed to compose the "middle-sized objects".
5. It may be objected that an ideal ruler, one we could not physically construct but which would be conceivable in pure geometry, could have "markings" or that are line segments. Line segments have no width (they are one geometrical point "wide", and points have no dimension).
One response to that would be to point out that the increments of measure between each two of these "markings" would still have some width, and so would the tip of the object being measured, so we would still have the problem of having to look seemingly forever for finer and finer divisions of distance.
Another response would be to note that the idea of a "ruler" or measuring line that is composed of dimensionless (sizeless) geometrical points was already shown to be incoherent on some level in part (a) above: a thing that is supposed to have positive magnitude cannot be coherently taken to be made up of pieces that have zero magnitude.
6. Many translators use the term 'infinite'
in her translation of fragments 1 and 3, but this is not
necessarily appropriate; the Greek word is 'apeiron',
which can mean "infinite", "indefinite", "indeterminate",
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An earlier version of these notes, keyed to
Freeman's Ancilla to the Pre-Socratic Philosophers