In Plato's Parmenides, the narrator is Cephalus of Clazomenae, who arrives in Athens and is greeted in the market place by Adimantus and his brother Glaucon. They go to meet Antiphon, the half-brother of Adimantus and Glaucon, in order to hear him recount a legendary conversation that Socrates once had with Zeno and Parmenides. Parmenides was about sixty-five years old at the time, Zeno was about forty, and Socrates was about twenty.
The conversation began with Zeno arguing that if there is a plurality of things, then they must be both like and unlike one another. Socrates responded, however, that if unlike things cannot be like and like things unlike, then there must not be a plurality of things. Since Zeno agreed with this conclusion, Socrates remarked that Zeno shared with Parmenides the view that all things are one, and not many. (128b)
Socrates proceeded to distinguish between being one thing and having many aspects, explaining that in this sense a given thing can be both one and many. (129d)
Parmenides responded by asking Socrates whether the abstract form of something (such as largeness or smallness or beauty or justice) as a whole can be shared by a plurality of things, and whether, if this were the case, it would mean that the form as a whole would be in a number of separate things and would therefore be separate from itself. (131b)
Socrates responded no, this would not mean that a form shared by many separate things would be separate from itself. The form itself would still be one and the same thing. (131b)
Parmenides then responded that a form shared by many separate things must therefore be divisible into parts, and that each thing that shares in that form will therefore only be able to share a part of it. (131c) But if equality, for example, were a form that things could share, then they would only be able to share a part of it, and would share in something less than equality itself, so they must actually not be able to share in equality either in part or as a whole.
Socrates admitted this would be a problem, if the abstract forms of things were taken to be divisible into parts. However, he explained that if a form covers all cases of a single thing, then it can still be one and the same in all cases of that thing.
Parmenides then told Socrates that if he wanted his theory of forms to be thoroughly exercised, he must examine the consequences both for a plurality of things (with reference to one another) and for a single thing (with reference to itself) of there being, or not being, a plurality of things. (136a) The One cannot have a plurality of parts and still be the One. (136c) Moreover, it cannot have a beginning, middle, or ending, because such stages would be parts of it, and it would no longer be the One. (136d) Nor can it be at rest, because then it would remain in the same place, and would be encompassed by something other than itself. Nor can it be in motion, because then it would be undergoing alteration, and would not be the One.
Furthermore, the One cannot be different from itself or the same as another. Nor can it be the same as itself or different from another, because it is simply the One. (139b) For the same reason, it cannot be equal or unequal to itself or to another. (140b) It cannot be older or younger than anything, or the same age as anything. It has nothing to do with time, and thus has never become, was never becoming, and never was, and will not become, be becoming, or be in the future. (141e)
Parmenides concluded there is no way in which the One can have being. How can this be the case with the One? In order to answer this question, Parmenides described the consequences of the contradictory proposition that the One can and does indeed have being. If the One has being, then number also has being. If number has being, then the One is both one and many, whole and parts, limited as well as indefinitely numerous. (145a) It is both at rest and in motion. It is both the same as, and different from, itself, and the same as, and different from, others. (146b) It is equal both to itself and to others, and it is both greater and smaller than others.
The One is also becoming older than something becoming younger, and in this sense it is becoming both younger and older than itself. (152b) But since in being and becoming it must take the same time as itself, it also becomes neither younger nor older than itself. (152e)
The One both is and is becoming older and younger than itself, and than others, and also neither is nor is becoming older or younger than itself, or than others. (155c)
Parmenides then described the consequences of the proposition contradictory to the proposition that the One is, the proposition that the One is not. If there is no One, but only things other than the One, then those things cannot be One or many. There cannot be many without a One. Parmenides concluded that without the One, there is in fact nothing at all.
The One, as defined by Parmenides, has an elusive and paradoxical nature, but this seems to be more a matter of definition than of cosmic or logical necessity. The dialogue between Socrates and Parmenides is perhaps more interesting for the philosophical and logical questions it raises than for the convoluted and often problematic arguments made by Socrates about the theory of forms and by Parmenides about the nature of unity and plurality.
Parmenides' method of inquiry is dialectical. He considers the consequences of a given proposition, and then considers the consequences of a contradictory proposition in order to get at the truth of the matter. In this way, he is able to reveal unrecognized inconsistencies and overlooked implications of previous assumptions made by his interlocutors.
Many of the logical arguments in the Parmenides fail to be persuasive or convincing. Nevertheless, they stimulate further examination of, and critical reflection upon, their meaning. For example, Socrates asks Zeno, "What does this statement mean, Zeno? 'If things are many,' you say, 'they must be both like and unlike. But that is impossible; unlike things cannot be like, nor like things unlike.'" (127e) Why exactly is it impossible that if there are many things, some may be like and others unlike one another? it is indeed possible. This fact wouldn't entail that the same things that are alike in some respect are simultaneously unlike in the same respect. That would indeed be contradictory. But Socrates does assume that this is what is entailed by Zeno's statement, and he then turns the statement around by saying, 'if things can't be both like and unlike, then there can't be a plurality of things.' His argument is formally valid (it takes the form of modus tollens or denying the consequent), but he commits the informal fallacy of equivocation by changing the implied meaning of Zeno's statement from 'among a plurality of things, some must be like and others unlike one another,' to 'in order for there to be a plurality of things, the same things that are alike in some respect must also be unlike in the same respect.'
Similarly, the assumption that Parmenides initially seems to make in describing the One is that there is nothing other than the One. The many are actually nothing other than the One, and the One is all there is. Thus the One has no beginning or ending, no size or shape, no location in time or space, no boundaries or limits, and no relation of likeness or unlikeness to anything else. But then Parmenides makes the contradictory assumption that there are indeed things other than the One, and that they are a plurality. Thus, the One may consist of a plurality of things, and it may be both One and many. (145a) It may be a whole consisting of many parts, have a beginning and ending, be at rest as well as in motion, and be like or unlike other things. Neither of these assumptions seems any more plausible or justifiable than the other.
REFERENCES
"Parmenides," translated by F.M. Cornford, in The Collected Dialogues of Plato, edited by Edith Hamilton and Huntington Cairns (New York: Random House, 1961).
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