Part I
Introduction
Outline of Argument
1. First Part of Dialogue
2. Second Part of Dialogue
2.1 First set of arguments (1a): Attributes and their contraries are impredicable of the One
2.2 Second set of arguments (1b): Attributes and their contraries are predicable of the One
Part II
2.3 Extension of (1b): Relations of the One to Other things
2.4 Third set of arguments (1c): Attributes and their contraries are predicable of the Others
2.5 Fourth set of arguments (1d): Attributes and their contraries are impredicable of the Others
2.6 Fifth set of arguments (2a): Attributes and their contraries are predicable of the One, though it is not
2.7 Sixth set of arguments (2b): Attributes and their contraries are impredicable of the One that is not
2.8 Seventh set of arguments (2c): The Others are neither one nor many, yet are apparently both like and unlike themselves and each other
2.9 Eighth set of arguments (2d): The Others are neither one nor many, and are apparently neither like nor unlike themselves and each other
3. Conclusion
The Parmenides dialogue has perplexed and divided interpreters of the highest caliber, not only in the specifics of its arguments, but in the overall design and intention of Plato, its author. What, if anything, was he trying to prove? Was he simply indicating his own perplexity, or urging readers to work through these problems for themselves? Our answers to these questions of design and intent may be informed in part by an assessment of the dialectical value of specific arguments within the dialogue. Here, however, we must take care to distinguish our own regard for the validity of these arguments from what Plato thought of them, for only the latter would inform his intent.
I propose interpreting the overall structure of the dialogue on the basis of the following assumptions: (1) Plato generally regards the specific arguments within the dialogue as valid applications of Eleatic dialectic, even if they lead to non-Eleatic conclusions; (2) he gives definite indications of his beliefs on the matter, through the voices of Parmenides and Socrates; i.e., he really means what he says. Both of these assumptions are highly dangerous when reading Plato, and open to serious objections. Nonetheless, I find that an interpretation following these assumptions yields a parsimonious result that agrees with the symmetries indicated in the structure of the dialogue, and with some important asymmetries that explain the conclusion. To support the credibility of these assumptions, I would note first that Plato, unlike Aristotle, never advanced a formal system of logic with distinctions in modes and categories of predication, so it would not be remarkable if he effectively adopted the Eleatic system of logic, with relatively little adaptation. Second, while he had too high respect for literary form to make his interlocutors mere parrots of his opinions, he always wrote with the purpose of revealing some truth, not for mere entertainment. His complex relationship with poetry, being at once a skilled lyricist and a disparager of art without knowledge, a critic of heroic epics and a proponent of pedagogical myth, does not authorize us to make him a sophist, someone who merely delights in arguing both sides in order to demonstrate his skill.
There are actually two dialogues in this work. The first is an apparently typical Socratic dialogue, except Socrates is cast in the junior role against the learned Parmenides. At first, Parmenides merely listens as his pupil Zeno defends his doctrine that all being is One or Unity, while it is impossible for Many or Plurality to exist, for the many would be both like and unlike. Socrates opposes this argument by positing a distinction between Ideas (Forms) and that which participates them. Thus the same thing may partake of likeness and unlikeness, though these two, considered as Ideas, cannot partake of each other.
Parmenides intervenes and lays waste to Socrates’s notion of Ideas and participation, in particular emphasizing paradoxes of unity and plurality, likeness and unlikeness, which result from this notion. These attacks seem to prove not only that the Ideas do not exist, but also that even if they did exist, they would be unknowable.
Surprisingly, Parmenides continues by affirming that these objections can be answered by a long and difficult process of demonstration. At the end of this process, one may learn that everything has a class and a knowable absolute essence. This process is a special form of dialectic, which has the appearance of useless talk, but draws out the consequences of affirming a hypothesis and also of denying it. If the same paradoxes arise whether we affirm or deny or hypothesis, the hypothesis is not to be blamed. If the hypothesis is the affirmation that a certain being exists, we should examine the consequences as to whether visible things are like and unlike themselves or in relation to other objects.
In the second part of the work, Parmenides agrees to perform this dialectic with respect to his hypothesis of the One. The young Aristoteles (presumably not the Stagirite) offers the responses without counterargument. The strangeness of this dialogue is in its relentlessly antinomic conclusions, proving on the one hand, that practically everything, including likeness and unlikeness, unity and plurality, is impredicable of the One, assuming that the One exists, and on the other hand all these things are predicable of the One. The same set of contrary conclusions is held of the Others (i.e., those beings that are not Unity), and then all four sets of conclusions are again affirmed under the counter-hypothesis that the One does not exist. Taken at face value, the argument concludes in perplexity.
Yet closer examination of parts of this second dialogue show certain asymmetries indicating a way out of these paradoxes, and at one point Parmenides even posits a notion of participation or partaking. While the overall argument does indeed show that we will have the same paradoxes under either hypothesis about the existence of Unity, these paradoxes arise only if we abstain from allowing any doctrine of participation that would allow us to speak of X being Y
in more than one sense. The paradoxes are so acute that they would require us to deny even that we can have opinions about the One, which is utterly contrary to experience. This leads us to recognize that there is a distinction between conceptual content and existence, i.e., between Ideas and objects, which is what a doctrine of participation attempts to articulate.
The dialogue’s literary introduction has the young Socrates engage with the aged Parmenides and his pupil Zeno, an encounter that is chronologically feasible. Zeno makes his classic arguments against the Many, but we get an unhistorical turn when Parmenides reveals that Zeno makes such arguments not so much to refute the Many, but to defend his master’s doctrine of the One against charges of absurdity. He accomplishes this by showing that the doctrine of plurality leads to greater absurdity. Zeno’s work was not meant for public consumption; it was published without his knowledge. Accordingly, it gives scandal to most people. This agrees with how Parmenides will characterize his own antinomic dialectic later, countering the absurd implication of a hypothesis with the even more absurd implication of its negation. This method scandalizes most people, who will regard it as idle and useless talk, and thus it should only be used among those who have much learning, experience and patience.
Zeno’s basic argument is as follows. If being is many, it must be both like and unlike. This is impossible, for the like cannot be unlike, nor can the unlike be like. Socrates tries to avoid the latter implication by advancing a distinction between Ideas (Forms) and that which participates in them. There is an Idea of likeness in itself and an Idea of unlikeness. Certainly, it is impossible for either of these Ideas to partake of the other. Nonetheless, particular things in our world may each participate in likeness and unlikeness to certain degrees, and there is no contradiction if the same thing should participate in likeness and unlikeness. So it could be that all is One insofar as all things partake of the idea Unity. Yet Unity as such cannot be Plurality, nor can Plurality as such be Unity.
This evasion of the paradox seems sensible enough, but Parmenides strictly scrutinizes the notion of particulars participating in universal Ideas. First, he asks which things have Ideas. Is it only beauty, the good, and other abstractions? Does it also include man and other sensible things presumed to partake of an essential nature, not identical with any individual? Or does it even include lowly sensible things like mud, where each part of mud seems to be nothing other than mud? Socrates admits that vile things are just as they appear to us. Sometimes I think there is nothing without an idea, but then I might fall into a bottomless pit of nonsense.
This hesitance is understandable. After all, if there is an idea of mud distinct from the mud we see, shall we say that some mud is more mud-like than other mud? Parmenides, however, finds fault with this reluctance to follow the implications of one’s thought, attributing it to the youth of Socrates. He is still too respectful of men’s opinions, and fears being mocked for admitting that there is an Idea of mud and other vile things. When he has a better grasp of philosophy, he will transcend the aristocratic sensibility of his culture, and learn not to despise even the meanest things.
Parmenides then proceeds to show various problems of the doctrine of participation. How can one Idea be in many things and yet still be undivided and one? If the individual resembles or is like the Idea, must not the converse be true? In that case, the Idea is part of a set of like things that must be accounted for with a further Idea, and so on in an infinite regress, so the Ideas explain nothing. Socrates offers various plausible sounding defenses or explanations, but all are shot down by Parmenides’ ruthless logic, showing the apparent incoherence and uselessness of any intelligible explanation of the doctrine of participation.
Supposing that there are Ideas of which other things partake, and from which they derive their names, we say that a thing becomes X because it partakes of the Idea of X. Does each individual thing partake of the whole Idea or only a part? If the whole Idea is one, yet is in each of the many, then the same thing (the Idea) will exist as a whole and in many individuals, and thus be in state of separation from itself. The problem is that if we say the Idea is in each individual, then it is no longer an idea or a unity.
Socrates proposes that an Idea could be like the day which is one and the same in many places at once, yet continuous with itself. Parmenides challenges this, for it is like saying a sail is spread over many men, so only a part of the sail is over each man, and likewise only a part of the day (whether we conceive of it spatially or temporally) covers different places. This implies that the Ideas are divisible. Yet if they are really divisible, they cannot remain one. Shall we say that a great thing is great in virtue of a portion of greatness rather than absolute Greatness?
Suppose you see a number of great objects, and there seems to be one and the same idea or nature in all, hence you conceive greatness as one. Since the idea of greatness is also great, then there should be another idea that includes this, and so forth in an infinite regress.
Socrates counters that perhaps Ideas are only thoughts in our minds, so each Idea is still one. Parmenides replies that a thought must be of something which is, i.e., a single form or nature that thought recognizes as attaching to all. So the problem is not avoided. Besides, if you say Ideas or Forms are thoughts, does this not imply that everything else participating in Ideas is made up of thoughts? This retort assumes that participation entails that the participant is somehow constituted by the Idea or takes on the nature of the Idea as Idea. We still have not precisely defined what can be meant by participation.
Socrates makes an attempt to define Ideas and participation: Ideas are patterns fixed in nature, and other things are like them, or resemblance to them, so what is meant by participation of other things in ideas, is really assimilation to them.
Here, the definition of participation is motivated by the definition of Ideas. So participation just means resemblance to a pattern. This is exposed to an Eleatic assault on the notion of likeness.
Parmenides asks: (1) If an individual is like the Idea, must not the Idea also be like the individual? (2) If two things are alike (the individual and the idea), surely they must partake of the same Idea? That which makes them alike must be the Idea itself. Some second instance of the Idea is needed, leading to the same infinite regress.
Question (1) assumes that likeness is reciprocal. This is generally unobjectionable when dealing with likeness between two individuals or two Ideas, but perhaps the likeness of an individual to an Idea is not the same as the likeness of the Idea to the individual. Yet surely for there to be any likeness, there must be something they hold in common. This thing held in common cannot be pure individuality, so it must be in the realm of universals or Forms. Surely the resemblance between this horse and the Idea of a horse must be in the Idea of a horse, and not in some other Idea. We have already accounted for the presence of this Idea in the individual via participation. Shall we not account for the resemblance of the Idea to the individual by the presence of this Idea in the Idea?
This proposed solution is impermissible if we treat the Ideas as though they were substantial, i.e., as absolutes, for it is then incoherent for an Idea to be in itself. As long as the Ideas are posited as absolutes, the objection of infinite regress stands, and some mode of participation besides assimilation must be devised.
There is also a difficulty in making each thing a single Idea, parted off from other things. This separation would seem to preclude participation, if object b participating in Idea B means that B is somehow in b. That would seem to contradict the notion that B is separate from individuals. Also, how would we deal with commonality among Ideas?
The greatest difficulty
is arguing against someone who maintains that Ideas, even if they exist, must remain unknown. There can be no hope of success unless that objector is of great ability and knowledge, and willing to follow a long and laborious demonstration. This indicates that Plato’s Parmenides holds that the Ideas really can be known, though this is difficult to prove.
The objection to the knowability of Ideas begins by observing that absolute essences cannot exist in us, for then they would not be absolute. So while the Ideas are what they are in relation to each other, their essence has nothing to do with resemblances in our sphere (i.e., of concrete particulars). Likewise, things in our sphere are relative to one another, not to ideas which have the same names as them. That is, concrete objects exist in relation to other concrete objects, and these relations are fundamentally different in kind from the conceptual similitudes among Ideas.
For example, there is nothing absolute (i.e., substantial rather than accidental or predicated) in the relation between a particular master and a particular servant. It is just a relation of one man to another: this man is the servant of that man; here being a servant
is a two-place predicate. This is distinct from servitude (or mastership) in the abstract, which is an absolute or substantive. Such a nature has nothing to do with us concrete individuals.
The conceptual relationship between mastership and servitude is certainly distinct from the particular relationship between two men, yet surely the latter is informed by the former, or we would not be able to characterize the relationship between two determinate men as that between master and servant. We must compare the particular relationship against its formal exemplar to see if the name of that form is applicable to this case. Yet this brings us back to the problem of defining participation as resemblance.
Each kind of absolute knowledge (i.e., knowledge of the absolute) will answer to each kind of absolute being. The knowledge we have will answer to each kind of truth we have, but we cannot possess Ideas themselves, since they cannot abide in concrete existents such as ourselves. Plato admits a distinction between our determinate, particular thought-acts and the universal Ideas to which they point.
Absolute natures or kinds can be shown only by an absolute Idea of knowledge, since a particular thought-act answers only to non-absolute being. Thus none of the Ideas can be known to us, because we have no share in absolute knowledge.
This argument works by denying that a particular, in this case a thought-act, can participate in a universal in a way such that the universal would somehow abide in the particular, or in this case be visible in the particular. Indeed, even with modern neuroscience and psychology, it remains a mystery how we are able to abstract knowledge of universals if our thought-life consists of nothing but phantasms of particulars. The best we can say is that somehow we discern patterns or regularities among particulars, which is similar to Plato’s notion that a form is a pattern. Yet how can the pattern be said to be in
each particular, while being distinct from all particulars? How can we pretend to know something that is distinct from all the direct objects of our thought?
Our hypothetical critic should consequently hold that the nature of the beautiful in itself, the good in itself, and other absolute ideas are unknown to us. Parmenides regards these as strange consequences,
i.e., counterintuitive. After all, we freely talk about these abstractions, which have nothing concrete about them, and this should be impossible if the critic’s argument were to hold. Note that Plato’s Parmenides, by implication, does believe that beauty and good in itself are knowable.
Absolute knowledge, if it exists, must be far more exact than our knowledge imperfectly gathered from particulars. This must be the case with knowledge of beauty, good, etc. No one is more likely than the Deity to have this most exact knowledge. But then how can the Deity have knowledge of human things? Ideas are not valid in relation to human things, and vice versa; each is limited to its sphere. Knowledge of universal Forms would tell us nothing about the determinate particulars that may resemble these forms. If God has this perfect authority and knowledge (of Goodness as such, Beauty as such), this knowledge cannot rule us, nor can his knowledge know us. This argument assumes that God can only have one kind of knowledge, not the other. Why would you have the less perfect if you have the more perfect?
Socrates exclaims, Surely to deprive God of knowledge is monstrous.
Parmenides replies, A man must be gifted with very considerable ability before he can learn that everything has a class and an absolute essence,
and this is even more difficult to teach others. Clearly, Parmenides (and likely Plato) really does believe that all things have a class and an absolute essence, but he recognizes the difficulty of proving this. He considers this to be a difficulty of pedagogy and of demonstration, not a doubt as to the reality of absolute essence. The denial of Ideas would lead to far more vicious consequences, since without the Ideas, the very power of dialectic, invoked by objectors, is destroyed. If each and every object were sui generis, we could not say anything intelligible about anything. No analysis would be possible.
The objector to the knowability of Ideas, whose refutation involves the greatest difficulty,
exploits the fact that the relations among Ideas are logical relations, while those among concrete things are not. In other words, he exploits the very distinction Socrates invoked in his attempt to circumvent Zeno’s argument. Parmenides does not fault Socrates for positing the Ideas, but for the naive way in which he attempted to define them.
To define the Ideas or teach about them requires special training, an art that most consider idle or useless talk. You need an art which the vulgar call idle talking… which you heard Zeno practising. I give you credit for saying to him that you did not care to examine the perplexity in reference to visible things… but only in reference to objects of thought….
Zeno’s discourse was an example of this art, so we are dealing with a kind of dialectic. Socrates also touched on this method when he insisted on focusing on objects of thought. Dialectic deals with laws of thought, and its principal objects are concepts rather than things in the sensible world. Accordingly, most see this as an impractical art. Parmenides will implement an especially abstruse form of dialectic, which explores all the contradictions that ensue from seemingly commonsense theses. Instead of neatly resolving the contradictions, he draws attention to them, which is precisely what common people might disregard as idle talk. The method involves considering not only the consequences of a hypothesis, but also the consequences of denying it. If Socrates would object to Zeno’s hypothesis about the many, he should consider not only the consequences of that hypothesis, but also the consequence of denying it.
In particular, this method of focusing on objects of thought enables us to show that visible things are both like and unlike and may experience any affectation. We should consider the consequences of Zeno’s hypothesis on the being of the many, and also the consequences of denying this hypothesis. What will be the consequence of affirming that likeness is or is not? In each case, we may ask what are the consequences for each visible thing in relation to itself and in relation to any other thing you may choose.
Parmenides reluctantly agrees to provide an example of this art, using his own hypothesis of the One. This constitutes the remainder of the dialogue, with the young Aristoteles giving replies. We can break down its structure as follows.
(1) Suppose that the One is.
(1a) The One is not many, so it is neither part nor whole, nor is it like or unlike itself or another.
(1b) The One partakes of being, so it is many and infinitely divisible; it is both like and unlike itself and others.
(1c) The Others (i.e., things that are not the One) are both like and unlike one another and themselves.
(1d) The Others are neither one nor many, neither whole nor part, neither like nor unlike the One; neither likeness nor unlikeness is in them. They are neither like nor unlike (absolutely).
Therefore the One is all things, and also nothing, both in relation to itself and Others.
(2) Suppose that the One is not.
(2a) Something that is not-One or Other exists (unless there were nothing at all), and these Others are many. The One, though it is not, nonetheless partakes of likeness and unlikeness, with respect to itself and the Others, and of greatness and smallness. The Others, having no unity, are neither like nor unlike themselves or each other.
(2b) The One does not partake of being, so nothing can be attributed to it, neither likeness nor unlikeness, nor greatness nor smallness.
(2c) The Others, having no unity, are neither One nor Many, though they appear as One or Many, and are apparently like or unlike themselves and each other.
(2d) The Others are neither One nor Many, nor do they appear as One or Many, since they can have nothing in common with any non-being (the One). Nor can they appear to be like or unlike.
The gist of the argument is this: Yes, it is true that the hypothesis that the One exists leads to paradoxical results regarding what is predicable or impredicable of it. Yet such criticism ignores the fact that these same paradoxical results ensue even if you suppose that the One does not exist, so you cannot invoke these paradoxes as disproving the existence of the One. The second half of the argument relies on our ability to discuss the One that is not,
that is, to analyze Unity logically as a concept, abstracting from its real existence.
Further, in (1c) we are given a doctrine of participation that may show a way out of these paradoxes. This leads to a method of defining universals that will be employed in the Sophist and the Statesman.
Lastly, it may be said under (2d) that the counter-hypothesis (2) leads to the most vicious conclusion, namely that things can not even appear to be like or unlike, so that if the One is not, then nothing exists. Yet surely things at least appear to be like or unlike.
It may seem to be an odd digression for Parmenides to apply this method to his doctrine of the One rather than to that of Ideas. Plato evidently considers this to be an easier or clearer example of the method. Moreover, it has important implications for the doctrine of participation under discussion. Before one may even attempt to define the Ideas or what is meant by participation, one should work out more fundamental problems. The problems of unity versus plurality and likeness versus unlikeness wrought havoc on young Socrates’ attempts to define participation. The doctrine of the One bears directly on the first problem, and indirectly on the second. Under the hypothesis and counter-hypothesis, Parmenides will consider the implications of likeness and unlikeness for the One and the Many both respect to themselves and each other. It will turn out that the paradoxes invoked against the doctrine of participation in Ideas were not peculiar to that doctrine, but exist even with respect to the doctrine of the One and its negation.
Suppose that the One is. Then it cannot be many. In both the hypothesis (1) and counter-hypothesis (2), we are considering the One or Unity as a concrete existent, without making any distinction between Ideas and that which participates in them. If the concrete existent One is nothing other than Unity, it surely cannot be many, since the concepts of One and Many (i.e., Unity and Plurality) are logically exclusive of each other.
Every part is a part of a whole. A whole is that of which no part is wanting. Thus the One can neither have parts nor be a whole, since in either case it would be made up of parts, and therefore be many.
If the One has no parts, it should have no beginning, middle or end, for these are parts (i.e., they are subsets whose intersection is null). All limits are beginnings or ends, so the One must be unlimited or infinite. If unlimited, it is formless, neither round nor straight. (Both of these are finite geometric constructs. Roundness, an arc, delimits extreme points equidistant from the center. A straight line is defined where the center intercepts the view of extremes. We can only be sure of straightness as long as we compare extremes of finite distance.)
Having no parts, it cannot be in any place, since it cannot be in itself or in another. If it were in another, it would be encircled by that other, touching it at many places with many parts. If it were in itself, it would be contained by itself. What contains must be other than what is contained. Then the One would be two. So the One cannot be anywhere.
The One cannot have rest or motion. If it were moved, it would either be moved in place or changed in nature, the only kinds of motion. Mere change in quantity (volume) is not considered a third kind of motion, perhaps because this is reducible to the local motion of parts. Change in nature apparently includes all qualitative change. If the One changes in nature and ceases to be itself, then it is no longer one. This is paradoxical only for the One, not for other changing things, since they might change in particular qualities and still be the same kind of thing, though we would need a theory of participation to work this out. Yet the One is absolutely unique, so any qualitative change whatsoever would make it other than the One.
If the One moved in place, it must either move round and round in the same place (change in situs) or from one place to another (change in locus). That which rotates upon itself must have a center of rotation within itself, and thus parts which are distinct from the center, but this is impossible for the One. The One cannot move from place to place, since it cannot even be in anything, much less come to be in anything. That is, if being X is impossible, then becoming X is impossible, since the endpoint of becoming is being. The converse does not hold, for it is conceivable that something can be X without becoming X, if that thing is eternal and unchanging.
Further, something that is coming to be in anything, i.e., in the process of going into something, cannot be yet fully in that other thing, nor completely out of it. So it must be made of parts, if it is partly in that thing (i.e., the place to which it is moving) and partly out of it. This argument relies on the notion that becoming is a process, applicable only to that which is made of parts of space, time, or substance. Thus the One cannot be in motion.
To show that the One cannot be at rest, consider first that the One cannot be in the Same, for then it would be in something. The definition of rest is to remain in the same place or thing over time. If the One cannot be in anything, much less can it remain in the same thing. The One cannot be in itself (for then it would be two), and it cannot be in another, as was shown. So it is never in the same place (indeed, not in any place). Therefore it is never at rest in the sense of opposition to local motion.
Plato here seems to ignore that the One might be at rest in the sense of opposition to change in nature. In fact, this will fall under a subsequent paradox, whether the One is the same or other than itself.
The One is not the same with itself or another, nor is it other than itself or another. If it were other than itself, it would be other than one, and thus not be one, which is contradictory. If it were the same as another, it would be that other, and thus other than One, so it would not have the nature of One. For we cannot allow that the One should be more than one thing, since that is contrary to the nature of unity. So the One cannot be the same with another or other than itself.
This last argument relies on ignoring any distinction between The One is X
in the sense of identity and the sense of predication, for that would invoke a theory of participation. We are following Eleatic logic to see the paradoxical implications of such denial.
Neither can the One be other than another, while remaining one. For only (an)other can be other than (an)other, and nothing else. Certainly not by virtue of being one (unity) can it be other (than another). That is to say, Otherness is essential to being other than something, and without a theory of participation, there is no distinction between exemplifying Otherness
and being Other(ness).
Clearly Unity does not contain the opposite notion of Otherness, so the One, if it is Other, certainly cannot be Other by virtue of Unity, i.e., by virtue of being one.
If the One is not other by virtue of being one, then it is not other by virtue of itself. If it is not other by virtue of itself, it is not itself other. That is to say, the One as such is not other. (If it were to become Other somehow, it would be by virtue of some external agent.) Yet if the One as such is not other at all, how can it be other than any thing? That is, the One essentially has no Otherness in it. So how can it be Other than anything?
The One cannot be the same with itself. The nature of the One is not identical with the nature of Same. Something does not become One merely when it becomes the same as anything. Being the same as something else entails some plurality. More problematically, whatever becomes the same as the Many necessarily becomes many and not one.
This argument treats Same as a two-place predicate, of the form X is the Same as Y.
It means, For all Z, if Y is Z, then X is Z.
If there is no distinction between identity and predication of being (i.e., participation), then X is the same as Y
implies X is Y
in the sense of identity. Without a theory of participation, our notion of Sameness relations become indistinguishable from identities.
If there were no difference between One and the Same, then when a thing became the Same, it would become one, and vice versa.
Clearly the notion of Unity is distinct from the notion of Sameness, even if we were to admit some overlap. Thus One is different from the Same. Yet if it has difference in it, in cannot be Same at all, for difference is opposed to sameness.
If the One is the same with itself, it will not be one with itself, which contradicts the assumption that the One is. Malcolm Schofield parses this argument as follows: If, then, the One is to be the same as itself, it will not be so through itself; and in this way, although it is one, it will not be one.
[M. Schofield. Plato on Unity and Sameness.
The Classical Quarterly (1974), 24(1):33-45.] That is to say, it is One insofar as it is the same as itself, but it is not one insofar as it is same not through itself, and therefore through another, thereby introducing non-unity into the One. Being One is the identifying characteristic
of the One, so if the One is X in virtue of something other than being one, then it is in virtue of something other than the One (i.e., not the One).
Schofield’s interpretation is basically sound, but I should remove language such as identifying characteristic
or anything else that supposes the possibility of participation. I instead parse the argument as follows: It was established (by the last quoted sentence, If there were no difference…
) that being one is not identical with being the same. So if the One is the same as itself, it is not by virtue of being one. It is therefore in virtue of something else, so the One is not one. Yet, by supposition, the One is, and therefore is one. So the One is both one and not one, which is impossible. Thus the One cannot be the same as itself.
Obviously the One cannot be other than itself, nor the same as Other. It has been proven (in the absence of participation) that the One cannot be other than other, nor the same with itself. This implies it cannot be same or other at all.
Likeness is a resemblance, less than total sameness. Likeness may be considered a sameness of affections; i.e., properties imposed by external agents or circumstances. The arguments regarding the sameness of the One may be applied to likeness: the One is neither like nor unlike itself or another. For the One to be like X would imply that the affections of One are the same as the affections of X. Yet One can have no affections other than itself and still be One. So this reduces to an identity between One and the affections of X, if we have no theory of participation to distinguish sameness from identity.
We had already established (2.1.5) that Sameness is distinct from Unity. If the One had any affection other than being one, it would be affected in such a way as to be more than one, which is impossible. Further, if, as previously established (2.1.4-2.1.5), it is absolutely impossible for the One to be the same as another or itself, much less can it be affected to become the same as another or itself. That is, if being X is impossible, becoming X must be impossible.
Yet this argument (2.1.4-2.1.5) about the impossibility of the One being the same as itself or other, if we take X is the Same as Y to mean X is Y, implies that it is impossible for the One to be anything. This contradicts the hypothesis that the One is. Again, all of this relies on the denial of any theory of participation.
So far we have confused results, as clearly these conclusions cannot all be true.
Returning to the original hypothesis (1) that the One is, Parmenides asks rhetorically: If the One is, can the One be, and not partake of being?
The first line of argument (1a) concluded that the One cannot partake of being, so we are clearly discarding that line and starting afresh.
It may be objected that we cannot take the existential statement The One is (exists)
as implying that being
can be treated as a predicate or attribute of which one may partake. Yet surely the statement, The One is,
means something more than merely declaring the concept of The One.
Nor are we merely declaring One
and Being
successively. Rather, by joining them in a sentence, we intend to say that Being in some way is joined to the One. Nor is the order of terms arbitrary, for we do not mean that Being has Unity, but rather that Unity has Being. We say nothing yet about the exact ontological relationship implied by this semantic predicability. Indeed, we cannot do so as long as we lack a doctrine of participation. However, the mere statement of this problem of how One partakes of Being already sets some constraints on such a doctrine.
Being has not the same significance as One, so Being and One are not identical. Yet our very hypothesis, The One is,
would be unintelligible or a disjointed non-statement unless we take it to mean that the One
somehow partakes of being, i.e., being must be somehow attributable to the One. This does not necessarily imply that being is a substance or accident, but it does require an ontology where a definite subject (the One
) can partake of that with which it is not identical.
Suppose, on the contrary, that the One is identical with Being, i.e., that unity is identical with existence. Then the statement The One is
would be identical with the statement The One is one.
Yet when we make an existential claim, we are positing something more than the formal truism that unity is unity. Even if existence is not some conceivable substance or property, it remains unavoidable that an existential proposition says something more than a mere declaration or description of the concept Unity. If to be
is just a grammatical placeholder, and The One is
simply means The One,
how can we admit a real distinction in significance between The One is
and The One is not
? Worse, if The One is
means the One is One,
then it is impossible for the One not to exist, since The One is not
would mean The One is not One.
Yet this reasoning could apply to any subject, so it is impossible for any subject not to exist, a sophism that would be taken up in the Sophist.
Granting that the One somehow partakes of Being, even though it is not identical with being, it seems to follow that the One must have parts. After all, if being is predicated of the one, per our hypothesis, and one is predicated of being (i.e., there is only one Being), and being and one are not the same, then the whole existent must consist of one and being as constituent parts. It matters not whether we consider either of these parts to be voluminous substances; we have already introduced plurality in the One.
Something is a part only relative to some whole. So when we say that Unity and Being are parts, there must be some whole of which they are parts. Since Unity is a constituent part of this whole, we may say that the whole is One, i.e., that One is predicable of the whole. Yet One is also merely a part of this whole. We have a further contradiction, for on the one hand, this whole has Unity, so it is One, but on the other hand it has two parts, so it is two, the One being merely a part.
Further, if we admit that being as such has unity, for to be X implies the unity of X, and we admit that unity has being (our hypothesis), then we cannot have one of these (unity, being) without the other. Thus each part (Unity and Being) must also be constituted of Unity and Being, and each of these are likewise constituted, so the number of parts doubles indefinitely. So the One, if it is, must be infinite in multiplicity.
This argument shows that, as soon as we admit existential claims about the One as distinct from the One as concept, we introduce a plurality of principles in the One, which cannot be resolved coherently as long as we make all attributions imply substantial identity. The only ways out would be either to deny the distinction between existential claims and declarations of concepts, a position discussed and refuted in the Sophist, or to admit that somehow the One can partake of Being without detriment to its Unity.
Another line of argument begins with the conclusion of the previous (2.2.1): The One partakes of being and therefore is. We saw from the last argument (2.2.2) that if the One has being, it turns out to be many. Let us abstract the One from the being of which it partakes, and consider it alone. The One as such is surely one, not many, or why should we even call it One? Then the being of the One must be other than One, for the One as such is not being, but only partakes of being. That is to say, Unity as such is not identical with being, as is obvious from the fact that many things besides Unity might conceivably be or exist. Being is only attributable to Unity under certain circumstances (i.e., it is at least conceivable for the One not to exist). Being is an attribute of Unity, but we do not know exactly what such attribution means, other than the fact that is not identity.
One is other than Being, but not by virtue of being one. Being is other than One, but not by virtue of being Being. Rather, they differ from one another by virtue of Otherness or difference. This may seem like a formalistic truism, but we are here considering One as such and Being as such, i.e., in terms of their conceptual significance. So Unity is different from Being, but not because the very notion of Unity means to be different from being, and Being is different from Unity, but not because the very notion of Being means to be different from unity. Saying that they are different by virtue of Otherness (Difference) does not seem to explain much, except to say that we can determine difference only by subjecting things to the two-place Otherness predication: X is Other than Y.
We could alternatively define Otherness in terms of the negation of identity: X is Other than Y
↔ X is not identical with Y.
Thus Otherness would be a sort of negative predication, where we declare the absence of identity. On the other hand, perhaps we mean a bit more than mere absence of identity, but rather the X that is Other than Y must also have some positive conceptual content. Otherwise, we could say that nothing is Other than Y. If we accept that difference involves making a comparison between two positive concepts, then Otherness may be considered a positive predicate in its own right.
Otherness is not the same as either One or Being. Thus, if you consider the One as such (ignoring that it partakes of being), you would have another thing, Otherness, that is attributable to the One, since the One is Other than Being, for example. Likewise, Other is attributable to Being, since Being is Other than One. No matter what, you will have at least two things: One and Being, One and Other, Being and Other. Here we have Eleatic logic leading to a non-Eleatic conclusion.
You can take any of these pairs of things and call them Both. That which is called both must be dual rather than singular. (Greek has distinct singular, dual and plural cases.) Each individual of the pair is one, so the two individuals and the pair or duality are three, i.e., a plurality (in the narrow Greek sense of three or more). This is not double counting, for our purpose is not to count only parts, but any entity admitted to be existent. We admit the two individuals exist, so the pair or duality must also exist. Since we ignore the possibility of participation, there is no distinction between duality as an Idea and as a concrete existent. Once you allow for the existence of the One or Unity, duality and plurality must likewise exist, in the univocal sense of an ontology without participation. Moreover, the existence of two and three implies the existence of even and odd numbers (for three was reckoned as the first odd number).
Once you have a duality or two, you must have the notion of twice, and if you have a plurality or three, you must have the notion of thrice. You can take two twice (e.g., the two of Both and the two of One and Being taken as individuals), giving four. You can take three thrice (One and Being and Both; One and the two of Both; Being and the two of Both), giving nine. If you have three and twice, there is twice three. If there are two and thrice, there is thrice two. Thus you have: even times even (2 x 2), odd times odd (3 x 3), even times odd (3 x 2), and odd times even (2 x 3), which represent all four species of number (in the Greek sense of integers greater than one). Thus all kinds of number are thereby shown to exist solely on the supposition that the One is.
The Greek arithmoi or numbers
may refer to the things counted or to that by which we count. Parmenides is not trying to generate numbers in either of these senses, as pluralities of units or as Number Forms. As R.E. Allen observes, if it were Plato’s purpose to generate all numbers arithmetically, he could have simply appealed to repeated addition of one. Far from generating numbers axiomatically, the argument presupposes the rules of arithmetic. We may summarize the argument as follows:
- The existence of
bothimplies the existence of duality.- The existence of
dualityimplies the existence of two, which implies the existence of even number.- The existence of
twoimplies the existence of twice, i.e., of multiplication by two.- The existence of
bothimplies the existence of two individuals, each of which are one. This assumes a rule of addition, i.e., that one and one are two.- The existence of
both,One and Being implies (assuming addition) the existence of a plurality of three.- The existence of
threeimplies the existence of odd number, and of thrice, i.e., of multiplication by three.- The existence of
twiceandthrice, coupled with the existence of odd and even number, implies the existence of odd-odd, even-odd, odd-even, and even-even number.
Again, existence
should be taken univocally, with no distinction between concrete existence and conceptual being or essence. By showing the existence of all four species of number, we establish that an odd number can be measured (multiplied) an odd number of times or an even number of times, and that an even number can be measured an odd number of times or an even number times, i.e., that any number can be measured any of number of times. Thus the conclusion is justified that we have an infinite multitude,
for number is defined as a multitude of units. [cf. Euclid Bk VII, Defs. i-xi.] Prime numbers are omitted here, because they do not measure numbers, but only the unit (one).
The categorization of number into odd and even, then odd-odd, even-even, even-odd, and odd-even, is not original to Plato, but it does exemplify his method for discerning natural species, which he will later expound in the Statesman, 262. We should try to divide a genus more or less evenly, rather than asymmetrically. A division of number into primes and composites would be highly asymmetric, and there is no clear characteristic by which we can identify primes. Even today it is notoriously difficult to predict which numbers will be prime, for we cannot generate the series of primes exactly by any efficient algorithm.
Odd and even, by contrast, have the defining characteristic of whether or not they can be split evenly, i.e., so that both parts are equal in number. Our choice to test division into two parts seems to entail an arbitrary preference for multiples of two, which is why the species of odd and even hold little import in modern number theory, even when dealing with the integers. Yet if we consider the act of dividing as a single cut, rather than in terms of two parts, then it seems there is something to be said for regarding how a number can be single-cut as something fundamental, and not dependent on an a priori preference for multiples of two.
The argument proves that, even on the Eleatic assumption that the One exists, concepts of multiplicity without limit are implied by this. This is contrary to the historical Eleatic belief that only the One exists and not any plurality. In Plato’s dramatic fiction, by contrast, the Eleatics poke holes in the hypothesis of plurality only to defend the existence of the One, but Parmenides now admits that the existence of the One would imply a sort of reality to plurality, which may even be predicable somehow of the One itself.
If all number partakes of Being (i.e., infinite multiplicity exists, as shown above), then every part of number likewise partakes of being, which is to say any arbitrarily small fraction may exist. We have conceded that any arbitrarily large integer may exist, and that these can all be multiplied with each other in any combination. If three exists, then there can be some whole with three equal parts, so one-third can exist. From this inference, along with the possibility of multiplication in any combination, we can derive arbitrarily small fractions. This means that Being is distributed over all things, i.e., even if there are many things, or many parts of things, they will all partake of being, since they partake of numbers or fractions, which partake of Being on the assumption that the One exists.
That is to say, the being of rational numbers establishes the being of all things, since all things that are can be so quantified, i.e., enumerated as wholes or parts. Thus all things that are
partake of Being, as would be trivially obvious anyway, for when we say that something is, it must partake of Being. The relevant point is that Being is distributed over all things, great or small, many or few, and down to smaller divisions without limit. Thus being has the greatest number of parts.
Any part of Being (i.e., the set of existents) must be a part, obviously. Since the part exists, it must be one rather than none. (The implication is that a non-existent part would be none. The paradoxes of non-existent things will be discussed in more detail in the Sophist.) Thus every single part of Being partakes of One, which is not absent in any part, great or small.
Yet the One in its entirety cannot be in many places at the same time. Then the only way it can be present in many parts is if it is divided. Note this is the same objection raised against Socrates’ defense of Ideas; it applies no less forcefully on the supposition that the One exists.
That which has parts will be as many as the parts are. Since the One, no less than Being, is distributed in all parts, it is not quite true that Being is distributed in the greatest number of parts. The One or Unity is coextensive with Being, distributed in the same number of parts (i.e., all parts that exist).
The One, then, by virtue of Being, is broken into as many parts as Being, so it is as numerous and infinitely divisible as Being. Thus the One is also many. Also, insofar as the parts are parts of the whole which is One, the One that is the whole acts as a limit, containing the parts. Thus the One as a whole likewise is limited, since the whole is fully constituted by its parts. The parts have limits in extension, though they are unlimited in number. Yet each of these parts is One, so the One is both limited and unlimited. Then the One, if it has being, is one and many, whole and parts, having limits and yet unlimited in number.
Since the One has limits, it must have extremes. Since it is a whole, it must have a beginning, middle and end; i.e., there must be something within the limits. Wholeness or fullness implies a continuity of content within the extremes. So there is a middle equidistant from the extremes. Such a notion is intelligible only on the supposition that the One has figure, whether rectilinear (where the middle is found by bisecting a line) or round (where the middle is the center of an arc) or some composition of these elements.
The next argument may be summarized:
(1) Every part is in the whole, and none is outside the whole.
(2) The One is all its parts, neither more nor less.
(3) The One is the whole.
(4) If all parts are contained by the whole [per (1)], and the One is all of the parts [per (2)], and also the whole [per (3)], then the One will be contained by the One.
Conclusion: So the One is in itself.
Note that (2) and (3) are distinct premises, for we do not assume that a whole is nothing more than a collection of parts. The One, however, has been shown to be all of its parts, for every part of being has Unity. Remember we are making no use of any doctrine of participation, so there is no distinction between something having Unity and being identical with the One. When we say the One is neither more nor less,
we mean only that we have not omitted any parts, nor does the One fail to be in any of its parts. It does not mean that the One cannot be something more than its parts, for in fact it is also the whole. The question of whether a whole, in general, is something distinct from an aggregation of parts can only be answered with reference to some notion of Ideas. What we do know, in the case of the One, is that each of its parts, without exception, is One, and that the whole containing all parts is One.
On the other hand, if the One is the whole, it cannot be in any of its parts, for we may also argue thus:
(1) If the whole were in all of its parts, it must be at least in some of them.
(2) If the whole were in some of its parts, the greater would be in the less, which is impossible.
(3) From (1) and (2), it follows that the whole cannot be in all of its parts.
(4) If the whole is not in some or all of its parts, it must be in something else, or cease to be anywhere.
(5) If it were nowhere, it would be nothing.
(6) The whole is not in some or all of its parts, by (2) and (3), so from (4) and (5) it follows that the whole must be in another if it exists.
Note that this argument applies to wholes in general, not just the One. So it is absolutely impossible for a whole to be in its parts. Therefore the One, insofar as it is a whole, cannot be in its parts, and if it exists, it must be in another. Yet we also showed earlier (2.2.5) that the One is in itself. That argument depended on the inference that, because every part of being is One, the One is all parts of being. Such an inference relies on a denial of any notion of participation, which would have enabled us to say that the parts partake of Unity without implying that Unity as such is identical with all the parts. As long as we deny the possibility of participation, we are left with the inescapable conclusion that the One is both in itself and in another.
If the One is both in itself and in another, it must be both at rest and in motion. It is at rest since it is in itself, being in Unity, and not passing out of this, so it is in the same thing, itself. Remaining in the same thing or place is the definition of rest. Thus the One is at rest insofar as it is always One. Yet we have also shown that the One is always in another, so it must never be in the same place or thing, and therefore is not at rest, but in motion. Thus the One is always at rest and always in motion.
This argument assumes that the One, being in another, must always be in a different other
from one moment to the next. It cannot be in the same other
because sameness and difference (otherness) are incompatible. Once again, this holds only if you deny a doctrine of participation that would allow you to treat sameness and difference as universal relations or qualities that admit participation, as distinct from concrete existents. If there is only a concrete existent Same
and a concrete existent Other,
it is impossible for there to be the same other
(i.e., something Other than One, but the Same as itself), for this requires a doctrine of participation where contraries can exist in the same subject.
Next, Parmenides shows that the One must be the same as itself and other than itself, the same as others and other than others. This argument consists of four parts.
First, to show that the One is the same as itself:
(1) Every X in relation to every other thing Y is one of the following:
(a) X is same as Y
(b) X is other than Y
(c) X is part of Y
(d) Y is part of X
(2) The One is not part of itself [excludes (c)].
(3) From (2), it follows that the One cannot be related to itself as whole to part [excludes (d)].
(4) The One is certainly not other than One, so it is not other than itself [excludes (b)].
(5) This leaves (a): the One is the same as itself
Premise (1) supposes a subject-only ontology, without participation, such as Boole proposed in 1854, and which may be represented set theoretically. The sameness relation between X and Y means that X and Y are identical, so that every element of X is an element of Y and vice versa, and there are no elements of X that are not in Y and vice versa. If X and Y share only some, but not all, elements, we may ask whether all the elements of X are contained in Y, all the elements of Y are contained in X, or neither is true. In the first two cases, we may say X is a part or subset of Y, or that Y is a part or subset of X. If neither is true, then we say X is other than Y. We may also say X is other than Y if there are no elements in common.
The second premise is affirmed without proof as self-evident. For the One to be a part of the One, we should have to say that there are some elements of the One that are not elements of the One, which is absurd. The rest of the argument follows easily, showing that the One is the same as itself.
The second part of this argument shows that the One is other than itself.
But then, again, a thing which is in another place fromitself,if thisitselfremains in the same place with itself, must be other thanitself,for it will be in another place?
For clarity let us call the thing X and let P be its proper place. If X should somehow be in a place other than P, X would be other than itself, since X as such must remain in its proper place P for as long as it is X. X as such must remain in P, so X insofar as it is in some place other than P is different from X as such. Since we have no participatory ontology by which to distinguish X from X as such,
we must say simply that X is other than X. This argument about X can be applied to the One, since the One has been shown in earlier arguments to be both in itself and in another. Thus the One is other than itself.
The third part of the argument will show that the One is other than others. We start from the conclusion of the second part, namely that the One is other than itself. Well, then, if anything be other than anything, will it not be other than that which is other?
Say X is other than at least one thing Y, where Y is some existent that could be the X itself or other than X. The proposed conclusion is that X is certainly other than some thing Z which is other than X. Naturally, if we suppose that Y is other than X, then this is true by substituting Z for Y. Even if Y is X itself, we have thereby established that X admits the (two-place) predicate other
with respect to at least one existent. Thus if we posit some existent Z that is other than X, the only way we could deny that X is other than Z, namely that X does not admit of being other than
something, is no longer available to us. Granted that it admits this predicate, we have only to show that, since Z is not the same as X, nor a whole or part of X, then neither is X the same as Z or a whole or part of Z. These possibilities being eliminated, and the applicability of being other than
to X being admitted, it follows that X is other than Z, where Z is some existent other than X.
This last conclusion can be used as the first premise of this syllogism.
If Z is other than X, then X is other than Z.
For every W that is not one, W is other than the One.
Therefore, the One is other than every W that is not one.
The syllogism is valid, and the second premise requires only the understanding that W is some existent distinct from the One. Note that there is no difference between is not one
and is not the One
, since we have no doctrine of participation or predication. As long as W is not identical with the One, it is not one.
At any rate, it is admitted that the One is one, so that which is not one cannot be the One, and is therefore an existent other than the One. By the previously established first premise, it follows conversely that the One is other than anything that is not one. So the One is other than the others,
i.e., those things that are not one.
The fourth part of the argument shows that the One is the same as the others. The Same as such and Other as such are opposites. Thus the Same as such cannot be in Other as such, nor can the Other as such be in the Same as such. If the Other is never in the Same, then the Other cannot be in anything for some finite duration of time, however small, for then it would be in the same thing. This last inference may seem invalid to us, because it conflates the same thing
with the absolute Same
(sameness as such). Yet we are not allowed to assume any doctrine of participation, so such a distinction is impossible. Thus the conclusion follows that the Other cannot be in anything that is, since every being is the same as itself. From this we infer that the Other cannot be in the One (which is the same as itself), nor in that which is not One
(which likewise is the same as itself).
Since Otherness cannot be in the One, we cannot say that the One is other than the not-one by virtue of otherness. Nor can we say that the not-One
(i.e., an existent which is not one) is other than One by virtue of otherness, since otherness cannot be in the not-One either.
The One cannot be other than the not-One by virtue of its oneness, for the not-One does not partake of oneness. The not-One cannot be other than the One by virtue of its non-unity, for the One does not partake of non-unity. Note here we assume the identities, the One = its oneness,
and the not-One = its non-unity,
since there is no participation other than identity or being a part.
The One cannot be other than the not-One by virtue of being itself (the One), for its own self has nothing of the not-One,
so it cannot refer to the not-One.
Otherness is a relation between two things. Yet a thing considered in itself cannot be understood to be in relation to some second thing of which it does not partake, since there is nothing of that second thing to be found in that first thing. This consideration is problematic for the otherness
relation, since it gives us no grounds for making something other in virtue of itself.
Since the One and the not-One cannot be other than each other, neither by virtue of themselves, nor by virtue of otherness, it is impossible for the One to be other than not-One, or for the not-One to be other than the One.
It remains to be shown that the One cannot be part of the not-One, or vice versa. If the not-One were part of the One, it would partake of the One. (This is the only kind of partaking admitted in our primitive, non-participatory ontology.) Yet the not-One cannot partake of the One, for then it would in some sense be one. Nor could it be any number, for as shown earlier, all number partakes of the One. Since the One and the not-One are distinct from each other in every sense, neither can partake of the other, and thus neither can be part of the other.
Having shown that the One is not other than the not-One, and vice versa, and that the One is not part of the not-One, and vice versa, it follows that the One is the same as the not-One, and vice versa. Thus the One is the same as others (i.e., things that are not one).
Combining the four conclusions (2.2.8.1-4), we say that the One is the same with itself and others, and also other than itself and others.
Consequently, the One will also be like and unlike itself and the others.
This can be shown as follows.
The first premise posits a reciprocity of the other
relation. As noted in the third part of the last argument (2.2.8.3), this presumes that the second subject also admits of being other than
something. It should be uncontroversial that others
admit of being other than something.
Reciprocity being established, a second premise holds that the One is other than the others to the same degree that the others are other than the One. What makes A other or different from B is (A1) A having qualities not held by B, and (A2) A lacking qualities held by B. What makes B different from A is (B1) B having qualities not held by A, and (B2) B lacking qualities held by A. It can be seen from a set diagram that the otherness of A from B (A1 + A2) is equal in degree to the otherness of B from A (B1 + B2).
Given that A is other than B to the same degree that B is other than A, it follows that A is other than B to a like degree that B is other than A, since likeness is a weaker condition than sameness (identity).
The fourth proposition speaks of affection, that which determines how a substance can be affected or acted upon. Here the act in question is being made other than something. The One (A) is made other than the Others by virtue of A1 and the privation of A2. The Others (B) are other than the One by virtue of B1 and the privation of B2, which is to say A2 and the privation of A1. Parmenides seems to say that these are the same affection (A1 and privation of A2; A2 and privation of A1), but in fact he only means that they are like affections. This likeness is shown by their symmetry.
Yet Parmenides expounds this proposition differently. The One is other
than Others, and the Other is other
than the One, so both are in the same condition, for we mean the same thing by other
in both instances (or we would be committing equivocation). (We have already established that both instances of is other
refer to the same degree of otherness.) It trivially follows that two things in the same condition are in like condition. So in virtue of the affection by which the One is other
(than Others), every thing will be like every thing, for every thing is other
(than something). Thus all things have this likeness in common, i.e., being other (than something).
Likeness is the opposite of unlikeness, and otherness (difference) is opposed to sameness. The One was previously shown (2.2.8.4) to be the same as others, and this is the opposite of being other than the others. Just now we have shown that the One, insofar as it is other,
is like other things.
I. The One, in that itis other,is like (other things).
II. The One, in that itis same,is unlike (other things).
The second proposition is inferred from the first by exploiting the fact same
and other
are opposite affections. If all things are other than something, then the One is like all things insofar as it is other.
For the One to be the same, however, opposes being others, so this is dissimilar to the common trait that all things have of being other. Thus the One is both like and unlike the Others.
Another argument shows the same conclusion that the One is like and unlike the Others.
Insofar as it is (in the same state | affected in the same way) [as the others], it is not (in another state | affected otherwise) [than the others], and thus is not unlike [the others], and not being unlike is like [the others]; but insofar as it is (in another state | affected by other) it is of (another sort | otherwise), and being of (another sort | otherwise affected) it is unlike.
Both translations are about as good as English allows, but we can capture the logic of the Greek grammar more accurately if we translate thus:
Insofar as it has been in the same (identical) state (tauton peponthe [adj.; vb. act. perf.]), it is not to have been another state (alloion peponthenai [adj.; vb. perf. inf. act.]); not in another (alloion) state, then it is not an unlike (anomoion), it is a like (homoion) state; but insofar as it is in another state, it is otherwise [adj.], being otherwise it is unlike.
The One is in the same state as the Others insofar as it is other than something. Insofar as it is in the same state as the Others, it is not in a different state. There can be no unlikeness (lack of resemblance) where there is no difference, so the One is not unlike the Others. Therefore the One is like the Others.
The One is in another state than the Others insofar as it is the same as something (i.e., identical with something, namely itself), for to be the same is other than being other. Being in another state, it is different from the Others, and to that extent unlike the Others.
Likewise, on both of these premises (I and II), whereby the One is other than itself and the same as itself, we may infer that the One is both unlike and like itself, for to be other implies some unlikeness (for otherness entails difference) and to be the same (identical) implies likeness (a weaker condition).
The One, insofar as it is in itself, would come in contact with or touch
itself, and it would touch others insofar as it is in others.
Yet, from another point of view, it is impossible that the One should touch itself, for that would require it to be situated next to itself, occupying the place next to it. This would make the One to be two, so this can never happen as long as it is One.
The One cannot touch Others, for that would require it to be adjacent to what it touches, with no third thing in between them. There must be at least two things for there to be any contact, and the number of contacts must be one less than the total number of things (the number of Others plus the One). Yet the Others, since they do not have one in them (for that belongs uniquely to the One), cannot have any number in them, so they can be neither one nor two nor any other number, so they cannot be in contact with One (for the number of contacts would be the number of adjacent Others, which have no number). Again, this argument exploits the absence of the predicability of number in more than one subject.
Further argument shows that the One is equal to itself and others by showing that it is impossible for it to be greater or lesser than itself or others. If the One were greater or less than Others, it would not be in virtue of being the One or being Other, but in virtue of greatness and smallness.
Smallness, if it is in the One, is either in the whole or only in a part of the whole. If it is in the whole, then it would be co-extensive with the One or contain the One. Then it would be equal to the One or greater than the One, but it is impossible for smallness to be greater than or equal to anything, for it would perform the functions of greatness and equality rather than its own. The same problem occurs if it is in a part, for it would be greater than or equal to that part. Again, this argument assumes that small
is an absolute, and ignores the possibility of participation, whereby the same thing may be small with respect to one thing and great with respect to another.
If greatness were in the One, or anything for that matter, there would be something greater than greatness, namely than in which greatness is. Moreover, if the One were great, it must exceed smallness, but this is impossible, since it has been shown that smallness must be absent. Since the One has neither greatness nor smallness, it cannot be greater nor less than the Others, so it is equal to the Others, and likewise to itself.
On the other hand, the One, being in itself, also surrounds itself, and so containing itself, is greater than itself. Being also contained in itself, it is less than itself. So the One is both greater and less than itself.
There cannot be anything that is not included in One and the Others. Since the One and the Others must be in something, for everything that exists is somewhere, then they must in one another, One in the Others and the Others in the One, if they are to be anywhere. Insofar as One is in the Others, it is less than the Others, and insofar as Others are in the One, it is greater than the others.
Thus the One is equal to, greater than, and less than itself and the Others. If greater and less and equal, it will be of equal and more or and less measures or divisions than itself and the Others, and if of measures, likewise of parts (corresponding to each measure), so it is equal and greater and lesser in number than itself and Others. This is the most acute antinomy, for the One can be greater or less than one!
Since the One is, it partakes of is
(einai), which is to be in the present time. Thus the one partakes of time. This exploits the fact that we are unable to conceive of definite existence without situating it in some present or time. Since the One is in time, and time moves forward, the One becomes older, which is to say older than the One. Since a thing is older with respect to that which is younger, the One in older than the One
becomes younger as the One in the first instance becomes older. Thus the One is also younger than the One. The One cannot go from the past to the future without passing through the present. When it reaches the present, it is no longer becoming, but simply is. Thus it is not becoming older, but now is older. Since it was becoming older than itself, it now is older than itself. Yet the present is always present with the One for as long as the one is, for whenever it is it is always now.
Thus the One always is older than itself, and is younger than itself. Likewise it becomes older and younger than itself. Yet it becomes for an equal amount of time with itself, so it is of the same age as itself.
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