Plato's Parmenides X, The One V: The End

This will be the final post on Plato's Parmenides. (If you missed the last one, take a moment to note that Parmenides gives a formulation of Newton's First Law of Motion in it). As in the last couple of posts, I'll give the text after the jump with occasional remarks. 

And greatness and smallness always stand apart? 
Then there is always something between them? 
There is. 
And can you think of anything else which is between them other than

As mentioned last time, there remains a scholarly debate about exactly what Plato meant by 'greatness and smallness.' "Aristotle treats it sometimes as a unitary principle (i.e., 'the great and the small' is something like the number line, with things getting smaller on one end and bigger on the other) and sometimes as a dyad, two different things that work to define reality between them. Here Plato (or Parmenides) is treating it clearly as the latter."

At first this looks like a three-place relation. Greatness is on one end, Smallness is on the other, and "Equality" stands between them. Now mathematically, at least, 'equality' means exact equality, so that would be a very small place between two very large extremities. 

No, it is equality which lies between them. 
Then that which has greatness and smallness also has equality, which
lies between them? 

I.e., if it has both. Say that a tree is bigger than many trees, but smaller than some of the largest ones. Then it must have both greatness and smallness, and since it has both, it must also in some sense have equality (even if there isn't, in fact, another tree of exactly equal proportion). We might say that it participates in both the Form of Greatness and the Form of Smallness, and it could therefore in principle be equal to something as well. 

But here this all looks like one thing; you might as well say that the tree participates in the Form of Dimension or of Extension, which is what gives it the ability to be greater or smaller or the same as another extended thing.

That is clear. 
Then the one, which is not, partakes, as would appear, of greatness
and smallness and equality? 

We're on the side of the ledger in which the one is not, having already run the traps in which 'the one is.' Here the paradox is that the one, which doesn't exist, must participate in a kind of being. The 'one tree' that does exist has to have it in order to 'be' greater or smaller or equal than others.

Further, it must surely in a sort partake of being? 
How so? 
It must be so, for if not, then we should not speak the truth in saying
that the one is not. But if we speak the truth, clearly we must say
what is. Am I not right? 

And since we affirm that we speak truly, we must also affirm that
we say what is? 

Then, as would appear, the one, when it is not, is; for if it were
not to be when it is not, but were to relinquish something of being,
so as to become not-being, it would at once be. 

Here the paradox is apparent. We're asking if it's coherent to say that the one is not, having already found it incoherent to say that the one is. It isn't coherent: the one must be, even if we treat it as not-being. Otherwise we cannot speak truly about anything, because 'any' 'thing' is also 'one thing.' 

Quite true. 
Then the one which is not, if it is to maintain itself, must have
the being of not-being as the bond of not-being, just as being must
have as a bond the not-being of not-being in order to perfect its
own being; for the truest assertion of the being of being and of the
not-being of not being is when being partakes of the being of being,
and not of the being of not-being-that is, the perfection of being;
and when not-being does not partake of the not-being of not-being
but of the being of not-being-that is the perfection of not-being.

Most true. 

This is a level of complexity of speech that you don't see again until the Germans get going on philosophy.

Step one: 'If the one is not,' it must 'have the bond of not being' because it 'is not.'

Step two: Contrast with a being, which must have as a bond 'the not-being of not-being.' 'Not-being' is being treated as a kind of property, rather than an absence of a property. To 'not be' you must have the property of 'not being.' But to be, you must have a kind of double negation: 'not-being not-being,' i.e., not being something that doesn't exist. 

Step three: The truest way to say that something is is to say what it is, and not what it isn't. This is the meaning of the very confusing talk about 'not-being-that.' The best way to talk about me is to say what I am, not that I am 'not a traffic signal.' 

Similarly, the truest way to say that something doesn't exist is to not to say 'it isn't that traffic signal,' but that it doesn't exist at all. (Of course, to say that you have to name it; and if you can apply a true name to it...)

Since then what is partakes of not-being, and what is not of being,
must not the one also partake of being in order not to be?


For one thing, anything that can be said to have the property of 'not-being' must be a thing in order to have a property. Unicorns don't exist; but to have the property of 'not-existing,' they must exist at least as an idea. Unicorns can be truly said to have one horn, too, even though none of them practically exist to have one (or not have one, or have two). 

It's not as easy as making the move from 'not-being' serving as an absence of a property rather than a property, either. Because I can accurately predicate these statements about unicorns, they must in some sense exist. Otherwise, there would be nothing to serve as a truthmaker for the claims about them. Thus, just as he says, they must be in order to not be.  

Then the one, if it is not, clearly has being? 
And has not-being also, if it is not? 
Of course. 

Paradox established. 

But can anything which is in a certain state not be in that state
without changing? 


Here Parmenides is suggesting we escape the paradox by introducing time. You could 'be un-tanned' at one time and 'be tanned' at another by changing; so you could 'be X' and then 'not be X' at another. 

Of course, it's been well established by earlier arguments in this dialogue that the one cannot change, whether it is or isn't. Parmenides walks us through that again.

Then everything which is and is not in a certain state, implies change?

And change is motion-we may say that? 
Yes, motion. 
And the one has been proved both to be and not to be? 
And therefore is and is not in the same state? 
Thus the one that is not has been shown to have motion also, because
it changes from being to not-being? 

That appears to be true. 
But surely if it is nowhere among what is, as is the fact, since it
is not, it cannot change from one place to another? 

Then it cannot move by changing place? 
Nor can it turn on the same spot, for it nowhere touches the same,
for the same is, and that which is not cannot be reckoned among things
that are? 

It cannot. 

Another thing we have learned about the one, which we will explore again, is that 'to be one' it has to be itself -- not something else.  

Then the one, if it is not, cannot turn in that in which it is not?

Neither can the one, whether it is or is not, be altered into other
than itself, for if it altered and became different from itself, then
we could not be still speaking of the one, but of something else?

But if the one neither suffers alteration, nor turns round in the
same place, nor changes place, can it still be capable of motion?

Now that which is unmoved must surely be at rest, and that which is
at rest must stand still? 

Then the one that is not, stands still, and is also in motion?

That seems to be true. 
But if it be in motion it must necessarily undergo alteration, for
anything which is moved, in so far as it is moved, is no longer in
the same state, but in another? 

Then the one, being moved, is altered? 
And, further, if not moved in any way, it will not be altered in any

Then, in so far as the one that is not is moved, it is altered, but
in so far as it is not moved, it is not altered? 

Then the one that is not is altered and is not altered? 
That is clear. 
And must not that which is altered become other than it previously
was, and lose its former state and be destroyed; but that which is
not altered can neither come into being nor be destroyed?

Very true. 
And the one that is not, being altered, becomes and is destroyed;
and not being altered, neither becomes nor is destroyed; and so the
one that is not becomes and is destroyed, and neither becomes nor
is destroyed? 


Thus, the attempt to escape the paradox by introducing time fails, because it produces more and worse paradoxes. 

Parmenides now suggests a new start: let's go back and try treating 'not-being' as absolute, rather than a property. Many of the same paradoxes emerge.

And now, let us go back once more to the beginning, and see whether
these or some other consequences will follow. 

Let us do as you say. 
If one is not, we ask what will happen in respect of one? That is
the question. 

Do not the words "is not" signify absence of being in that to which
we apply them? 

Just so. 
And when we say that a thing is not, do we mean that it is not in
one way but is in another? or do we mean, absolutely, that what is
not has in no sort or way or kind participation of being?

Quite absolutely. 
Then, that which is not cannot be, or in any way participate in being?

It cannot. 
And did we not mean by becoming, and being destroyed, the assumption
of being and the loss of being? 

Nothing else. 
And can that which has no participation in being, either assume or
lose being? 

The one then, since it in no way is, cannot have or lose or assume
being in any way? 

Then the one that is not, since it in no way partakes of being, neither
nor becomes? 

Then it is not altered at all; for if it were it would become and
be destroyed? 

But if it be not altered it cannot be moved? 
Certainly not. 
Nor can we say that it stands, if it is nowhere; for that which stands
must always be in one and the same spot? 

Of course. 
Then we must say that the one which is not never stands still and
never moves? 

Nor is there any existing thing which can be attributed to it; for
if there had been, it would partake of being? 

That is clear. 
And therefore neither smallness, nor greatness, nor equality, can
be attributed to it? 

Nor yet likeness nor difference, either in relation to itself or to

Clearly not. 
Well, and if nothing should be attributed to it, can other things
be attributed to it? 

Certainly not. 
And therefore other things can neither be like or unlike, the same,
or different in relation to it? 

They cannot. 
Nor can what is not, be anything, or be this thing, or be related
to or the attribute of this or that or other, or be past, present,
or future. Nor can knowledge, or opinion, or perception, or expression,
or name, or any other thing that is, have any concern with it?

Then the one that is not has no condition of any kind? 
Such appears to be the conclusion. 

In addition to the paradoxes above, what happens to everything else if the one really 'is not' in an absolute sense?  

Yet once more; if one is not, what becomes of the others? Let us determine

Yes; let us determine that. 
The others must surely be; for if they, like the one, were not, we
could not be now speaking of them. 


NB that this is Parmenides using the objection I raised about unicorns above: it's not enough to dispose of the idea that 'not being' is a property, because the ability to predicate true statements about things already shows that they exist. 

It doesn't say much about the nature of their existence. Gandalf, for example, exists as an idea that was first in the mind of J.R.R. Tolkien, and later came to be in the minds of his readers. (Is it the same idea? That's an interesting question too. Both positions can be argued for plausibly.) We can predicate true statements about him, which shows that he exists: e.g., he was a wizard, he fought at least one Balrog, he remained faithful to his mission. I can also tell you true things about my neighbor, e.g., that he is a welder. The truthmaker for the one set of claims is walking around in the world and you can meet him; the other one is not. Both exist, in a way. That fuzziness will not be resolved here.

But to speak of the others implies difference-the terms "other" and
"different" are synonymous? 

Other means other than other, and different, different from the different?

Then, if there are to be others, there is something than which they
will be other? 

And what can that be?-for if the one is not, they will not be other
than the one. 

They will not. 
Then they will be other than each other; for the only remaining alternative
is that they are other than nothing. 


The problem here is that we might well distinguish two horses by saying, "Well, that's one horse, and that's the other." But if 'the one' is not (absolutely), we can't do that. We just have two horses that are other than each other. They are collections of properties that become hard to distinguish; for example, we can't say 'they each have two ears' because 'each' is just another way of trying to collect them into 'one horse' and 'the other (one) horse.'  

And they are each other than one another, as being plural and not
singular; for if one is not, they cannot be singular but every particle
of them is infinite in number; and even if a person takes that which
appears to be the smallest fraction, this, which seemed one, in a
moment evanesces into many, as in a dream, and from being the smallest
becomes very great, in comparison with the fractions into which it
is split up? 

Very true. 

They are infinite if the one is not because there is nothing to hold them together as a unity, since 'the one' absolutely does not exist. So, any part of them can be divided from the rest, and infinitely so given the nature of division. Once again, 'the one' proves indispensable to our ability to think. 

And in such particles the others will be other than one another, if
others are, and the one is not? 

And will there not be many particles, each appearing to be one, but
not being one, if one is not? 

And it would seem that number can be predicated of them if each of
them appears to be one, though it is really many? 

It can. 

You can see how this would be a problem. If I can't count, "one, two, three" because I can't conceptually distinguish the units, how can I apply number to the things? Likewise, if each thing turns out to be infinitely divisible (at least conceptually), without the unity of 'one' there's nothing to hold onto. We can't give up the Form of One because we need it for everything. 

And there will seem to be odd and even among them, which will also
have no reality, if one is not? 

And there will appear to be a least among them; and even this will
seem large and manifold in comparison with the many small fractions
which are contained in it? 

And each particle will be imagined to be equal to the many and little;
for it could not have appeared to pass from the greater to the less
without having appeared to arrive at the middle; and thus would arise
the appearance of equality. 


The fact that everything dissolves into infinite divisions means that all our conceptual approaches -- lesser, greater, equal, odd, even, etc -- turn out to be meaningless. All of these things we use to conceive of and work with the world end up depending on the idea of establishing unities, i.e., 'oneness' of things. 

And having neither beginning, middle, nor end, each separate particle
yet appears to have a limit in relation to itself and other.

How so? 
Because, when a person conceives of any one of these as such, prior
to the beginning another beginning appears, and there is another end,
remaining after the end, and in the middle truer middles within but
smaller, because no unity can be conceived of any of them, since the
one is not. 

Very true. 

That sounds confusing but it's just another infinite-divisions-without-unities problem. If I can't say that two things (one and one) are next to each other, then the divisions 'between' the 'things' multiply our of measure. 

And so all being, whatever we think of, must be broken up into fractions,
for a particle will have to be conceived of without unity?

And such being when seen indistinctly and at a distance, appears to
be one; but when seen near and with keen intellect, every single thing
appears to be infinite, since it is deprived of the one, which is

Nothing more certain. 
Then each of the others must appear to be infinite and finite, and
one and many, if others than the one exist and not the one.

They must. 

Having established that we need unity for our most basic conceptual approaches to the world, Parmenides finds more paradoxes yet. 

Then will they not appear to be like and unlike? 
In what way? 
Just as in a picture things appear to be all one to a person standing
at a distance, and to be in the same state and alike? 

But when you approach them, they appear to be many and different;
and because of the appearance of the difference, different in kind
from, and unlike, themselves? 

And so must the particles appear to be like and unlike themselves
and each other. 

And must they not be the same and yet different from one another,
and in contact with themselves, although they are separated, and having
every sort of motion, and every sort of rest, and becoming and being
destroyed, and in neither state, and the like, all which things may
be easily enumerated, if the one is not and the many are?

Most true. 
Once more, let us go back to the beginning, and ask if the one is
not, and the others of the one are, what will follow. 

Let us ask that question. 
In the first place, the others will not be one? 
Nor will they be many; for if they were many one would be contained
in them. But if no one of them is one, all of them are nought, and
therefore they will not be many. 

If there be no one in the others, the others are neither many nor

They are not. 
Nor do they appear either as one or many. 
Why not? 
Because the others have no sort or manner or way of communion with
any sort of not-being, nor can anything which is not, be connected
with any of the others; for that which is not has no parts.

Nor is there an opinion or any appearance of not-being in connection
with the others, nor is not-being ever in any way attributed to the

Then if one is not, the others neither are, nor any of the others
either as one or many; for you cannot conceive the many without the

You cannot. 
Then if one is not, there is no conception of can be conceived to
be either one or many? 

It would seem not. 
Nor as like or unlike? 
Nor as the same or different, nor in contact or separation, nor in
any of those states which we enumerated as appearing to be;-the others
neither are nor appear to be any of these, if one is not?


So Parmenides concludes 'the one is not' inquiry by finding that, if the one isn't, nothing else is in any meaningful sense.  

Then may we not sum up the argument in a word and say truly: If one
is not, then nothing is? 


And now the final conclusion: neither 'the one is' nor 'the one is not' makes any sense. Paradoxes and contradictions abound no matter which road you take. The inquiry into this simplest and most basic of concepts, one, proves inconclusive. 

Let thus much be said; and further let us affirm what seems to be
the truth, that, whether one is or is not, one and the others in relation
to themselves and one another, all of them, in every way, are and
are not, and appear to be and appear not to be. 

Most true. 


And yet it is not really inconclusive. We have learned, without question, that unities can't be a feature of the physical world; and also that they are an absolutely essential feature of the mental world. What remains to be determined is if they are real in an immaterial, non-psychological sense. 

But of course they must be, right? Aren't you 'one'? Well, your body is extended in space and time, and so is your consciousness; but there is a unity holding it together. This unity isn't merely psychological, since it extends to the non-mental body and allows you to distinguish the things that do and don't belong to it. And it isn't material. since immediately after death the material will all still be there, and yet the 'you' that is the 'one you' won't be any more. 

Things like that imply that the one has metaphysical status in at least some cases. The Form of the One thus seems to exist, at least in us; and since the other people are not you, it doesn't seem to be a feature of just an individual's consciousness. It is part of what allows us to conceive of the world and operate in it, which we have done successfully for untold thousands of years. That very success at manipluating the world implies that the concept has a truth to it, too, since we would not expect to profit by applying an untrue or inaccurate concept to reality. Usually when you're badly wrong about a basic assumption, it ends up hurting you. 

So maybe the One is real after all. Plato doesn't lead you by a nose to that conclusion, and in fact gives you his best arguments against the One or even 'one' being a thing that could exist in the world. In the end, he expects you to think it through yourself. Plato teaches philosophy, not just philosophical theories. He teaches the practice, and his product is not people who think the 'right way,' but people who think for themselves. 


james said...

I hope I will not be thrown out of school here if I confess that it is hard to keep track of the ways he is using one and other. If I can be simple-minded here:
In this world exist many things; it is not hard to abstract this back to some oneness of existence of a thing-in-itself.
In this world exist many relationships too. You can abstract this back to a oneness of generic relationship.

Can those two oneness's be the same?

Grim said...

So, this an opportunity to introduce Aristotle’s favorite mode: “In a way / In another way.”

In a way, of course they’re not the same. One is derived from things (“substances”), another from relationships (a kind of attribute). Attributes belong to substances, so the way in which all of my relations are unified is categorically different from the way that I am one (and you are ‘another one’).

Parmenides is after a different way of understanding. He’s after a much more basic notion of what it is for anything to be one. His argument is that you can’t actually conceive of a world of ‘many things’ or ‘many attributes’ without being able to separate the many into ‘one’s.

In that way the concept is the same: it’s the ability to treat what we encountered in the world as discrete, and thus countable, entities. You don’t reason from the many to an idea of oneness: you have to be able to think of things as ‘one’ and ‘one’ and ‘one’ to get to many.

Ontologically we blur right over what Aristotle wants to say about substances anyway. We reason just as readily about “one tree” (a substance in Aristotle’s sense, and plausibly a true ‘one’ because it is a single entity) as we do “one day” (a timeframe that unites all of our experiences of every kind of thing for its length). There’s still a lot of work to be done.

But without “one” in Parmenides’ sense, we can’t do any of it.

james said...

My thoughts about "relationship" have to do with his use of "other" In a universe of one, "other" is not defined, unless "one" in some way also includes relationship. If it does, then of course you can define "nothing" in opposition to the "one", and go on from there--as he does.

It is fair to assert that "otherness" is possible, because we observe it--and even if it is illusion it still includes difference. I wish he had been more explicit about that.

Grim said...

When interpreting this part, it’s important to remember that the hypothesis being explored is ‘the one is not.’ There are good reasons to think it might not be from the first part of the long argument, when the hypothesis was ‘the one is.’ Paradoxes arise everywhere; but it turns out that they do on this horn too.

The alternative you’re raising is, if I understand you correctly, similar to the arguments about whether space is absolute or relative. In the older absolute view, space is ‘real’ and every thing has a location in it. In the relative view (Leibniz’s if I recall), space is not real; objects define space relative to each other.

So you might be able to just have ‘other’ defined relative to ‘other others,’ without any of them being ‘one.’ Then the only alternative to otherness is nothing.

But it turns out that otherness depends on oneness as well. What makes the other another rather than part of the same thing? It’s only when you can define a discrete entity that ‘other’ comes to make sense.

james said...

Sorry, I wasn't clear--I was thinking of the whole dialog, not just this part.

Grim said...

So there's a famous thought experiment by Max Black that is meant to undermine Leibniz's 'principle of the identity of indiscernables.'

Black asks you to imagine a universe with only two objects in it, which are exactly alike in all properties but not the same. So, then, they're not quite alike in all properties if space is absolute, because you can discern them by their location differences. But if space is relative (as Leibniz thought), then you can't do that anymore; both of them are exactly as far from each other as the other is from the other.

Leibniz's principle is that we should assume the identity of the two things, because all of their properties seem to be the same (including, in this case, their relationship of being X meters distant from the only other object in the universe). Black's intuition is that it's conceivable that they're not the same, even though they are indiscernable, and therefore that Leibniz's principle fails.

That may be helpful for talking through your objection. If we can define one of them as "one," then the other one is the other. But if we can't define either of them as "one," they are both other to each other; they become apparently even more indiscernable. They exist, so they are both distinct from nothing.

Yet it proves that "otherness" has a description, at least, which is that the other is separate and distinct -- it's X meters distant, surrounded by nothing. In saying that 'the other is distinct,' we are saying that the first object is one, i.e., also distinct. And so is the other, a distinct individual that can be discussed as such.

james said...

As side-notes:

WRT Black's thought experiment: if space is "absolute" you have 3 things in the universe, not two.

In the relative case you have to supply your own coordinate system, and you have no yardstick except the size of the objects, e.g. they are N diameters apart.

If the objects are truly identical in all respects, and may be described by wave equations, we could call them bosons. (If they were fermions they would have to differ in some respect.)

In geometry we have points and lines, but if you talk about putting points together to make a line you run into inconsistencies. One problem is the language--the word "putting" points to an algorithmic operation: do a, do a+1, do a+2...--but that never gets you beyond the "countably infinite" It's kind of hard to twist the language in ways that make the uncountable infinite part of the description. You can't define a "next" point on the line.

Grim said...

WRT Black's thought experiment: if space is "absolute" you have 3 things in the universe, not two.

Yes, right. It's important that he was responding to Leibniz (whose PII and 'relative space' are both concerned).

You can't define a "next" point on the line.

One of the problems is that you can't definite it, but you can refer to it. (You just did.) It seems as if you know what you mean by saying it, and I understand what you mean by referencing it. But of course it's impossible to conceive of it, even, except as an abstract idea; there is no 'next point on the line,' only an uncountably infinite series of points.

There's something similar at work here, which may be why Plato elected to start with Zeno. We want to talk about things as discrete units, and indeed apparently have to conceive of things that way. Yet it seems as if the world doesn't contain such things, just as the number line both 'has to' and cannot hold a 'next point.'

james said...

"Next" hides algorithmic ideas under the hood, and it fails when applied to something that isn't countable. You have to carefully define and limit the use of terms when you want to talk about such things without running into contradictions.
I probably should have written "Next" is a word that doesn't apply to a point on a line.

Grim said...

It's helpful, really, because it points to a similar problem. The fact is that we can conceive of something that can't, properly, exist. Many philosophers would insist that was impossible, but you've given a nice example of it. Of course there 'must' be a 'next point,' because if there isn't you're back to Zeno's paradox of motion: you really do have to traverse an infinite series of points even to get to a nearby point, let alone the final point. Yet there is not.

We've already worked through Zeno, so enough has been said about ways to solve or think about that paradox. Still, it's easy to see the plausibility of it given your example. If there literally 'is no next point,' how do I get anywhere? Where is there to go from wherever I am?

james said...

If you're interested in the infinities and an application to music, you might like this: