16 Jun 2009

Plato, Parmenides, 149-152

Corry Shores
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[The following material is complex, and it serves only as background for a more important entry that will be posted later. The original text is given below. Perhaps it's best to begin there.]

We will be discussing arguments regarding the One and the many. Our senses tell us there are many things. But there are rational ways that we might conclude that everything in the cosmos is really just one thing, which we call 'the One.' Let's consider some ways that we might conclude there is only the One.

For long there was debate over whether there could be empty space. Perhaps only things can extend. And if there is no thing, there would be no extension or distance. The basic intuition runs like this.

If there is nothing there, how can it extend? For, there is nothing there to begin with that could extend. And, distance is extension. And also, space is distance. So if there is no thing, there is no extension, which means there is no distance, and hence no empty space. Therefore, there is no empty space between things that are at a distance apart.

Because this was a dominating assumption, many held that there is not void between the heavenly bodies. Instead, there is an airy stuff, called aether. It might as well be empty. Really it is just barely matter that fills the space so that there can be distance between things.

Now, if there is no empty space between things, then there is nothing that separates things. And if nothing separates things, then things are not distinct from one another. Hence there is really only just one thing, because there are really no absolutely distinct parts in the cosmos. From this perspective, we can see why some argued that there is only the One and not the many.

Of course the sciences have established that there is void between the heavenly bodies. As well, there is mostly void on the atomic level.

So there seems to be grounds to deny that there is just the One. However, we could also note that we consider such solid bodies as diamonds to be unitary objects, even though they are made-up mostly of void on the molecular level. In the same way that there is mostly void between atomic particles, we might note that there is mostly void between heavenly bodies. So we might consider a galaxy like a diamond. Even though it is made up mostly of void, it is a unitary object. Then, we could consider all the galaxies in the cosmos together as a unitary object, even if truly the cosmos is made-up mostly of void. So in the same way we might ascribe oneness to an atom, we can do the same for the whole cosmos. Thus even if we presume void rather than aether, we can consider everything as the One.

So we have reason to wonder if all is one, or if there is a multiplicity of things. If there is one, than there is not many. But if there is many, than there is not one. However, we have ideas for both. So can both the One and the many co-exist? But when we combine these ideas, we obtain contradictions. These absurdities can tell us what is not the case. So we will uncover the contradictory implications of these ideas so we can clarify what we know about them.

Absurd claim: The One is equal and unequal to itself, and it is equal and unequal to the many others.

When we compare the One with itself, we cannot say that it is greater or lesser than itself. Likewise, we might consider the quantity of the many taken all together. If we compare the many to itself, we do not find that the many is greater or lesser than itself.

However, we might also compare the One with the many. We might find that the One is greater or lesser in magnitude than the many taken altogether. So before when we compared the One with itself, and the many with itself, determinations like greater and smaller could not possibly apply. However, now that we are comparing two different things, the One and the many, it becomes possible to attribute to them greatness or smallness.

We know that greatness and smallness are opposites. But only existing things can be opposites. So greatness and smallness are ideas that exist. Also, we know that things exhibit greatness and smallness. The only way greatness and smallness could "come into being in things" is if they existed.

So let's first suppose that smallness is present in the One. If this is so, then either the One as a whole exhibits smallness, or some part of it exhibits smallness.

Let's first suppose that the One as a whole exhibits smallness. This could be so in two ways. First imagine everything as one. If everything throughout the one is small, then the One as a whole is small.

Now let's consider a different perspective. We suppose there is the One. It has the property of oneness. Let's say that it is small. Now consider the fact that we can also imagine many things, even if truly there is only one. We can also conceive these many things as being small too. So, the idea of smallness can apply to both the One or the many. From this perspective, smallness is a broader concept, because it extends to more possible objects, namely, the one or the many. Now also consider that we might imagine the many things to be small, but we cannot imagine the many things to be one, so long as we mean that they are truly many. Hence the idea of oneness does not include the idea of manyness. But, the idea of smallness can include both ideas. So in a way, smallness may include oneness, but oneness cannot totally include smallness, because the many can be small too.

So we are considering two ways that the One may be entirely small. One way is if it is pervasively small throughout. The other way is if the one is completely included as a subcategory of the small.

Now, if the One is pervasively small, that means that smallness extends equally throughout it. In other words, smallness would be co-extensive with the One.

However, if the One is included under the broader category of smallness, then smallness is greater than the One. [150a]

In the first case, smallness was co-extensive with oneness. Also, we are comparing this to another perspective that considers other things that might be small. But here we are just considering the One as small. Because there are no comparisons, there is no way to establish something as small. So in other words, we could say that something is homogeneously small, but that is no different than saying it is homogenously large. For, in neither case is there something else to compare it to. Hence what we call small is something whose magnitude can only be compared with itself. So in way, we are saying that something that is small is equal to the only thing that it can be smaller than, which would be itself. But it is absurd to say that the small is equal.

In the second case, we are saying that the property of smallness includes oneness, and can apply to other things as way, namely, the many. In this way, smallness is more inclusive than oneness. So more things are unified under smallness than under oneness. This means that smallness is a greater unity than oneness. Or in other words, if we say that oneness is included in smallness, then we are saying that smallness is truly the real oneness, and within it is a smaller oneness. The smaller oneness would be smaller than smallness. But here are two absurdities:

a) there is more than one oneness, and

b) smallness as a property extends beyond oneness (to the many). But we call the One small. So the One's smallness is greater than itself, which is absurd.

Hence there is no way that smallness exists in the whole of the One. It must instead only exist in part of the One.

But if we say that smallness exists in the whole of the part, then we encounter the same absurdities. But if smallness cannot exist in the whole, nor in a part, nor in a part of a part, and so on, smallness could only exist in what does not have either wholeness or parts. Such a thing could only be absolute smallness itself. [150b]

If instead we said that the one is absolutely great, we are saying that the One has both oneness and absolute greatness. But then we are saying that the One is a broader thing than the absolutely great, because the One contains more than the absolutely great does. But then we are saying that there is something greater than the absolutely great, which is absurd. And we said that there can be no smallness in the one, so if the One is absolutely greater, it would have nothing smaller to it to be greater than. [150c]

And the only way that something else could be greater or smaller than the One is if they had greatness or smallness.

And since the One cannot have greatness or smallness, we cannot say that the one is greater or smaller than anything else. Great and small things can be greater or lesser to other great and small things.

So if the One cannot be greater or smaller than anything else, then it cannot be exceeded by anything else in magnitude (exceeded in smallness or greatness). Likewise, since the One has no greatness or smallness, it cannot exceed anything else (in greatness or smallness). Whatever neither exceeds nor is exceeded is an equality. So the One is equal. [150d] And it is equal with itself. So if can be not be unequal to itself, and if it cannot be unequal to anything else, then it is equal to both itself and to anything else.

However, the One does not exist outside itself. Rather, the One exists within itself. But that means it is larger than itself, in the sense that it is larger the self contained within itself. But also, that One contained within itself is itself. So the One is also smaller than itself. [151a]

So we are considering two possibilities: the One and the others (the many). There can exist nothing outside these two things. But, if something does exist, it must exist somewhere. And, wherever that thing exists, it will be smaller that what it exists-in.

Now if there were both the One and the others, then they would have to exist in each other. For, there is nowhere else for them to exist. Hence the One must exist in the others, and the others must exist in the One.

But if the One is in the others, than the others will be greater than the One. For, they contain it. But then, because the others are in the One, they will also be smaller than the One. [151b]

So from everything so far we can say that the One is

a) equal to itself and equal to the others

b) greater than itself and greater than the others, and

c) less than itself and less than the others. [151c]

If something is greater, it has greater measure, and hence more divisions. Likewise if it were smaller, it would have fewer divisions. And if equal, it would have equal divisions. So it will have greater, lesser, and equal number of parts as itself and the others. [151d]

From the original Fowler translation:

“And is the one both equal and unequal to itself and the others?” “How is that?” “If the one were greater or less than the others, [149e] or, again, the others greater or less than the one, is it not true that the one, considered merely as one, and the others, considered merely as others, would be neither greater nor less than one another, so far as their own natures are concerned; but if in addition to their own natures, they both possessed equality, they would be equal to one another or if the others possessed greatness and the one smallness, or vice versa, that class to which greatness was added would be greater, and that to which smallness was added would be smaller?” “Certainly.” “These two ideas, greatness and smallness, exist, do they not?” “For if they did not exist, they could not be opposites of one another and could not come into being in things.” “That is obvious.” [150a] “Then if smallness comes into being in the one, it would be either in a part or in the whole of it.” “Necessarily.” “What if it be in the whole of one?” “Will it not either be on an equality with the one, extending throughout the whole of it, or else contain it?” “Clearly.” “And if smallness be on an equality with the one, will it not be equal to the one, and if it contain the one, greater than the one?” “Of course.” “But can smallness be equal to anything or greater than anything, performing the functions of greatness or equality and not its own functions?” [150b] “No, it cannot.” “Then smallness cannot exist in the whole of the one, but, if at all, only in a part of it.” “Yes.” “And neither can it exist in a whole part, for then it will behave just as it did in relation to the whole; it will be equal to or greater than the part in which it happens to exist.” “Inevitably.” “Then smallness will never exist in anything, either in a part or in a whole, nor will anything be small except absolute smallness.” “So it appears.” “Nor will greatness exist in the one. [150c] For in that case, something other than absolute greatness and differing from it, namely that in which greatness exists, would be greater, and that although there is no smallness in it, which greatness must exceed, if it be great. But this is impossible, since smallness exists nowhere.” “True.” “But absolute greatness is not greater than anything but absolute smallness, and absolute smallness is not smaller than anything but absolute greatness.” “No.” “Then other things are neither greater nor smaller than the one, if they have neither greatness nor smallness, [150d] nor have even these two the power of exceeding or being exceeded in relation to the one, but only in relation to each other, nor can the one be greater or less than these two or than other things, since it has neither greatness nor smallness.” “Evidently not.” “Then if the one is neither greater nor smaller than the others, it can neither exceed them nor be exceeded by them?” “Certainly not.” “Then that which neither exceeds nor is exceeded must be on an equality, and being on an equality, must be equal.” “Of course.” [150e] “And the one will be in the same relation to itself also; if it have in itself neither greatness nor smallness, it cannot be exceeded by itself or exceed itself; it would be on an equality with and equal to itself.” “Certainly.” “The one is, then, equal to itself and to the others.” “Evidently.” “But the one, being within itself, would also be contained by itself, and since it contains itself it would be greater than itself, and since it is contained by itself it would be less than itself; [151a] thus the one would be both greater and less than itself.” “Yes, it would.” “And is it true, moreover, that nothing can exist outside of the one and the others?” “Of course.” “But that which exists must always exist somewhere.” “Yes.” “And that which exists in anything will be smaller and will exist in the greater? One thing cannot exist in another in any other way, can it?” “No, it cannot.” “But since there is nothing else apart from the one and the others, and they must be in something, must they not be in one another, the others in the one and the one in the others, [151b] or else be nowhere at all?” “Clearly.” “And because the one is in the others, the others will be greater than the one, since they contain it, and the one less than the others, since it is contained; but because the others are in the one, the one will by the same reasoning be greater than the others, and the others less than the one.” “So it appears.” “Then the one is equal to and greater and less than itself and the others.” “Evidently.” “And if equal and greater and less, it will be of equal and more and [151c] less measures with itself and the others, and since of equal, more, and less measures, of equal, more, and less parts.” “Of course.” “And being of equal and more and less measures, it will be less and more in number than itself and the others and likewise equal in number to itself and the others.” “How is that?” “If it is greater than any things, it will be of more measures than they; and of as many parts as measures. Similarly if it is less or equal, the number of parts will be less or equal.” “True.” “Then one, being greater and less than itself [151d] and equal to itself, will be of more and less measures than itself and of equal measures with itself, and if of measures, of parts also?” “Of course.” “And being of equal parts with itself, it will also be equal in number to itself, and if of more parts, more in number, and if of less parts, less in number than itself.” “Clearly.” “And will not the one possess the same relation towards other things?” “Because it is shown to be greater than they, must it not also be more in number than they and because it is smaller, less in number? And because it is equal in size, must it not be also, equal in number to the others?” “Yes, it must.” [151e] “And so once more, as it appears, the one will be equal to, greater than, and less than itself and other things in number.” “Yes, it will.”

Text from Perseus Project
Parmenides. in Plato in Twelve Volumes, Vol. 9. Transl. Harold N. Fowler.
Cambridge, MA: Harvard University Press; London: William Heinemann Ltd., 1925. OCLC: 20083931, 19433521, 377367, 21777623 ISBN: 0674990404, 0674991842, 0674991850, 0674991826 Available online at: http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus:text:1999.01.0174&query=head%3D%231

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