Thursday, June 09, 2005

Plato and the Sorities Paradox

Looking over Plato's Phaedo, it appears to me that Socrates could not have taken the sorities paradox seriously.

Here is a quote from the Phaedo(100e-101b):
And that by Greatness only great things become great and greater greater, and by Smallness the less becomes less.


Then if a person remarks that A is taller by a head than B, and B less by a head than A, you would refuse to admit this, and would stoutly contend that what you mean is only that the greater is greater by, and by reason of, greatness, and the less is less only by, or by reason of, smallness; and thus you would avoid the danger of saying that the greater is greater and the less by the measure of the head, which is the same in both, and would also avoid the monstrous absurdity of supposing that the greater man is greater by reason of the head, which is small. Would you not be afraid of that?

So if I am reading this right, Plato's theory of the Forms suggests that things that have a certain property, let us say being large, are large because they participate in the form of Largeness. They are not large because they have some extra size.

Plato here of course does not have to worry about the question of "larger than what?" because for him there is some absolute standard which is exemplified by participating in the form.

Though I still find it odd that there are Forms of adjectives or adverbs, even if there can be forms of nouns. I can imagine the perfect woman, or perfect apple - but the perfect running, or the perfect redness? I am not sure.

But regardless. If there are Forms, and objects are what they as because they participate in them then could Socrates have bought in to the Sorities paradox? Could he have bought in to the paradox of the heap?

The paradox states that a grain of sand is just that - a grain. A pile of sand is a pile. For any amount of grains of sand, if it is not a pile, then adding just one grain will not make it one. Yet, there is an easy procedure for getting from grains to piles, and that is by continuously adding grains - hence the paradox.

More colloquially, there is no straw that breaks camels' backs. If a certain amount of straws will not break a camel's back, then it should be able to hold one more too. Yet, if we keep adding straws, the camel's back will be broken.

More formally there is a base case, namely that one grain is not a heap. And there is an induction premise: for any n such that n grains is not a heap, n+1 grains will not be a heap either. But enough instantiations of this rule will create a heap.

There are various solutions to this problem, and it seems that Plato too had a solution. Plato's solution is to deny that the induction premise works at all. Plato claims that you cannot get something that was big by simply adding small things. For Plato, it was patently absurd to try to reach something that participated in the form of the Large, by repeated instantiations of something that participated in the Form of the Small. For Plato, some things participated in the Large, and others in the Small, and that determined what something was.

It is unclear how many forms there are. (That is an interesting question!) But is there a form of the in-between? And Plato would have to account for these aggregation effects. After all, things do get bigger without having to go through the forms. They get bigger by addition. But Plato would claim is that what makes them bigger is their participation in the forms, and not the aggregation effect.

In more modern terms we might have to say that Plato would have to have taken an epistemological stand toward the origin of the problem. He would have to say that there is are grains and there are heaps, and there is a sharp boundary, only we don't know where it is.

This is a version of Timothy Williamson's approach. His claim, if memory serves, is that we get a contradiction out of denying bivalence, so everything is clearly in a category but we may be ignorant of it. Plato too, I would assume had no problem assuming human ignorance of the forms. Overcoming this ignorance is of course the goal of philosophy.

This of course sounds obvious, but I do not recall seeing the Phaedo mentioned in connection with the paradox in such an obvious way.

(This sounds like an undergraduate paper. Damn!)


bec said...

did you hear that thud?
i think i just passed out.

Shosh said...

Isn't there something in Plato about appearances vs. essences and the problem of how we perceive things? So if a thing is large and participates in largeness but it's largeness has only come about as a result of constant small additions, couldn't it be that the small things were never small at all, they just looked that way, or we thought of them that way? Because ultimately they participate in the form of largeness so their once having appeared "small" is a bit of a trick that we need to get our minds around? I'm not saying that it's not a paradox, just that Plato likely didn't care because the form's the thing.
I'm very open to the possibility that I'm completely wrong and that this is a stupid comment.

Karl said...

Plato was certainly aware of Zeno's Paradoxes emerging from Parmenides' philosophy. The paradoxes attempt to defend the notion that nothing changes. But there is no reason to think that he agrees. Thus, we can say that Plato did think about change, but I am not sure how. I doubt he meant to imply that the change was merely a change in perception. He certainly did nt believe int eh reliablility of sense perception, but he would have to acount for each particular change in perception, which I do not see him doing.

30 something said...

Bigness and Smallness do not exist as objective entities. Something is only big in relation to something else that is smaller than itself. For example, planet earth seems pretty big to me. Its much bigger than I am and so it seems reasonable to say that Earth is big. However, relative to our solar system earth is pretty small. If someone asked our solar system what is big it would say that the galaxy is big while the earth is small. Of course if someone was to ask our galaxy what is big it would say the globular cluster is big while the solar system is small. This can go on infinitely of course. Bigness and Smallness do not exist in and of themselves, and the same is true for most adjectives like fast and slow, or hot and cold (with the exception of speed and temperature equaling absolute zero). Adjectives of this type are relational allowing us to describe some property of objects relative to other objects.

You said:

"Plato here of course does not have to worry about the question of "larger than what?" because for him there is some absolute standard which is exemplified by participating in the form."

I don’t really know what you mean by that. To me, the objection that I raise above seems to render the idea of Platonic Bigness incoherent. That is, unless we claim that the idea of Platonic Bigness and Smallness holds only within, but not between categories. For example, a man can be big relative to other men. Likewise, a planet can be big or small relative to other planets. A small planet can be bigger than a big man, and if we assume that Platonic bigness and smallness only hold within a particular category, this is not a contradiction despite the obvious fact that a big man is smaller than a small planet. According to this way of seeing it, there is no such thing as Platonic Bigness but there does exist a platonic form of a Big Man and a platonic form of a Small Planet.

Karl said...

I am here, of course not defending Plato, or his version of the Forms. I can't imagine why anyone would, at least in the metaphysical way that Plato had it. The whole idea seems to make little sence. (If you want to defend a more semantical account of the forms, then I can probably tolerate listening to it on a patient day.)

Here however, I just point out a consequence for the Sorities paradox.

I also realize that I spoke a bit hastily. Plato apparently believes that bigness and smallness DO exist in themselves, and objects do take on the property of bigness and smallness. However, Bigness and Smallness, for example, can leave an object, when they are compared to (put next to) a much bigger object, so that Bigness and Smallness are not relational for Plato, but absolute. However what is big and what is small can still change.

Again, I am not saying it makes sense or I agree with it, but it does seem to be a coherent theory, and perhaps there is some semantical way of making working it out.

Shosh said...

Are we talking about bigness and smallness or "the big" and "the small?" Because I think they would be forms like the beautiful and the good.
And something big has a share in or participates in the form of the big. So I think it wouldn't matter if you crossed categories because the apprehension of or participation in the form is what counts. It doesn't matter that it's relative because the child who says that her dad is big or has bigness and a physicist who says that the universe is big have both apprehended the form of the big, equally. Because the form of the big is bigger than big. You dig?

30 something said...

I have no problem with their being a completely abstract Form of Bigness that is independent of any physical object. This point of view takes the Form of Bigness out of the Universe and places it directly in your mind—which is exactly where I like things to be. The form of the big can be apprehended though a molecule or through the universe. When apprehending the form of bigness the size of objects doesn’t matter. What matters is whether something is perceived as big. As shosh said, it doesn’t matter whether a child is looking at her dad or the physicist is looking at the universe, the form of Big can be apprehended through either of them. And, conversely, so can smallness. If Andre the Giant was looking at that girl’s dad and apprehended the form of smallness at the same time as the little girl was apprehending the form of bigness this would not be contradictory for the form of the small is also bigger than big.

The forms of the beautiful and the good are different because one can make the case that they are not relational. In other words an object cannot be beautiful and not beautiful at the same time no matter how it is perceived. Beauty goodness and truth are objective properties of certain objects. I think that Plato actually believed this.

However, according to this point of view it would be impossible, I think, to claim that A is greater than B because A is one head taller than B. The form of big exists in the platonic universe and it can be apprehended through any object—big or small. No object however participates in the form of bigness. If the kind of big that Plato is talking about here is abstract, independent of the size of objects, and purely the result of out perception of objects, then A can be greater than B only when it is apprehended by someone to be greater than B. At the same time no form of Big A can exist because it is possible to apprehend Big A as being Small A.