Now, as this Stanford article points out, we don't actually know what Zeno said in his own words for the most part. Only one of the paradoxes comes down to us in what seems like a verbatim form. Socrates was a young man when the discussion with Zeno happened; Plato was Socrates' student; Aristotle was Plato's. So it's possible that Zeno might have rejected Aristotle's settlement, or even Aristotle's reconstruction of what he was trying to say.
Zeno is tackling some of the hardest parts of philosophy, which are the parts that treat the most basic things. You might think the basic things would be the easiest parts, but it's the other way around. Everyone can carry off an opinion about politics, which is enwrapped in ethical concerns and practical concerns and emotions and reasons and all of that. Any two old men sitting on a bench outside the bait store can do it, and make themselves understood to each other and any listeners. We deal with these complex and composite objects all the time, and think we know how to do it -- certainly we know a way of talking about it that will make sense to everyone. Reading Plato's Laws was relatively easy, because it is a conversation like that -- three old men on the road, chewing the fat about how things ought to be.
It's really difficult to talk about the basic things. For example, here's the one Zeno antimony that we seem to have intact:
If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus the things that are are unlimited.
If that didn't make sense on the first pass, it's because the ideas are so basic and fundamental that they seem to lack context that helps you understand what is going on. So let's walk through it.
"If there are many things..."
Possibly there aren't many things. That's actually what Zeno is going to be getting at, and Parmenides as well when we get there.
It seems like there are obviously many things, though. You can look around you and see what appear to be many different things. In my vision right now are this computer, a coffee cup with a skull and crossbones on it, and a Gerber Applegate-Fairbairn combat knife. It seems like these are several separate things, not just because they don't appear to be touching, but because my mind knows what each of these artifacts is for and it's not the same thing. Since each artifact has a distinct purpose, it must have a distinct reason for having come into being; and thus, since each thing was made at a different time for a different reason, it follows that they must be different things.
So Zeno is raising a problem for an idea that seems extremely plausible, and suggesting that it's impossible.
"If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer."
Say that there are many different things. Then, necessarily, there is a number of such things. We may not actually be able to count to it, but in principle the number of things could be known. This is true even if you move from counting knives and cups to counting atoms, or electrons, or whatever you decide qualifies as a thing -- that is itself a very important question in metaphysics. Whatever that number is, we shall call it a.
Now, all Zeno is saying is that a=a. Whatever a is, it must be equal to itself. It cannot be greater nor less than itself.
But if they are just as many as they are, they will be limited.
Whatever a is, it isn't greater than a; for example it isn't equal to a+1. It is a limited number, too, not an infinite number, because each of the individual things is finite (specifically, each part is one thing). You can't get to the infinite by adding up a finite number of things, no matter how big that finite number might be.
If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those.
An aside that non-mathematicians might skip: some of you are mathematicians, and you probably recognize this move as the shift from countable to uncountable infinities. That's bringing in a lot of furniture from mathematical theory that doesn't quite belong here; it would be easy to get confused if you think of this as integers being 'things' and the infinitely divisible 'space' on the number line between the integers as being the uncountably infinite. One reason that may confuse you is that there really are an infinite number of integers, but there really aren't an infinite number of things in the world. If you make that analogy (all analogies always break) the breaking point will be there. Zeno is talking about the physical world, not about the conceptual number line.
Aside over: is it reasonable to say that, in the physical world, the 'things' between objects are infinitely divisible? Well, yes, it is: and not just because we can't really say much about the levels of it that are too small to see with an electron microscope. Even if you can't divide a pea practically into much more than eighths, and that with a sharp knife, we know you could continue to divide it down to the level of atoms in theory; and even those can be split; and even there, there's a lot of 'empty space' that could be divided conceptually into parts. This is different from the purely conceptual division of the number line, because it is a conceptual division of a physical space; but it is, conceptually, just as infinite. (Indeed uncountably so, for those who followed the aside.)
But if they are just as many as they are, they will be limited.... [yet] thus the things that are are unlimited.
So here we have a paradox. Just because of the nature of there being 'many things,' those things must have a limited number. Yet because there will also be 'things' between them, and those things can be divided infinitely, a world of many things must have an unlimited number of things in it.
If the world has many things in it, then, they must be both limited and unlimited in number. Since that appears to be a logical contradiction arising from consequences of the antecedent "if there are many things," we must reject the antecedent. Therefore: There are not many things.
Thus, there are either no things at all, or only one thing.
There are several ways to attack this paradox. I think they're fun, so mostly I'll leave it as an exercise for you to do in the comments if you want. My personal favorite is to attack the idea that a conceptual division (which gives rise to the infinity) actually gives rise to additional things. Probably a better method, but a harder one, is to be precise about the metaphysical stature required to constitute a thing at all.
On the other hand, you could accept the argument. It's quite logical. If a material conditional leads to a contradiction in formal logic, we do indeed reject the antecedent in just the way Zeno proposes.
3 comments:
My take is pretty much like yours: he's using "things" in different ways. If he is including "arbitrary subdivisions of space between" as "things", then the arbitrariness of it means that their count is not well defined.
So, the Stanford author doesn't like reading it the way I do, and he gives a different account of what he thinks Zeno was doing there.
"In fact, the argument depends on a postulate specifying a necessary condition upon two things being distinct, rather than on division per se, and it may be reconstructed as follows: If there are many things, they must be distinct, that is, separate from one another. Postulate: Any two things will be distinct or separate from one another only if there is some other thing between them. Two representative things, x1 and x2, will be distinct only if there is some other thing, x3, between them. In turn, x1 and x3 will be distinct only if there is some other thing, x4, between them. Since the postulate can be repeatedly applied in this manner unlimited times, between any two distinct things there will be limitlessly many other things. Therefore, if there are many things, then there must be limitlessly many things."
Now, from where I sit, saying that 'two things are distinct iff there is a thing separating them' is only different from a division in that it specifies a dividing object. Since most of physical reality -- even at our scale -- is empty space, that dividing object is often going to be 'divided' internally only (a) conceptually, or (b) by subdivisions of 'empty space,' whatever that means (e.g. open air, or the distance from the nucleus to the electron shell).
The alternative approach might be intended to overcome that problem of ambiguity as to what constitutes a 'thing' by making the dividing thing what the Stanford author calls 'an x' of some subset. But now we're confusing mathematical objects with physical ones again, I think; it only sounds right to say that 'every x is an x.' Really, all the things that might be x are different: perhaps a pea, or an atom, or a set of empty space a foot wide, or a set of empty space half a foot wide.
Well, a is a variable that can be assigned and reassigned. So, if you divide a pea, now you've just added 1 to what a was (a = a + 1). The value of a changing doesn't actually mean that a != a.
The point that if x1 and x2 are separate only if x3 is between them doesn't necessarily make sense. imagine a knife splitting a pea. Certainly, the two halves of the pea are now separate, but the knife can be touching both halves of the pea. That is, things that are separate can touch, needing nothing between them. Quite the opposite, it is not something that is required to separate, but something absent that separates, I think.
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