Zeno's Paradoxes: some notes

Zeno's Paradoxes

First off, let's get some distance from the particular paradoxes.
What is Zeno trying to do generally?

Prove that Parmenides is right: everything is unified: there is no movement, no time, no alteration
Prove that reality is one thing, unchanging, and indivisible.
There is no part that is more or less real nor is there more than one reality or truth or being

As an aside, there are those who think that quantum general relativity is on Parmenides' side: they are much smarter than I and I cannot judge their arguments, but it's interesting.

Many people criticized Parmenides because what he claimed seemed so preposterous
Zeno wants to show that assuming that the world is the way people claim is also absured

Claiming that things move has absurd consequences
Claiming that there are many things has absurd consequences
etc.

Digression: Argument form:

Ad absurdum:

You claim one thing. You prove it by assuming the contrary of that claim, just for the sake of argument. Then you show that absurd, unacceptable, obviously false, or contradictory results follow.

You need light to see, otherwise you could see in the dark.
You have to brush your teeth, otherwise your teeth will fall out.

Ad hominem:

an argument addressed to a particular position or a particular person.

In this case, the argument only really works against that position or person, but may have huge flaws otherwise.
I can argue against racism by arguing that Zorgunthol's particular racist theory is wrong: that is an ad hominem argument.

This is the sort Zeno uses: he is trying to take people's own beliefs and defeat them. He is not concerned with whether those beliefs are right or not.

Another way to make an ad hominem argument is less respectable: it is downright false

You know nothing about fashion: just look at your car.
If your mother is such a good pool player, why does she miss every basket she shoots?
The argument that what they discharge into the river is harmless is wrong: that company is a for profit entity.
who are you to say I stink: have you sniffed your own shoes?

Paradox.

A paradox is what results from taking reasonable assumptions and deducing a contradiction or unacceptable consequence from them.

Dilemma.

A dilemma is an argument in which you claim that there are two possible ways to go, but both have unacceptable consequences.

Damned if you do, damned if you don't.
A catch-22.

Particular absurdities which Zeno pointed out:

If you think that motion is infinitely divisible, then you must believe that nothing moves (the arrow).
If you think there are many things, then Zeno points out that you must conclude that things are both infinitely large and infinitely small (dichotomy)

If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. (Simplicius(a) On Aristotle's Physics, 140.29)

Claims that claiming there is more than one thing involves contradiction.
Think of birds on a telephone wire: 10 of them.

There is no bird between bird #3 and bird #4. Zeno's argument does not work.
But we want to see what he meant, and this is so obvious that he cannot have meant that!

There is air between the birds!

But between bird #1 and the air that touches it, there is no thing.
So there is an easy limit: there are 10 birds and 9 chunks of air between them.
But we want to see what he meant, and this is so obvious that he cannot have meant that!

What of points on a line?

Between every single pair of points, there is a point half way between them, so there are always more points!
This works like Zeno's argument!
But wait: how are the things in this world points?

Are things really point-like?
Well, no. But modern mathematicians are curious about the points point too, and they think that in fact some infinities are bigger than others. A man named Cantor worked out in the 19th century a way to treat infinite numbers as definite, believe it or not, and so Zeno's paradox may not even apply to points on a line!

… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (Simplicius(a) On Aristotle's Physics,139.9)
But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. (Simplicius(a) On Aristotle's Physics, 141.2)

suppose there are more than one things.
since he's just argued that things with no spatial extent do not exist, we must suppose that at least one spatially extended object exists.

If spatially extended, it has a front and a back and parts, and each part has extension too. And each part has parts that have spatial extension, and each part of a part of a part ...
If you add up all those parts, you get an infinitely large object.
Because adding up an infinite number of objects each of which has spatial extension yields an object of ... infinite spatial extension!
Right? No, wrong.
Because there is a limit.

1/2+1/4+1/8+1/16 .... = (wait for it...) 1

But now come the modern mathematicians:

What about the series 1-1+1-1+1-1...
You can rewrite it as (1-1)+(1-1)+(1-1)...
And in that case, it seems that (1-1)+(1-1)+(1-1)...=0
But then you can rewriteit as 1+ ((1-1)+(1-1)+(1-1)...)=1
Seems preposterous!
Cauchy worked it out and said that such a series is simply "undefined"!

And what if Zeno were clever and said that he was not just dividing one half in half, but rather both halves in half each time?

Then he would be summing up an infinite number of equally-sized itty-bitty bits, and that sum does indeed seems to be infinitely large.
If we say that he has divided them into not just itty-bitty bits, but bits so small that they have no extension at all, then we have another problem, because then they add up to a sum that has no extension either!

… whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided … though perhaps nobody in fact could so divide it.
What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude. (Aristotle On Generation and Corruption, 316a19)

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

The [second] argument was called “Achilles,” accordingly, from the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. (Simplicius(b) On Aristotle's Physics, 1014.10)