Zeno's Paradoxes: Misunderstanding or Misconception?

[quantumdude]

OK, it goes something like this:

1. If the arrow occupies a space its own size, then it is at rest.
2. The arrow always occupies a space its own size.
3. Therefore, the arrow is always at rest.

As far as I can see, the error in this argument is that the first premise is false. It basically says that if you know the position of the arrow, you know its state of motion. But every student of Physics I knows that you have to specify both the initial position and the initial velocity to get the state, because motion is described by a 2nd order differential equation.

So, in order to conclude that motion is impossible, Zeno had to assume that motion is impossible!

 

[wuliheron]

Zeno's paradoxes all make extensive use of reductio ad absurdum. Formal logic had not been invented yet and his were all inductive arguments. Instead of attempting to prove motion was impossible or false, he was really attempting to show that the idea of motion was just as absurd as his own idea that nothing moved. Therefore the real issue on the table is not whether or not he proved motion is impossible, but did he prove it was equally absurd, illogical, irrational...

 

[quantumdude]

Originally posted by wuliheron
Zeno's paradoxes all make extensive use of reductio ad absurdum.

For sure that's true of The Dichotomy, but I don't see how The Arrow is reductio ad absurdum.

Formal logic had not been invented yet and his were all inductive arguments.

It seems to me that they are all attempts at deductive arguments. The only problem with them is that they all use false premises. If Zeno were right about the infinite series or one parameter specifying a dynamical state, he would have two sound deductive arguments here!

 

[wuliheron]

Originally posted by Tom

For sure that's true of The Dichotomy, but I don't see how The Arrow is reductio ad absurdum.

Zeno's argument revolves around the concept of infinitely divisible space and an "instant" in time which is not divisible by definition. Hence the paradox of our ordinary perception of time. He was pointing out that people tacitly accept some things are infinitely divisible while others are not.

Can you have half a human being, a half pregnent woman, etc? Why should space be infinitely divisible and not time? On the other hand, if you accept that nothing is infinitely divisible it leads to an equally absurd paradox, motion is impossible.

It seems to me that they are all attempts at deductive arguments. The only problem with them is that they all use false premises. If Zeno were right about the infinite series or one parameter specifying a dynamical state, he would have two sound deductive arguments here!

Nah, his style of absurity was the common fare among ancient Greeks at the time for entertainment purposes as much as anything else. Another more pointed purpose was political and religious. Criticism of the established faith and political leaders was punishable by death. A way around this was to create "puzzles" which did not directly criticize religion and politics, but humorously pointed out the absurdity of the situation as indirectly as possible.

 

[Lifegazer]

Originally posted by Tom
But every student of Physics I knows that you have to specify both the initial position and the initial velocity to get the state, because motion is described by a 2nd order differential equation.

That just implies that nothing can exist unless it moves. As observed by the eye. But you cannot impose physical-law upon all existence. There is no solid argument of reason which can make that claim. So when you deconstruct Zeno's argument by-way of reason deducted via physics, your argument has little merit Tom.

Edit: I expect you to reply that the paradox is about 'matter', and that therefore your reason is valid. But your reason is actually about fundamental-matter (fundamental energy). You are discussing the base-energy of existence itself. And so, as mentioned above, your logic is invalid.

 

[quantumdude]

Lifegazer, you have a real knack for completely missing the point.

Let's look at what I wrote again:

1. If the arrow occupies a space its own size, then it is at rest.
2. The arrow always occupies a space its own size.
3. Therefore, the arrow is always at rest.

There is Zeno's argument. The first premise says that if the arrow has a location at some time, then it is at rest. In other words, he is saying that specifying x(t₀) at some time logically implies[/color] that it is not moving. If we take the correct specification of a state, namely that one must specify both x(t₀) and v(t₀), then we see that Zeno tacitly assumes that v(t₀)=0.

When the tacit assumption is recognized and inserted into the argument, we have "If an arrow is at a location and not moving, then it is not moving", which is trivial.

Originally posted by Lifegazer
That just implies that nothing can exist unless it moves.

Nothing I wrote implies that. You just pulled it out of the air.

As observed by the eye. But you cannot impose physical-law upon all existence. There is no solid argument of reason which can make that claim. So when you deconstruct Zeno's argument by-way of reason deducted via physical-laws, your argument has little merit Tom.

Again, you completely misunderstand both me and Zeno. Zeno is starting from what he believes are physical laws. He just got it wrong is all.

Edit: I expect you to reply that the paradox is about 'matter', and that therefore your reason is valid. But your reason is actually about fundamental-matter (fundamental energy). You are discussing the base-energy of existence itself. And so, as mentioned above, your logic is invalid.

There is nothing in either Zeno's argument or my argument about "fundamental energy" or "existence". You are seeing something that is not there.

 

[Lifegazer]

Originally posted by Tom
When the tacit assumption is recognized and inserted into the argument, we have "If an arrow is at a location and not moving, then it is not moving", which is trivial.

I disagree with this. Zeno says more than this. He also says the arrow cannot exist in any other state (premise 2). Therefore, the conclusion (premise 3), still seems to have validity.

There is nothing in either Zeno's argument or my argument about "fundamental energy" or "existence". You are seeing something that is not there.

Okay, I apologise for that. I thought you was hinting at QM.

 

[quantumdude]

Nope, no QM is necessary here.

Originally posted by Lifegazer
I disagree with this. Zeno says more than this. He also says the arrow cannot exist in any other state (premise 2). Therefore, the conclusion (premise 3), still seems to have validity.

Premise 2 only says that the arrow always has to have a location, which is true. However, that has no bearing on Premise 1, in which Zeno says that the position alone specifies the state. Premise 1 is false, and the argument is deductively valid, so we know that the conclusion is false.

 

[quantumdude]

Incidentally, it is said that Zeno's reason for forwarding Premise 1 is that, at any instant, there is no physical difference between a stationary arrow and a moving arrow. I don't see the logical connection there, but we now know that there is a physical difference: the moving arrow would be length contracted as per Special Relativity.

If SR were known at the time, would Zeno have offered this argument?
Did Zeno forsee SR?

Hmmm...

 

[wuliheron]

Originally posted by Tom
Incidentally, it is said that Zeno's reason for forwarding Premise 1 is that, at any instant, there is no physical difference between a stationary arrow and a moving arrow. I don't see the logical connection there, but we now know that there is a physical difference: the moving arrow would be length contracted as per Special Relativity.

If SR were known at the time, would Zeno have offered this argument?
Did Zeno forsee SR?

Hmmm...

He did in the sense that special relativity proposes an indivisible spacetime continuum, and Zeno believed everything was indivisible. Again, the argument is that having half a human being or a half pregnent woman or whatever is impossible. The alternative which he makes fun of in his paradoxes is that everything is infinitely divisible and, at that point, the Sorites heap paradox kicks in.

 

[ahrkron]

Originally posted by Tom
[...]Zeno's reason for forwarding Premise 1 is that, at any instant, there is no physical difference between a stationary arrow and a moving arrow. I don't see the logical connection there, but we now know that there is a physical difference: the moving arrow would be length contracted as per Special Relativity.

If SR were known at the time, would Zeno have offered this argument?
Did Zeno forsee SR?

I don't see the connection either, but if there is one, we can also ask the question in a slightly different way: would motion be possible in a universe with no length contraction?

This reminds me of the fact that Poincare was extremely close to the gist of SR. He even mentioned that there is no experimental backup for the idea that simultaneity is absolute (which is closely related to length contraction).

Of course, Poincare had the context of Maxwell equations and the struggle to interpret the "Lorentz-Fitzgerald contraction", but anyway, it strikes me as impressive that he made the connection, and the sole idea that Zeno may have glanced at it so long ago is just mind-boggling.

 

[heusdens]

Originally posted by Tom
Incidentally, it is said that Zeno's reason for forwarding Premise 1 is that, at any instant, there is no physical difference between a stationary arrow and a moving arrow. I don't see the logical connection there, but we now know that there is a physical difference: the moving arrow would be length contracted as per Special Relativity.

If SR were known at the time, would Zeno have offered this argument?
Did Zeno forsee SR?

In SR we would just state that the arrow, in it's own space-time frame is at rest. It shows to be moving, according to a different space-time frame, and also is length contracted and has a little more mass. These are observational things though. As seen from the arrow itself, physical properties don't change.

 

[heusdens]

Originally posted by ahrkron
I don't see the connection either, but if there is one, we can also ask the question in a slightly different way: would motion be possible in a universe with no length contraction? [/b/]

This is a weird interpretation, you better focus on the causes of length contraction and other relativistic phenomena, which are of course a consequence of the speed of light.
We could state it differently, what if the speed of light were infinite? Then every measurement would be instantaniously, since no time elapse between the happening of the event and the observation of it from a distant. Hence, no length contraction or any other relativistic phenomena would occur.

I would reformulate this then as:

Suppose the light speed would be infinite, would there still be motion possible?

What would the universe look like if the speed of light were infinite?


But let us look at the problem again, the arrow in motion.
What is the difference between an arrow, that is in motion and one that is at rest (relative to some default intertial frame of reference that is).
Both arrows are at some point in time T at some point in space S.
Yet, the moving arrow is at a different point in space at a different time, and the arrow in rest, does not change it's point in space at a different time. So obviously, the arrows differ.

We have to be aware of the fact though that the moving arrow, did not start it's motion out of nothing, but it was accelerated due to some force. This force has transferred energy onto the arrow.
That makes the arrows physically different. The moving arrow carries kinetic energy, the arrow at rest not.

The speed of light in this experiment is of no theoretical consequence. Hence this would not be of any influence to the fact that one arrow is moving, and the other is at rest.

 

[heusdens]

Originally posted by Lifegazer
That just implies that nothing can exist unless it moves.

This is fairly true, there is no existence without motion or change.

But you cannot impose physical-law upon all existence

Since all existence is physical existence, it is obvious that all physical existence is subject to physical laws.

 

[Mentat]

Originally posted by Tom
OK, it goes something like this:

1. If the arrow occupies a space its own size, then it is at rest.
2. The arrow always occupies a space its own size.
3. Therefore, the arrow is always at rest.

As far as I can see, the error in this argument is that the first premise is false. It basically says that if you know the position of the arrow, you know its state of motion. But every student of Physics I knows that you have to specify both the initial position and the initial velocity to get the state, because motion is described by a 2nd order differential equation.

So, in order to conclude that motion is impossible, Zeno had to assume that motion is impossible!

Relativity also contradicts the first premise, doesn't it? In Relativistic reasoning, any frame of reference could be said to be "at rest".