Zeno's arrow paradox and general relativity?

[Andraz Cepic]

I don't know much about general relativity, but I'm curious if Zeno somehow foresaw the reallty of the universe many many years before Einstein did(without any rigour ofc, but still the idea holds?).

Understanding the Zeno's arrow paradox, stating that an arrow is motionless at a certain moment meaning that time is just a collection of moments(dimension?), I came to a conclusion that this is how "time" actually works, since as far as I know the notion of time is just the way we experience the 4D mathematical space that the universe is. Thus there is no actual "movement", but rather an array of points in this space represent the motion that we percieve. Since I haven't yet studied physics I do not yet fully understand such things, so here I am asking out of pure curiosity and impatience :D.

Did Zeno foresee the reality of time, or did he just think of his mental experiment as a paradox in a sense that such a claim doesn't make sense at all, even though it has been shown through general relativity to be true?

Also how did Einstein come up with this idea of the universe as a 4D space?(if it's not too much to ask xD)
(I probably did many stupid claims, but still I'm here to learn :P )

 

[Dale]

Zeno's arrow paradox, stating that an arrow is motionless at a certain moment

Zeno was flat out wrong about this point.

He did not have a similar insight as Einstein, and his paradoxes are not particularly relevant to GR. Their only value is as a historical curiosity which has been superceded since the development of limits.

 

[Andraz Cepic]

I figured, but how exactly was his wrong, I'm sorry but i don't understand. Would It require more knowledge to understand?

 

[PeroK]

I figured, but how exactly was his wrong, I'm sorry but i don't understand. Would It require more knowledge to understand?

He was also wrong empirically, because time passes and things move. Science explains what we observe, rather than using (in this case flawed) logic to suggest that what we observe must be an illusion.

If you have proved that something we observe logically cannot happen, then your logic is at fault, not nature.

 

[PeterDonis]

how exactly was his wrong

There are two answers to this:

(1) He did not understand the concept of a continuum. His argument assumes that for any given moment, there is a "next moment". But our current theories treat time and space as a continuum, and in a continuum, there is no "next moment" after a given moment: no matter which pair of moments you pick, and how close together they are, there will be other moments between them. He did not even consider this possibility.

(2) He assumed that "motionless" vs. "motion" was a concept that could be applied at a single point. He never considered the possibility that this doesn't make sense: that "motion" vs. "motionless" can only be defined if you consider multiple points in time and space.

 

[Dale]

I figured, but how exactly was his wrong, I'm sorry but i don't understand. Would It require more knowledge to understand?

Physically the state of a system is not given by the positions only, but rather by both the positions and the momenta. So an arrow is not motionless at a given instant, but posses a well defined and physically important momentum as well.

A series of arrows at the same location but with different momenta would behave differently. So Zenos idea that motion disappears simply by looking at a single moment in time is philosophically appealing, but physically wrong.

The moral of the story is to check your theory against the facts. Experiment trumps philosophy.

 

[Jeronimus]

I am not sure you can dismiss Zeno's paradoxes that easily.

For example the one with the tortoise and Achilles.

"In his Achilles Paradox, Achilles races to catch a slower runner–for example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones,"

No matter how many steps where Achilles runs to the place the tortoise was before, the tortoise will always be a little bit ahead. I would venture to say that even with infinite steps, Achilles could not reach the tortoise.

 

[russ_watters]

I am not sure you can dismiss Zeno's paradoxes that easily.

For example the one with the tortoise and Achilles.

You didn't provide any analysis of the "paradox" or acknowledge you recognize the explanations that were given. The flaws in the "paradoxes" are the same for all of them. Can you apply what has been explained above yourself?

 

[PeroK]

I am not sure you can dismiss Zeno's paradoxes that easily.

For example the one with the tortoise and Achilles.

"In his Achilles Paradox, Achilles races to catch a slower runner–for example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones,"

No matter how many steps where Achilles runs to the place the tortoise was before, the tortoise will always be a little bit ahead. I would venture to say that even with infinite steps, Achilles could not reach the tortoise.

The "paradox" is nonsense. You don't need a tortoise, you just need Achilles running towards a line, say:

At some time he is half way to the line, later he is 3/4 to the line, then 7/8 of the way and so on. So, he never reaches the line.

The paradox is solved by the fact that there is such a thing as an infinite sum that converges to a finite number. Adding 1/2 + 1/4 + 1/8 never gets you to 1, let alone past it. So, Zeno is simply restricting himself to a finite time interval in which Achilles "never" reaches the tortoise.

 

[Jeronimus]

The "paradox" is nonsense. You don't need a tortoise, you just need Achilles running towards a line, say:

At some time he is half way to the line, later he is 3/4 to the line, then 7/8 of the way and so on. So, he never reaches the line.

The paradox is solved by the fact that there is such a thing as an infinite sum that converges to a finite number. Adding 1/2 + 1/4 + 1/8 never gets you to 1, let alone past it. So, Zeno is simply restricting himself to a finite time interval in which Achilles "never" reaches the tortoise.

I don't think that this can be reduced to being a time interval issue, but the issue being that you require more than infinite steps to reach the tortoise if you break his journey down into steps as described in the paradox.

 

[PeroK]

I don't think that this can be reduced to being a time interval issue, but the issue being that you require more than infinite steps to reach the tortoise if you break his journey down into steps as described in the paradox.

If, at t=0, the tortoise is 1m ahead of Achilles and Achilles is moving 1m/s faster than the tortoise, where is Achilles at t=1 seconds and t=2 seconds?

 

[russ_watters]

I don't think that this can be reduced to being a time interval issue, but the issue being that you require more than infinite steps to reach the tortoise if you break his journey down into steps as described in the paradox.

Again: you haven't acknowledged the explanations already given. "Infinite steps" doesn't mean/do anything. 1/1, 2/2, 3/3, etc. are all the same duration: 1.

 

[Jeronimus]

If, at t=0, the tortoise is 1m ahead of Achilles and Achilles is moving 1m/s faster than the tortoise, where is Achilles at t=1 seconds and t=2 seconds?

But that is not what the paradox is about, as i see it. You are simply doing a big jump here, ignoring the paradox completely, which is about doing the journey in many little steps, resulting in even infinite steps not being enough to reach the tortoise.

 

[russ_watters]

But that is not what the paradox is about, as i see it. You are simply doing a big jump here, ignoring the paradox completely, which is about doing the journey in many little steps, resulting in even infinite steps not being enough to reach the tortoise.

Nonsense: every journey contains infinite steps. Again: you are ignoring what has been explained already.

 

[PeroK]

But that is not what the paradox is about, as i see it. You are simply doing a big jump here, ignoring the paradox completely, which is about doing the journey in many little steps, resulting in even infinite steps not being enough to reach the tortoise.

Yes, that is precisely what the paradox is not about. The paradox is nonsense, because it denies that 1s ever passes.

Anyway, Zeno was wrong. Achilles caught the tortoise long ago. Time marches on and a few lost souls are stuck in the 5th century BC!

 

[Jeronimus]

Nonsense: every journey contains infinite steps. Again: you are ignoring what has been explained already.

Well, this journey when broken down as the paradox requires it, does not complete even with infinite steps. I am not sure this is really nonsense.

 

[PeroK]

Well, this journey when broken down as the paradox requires it, does not complete even with infinite steps. I am not sure this is really nonsense.

What if Achilles got 1m behind the tortoise and then took a really big step? Or jumped right over the tortoise?

 

[Dale]

Therefore, good reasoning shows that fast runners never can catch slow ones,"

No matter how many steps where Achilles runs to the place the tortoise was before, the tortoise will always be a little bit ahead. I would venture to say that even with infinite steps, Achilles could not reach the tortoise.

And yet experiments clearly show that fast runners can overtake slow runners in a finite number of strides and a finite amount of time. A good experiment trumps "good reasoning".

All of Zenos paradoxes have been resolved logically with the development of limits.

 

[Jeronimus]

What is Achilles got 1m behind the tortoise and then took a really big step? Or jumped right over the tortoise?

Doesn't matter how big of a step he takes. His foot would certainly have to pass through the point the tortoise was before, midair. That is a very weak argument there. We do not have to wait for his foot to hit the ground.

 

[Jeronimus]

And yet experiments clearly show that fast runners can overtake slow runners in a finite number of steps and a finite amount of time. A good experiment trumps good reasoning.

All of Zenos paradoxes have been resolved logically with the development of limits.

You are wrong about the finite number of steps part. But yes, experiments show that a fast runner can overtake a slow runner, which this paradox does not deny. The paradox is about if space and time is as we imagine it to be or not.

 

[Dale]

You are wrong about the finite number of steps part

No, I am not. You can count each time his foot hits the ground. Each time his foot hits the ground is one step, by definition. It is a finite number.

 

[russ_watters]

Well, this journey when broken down as the paradox requires it, does not complete even with infinite steps. I am not sure this is really nonsense.

Enough. You're just repeating what the paradox says without making any attempt to incorporate the explanations given into your understanding. Thread locked. If any moderator wants to reopen this and try again, feel free to try again to make this CD stop skipping.

Anyway, Zeno was wrong. Achilles caught the tortoise long ago. Time marches on and a few lost souls are stuck in the 5th century BC!

I wonder if Zeno was trolling us or if he knew his paradox was wrong? I would think history would have recorded if he put his money where his mouth was and stood in front of an arrow someone shot at him! I think I'll give him the benefit of the doubt and assume he was bad at math and posed the questions as an attempt to learn/invent new math, but didn't actually believe them.