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Zeno's Paradox Solutions

DanT
Posts: 5,693
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12/10/2013 6:45:32 AM
Posted: 2 years ago
Achilles and the Tortoise & Dichotomy paradox
In this paradox Achilles and the Tortoise race at a constant speed. The tortoise, having a slower speed, is given a head start of 100 meters, but by the time Achilles runs 100 meters the tortoise has reached another 10 meters. According to Zeno Achilles can never reach the tortoise because the tortoise will continue to gain ground as he is catching up.
We know the tortoise moves at 1/10th the speed of Achilles, so let's assume the tortoise is moving at 0.5 meters per second, and Achilles is moving at 5 meters per second.
It takes 200 seconds for the tortoise to achieve the 100 meter head start, and 20 seconds for Achilles to catch up.
220 seconds into the race Achilles has run 100 meters, and the Tortoise has run 110
221 seconds into the race Achilles has run 105 meters, and the Tortoise has run 110.5
222 seconds into the race Achilles has run 110 meters, and the Tortoise has run 111
223 seconds into the race Achilles has run 115 meters, and the Tortoise has run 111.5

Therefore after 222 seconds Achilles has passed the Tortoise. The error in Zeno's logic was that as he divided the distance needed to surpass the tortoise he also divided the time it took to surpass the Tortoise, so at 222 seconds into the race Achilles and the Tortoise slowed to a crawl as Achilles attempted to surpass the Tortoise. Because the distance between two objects are finite, there is a point at which you can no longer cut the distance in half. For argument's sake, let's say this distance is a Planck length; once the tortoise's lead was reduced to a Planck length, Achilles was able to overpower the tortoise.

This solution can also explain away the Dichotomy paradox, which is just a variation of the 1st paradox.

Arrow paradox

The arrow paradox states that an arrow fired during a single durationless moment in time, it cannot move to where it is nor to where it is not, therefore because time is made up of moments, and everything is motionless in instants, everything is motionless throughout time. This of course is a fallacy of composition. If we take a string of motionless frames, and we flip through them at 24 frames per second, we get a moving image from a series of motionless frames. Thus, while one timeless instant may be motionless, a series of instances creates motion.
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cybertron1998
Posts: 5,818
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12/11/2013 10:56:38 AM
Posted: 2 years ago
At 12/10/2013 6:45:32 AM, DanT wrote:
Achilles and the Tortoise & Dichotomy paradox
In this paradox Achilles and the Tortoise race at a constant speed. The tortoise, having a slower speed, is given a head start of 100 meters, but by the time Achilles runs 100 meters the tortoise has reached another 10 meters. According to Zeno Achilles can never reach the tortoise because the tortoise will continue to gain ground as he is catching up.
We know the tortoise moves at 1/10th the speed of Achilles, so let's assume the tortoise is moving at 0.5 meters per second, and Achilles is moving at 5 meters per second.
It takes 200 seconds for the tortoise to achieve the 100 meter head start, and 20 seconds for Achilles to catch up.
220 seconds into the race Achilles has run 100 meters, and the Tortoise has run 110
221 seconds into the race Achilles has run 105 meters, and the Tortoise has run 110.5
222 seconds into the race Achilles has run 110 meters, and the Tortoise has run 111
223 seconds into the race Achilles has run 115 meters, and the Tortoise has run 111.5

Therefore after 222 seconds Achilles has passed the Tortoise. The error in Zeno's logic was that as he divided the distance needed to surpass the tortoise he also divided the time it took to surpass the Tortoise, so at 222 seconds into the race Achilles and the Tortoise slowed to a crawl as Achilles attempted to surpass the Tortoise. Because the distance between two objects are finite, there is a point at which you can no longer cut the distance in half. For argument's sake, let's say this distance is a Planck length; once the tortoise's lead was reduced to a Planck length, Achilles was able to overpower the tortoise.

This solution can also explain away the Dichotomy paradox, which is just a variation of the 1st paradox.

Arrow paradox

The arrow paradox states that an arrow fired during a single durationless moment in time, it cannot move to where it is nor to where it is not, therefore because time is made up of moments, and everything is motionless in instants, everything is motionless throughout time. This of course is a fallacy of composition. If we take a string of motionless frames, and we flip through them at 24 frames per second, we get a moving image from a series of motionless frames. Thus, while one timeless instant may be motionless, a series of instances creates motion.

lost on the first one but i get the second
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ADreamOfLiberty
Posts: 1,570
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12/11/2013 11:38:27 AM
Posted: 2 years ago
Aren't these solvable using converging series?
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Subutai
Posts: 3,213
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12/12/2013 10:34:01 PM
Posted: 2 years ago
At 12/11/2013 11:38:27 AM, ADreamOfLiberty wrote:
Aren't these solvable using converging series?

Yes, and that's the point he's trying to make.
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Enji
Posts: 1,022
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12/12/2013 10:59:09 PM
Posted: 2 years ago
At 12/12/2013 10:34:01 PM, Subutai wrote:
At 12/11/2013 11:38:27 AM, ADreamOfLiberty wrote:
Aren't these solvable using converging series?

Yes, and that's the point he's trying to make.

I thought he was going with the time and space are discrete and not infinitely divisible approach rather than with convergent infinite series (which entails that space and time are continuous and hence infinitely divisible).
chui
Posts: 507
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12/13/2013 5:23:09 AM
Posted: 2 years ago
At 12/10/2013 6:45:32 AM, DanT wrote:


Arrow paradox

The arrow paradox states that an arrow fired during a single durationless moment in time, it cannot move to where it is nor to where it is not, therefore because time is made up of moments, and everything is motionless in instants, everything is motionless throughout time. This of course is a fallacy of composition. If we take a string of motionless frames, and we flip through them at 24 frames per second, we get a moving image from a series of motionless frames. Thus, while one timeless instant may be motionless, a series of instances creates motion.

But the motion seen in a motion picture is an illusion. This, I think, was Zeno's point. If in reality the arrow jumps from position to position, then the smooth motion we see is not reality but an illusion. Ideas from quantum physics would seem to back up the idea that motion actually occurs as a series of finite jumps (Planck length etc.) although the wave function that corresponds to a moving particle evolves smoothly with time. Relativity would say that motion at a steady velocity is not an absolute and is identical to being stationary. Perhaps in future we will develop cameras that can resolve this issue but for now I feel our ideas of space and time are not complete.
AlbinoBunny
Posts: 3,781
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12/23/2013 2:05:23 PM
Posted: 2 years ago
Doesn't it just highlight that we don't really understand infinity? If there can be an infinitesimal amount of time, what does that mean, zero time? Would an infinite amount of them create a certain amount of time, how much? An infinite amount of time? Then is it truly infinitesimal?

Basically we understand the finite; our surroundings, but not so much the infinite; (our container?).
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RoyLatham
Posts: 4,488
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12/24/2013 2:01:33 AM
Posted: 2 years ago
The direct calculation of position shows that Achilles in fact passes the tortoise. But we all knew that. That sets up the paradox, which is that no matter how close Achilles gets, he still has further to go. The paradox is resolved only by showing that summing the infinite series actually get Achilles even with the tortoise at the moment he should be, as calculated directly. Another way to look at it is to note that an infinite number of terms of the series is actually summed in real life, contrary to the intuition that an infinite number of anything cannot be realized.

The arrow paradox depends upon time consisting of an infinite number of "moments." The infinity is moments is continuous.