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Zeno's Paradox

PeacefulChaos
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5/11/2014 11:23:25 AM
Posted: 2 years ago
I'm sure many of us have heard of the famous paradox, but for those who haven't, I'll give a brief summary below.

Suppose Achilles and a tortoise participated in a race, and the tortoise got a slight head start of about 5 or 10 seconds. Despite the small head start, one would think that Achilles should easily be able to overcome the slow and cumbersome tortoise, but the reality is that Achilles will never be able to catch up to it, no matter how fast he travels. Let us say that the distance between him and the tortoise is "x." To first reach "x," he must travel 1/2 x, and to reach 1/2 x, he must travel 1/4 x, and so on. In other words, he must travel an infinite amount of points or distance in order to reach the tortoise; however, this is obviously impossible and cannot be done; consequentially, movement is an illusion. I suppose it would be similar to an asymptote.

The primary refutation to this paradox that I have heard thus far is this:

An infinite series such as Zeno has described (i.e. 1/2, 1/4, 1/8, 1/16 ...) will ultimately add up to one, meaning it is possible for Achilles to catch up to the tortoise in a measurable amount of time.

I have several problems with this refutation, as it doesn't really leave me satisfied.

1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

2. The refutation to Zeno's paradox only takes into account a geometric sequence in which r = 1/2. What of all the numbers between 1/2 and 1/4, or 1/4 and 1/8, and so on? For example, before Achilles can reach 0.5x, he must get to 0.49x, and before he can get to 0.49x, he must reach 0.48x, and so on. The sum of all these numbers goes on toward infinity, and is certainly greater than 1.

3. Even if the refutation was valid, I still fail to see how it proves that Achilles can catch up to the tortoise. Although the numbers would add up to one, what of it? How does this prove that Achilles can traverse an infinite amount of distance himself?

Thanks for reading.
Stephen_Hawkins
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5/11/2014 12:25:58 PM
Posted: 2 years ago
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.
Give a man a fish, he'll eat for a day. Teach him how to be Gay, he'll positively influence the GDP.

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philochristos
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5/11/2014 12:43:18 PM
Posted: 2 years ago
Even if Achilles could reach the 1 distance (or x, or whatever it is), it wouldn't matter because by the time he got there, the tortoise will have moved ahead. So Achilles will never be able to catch it because every time he gets to where the tortoise was before, it will have moved ahead.
"Not to know of what things one should demand demonstration, and of what one should not, argues want of education." ~Aristotle

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Envisage
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5/11/2014 12:46:42 PM
Posted: 2 years ago
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
I'm sure many of us have heard of the famous paradox, but for those who haven't, I'll give a brief summary below.

Suppose Achilles and a tortoise participated in a race, and the tortoise got a slight head start of about 5 or 10 seconds. Despite the small head start, one would think that Achilles should easily be able to overcome the slow and cumbersome tortoise, but the reality is that Achilles will never be able to catch up to it, no matter how fast he travels. Let us say that the distance between him and the tortoise is "x." To first reach "x," he must travel 1/2 x, and to reach 1/2 x, he must travel 1/4 x, and so on. In other words, he must travel an infinite amount of points or distance in order to reach the tortoise; however, this is obviously impossible and cannot be done; consequentially, movement is an illusion. I suppose it would be similar to an asymptote.

The primary refutation to this paradox that I have heard thus far is this:

An infinite series such as Zeno has described (i.e. 1/2, 1/4, 1/8, 1/16 ...) will ultimately add up to one, meaning it is possible for Achilles to catch up to the tortoise in a measurable amount of time.

I have several problems with this refutation, as it doesn't really leave me satisfied.

1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

2. The refutation to Zeno's paradox only takes into account a geometric sequence in which r = 1/2. What of all the numbers between 1/2 and 1/4, or 1/4 and 1/8, and so on? For example, before Achilles can reach 0.5x, he must get to 0.49x, and before he can get to 0.49x, he must reach 0.48x, and so on. The sum of all these numbers goes on toward infinity, and is certainly greater than 1.

3. Even if the refutation was valid, I still fail to see how it proves that Achilles can catch up to the tortoise. Although the numbers would add up to one, what of it? How does this prove that Achilles can traverse an infinite amount of distance himself?

Thanks for reading.

Remember, the distance is halved each time, and velocity remains constant. The time to reach each 'checkpoint' is also halved, and when you do the infinite series of each subdivision of time taken, the time taken is a finite quantity.

You could also think of it this way, the number of 'checkpoints' crossed each second increases inversely proportionally to the size of the checkpoints... and therefore cancel each other out.
dylancatlow
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5/11/2014 1:16:45 PM
Posted: 2 years ago
The paradox is resolved by acknowledging the fact that it is impossible to describe or define a change in any way that does not involve a sudden finite jump in some parameter.
PeacefulChaos
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5/11/2014 5:24:42 PM
Posted: 2 years ago
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.
PeacefulChaos
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5/11/2014 5:28:15 PM
Posted: 2 years ago
At 5/11/2014 12:46:42 PM, Envisage wrote:

Remember, the distance is halved each time, and velocity remains constant. The time to reach each 'checkpoint' is also halved, and when you do the infinite series of each subdivision of time taken, the time taken is a finite quantity.

Okay, but how does this explain the fact that Achilles is able to move across an infinite amount of "checkpoints?" I kind of see what you're saying, but it isn't exactly sticking. Regardless of time, it is impossible to move across an infinite number of points and go beyond that infinite number of points, isn't it?


You could also think of it this way, the number of 'checkpoints' crossed each second increases inversely proportionally to the size of the checkpoints... and therefore cancel each other out.
PeacefulChaos
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5/11/2014 5:32:42 PM
Posted: 2 years ago
At 5/11/2014 1:16:45 PM, dylancatlow wrote:
The paradox is resolved by acknowledging the fact that it is impossible to describe or define a change in any way that does not involve a sudden finite jump in some parameter.

I'm not seeing how acknowledging that fact resolves the paradox. The only way it would resolve the paradox is if you accept that change is possible yet at the same time support the paradox. This would be a contradiction, and your arguments would fall apart.

I believe, however, that Zeno argued that not only did motion not exist, but neither did change; thus, it is compatible with his views.
Stephen_Hawkins
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5/11/2014 6:07:34 PM
Posted: 2 years ago
At 5/11/2014 5:24:42 PM, PeacefulChaos wrote:
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.

Consider it as a limit equivalent if you'd like. Ultimately, the two are equivalent though. It's just one of those things about how maths works - there are a lot of unintuitive things (like 1+2+3+4+... to infinity equals -1/12). Maths is deductive and not intuitive unfortunately.
Give a man a fish, he'll eat for a day. Teach him how to be Gay, he'll positively influence the GDP.

Social Contract Theory debate: http://www.debate.org...
dylancatlow
Posts: 12,245
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5/11/2014 6:36:10 PM
Posted: 2 years ago
At 5/11/2014 6:07:34 PM, Stephen_Hawkins wrote:
At 5/11/2014 5:24:42 PM, PeacefulChaos wrote:
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.

(like 1+2+3+4+... to infinity equals -1/12).

That's just fking bizarre lol
PeacefulChaos
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5/11/2014 6:44:58 PM
Posted: 2 years ago
At 5/11/2014 6:07:34 PM, Stephen_Hawkins wrote:
At 5/11/2014 5:24:42 PM, PeacefulChaos wrote:
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.

Consider it as a limit equivalent if you'd like. Ultimately, the two are equivalent though. It's just one of those things about how maths works - there are a lot of unintuitive things (like 1+2+3+4+... to infinity equals -1/12). Maths is deductive and not intuitive unfortunately.

Wait, what the ...

Could you explain the 1 + 2 + 3 + 4 ... = -1/12?
xXCryptoXx
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5/11/2014 7:14:22 PM
Posted: 2 years ago
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

This would correctly mean that we can travel an "infinite" fractional distance.
Nolite Timere
PeacefulChaos
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5/11/2014 7:16:40 PM
Posted: 2 years ago
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.


You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.


This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.
xXCryptoXx
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5/11/2014 7:19:19 PM
Posted: 2 years ago
At 5/11/2014 7:16:40 PM, PeacefulChaos wrote:
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.

I don't think it is an illusion on our part. I think the paradox itself is an illusion, as in that is how it would appear but it is actually false since reality would dictate that we can travel from point A to point B.

You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.

But by traveling to 0.5X you have already traveled all other fractional distances.

This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.

I know there is a refutation to the paradox, and it probably involves physics but I can'tp lace my finger on it.
Nolite Timere
PeacefulChaos
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5/11/2014 7:29:31 PM
Posted: 2 years ago
At 5/11/2014 7:19:19 PM, xXCryptoXx wrote:
At 5/11/2014 7:16:40 PM, PeacefulChaos wrote:
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.


I don't think it is an illusion on our part. I think the paradox itself is an illusion, as in that is how it would appear but it is actually false since reality would dictate that we can travel from point A to point B.

What is reality, though? Once again, you are basing reality based upon your own perspective of traveling from point A to B. "Reality" does not dictate anything. Zeno instead bases his argument off of objective facts.


You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.

But by traveling to 0.5X you have already traveled all other fractional distances.

Except you cannot ever reach 0.5x. Or 0.25x. Or any distance at all, because any movement whatsoever would require an infinite amount of distance to cover.


This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.

I know there is a refutation to the paradox, and it probably involves physics but I can'tp lace my finger on it.
Ore_Ele
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5/11/2014 7:46:52 PM
Posted: 2 years ago
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

It (being point #3) is also not true because that is simply not how you treat decimals. 10 * 0.1 =/= 0.10.
"Wanting Red Rhino Pill to have gender"
Ore_Ele
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5/11/2014 7:47:37 PM
Posted: 2 years ago
At 5/11/2014 12:43:18 PM, philochristos wrote:
Even if Achilles could reach the 1 distance (or x, or whatever it is), it wouldn't matter because by the time he got there, the tortoise will have moved ahead. So Achilles will never be able to catch it because every time he gets to where the tortoise was before, it will have moved ahead.

Please tell me you are applying devil's advocate.
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xXCryptoXx
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5/11/2014 7:50:35 PM
Posted: 2 years ago
At 5/11/2014 7:29:31 PM, PeacefulChaos wrote:
At 5/11/2014 7:19:19 PM, xXCryptoXx wrote:
At 5/11/2014 7:16:40 PM, PeacefulChaos wrote:
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.


I don't think it is an illusion on our part. I think the paradox itself is an illusion, as in that is how it would appear but it is actually false since reality would dictate that we can travel from point A to point B.

What is reality, though? Once again, you are basing reality based upon your own perspective of traveling from point A to B. "Reality" does not dictate anything. Zeno instead bases his argument off of objective facts.

Reality is how thing objectively are. I am of course, basing reality off of human senses and perception. Human perception would dictate that this paradox is false. Now whether or not the paradox is objectively true according to reality is a different question and one that I cannot answer.

You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.

But by traveling to 0.5X you have already traveled all other fractional distances.

Except you cannot ever reach 0.5x. Or 0.25x. Or any distance at all, because any movement whatsoever would require an infinite amount of distance to cover.

Our perception of reality would contradict that.

This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.

I know there is a refutation to the paradox, and it probably involves physics but I can'tp lace my finger on it.
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Ore_Ele
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5/11/2014 7:53:28 PM
Posted: 2 years ago
At 5/11/2014 6:07:34 PM, Stephen_Hawkins wrote:
At 5/11/2014 5:24:42 PM, PeacefulChaos wrote:
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.

Consider it as a limit equivalent if you'd like. Ultimately, the two are equivalent though. It's just one of those things about how maths works - there are a lot of unintuitive things (like 1+2+3+4+... to infinity equals -1/12). Maths is deductive and not intuitive unfortunately.

"like 1+2+3+4+... to infinity equals -1/12"

It does not "equal" -1/12, it can be represented as -1/12 for certain mathematical applications. That would be a debate that I'd be willing to have if needed.
"Wanting Red Rhino Pill to have gender"
PeacefulChaos
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5/11/2014 7:55:00 PM
Posted: 2 years ago
At 5/11/2014 7:50:35 PM, xXCryptoXx wrote:
At 5/11/2014 7:29:31 PM, PeacefulChaos wrote:
At 5/11/2014 7:19:19 PM, xXCryptoXx wrote:
At 5/11/2014 7:16:40 PM, PeacefulChaos wrote:
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.


I don't think it is an illusion on our part. I think the paradox itself is an illusion, as in that is how it would appear but it is actually false since reality would dictate that we can travel from point A to point B.

What is reality, though? Once again, you are basing reality based upon your own perspective of traveling from point A to B. "Reality" does not dictate anything. Zeno instead bases his argument off of objective facts.

Reality is how thing objectively are. I am of course, basing reality off of human senses and perception. Human perception would dictate that this paradox is false. Now whether or not the paradox is objectively true according to reality is a different question and one that I cannot answer.

Zeno's paradox is pretty much saying that our perceptions of reality are just illusions, and he shows this through his basic mathematical argument; thus, trying to refute his argument based off of perceptions doesn't really work, since you'd have to come at it from a different angle.


You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.

But by traveling to 0.5X you have already traveled all other fractional distances.

Except you cannot ever reach 0.5x. Or 0.25x. Or any distance at all, because any movement whatsoever would require an infinite amount of distance to cover.

Our perception of reality would contradict that.

This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.

I know there is a refutation to the paradox, and it probably involves physics but I can'tp lace my finger on it.
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5/11/2014 7:55:26 PM
Posted: 2 years ago
At 5/11/2014 7:53:28 PM, Ore_Ele wrote:
At 5/11/2014 6:07:34 PM, Stephen_Hawkins wrote:
At 5/11/2014 5:24:42 PM, PeacefulChaos wrote:
At 5/11/2014 12:25:58 PM, Stephen_Hawkins wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

1) The error is just ludicrous. PBG says "1-0.9... = 0.1..." which is just flat out wrong. The elipsis clearly referred to more of the previous digit. So that means, by PBG's estimation, 1-0.999... = 0.111 ... which is false.

2) He claims that 1/3 is an approximation of 0.333... which is again false. The BoP would be to show this, but ultimately we know that this is not an approximation but true. 0.333... is equal to 1/3, as we can show with simple (or bus-stop) division.

3) the concept you have to "add a zero to the right" is very interesting, but just false. As it is an infinite series, you do not have an "end" to put the 0 on. A novel response, but just not true.

1. Yeah, I noticed that, too. I agree that is not a valid rebuttal.

2. He provided a source, and I just skimmed over it, but I didn't find anything about 1/3 not being 0.3333.

3. Okay. I also tried looking at the source he provided to back this up, but again I didn't find anything about it.

Nevertheless, it seems odd that 0.999... = 1. After all, we can agree 0.9 isn't equal to one. We can agree 0.99 isn't equal to one. We can agree that 0.999...0 (the number of nines would be equal to the number of atoms on earth) isn't equal to one, and so on. Wouldn't this just be like a line approaching an asymptote? Even though it can get infinitesimally close to the line, it will never reach it.

Consider it as a limit equivalent if you'd like. Ultimately, the two are equivalent though. It's just one of those things about how maths works - there are a lot of unintuitive things (like 1+2+3+4+... to infinity equals -1/12). Maths is deductive and not intuitive unfortunately.

"like 1+2+3+4+... to infinity equals -1/12"

It does not "equal" -1/12, it can be represented as -1/12 for certain mathematical applications. That would be a debate that I'd be willing to have if needed.

Yeah. Could anyone explain that to me?
xXCryptoXx
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5/11/2014 7:56:28 PM
Posted: 2 years ago
At 5/11/2014 7:55:00 PM, PeacefulChaos wrote:
At 5/11/2014 7:50:35 PM, xXCryptoXx wrote:
At 5/11/2014 7:29:31 PM, PeacefulChaos wrote:
At 5/11/2014 7:19:19 PM, xXCryptoXx wrote:
At 5/11/2014 7:16:40 PM, PeacefulChaos wrote:
At 5/11/2014 7:14:22 PM, xXCryptoXx wrote:
Well obviously the paradox isn't sound considering that travel does not work that way in reality.

I can travel from point A to point B even though I must first travel the infinite number of fractions between the two points.

The problem is that this is based upon your own perceptions. It is, as Zeno states, an illusion. You believe you can travel from point A to point B, but that is not the reality of the situation; thus, personal accounts are not sufficient rebuttals to the paradox.


I don't think it is an illusion on our part. I think the paradox itself is an illusion, as in that is how it would appear but it is actually false since reality would dictate that we can travel from point A to point B.

What is reality, though? Once again, you are basing reality based upon your own perspective of traveling from point A to B. "Reality" does not dictate anything. Zeno instead bases his argument off of objective facts.

Reality is how thing objectively are. I am of course, basing reality off of human senses and perception. Human perception would dictate that this paradox is false. Now whether or not the paradox is objectively true according to reality is a different question and one that I cannot answer.

Zeno's paradox is pretty much saying that our perceptions of reality are just illusions, and he shows this through his basic mathematical argument; thus, trying to refute his argument based off of perceptions doesn't really work, since you'd have to come at it from a different angle.

Ah. Well I'll ponder a bit and get back to you. Hopefully someone else here will come up with some scientific/philosophical rebuttal because I'm curious to know.

You see, if I travel half the distance between A and B I must not first travel 1/4, 1/8, 1/16, ect. because I already traveled those distances by traveling half way to B.

To get to 0.5x, however, you must have first gotten to 0.25x. This is basic logic. You couldn't have simply teleported to 0.5x.

But by traveling to 0.5X you have already traveled all other fractional distances.

Except you cannot ever reach 0.5x. Or 0.25x. Or any distance at all, because any movement whatsoever would require an infinite amount of distance to cover.

Our perception of reality would contradict that.

This would correctly mean that we can travel an "infinite" fractional distance.

Based upon our own perceptions, yes.

I know there is a refutation to the paradox, and it probably involves physics but I can'tp lace my finger on it.
Nolite Timere
Mhykiel
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5/11/2014 7:56:29 PM
Posted: 2 years ago
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
I'm sure many of us have heard of the famous paradox, but for those who haven't, I'll give a brief summary below.

Suppose Achilles and a tortoise participated in a race, and the tortoise got a slight head start of about 5 or 10 seconds. Despite the small head start, one would think that Achilles should easily be able to overcome the slow and cumbersome tortoise, but the reality is that Achilles will never be able to catch up to it, no matter how fast he travels. Let us say that the distance between him and the tortoise is "x." To first reach "x," he must travel 1/2 x, and to reach 1/2 x, he must travel 1/4 x, and so on. In other words, he must travel an infinite amount of points or distance in order to reach the tortoise; however, this is obviously impossible and cannot be done; consequentially, movement is an illusion. I suppose it would be similar to an asymptote.

The primary refutation to this paradox that I have heard thus far is this:

An infinite series such as Zeno has described (i.e. 1/2, 1/4, 1/8, 1/16 ...) will ultimately add up to one, meaning it is possible for Achilles to catch up to the tortoise in a measurable amount of time.

I have several problems with this refutation, as it doesn't really leave me satisfied.

1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

2. The refutation to Zeno's paradox only takes into account a geometric sequence in which r = 1/2. What of all the numbers between 1/2 and 1/4, or 1/4 and 1/8, and so on? For example, before Achilles can reach 0.5x, he must get to 0.49x, and before he can get to 0.49x, he must reach 0.48x, and so on. The sum of all these numbers goes on toward infinity, and is certainly greater than 1.

3. Even if the refutation was valid, I still fail to see how it proves that Achilles can catch up to the tortoise. Although the numbers would add up to one, what of it? How does this prove that Achilles can traverse an infinite amount of distance himself?

Thanks for reading.

That debate is funny. Because it is a mathematical fact .9 repeating equals 1.

Anyways, I'm interested in hearing what you guys may think.

The smallest distance between two points that is indivisible is a planck length. Due to the underlining structure of spacetime this is smallest length a quantum state can differ from another state, but not actual traverse the distance.

The smallest amount time to elapse is a planck second. Due to physical laws are unable to discern a half of a planck second.

So eventually after a long series of halves, Artemis will be 1 planck length away from the turtle. The turtle will be moving forward as well but slower. The next possible tick of the clock is 1 planck second. In 1 planck second Artemis will move the one planck length needed to reach the turtle.

The turtle moving slower will not move the planck length in that planck second. But will have to wait 1 more planck second to move the planck distance away from Artemis. Turtle needs 2 planck seconds to move 1 planck length. By this time Artemis has already caught up to the turtle.
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5/11/2014 8:01:42 PM
Posted: 2 years ago
At 5/11/2014 7:56:28 PM, xXCryptoXx wrote:

Ah. Well I'll ponder a bit and get back to you. Hopefully someone else here will come up with some scientific/philosophical rebuttal because I'm curious to know.

Same here, actually. It's quite an odd paradox, because you think it's obviously wrong because it's so counter-intuitive and you feel like you've got some kind of rebuttal that's just lying there waiting for you to find it, but you just can't grasp it.

Ah, well.
Envisage
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5/11/2014 8:04:20 PM
Posted: 2 years ago
At 5/11/2014 8:01:42 PM, PeacefulChaos wrote:
At 5/11/2014 7:56:28 PM, xXCryptoXx wrote:

Ah. Well I'll ponder a bit and get back to you. Hopefully someone else here will come up with some scientific/philosophical rebuttal because I'm curious to know.

Same here, actually. It's quite an odd paradox, because you think it's obviously wrong because it's so counter-intuitive and you feel like you've got some kind of rebuttal that's just lying there waiting for you to find it, but you just can't grasp it.

Ah, well.

The rebuttal I gave is pretty much it. You are just overthinking it, and attempting to quantify an infinite number of infinitesimally small events.
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5/11/2014 8:04:22 PM
Posted: 2 years ago
At 5/11/2014 7:56:29 PM, Mhykiel wrote:
At 5/11/2014 11:23:25 AM, PeacefulChaos wrote:
I'm sure many of us have heard of the famous paradox, but for those who haven't, I'll give a brief summary below.

Suppose Achilles and a tortoise participated in a race, and the tortoise got a slight head start of about 5 or 10 seconds. Despite the small head start, one would think that Achilles should easily be able to overcome the slow and cumbersome tortoise, but the reality is that Achilles will never be able to catch up to it, no matter how fast he travels. Let us say that the distance between him and the tortoise is "x." To first reach "x," he must travel 1/2 x, and to reach 1/2 x, he must travel 1/4 x, and so on. In other words, he must travel an infinite amount of points or distance in order to reach the tortoise; however, this is obviously impossible and cannot be done; consequentially, movement is an illusion. I suppose it would be similar to an asymptote.

The primary refutation to this paradox that I have heard thus far is this:

An infinite series such as Zeno has described (i.e. 1/2, 1/4, 1/8, 1/16 ...) will ultimately add up to one, meaning it is possible for Achilles to catch up to the tortoise in a measurable amount of time.

I have several problems with this refutation, as it doesn't really leave me satisfied.

1. The infinite series of 1/2, 1/4, 1/8 ... should theoretically add up to 1, but isn't the reality that it will only get infinitesimally close to 1? So, in order for the refutation to Zeno's paradox to be valid, would we not have to accept that 0.999...= 1? Certainly, there have been arguments for this; however, I've observed multiple debates in which PotBelliedGeek manages to demonstrate that 0.999... = / = 1.

http://www.debate.org...

If there would be any refutations to his arguments, I'd gladly like to hear them.

2. The refutation to Zeno's paradox only takes into account a geometric sequence in which r = 1/2. What of all the numbers between 1/2 and 1/4, or 1/4 and 1/8, and so on? For example, before Achilles can reach 0.5x, he must get to 0.49x, and before he can get to 0.49x, he must reach 0.48x, and so on. The sum of all these numbers goes on toward infinity, and is certainly greater than 1.

3. Even if the refutation was valid, I still fail to see how it proves that Achilles can catch up to the tortoise. Although the numbers would add up to one, what of it? How does this prove that Achilles can traverse an infinite amount of distance himself?

Thanks for reading.

That debate is funny. Because it is a mathematical fact .9 repeating equals 1.

Anyways, I'm interested in hearing what you guys may think.

The smallest distance between two points that is indivisible is a planck length. Due to the underlining structure of spacetime this is smallest length a quantum state can differ from another state, but not actual traverse the distance.

The smallest amount time to elapse is a planck second. Due to physical laws are unable to discern a half of a planck second.

So eventually after a long series of halves, Artemis will be 1 planck length away from the turtle. The turtle will be moving forward as well but slower. The next possible tick of the clock is 1 planck second. In 1 planck second Artemis will move the one planck length needed to reach the turtle.

The turtle moving slower will not move the planck length in that planck second. But will have to wait 1 more planck second to move the planck distance away from Artemis. Turtle needs 2 planck seconds to move 1 planck length. By this time Artemis has already caught up to the turtle.

http://en.wikipedia.org... infinity in 2 minutes.

Numberphile vids:
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5/11/2014 8:04:23 PM
Posted: 2 years ago
At 5/11/2014 7:56:29 PM, Mhykiel wrote:

The smallest distance between two points that is indivisible is a planck length. Due to the underlining structure of spacetime this is smallest length a quantum state can differ from another state, but not actual traverse the distance.

So you are saying that there is no such thing as an infinite amount of space?

Why can't you simply divide that planck length by 2? You justify it by saying that it is the "smallest length of a quantum state [that] can differ from another state, but not actual[ly] traverse the distance."

Could you explain this more in depth? I


The smallest amount time to elapse is a planck second. Due to physical laws are unable to discern a half of a planck second.

So eventually after a long series of halves, Artemis will be 1 planck length away from the turtle. The turtle will be moving forward as well but slower. The next possible tick of the clock is 1 planck second. In 1 planck second Artemis will move the one planck length needed to reach the turtle.

The turtle moving slower will not move the planck length in that planck second. But will have to wait 1 more planck second to move the planck distance away from Artemis. Turtle needs 2 planck seconds to move 1 planck length. By this time Artemis has already caught up to the turtle.
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5/11/2014 8:06:48 PM
Posted: 2 years ago
At 5/11/2014 8:04:20 PM, Envisage wrote:
At 5/11/2014 8:01:42 PM, PeacefulChaos wrote:
At 5/11/2014 7:56:28 PM, xXCryptoXx wrote:

Ah. Well I'll ponder a bit and get back to you. Hopefully someone else here will come up with some scientific/philosophical rebuttal because I'm curious to know.

Same here, actually. It's quite an odd paradox, because you think it's obviously wrong because it's so counter-intuitive and you feel like you've got some kind of rebuttal that's just lying there waiting for you to find it, but you just can't grasp it.

Ah, well.

The rebuttal I gave is pretty much it. You are just overthinking it, and attempting to quantify an infinite number of infinitesimally small events.

Yeah, I asked some questions about your rebuttal, but you didn't respond to them.

Did I just completely miss the point or something and now I'm asking irrelevant questions?
Mhykiel
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5/11/2014 8:10:18 PM
Posted: 2 years ago
At 5/11/2014 8:04:23 PM, PeacefulChaos wrote:
At 5/11/2014 7:56:29 PM, Mhykiel wrote:

The smallest distance between two points that is indivisible is a planck length. Due to the underlining structure of spacetime this is smallest length a quantum state can differ from another state, but not actual traverse the distance.

So you are saying that there is no such thing as an infinite amount of space?

Why can't you simply divide that planck length by 2? You justify it by saying that it is the "smallest length of a quantum state [that] can differ from another state, but not actual[ly] traverse the distance."

Could you explain this more in depth? I


The smallest amount time to elapse is a planck second. Due to physical laws are unable to discern a half of a planck second.

So eventually after a long series of halves, Artemis will be 1 planck length away from the turtle. The turtle will be moving forward as well but slower. The next possible tick of the clock is 1 planck second. In 1 planck second Artemis will move the one planck length needed to reach the turtle.

The turtle moving slower will not move the planck length in that planck second. But will have to wait 1 more planck second to move the planck distance away from Artemis. Turtle needs 2 planck seconds to move 1 planck length. By this time Artemis has already caught up to the turtle.

I posted vids and about Thompson's lamp.

It goes back to what dylancatlow was saying. You are assuming you can divide the distance infinitesimally smaller. You can not divide a planck length in half. Nor a planck secound.

Cause nothing can go faster than light, 1 planck second is how long it takes light in a vacuum to go 1 planck length. It is the shortest measurable thing.