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Not Satisfied With Answers to Zeno's Paradox

PeacefulChaos
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11/23/2014 9:47:55 PM
Posted: 2 years ago
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?
Unitomic
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11/23/2014 10:21:53 PM
Posted: 2 years ago
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

The misrepresented logic here is that it makes us view time and distance as going inward instead of outward. with each new number we simply go inward to it's individual parts. It's hard to explain, but the problem lies in us going inward instead of forward.
PeacefulChaos
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11/24/2014 11:14:37 AM
Posted: 2 years ago
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.


The misrepresented logic here is that it makes us view time and distance as going inward instead of outward. with each new number we simply go inward to it's individual parts. It's hard to explain, but the problem lies in us going inward instead of forward.

Can it not do both?
Wocambs
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11/24/2014 1:02:06 PM
Posted: 2 years ago
At 11/24/2014 11:14:37 AM, PeacefulChaos wrote:
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.

If I said: 'Movement is an arbitrary notion, only existing in the context of arbitrary distinctions being made, i.e. 'parts' being carved from the undifferentiated 'whole' that is reality', is that something Parmenides would agree with? I'm pretty ignorant of Ancient Greek philosophy. Help me out :p


The misrepresented logic here is that it makes us view time and distance as going inward instead of outward. with each new number we simply go inward to it's individual parts. It's hard to explain, but the problem lies in us going inward instead of forward.

Can it not do both?
PeacefulChaos
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11/24/2014 3:15:05 PM
Posted: 2 years ago
At 11/24/2014 1:02:06 PM, Wocambs wrote:
At 11/24/2014 11:14:37 AM, PeacefulChaos wrote:
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.

If I said: 'Movement is an arbitrary notion, only existing in the context of arbitrary distinctions being made, i.e. 'parts' being carved from the undifferentiated 'whole' that is reality', is that something Parmenides would agree with? I'm pretty ignorant of Ancient Greek philosophy. Help me out :p

It's not quite like that, because Parmenides believed that reality was uniform and indivisible, so there were no "parts" of the "whole," there was only reality (a.k.a. Being).

He believes that non-being or non-existence does not exist, and motion requires the existence of non-existence, because you can't move without there being some form of non-existence (e.g. empty space). If there were no empty space, motion wouldn't be possible, and since he believed that non-existence didn't exist, then he also believed that motion was an illusion.

Just typing that is confusing lol
Wocambs
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11/24/2014 3:59:12 PM
Posted: 2 years ago
At 11/24/2014 3:15:05 PM, PeacefulChaos wrote:
At 11/24/2014 1:02:06 PM, Wocambs wrote:
At 11/24/2014 11:14:37 AM, PeacefulChaos wrote:
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.

If I said: 'Movement is an arbitrary notion, only existing in the context of arbitrary distinctions being made, i.e. 'parts' being carved from the undifferentiated 'whole' that is reality', is that something Parmenides would agree with? I'm pretty ignorant of Ancient Greek philosophy. Help me out :p

It's not quite like that, because Parmenides believed that reality was uniform and indivisible, so there were no "parts" of the "whole," there was only reality (a.k.a. Being).

He believes that non-being or non-existence does not exist, and motion requires the existence of non-existence, because you can't move without there being some form of non-existence (e.g. empty space). If there were no empty space, motion wouldn't be possible, and since he believed that non-existence didn't exist, then he also believed that motion was an illusion.

Just typing that is confusing lol

Yeah, isn't that what I'm saying? If A, B and X are all imaginary constructs, then 'X moves from A to B' is an imaginary event.
Wocambs
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11/24/2014 4:10:04 PM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I mean, I think I agree with Parmenides, I think, though I was reading Schopenhauer when I came to that conclusion. I think the 'solution' is to acknowledge that, since this is all arbitrary, or imaginary, that if we imagine the distance being continually halved, then we've defined it as never ending, since we're reducing the distance to a portion of that distance, making it impossible that the event will ever terminate. X x 1/2 could only result in zero if the value of X is already zero. But that isn't how reality actually works.
PeacefulChaos
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11/24/2014 5:52:17 PM
Posted: 2 years ago
At 11/24/2014 3:59:12 PM, Wocambs wrote:
At 11/24/2014 3:15:05 PM, PeacefulChaos wrote:
At 11/24/2014 1:02:06 PM, Wocambs wrote:
At 11/24/2014 11:14:37 AM, PeacefulChaos wrote:
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.

If I said: 'Movement is an arbitrary notion, only existing in the context of arbitrary distinctions being made, i.e. 'parts' being carved from the undifferentiated 'whole' that is reality', is that something Parmenides would agree with? I'm pretty ignorant of Ancient Greek philosophy. Help me out :p

It's not quite like that, because Parmenides believed that reality was uniform and indivisible, so there were no "parts" of the "whole," there was only reality (a.k.a. Being).

He believes that non-being or non-existence does not exist, and motion requires the existence of non-existence, because you can't move without there being some form of non-existence (e.g. empty space). If there were no empty space, motion wouldn't be possible, and since he believed that non-existence didn't exist, then he also believed that motion was an illusion.

Just typing that is confusing lol

Yeah, isn't that what I'm saying? If A, B and X are all imaginary constructs, then 'X moves from A to B' is an imaginary event.

In Parmenides' world, there is no A, B, and X. There is only A, because reality is uniform and indivisible.

But yes, you are correct that movement would be imaginary from our perceived points of A and B.
PeacefulChaos
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11/24/2014 5:55:06 PM
Posted: 2 years ago
At 11/24/2014 4:10:04 PM, Wocambs wrote:
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I mean, I think I agree with Parmenides, I think, though I was reading Schopenhauer when I came to that conclusion. I think the 'solution' is to acknowledge that, since this is all arbitrary, or imaginary, that if we imagine the distance being continually halved, then we've defined it as never ending, since we're reducing the distance to a portion of that distance, making it impossible that the event will ever terminate. X x 1/2 could only result in zero if the value of X is already zero. But that isn't how reality actually works.

That's the thing, though. Saying that it's not how reality actually works is based on our perception of reality, which Zeno argues is an illusion.

So trying to refute his argument by using our perceptions of reality isn't going to work, because that's the very thing he's addressing.
Wocambs
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11/24/2014 6:09:24 PM
Posted: 2 years ago
At 11/24/2014 5:52:17 PM, PeacefulChaos wrote:
At 11/24/2014 3:59:12 PM, Wocambs wrote:
At 11/24/2014 3:15:05 PM, PeacefulChaos wrote:
At 11/24/2014 1:02:06 PM, Wocambs wrote:
At 11/24/2014 11:14:37 AM, PeacefulChaos wrote:
At 11/23/2014 10:21:53 PM, Unitomic wrote:
There are two types of Paradoxes. This one falls under logical Paradox. What that means is that the paradox is built upon faulty or misrepresented logic, and therefore only SOUND illogical, when in reality it's simply fallical.

But Zeno "resolved" this paradox himself by reaching the conclusion that motion is an illusion and that, consequently, reality is constant.

He simply didn't propose a paradox and leave it as that. He used it to support the conclusions of Parmenides, thus eliminating the contradiction of premises.

If I said: 'Movement is an arbitrary notion, only existing in the context of arbitrary distinctions being made, i.e. 'parts' being carved from the undifferentiated 'whole' that is reality', is that something Parmenides would agree with? I'm pretty ignorant of Ancient Greek philosophy. Help me out :p

It's not quite like that, because Parmenides believed that reality was uniform and indivisible, so there were no "parts" of the "whole," there was only reality (a.k.a. Being).

He believes that non-being or non-existence does not exist, and motion requires the existence of non-existence, because you can't move without there being some form of non-existence (e.g. empty space). If there were no empty space, motion wouldn't be possible, and since he believed that non-existence didn't exist, then he also believed that motion was an illusion.

Just typing that is confusing lol

Yeah, isn't that what I'm saying? If A, B and X are all imaginary constructs, then 'X moves from A to B' is an imaginary event.

In Parmenides' world, there is no A, B, and X. There is only A, because reality is uniform and indivisible.

But yes, you are correct that movement would be imaginary from our perceived points of A and B.
That's the thing, though. Saying that it's not how reality actually works is based on our perception of reality, which Zeno argues is an illusion.

So trying to refute his argument by using our perceptions of reality isn't going to work, because that's the very thing he's addressing.

I'm hardly trying to refute his argument when I'm trying to work out to what extent our views are compatible, or identical. To think in terms of A and B in no way implies that you believe that reality is actually divided in such a way.
PeacefulChaos
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11/25/2014 12:56:59 PM
Posted: 2 years ago
At 11/24/2014 6:09:24 PM, Wocambs wrote:

I'm hardly trying to refute his argument when I'm trying to work out to what extent our views are compatible, or identical.

Okay. I was just trying to point out that our perception of reality is in conflict with what Zeno believes, and as it stands, it appears as though his argument and conclusion hold true.

To think in terms of A and B in no way implies that you believe that reality is actually divided in such a way.

Just making sure we're on the same page :)
Sidewalker
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11/25/2014 1:54:25 PM
Posted: 2 years ago
At 11/25/2014 12:56:59 PM, PeacefulChaos wrote:
At 11/24/2014 6:09:24 PM, Wocambs wrote:

I'm hardly trying to refute his argument when I'm trying to work out to what extent our views are compatible, or identical.

Okay. I was just trying to point out that our perception of reality is in conflict with what Zeno believes, and as it stands, it appears as though his argument and conclusion hold true.

The key is that it only appears to hold true. It"s not a real paradox, its Sophistry at best and Zeno knew it, he used a bait and switch technique to lull you into thinking there is a paradox where there just isn"t one.

Here"s how it works, Zeno starts out talking about speed, which is motion over time, the premise is that Achilles is faster than the tortoise, so the formula for the initial conditions includes time in order to establish relative speeds. But in the next step, he drops time from the formula and presents an analysis that doesn"t include the dimension of time, and from that bait and switch he concludes that motion is impossible, well duh, without time, there is no motion, we already knew that.

If you keep time in the analysis, and you know the relative speeds and location in your initial conditions, it"s a matter of relatively simple mathematics to determine when and where Achilles will pass the tortoise, we all solved word problems like that in the third grade, it tended to be word problems with cars or trains back then and there were no infinities involved to make us completely irrational.

Zeno"s trick was to start with a real world example, the race between Achilles and the tortoise, then move to a complete abstraction where you can, conceptually at least, create an infinite series of dividing operations, and because it would take an infinite amount of time to complete an infinite number of such operations, you incorrectly assume that would apply to the real world example, but it doesn"t have any bearing on the real world example, Achilles and the tortoise don"t slow down and come to a stop because you have contrived a conceptual mathematics operation that is infinitely repeating. Achilles passes the tortoise at the same time and place he would have if you weren"t contriving an infinitely repeating dividing operation, and as mentioned, we learned how to calculate that time and place in the third grade.

In the end, his conclusion is the direct opposite of the actual argument he has made, it"s a lot like starting with a yard stick, mentally cutting it in half, then that piece in half, and so on, conceptually this can certainly go on forever, but from that fact you wouldn"t conclude that the yardstick must be infinitely long, no, you"d only conclude that the subsequent pieces become infinitely small over time.

Better yet, it"s like trying to determine if .999" actually equals 1 (it does), and because it is a repeating operation concluding that .999" must equal infinity, but no, that isn"t how mathematics works here, not in this word problem or in the Achilles word problem.

Here"s another one, it"s similar along the same lines, there"s a bait and switch that makes it look paradoxical, see if you can solve it:

Three men check into a hotel with the bellhop because the manager is on break, he tells them the room is $30, they each pay ten dollars and they go to their room. The manager comes back, tells the bellhop he was mistaken, the room is only $25, and he sends the bellhop to their room to give them their $5 change.

When the bellhop explains, the men realize they can"t evenly divide the $5, so they each take $1 and give the bellhop a $2 tip.

Now"the three men have paid $9 each for the room, or a total of $27, and they gave the bellhop a $2 tip, so they have paid $29 for the room.

But we started with $30, so what happened to the other dollar?
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
PeacefulChaos
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11/25/2014 6:50:37 PM
Posted: 2 years ago
At 11/25/2014 1:54:25 PM, Sidewalker wrote:

Zeno"s trick was to start with a real world example, the race between Achilles and the tortoise, then move to a complete abstraction where you can, conceptually at least, create an infinite series of dividing operations, and because it would take an infinite amount of time to complete an infinite number of such operations, you incorrectly assume that would apply to the real world example, but it doesn"t have any bearing on the real world example, Achilles and the tortoise don"t slow down and come to a stop because you have contrived a conceptual mathematics operation that is infinitely repeating. Achilles passes the tortoise at the same time and place he would have if you weren"t contriving an infinitely repeating dividing operation, and as mentioned, we learned how to calculate that time and place in the third grade.

You state that "Achilles would pass the tortoise at the same time and place he would have if you weren't contriving an infinitely repeating dividing operation," but we only "know" this because we see it happening, which is not an adequate rebuttal against Zeno's viewpoints.

And, theoretically speaking, it should indeed be impossible to cross an infinite amount of points, regardless of whether or not time is included. Practically speaking, we can see with our very eyes that this is not the case, but that's the very thing that Zeno is attacking - our perceptions and practicality.


In the end, his conclusion is the direct opposite of the actual argument he has made, it"s a lot like starting with a yard stick, mentally cutting it in half, then that piece in half, and so on, conceptually this can certainly go on forever, but from that fact you wouldn"t conclude that the yardstick must be infinitely long, no, you"d only conclude that the subsequent pieces become infinitely small over time.

Zeno would not argue that the yard stick is infinitely long, but that it contains an infinite amount of points within a limited space. An "inside" infinity, if you will.


Better yet, it"s like trying to determine if .999" actually equals 1 (it does), and because it is a repeating operation concluding that .999" must equal infinity, but no, that isn"t how mathematics works here, not in this word problem or in the Achilles word problem.

Again, I don't think Zeno would conclude that 0.999... equals infinity, but rather that there are an infinite amount of points between 0 and 1. I'm not sure how the analogy here applies.


Here"s another one, it"s similar along the same lines, there"s a bait and switch that makes it look paradoxical, see if you can solve it:

Three men check into a hotel with the bellhop because the manager is on break, he tells them the room is $30, they each pay ten dollars and they go to their room. The manager comes back, tells the bellhop he was mistaken, the room is only $25, and he sends the bellhop to their room to give them their $5 change.

When the bellhop explains, the men realize they can"t evenly divide the $5, so they each take $1 and give the bellhop a $2 tip.

Now"the three men have paid $9 each for the room, or a total of $27, and they gave the bellhop a $2 tip, so they have paid $29 for the room.

I think the error occurs here, and is just a conceptual error. They didn't pay $29 for their room, they paid $25.
Subutai
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11/26/2014 12:12:44 AM
Posted: 2 years ago
The geometric series is usually obtained by saying that Achilles gets halfway to the tortoise continually. So, let's say they start 1 meter apart. When Achilles, gets halfway, he'll be 1/2 meters away from the tortoise. Again, 1/4, 1/8, 1/16, etc... What we obtain is the series 1/n^2. This series converges, so the sum of this infinite set of terms in finite.

It's really arbitrary. Say we define the series such that Achilles gets a third of the way to the tortoise. The sequence is 1, 2/3, 4/9, 8/27, etc... This series is 2^n/3^n. This is also convergent.

In other words, no matter what distance we set, the sum will always lead to a finite number because the series will always be convergent.
I'm becoming less defined as days go by, fading away, and well you might say, I'm losing focus, kinda drifting into the abstract in terms of how I see myself.
Sidewalker
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11/26/2014 6:11:50 AM
Posted: 2 years ago
At 11/25/2014 6:50:37 PM, PeacefulChaos wrote:
At 11/25/2014 1:54:25 PM, Sidewalker wrote:

Zeno"s trick was to start with a real world example, the race between Achilles and the tortoise, then move to a complete abstraction where you can, conceptually at least, create an infinite series of dividing operations, and because it would take an infinite amount of time to complete an infinite number of such operations, you incorrectly assume that would apply to the real world example, but it doesn"t have any bearing on the real world example, Achilles and the tortoise don"t slow down and come to a stop because you have contrived a conceptual mathematics operation that is infinitely repeating. Achilles passes the tortoise at the same time and place he would have if you weren"t contriving an infinitely repeating dividing operation, and as mentioned, we learned how to calculate that time and place in the third grade.

You state that "Achilles would pass the tortoise at the same time and place he would have if you weren't contriving an infinitely repeating dividing operation," but we only "know" this because we see it happening, which is not an adequate rebuttal against Zeno's viewpoints.

Sure it is, mathematics is an abstraction from the real world, if you apply it to the real world incorrectly, then it's your mathematics that is wrong, not the real world. Reality is under no obligation to conform to the way we think about it, if we are looking for truth, we need to make our thinking conform to reality.

And, theoretically speaking, it should indeed be impossible to cross an infinite amount of points, regardless of whether or not time is included.

Nonsense, that only appears to be the case because you are conceptually disregarding time in claiming there is an infinite amount of points and then sneaking time back into the equation and assuming some arbitrary amount of time is necessary to travel from one point to another point. Our initial conditions are rates of speed, which is distance traveled over a period of time, let's call it a yard a minute. If in the abstract, you conceptually determine that there are an infinite number of points in the yard, then to be consistent mathematically, you also have to recognize that this corresponds to an infinite number of moments in the minute, you need to multiply the same infinity times both the numerator and denominator, and when you do, the rate of speed doesn't change, it is still one yard per minute.

Practically speaking, we can see with our very eyes that this is not the case, but that's the very thing that Zeno is attacking - our perceptions and practicality.

I don't think so, he's attacking the mathematically incorrect idea that the sum of an infinite number of terms is an infinite sum with an infinite result, and that just isn't the case. You agree that an infinite number of subdivisions of a yard doesn't make the yard longer, as is the case with the minute, an infinite number of subdivisions of a minute doesn't make the minute longer. In Zeno's example, he starts with a rate so his mathematical operation of infinite subdivision must be applied to both numerator and denominator, so the rate remains the same.

In the end, his conclusion is the direct opposite of the actual argument he has made, it"s a lot like starting with a yard stick, mentally cutting it in half, then that piece in half, and so on, conceptually this can certainly go on forever, but from that fact you wouldn"t conclude that the yardstick must be infinitely long, no, you"d only conclude that the subsequent pieces become infinitely small over time.

Zeno would not argue that the yard stick is infinitely long, but that it contains an infinite amount of points within a limited space. An "inside" infinity, if you will.

Yeah, and that's my point, he also wouldn't argue that a minute is infinitely long, but that it contains an infinite number of "moments" within a limited amount of time, an infinity "inside" the minute, I really like the way you phrased that concept.

Better yet, it's like trying to determine if .999... actually equals 1 (it does), and because it is a repeating operation concluding that .999... must equal infinity, but no, that isn't how mathematics works, not in this word problem or in the Achilles word problem.

Again, I don't think Zeno would conclude that 0.999... equals infinity, but rather that there are an infinite amount of points between 0 and 1. I'm not sure how the analogy here applies.

The analogy applies directly to the point, and that's why I used it, I thought you'd make the mistake you just made.

We both know mathematics is an abstraction; problems come in when you apply it to realty, specifically when you apply it to reality in an inconsistent manner. You just admitted that he and you would only conclude that there are an infinite amount of points between 0 and 1, but 0 and 1 what? Yards or minutes? Aren't you actually arguing that .999... of a yard equals a yard, but .999... of a minute equals an infinite amount of time? You are, even though both .999...s are "inside" the unit of measurement.

In the end, the "inside infinity" as you called it, was Zeno's actual bait and switch, conceptually speaking his trick was a slight of hand trick, he played the shell game with the concept of an "inside infinity". He starts with a ratio, speed, which has two mathematical terms, time and distance, then he drops time from the analysis to deal only with distance and shows that the yard is still a yard, which is logically concrete and easy to visualize. Once he has you thinking in infinites, he brings time back in to the analysis and because time is conceptually much less concrete than distance, we get confused and think .999" of a unit of time actually equals infinity. To use your words, the infinity he is talking about is "inside" the unit of time, just like it is "inside" the unit of distance, the problem is we can easily "visualize" distance, but time is invisible and harder to visualize, so an infinite amount of inside points seems like a lot less than an infinite amount of "inside" moments. Like I said before, it would take an infinite amount of time to perform the infinite number of division operations on the minute, so we are fooled into thinking .999... of a minute equals infinity, but it doesn't, it equals a minute, it would just take an infinite amount of time to do the calculations, but the length of the minute doesn't change.

Here"s another one, it"s similar along the same lines, there"s a bait and switch that makes it look paradoxical, see if you can solve it:

Three men check into a hotel with the bellhop because the manager is on break, he tells them the room is $30, they each pay ten dollars and they go to their room. The manager comes back, tells the bellhop he was mistaken, the room is only $25, and he sends the bellhop to their room to give them their $5 change.

When the bellhop explains, the men realize they can"t evenly divide the $5, so they each take $1 and give the bellhop a $2 tip.

Now"the three men have paid $9 each for the room, or a total of $27, and they gave the bellhop a $2 tip, so they have paid $29 for the room.

I think the error occurs here, and is just a conceptual error. They didn't pay $29 for their room, they paid $25.

Well done, I added the two dollars to get 29 when I should have subtracted it to get 25, most people fall for it and think a dollar has gone missing. I probably shouldn't have included "for their room", anyway, very few people catch the error and spend a lot of time trying to find that missing dollar, nice catch.
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
dhardage
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11/26/2014 9:42:36 AM
Posted: 2 years ago
This is only an apparent paradox since it relies on a fixed period of time subdivided infinitely. Time does not work that way and any footrace will show that the paradox is indeed unfounded and untrue. This only goes to show that just because something appears logical, that does not necessarily make it true or valid.
xXCryptoXx
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11/26/2014 11:29:48 AM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

Why I can't a transcend an infinitely small distance?
Nolite Timere
PeacefulChaos
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11/30/2014 6:52:42 PM
Posted: 2 years ago
At 11/26/2014 6:11:50 AM, Sidewalker wrote:

Sure it is, mathematics is an abstraction from the real world, if you apply it to the real world incorrectly, then it's your mathematics that is wrong, not the real world. Reality is under no obligation to conform to the way we think about it, if we are looking for truth, we need to make our thinking conform to reality.

I'd say that it's debatable if we should consider mathematics an abstraction of the real world (which, again, we base on our perceptions). But I agree to a certain degree.


Nonsense, that only appears to be the case because you are conceptually disregarding time in claiming there is an infinite amount of points and then sneaking time back into the equation and assuming some arbitrary amount of time is necessary to travel from one point to another point. Our initial conditions are rates of speed, which is distance traveled over a period of time, let's call it a yard a minute. If in the abstract, you conceptually determine that there are an infinite number of points in the yard, then to be consistent mathematically, you also have to recognize that this corresponds to an infinite number of moments in the minute, you need to multiply the same infinity times both the numerator and denominator, and when you do, the rate of speed doesn't change, it is still one yard per minute.

Ah, I now see what you mean. It makes more sense to me now, thanks.
twocupcakes
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12/7/2014 9:59:58 AM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I don't see a paradox at all?

Yeah, you can divide length up into infinite points. But, it is easy to see how someone can catch the turtle.
PeacefulChaos
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12/7/2014 11:50:13 AM
Posted: 2 years ago
At 12/7/2014 9:59:58 AM, twocupcakes wrote:
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I don't see a paradox at all?

Yeah, you can divide length up into infinite points. But, it is easy to see how someone can catch the turtle.

That was the point of the paradox. You see motion taking place, but theoretically it's impossible. So, Zeno concluded that motion was an illusion.
tabularasa
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12/7/2014 2:45:27 PM
Posted: 2 years ago
Paradox says: Achilles will never reach the finish if he moves only half the distance between current position and finish every time he moves.

This is true, and not a paradox. If he moves only half the distance each move, he will never reach finish. If he moves the whole distance he will reach the finish.
1. I already googled it.

2. Give me an argument. Spell it out. "You're wrong," is not an argument.
tabularasa
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12/7/2014 2:57:33 PM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, the

It is a mistake to add each distance covered as the total distance moved in each move plus the cumulative distance moved from start in each subsequent move. If you do this, the sum is incalculable. You should be adding the distance from move 1 to the distance moved from the second starting point to the second stopping point. The sum is 1, the total number of moves is infinite.

d is the distance covered in one move.

d1 is the distance covered from start to stop in first move.

d2 is the distance covered from first stop to second stop, etc.

If the distance from start to finish is 1, then the total set of moves from start to finish (d1+d2+d3, continuing) will add up to 1 upon finish.

The total number of subset starts and stops from which ever point you want to start and stop is incalculable.

This does not actually produce a paradox.
1. I already googled it.

2. Give me an argument. Spell it out. "You're wrong," is not an argument.
twocupcakes
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12/7/2014 7:23:43 PM
Posted: 2 years ago
At 12/7/2014 11:50:13 AM, PeacefulChaos wrote:
At 12/7/2014 9:59:58 AM, twocupcakes wrote:
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I don't see a paradox at all?

Yeah, you can divide length up into infinite points. But, it is easy to see how someone can catch the turtle.

That was the point of the paradox. You see motion taking place, but theoretically it's impossible. So, Zeno concluded that motion was an illusion.

I don't see how any reasonable person would conclude that.

It definitely is possible.
MettaWorldPeace
Posts: 27
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12/9/2014 11:45:39 PM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

I think you are confusing his paradox, for it is not that those numbers add to any particular number, but that between any two numbers there is an infinite number of halfway points (or any fractional division).

For instance between 0 and 1 there is 3/4 and then 3/16 and then 3/64. By this logic we should never reach any number from any number. I would say we might be able to answer this by chunking or rather saying that there is some minimum constituent of numbers. For instance if this was 1/32, then we would go 1/2,1/4,1/8,/1/16,1/32,1/32,1/32. If there is no minimum constituent, then it would seem one would have to create an arbitrary one.

It is rather like assigning 0 to be the origin or the length of 1 meter. To have a half of a meter, you have to have a whole one. To have a calendar, you must have an original date.

Now if we can move one meter in half a second, we can move a half meter in a quarter of a second. So when we halve our movement through space, we halve our movement through time. Thus his paradox itself never realizes manifestation because smaller distances travelled are associated with smaller time frames when those distances are crossed at the same speed.
I awakened to another dream.
phiLockeraptor
Posts: 233
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12/10/2014 1:15:34 AM
Posted: 2 years ago
At 11/23/2014 9:47:55 PM, PeacefulChaos wrote:
The specific paradox I'm referring to is the one concerning Achilles and the tortoise, and the fact that in order to catch up to the tortoise, Achilles must pass through an infinite number of points (e.g. To get to "x," you must pass "1/2 x," and to get to 1/2 x, you must pass 1/4 x, and so on).

The typical answer to this is that the sum of the geometric series: 1, 1/2, 1/4, 1/8 ... is equal to 2. I understand and accept this, but it doesn't solve Zeno's paradox fully.

If we consider that there are infinite amount of points between two numbers, such as (0,1), then we should also realize that this amount of numbers is not limited to the above geometric series, since there are an infinite amount of numbers between 1 and 1/2, too.

For example, there is 1, 0.9999, 0.999, 0.99, 0.9, and so on. The sum of all these numbers is not 2. It's infinity. It's only 2 when you consider that particular geometric series, not when you consider every single number between 0 and 1.

What is the answer to this, then?

Okay so imagine a segment This segment is what Achillees has to cross. Let's call the ends Point A and Point B. Now, let's say he can get there in one step, for simplicities sake.

Now let's say you consider the one half point, right in the middle. We'll call that Point C.

We can do this all the way down, to infinity, just like you said.

The points, then, are infinite. However, the distance between them is not.

The distance between .99 and .999, for example, is much smaller than the distance between 1 and 2. All you're doing when you add more of these decimals is cutting the same total into smaller and smaller pieces, with more and more points to seperate it into segments.

The original distance, that is, Point A to Point B, remains unchanged.

Another way to put it is to imagine 3 separate sticks, all put together. I have to cross three sticks, but the actual distance is the same.

Heck, I could cut those sticks in half, in quarters, in thousandths! But when I put them together, end to end, it's still the same distance.

However, I could not cut them infinitely small, because I'd reach atoms eventually. This is where the "infinity" ends.

To address the paradox as a mathematical concept, I give you this answer: So what?

Seriously. Is there not an infinite amount of whole numbers, to count to to infinity? We simply choose to name and work with the ones that are relevant to us. That's why there's probably a 869-digit decimal that we've never even seen!
"Philosophy is a great conversation that never ends"

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dhardage
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12/11/2014 10:47:03 AM
Posted: 1 year ago
Zeno's Paradox is nothing of the sort. It demands that we keep dividing a specific period of time into smaller and smaller increments WITHOUT allowing time to pass. Time does not stop, it moves on so Achilles will easily outdistance Zeno with that passage of time. It's not nearly as abstruse and deep as it seems.