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 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/23/2014 9:47:55 PMPosted: 3 years agoThe 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?
 Posts: 596 Add as FriendChallenge to a DebateSend a Message 11/23/2014 10:21:53 PMPosted: 3 years agoThere 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.
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/24/2014 11:14:37 AMPosted: 3 years agoAt 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?
 Posts: 1,505 Add as FriendChallenge to a DebateSend a Message 11/24/2014 1:02:06 PMPosted: 3 years agoAt 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 :pThe 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?
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/24/2014 3:15:05 PMPosted: 3 years agoAt 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 :pIt'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
 Posts: 1,505 Add as FriendChallenge to a DebateSend a Message 11/24/2014 3:59:12 PMPosted: 3 years agoAt 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 :pIt'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 lolYeah, 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.
 Posts: 1,505 Add as FriendChallenge to a DebateSend a Message 11/24/2014 4:10:04 PMPosted: 3 years agoAt 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.
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/24/2014 5:52:17 PMPosted: 3 years agoAt 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 :pIt'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 lolYeah, 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.
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/24/2014 5:55:06 PMPosted: 3 years agoAt 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.
 Posts: 1,505 Add as FriendChallenge to a DebateSend a Message 11/24/2014 6:09:24 PMPosted: 3 years agoAt 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 :pIt'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 lolYeah, 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.
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/25/2014 12:56:59 PMPosted: 3 years agoAt 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 :)
 Posts: 2,649 Add as FriendChallenge to a DebateSend a Message 11/25/2014 6:50:37 PMPosted: 3 years agoAt 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.
 Posts: 4,211 Add as FriendChallenge to a DebateSend a Message 11/26/2014 12:12:44 AMPosted: 3 years agoThe 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.Lying on the hill, crawling over the windowsill into your living room, they stare out, glass-eyed aimless heads, bodies torn by vultures. You are the man whose hands are rank with the smell of death. - Peter Hammill, "The Emperor in His War Room"