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Paradox of the Hare

 Posts: 12,788 Add as FriendChallenge to a DebateSend a Message 7/21/2012 2:17:36 PMPosted: 5 years ago"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?"Easy is the descent to Avernus, for the door to the Underworld lies upon both day and night. But to retrace your steps and return to the breezes above- that's the task, that's the toil."
 Posts: 3,667 Add as FriendChallenge to a DebateSend a Message 7/21/2012 2:49:56 PMPosted: 5 years agoIt's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.
 Posts: 25,980 Add as FriendChallenge to a DebateSend a Message 7/21/2012 2:51:17 PMPosted: 5 years agoAt 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?It's basic calculus dude. Actually, it the limit function that you learn in the first few weeks of calculus."Wanting Red Rhino Pill to have gender"
 Posts: 25,980 Add as FriendChallenge to a DebateSend a Message 7/21/2012 2:52:05 PMPosted: 5 years agooh, just say age = 16."Wanting Red Rhino Pill to have gender"
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 8:02:21 AMPosted: 5 years agoAt 7/21/2012 2:52:05 PM, Ore_Ele wrote:oh, just say age = 16.The Fool: you guys havent solved it. Infinite is not never completes. Its indefinite. IT doesn;t ever reach the limit. There is an answer but those are not it.IN fact calculus was born out of that problem not the other way around."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 8:13:12 AMPosted: 5 years agoAt 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?The Fool:This is the answer to your version:So let's say Achilles runs 10 M per second, and Tortoise runs as 1 M per second, then at then at exactly 11 seconds Achilles will have ran 110 m and the tortoise will have ran 110. And that is the precise moment Achilles will pass the tortoise.Q.E.D."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 8:18:38 AMPosted: 5 years agoAt 7/22/2012 8:13:12 AM, The_Fool_on_the_hill wrote:At 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?Opps correction.The Fool:This is the answer to your version:So let's say Achilles runs 10 M per second, and Tortoise runs as 1 M per second, then at then at exactly 11 seconds Achilles will have ran 110 m and the tortoise will have ran 11. So by 12 second achilles will have ran passed the Tortoise. because he will be at 20 M from the Tortoises beginning and the Tortoise will be only at 12 M."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 7,102 Add as FriendChallenge to a DebateSend a Message 7/22/2012 9:52:31 AMPosted: 5 years agoAt 7/21/2012 2:49:56 PM, Kinesis wrote:It's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.Brilliant, but the story indicates a constant speed. In the model you presented, he'd be slowing down significantly with each step or unit of measure or whatever. From sprinting to crawling in a matter of moments, so that by the time he's halfway to the finish line, he wouldn't seem as though he's moving at all.That doesn't seem very likely. He'd actually start decelerating at a rather high rate 20 measures in.Well, in any case, the point is that you're comparing it to the dichotomy paradox, and neither paradox makes sense to me. But, that's because I understand modern physics, while the men that contrived these paradoxes do not. Therefore, they committed a fallacy -- specifically, they assumed that any mathematical extrapolation of a situation automatically denotes the physicality of that situation. That, of course, is untrue. You must travel the first inch before walking down the street, but that first inch is largely irrelevant to walking in general, given most steps are about 12 times that, and a walk down the street actually begins with a step, not an inch.These sort of ideas, though, actually led us down the path of hyperbolic geometry, which does not abide by Euclidean Laws (all lines converge into a point known as an angle), opening the potential for us to explore form without boundaries, such as which the Universe itself seems to express (as it is a hypersphere which, from the perspective of an observer, always rests at the center by perspective, although it has a margin).Here on earth, though, all lines converge to a point and abide by spatial laws. In a 200 meter race, Achilles would have overtaken the tortoise.
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 10:17:19 AMPosted: 5 years agoAt 7/22/2012 9:52:31 AM, Ren wrote:At 7/21/2012 2:49:56 PM, Kinesis wrote:It's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.Brilliant, but the story indicates a constant speed. In the model you presented, he'd be slowing down significantly with each step or unit of measure or whatever. From sprinting to crawling in a matter of moments, so that by the time he's halfway to the finish line, he wouldn't seem as though he's moving at all.That doesn't seem very likely. He'd actually start decelerating at a rather high rate 20 measures in.Well, in any case, the point is that you're comparing it to the dichotomy paradox, and neither paradox makes sense to me. But, that's because I understand modern physics, while the men that contrived these paradoxes do not. Therefore, they committed a fallacy -- specifically, they assumed that any mathematical extrapolation of a situation automatically denotes the physicality of that situation. That, of course, is untrue. You must travel the first inch before walking down the street, but that first inch is largely irrelevant to walking in general, given most steps are about 12 times that, and a walk down the street actually begins with a step, not an inch.These sort of ideas, though, actually led us down the path of hyperbolic geometry, which does not abide by Euclidean Laws (all lines converge into a point known as an angle), opening the potential for us to explore form without boundaries, such as which the Universe itself seems to express (as it is a hypersphere which, from the perspective of an observer, always rests at the center by perspective, although it has a margin).Here on earth, though, all lines converge to a point and abide by spatial laws. In a 200 meter race, Achilles would have overtaken the tortoise.The Fool: you didnt get mine???"The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 7,102 Add as FriendChallenge to a DebateSend a Message 7/22/2012 10:22:14 AMPosted: 5 years agoAt 7/22/2012 10:17:19 AM, The_Fool_on_the_hill wrote:At 7/22/2012 9:52:31 AM, Ren wrote:At 7/21/2012 2:49:56 PM, Kinesis wrote:It's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.Brilliant, but the story indicates a constant speed. In the model you presented, he'd be slowing down significantly with each step or unit of measure or whatever. From sprinting to crawling in a matter of moments, so that by the time he's halfway to the finish line, he wouldn't seem as though he's moving at all.That doesn't seem very likely. He'd actually start decelerating at a rather high rate 20 measures in.Well, in any case, the point is that you're comparing it to the dichotomy paradox, and neither paradox makes sense to me. But, that's because I understand modern physics, while the men that contrived these paradoxes do not. Therefore, they committed a fallacy -- specifically, they assumed that any mathematical extrapolation of a situation automatically denotes the physicality of that situation. That, of course, is untrue. You must travel the first inch before walking down the street, but that first inch is largely irrelevant to walking in general, given most steps are about 12 times that, and a walk down the street actually begins with a step, not an inch.These sort of ideas, though, actually led us down the path of hyperbolic geometry, which does not abide by Euclidean Laws (all lines converge into a point known as an angle), opening the potential for us to explore form without boundaries, such as which the Universe itself seems to express (as it is a hypersphere which, from the perspective of an observer, always rests at the center by perspective, although it has a margin).Here on earth, though, all lines converge to a point and abide by spatial laws. In a 200 meter race, Achilles would have overtaken the tortoise.The Fool: you didnt get mine???Lol, I saw it, but had nothing to say, because you're right. ^_^I've been crying about being able to edit our own posts forever, but I would settle for +1/-1 buttons.
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 10:25:36 AMPosted: 5 years agoThe Fool:The fallacy in the original version is caused by defining achilles progress in less then complete Units, so since the distance of the line is a complete unit. He could never complete it. IN the orginal case he progressed in have units."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/22/2012 10:29:08 AMPosted: 5 years agoAt 7/22/2012 10:22:14 AM, Ren wrote:At 7/22/2012 10:17:19 AM, The_Fool_on_the_hill wrote:At 7/22/2012 9:52:31 AM, Ren wrote:At 7/21/2012 2:49:56 PM, Kinesis wrote:It's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.Brilliant, but the story indicates a constant speed. In the model you presented, he'd be slowing down significantly with each step or unit of measure or whatever. From sprinting to crawling in a matter of moments, so that by the time he's halfway to the finish line, he wouldn't seem as though he's moving at all.That doesn't seem very likely. He'd actually start decelerating at a rather high rate 20 measures in.Well, in any case, the point is that you're comparing it to the dichotomy paradox, and neither paradox makes sense to me. But, that's because I understand modern physics, while the men that contrived these paradoxes do not. Therefore, they committed a fallacy -- specifically, they assumed that any mathematical extrapolation of a situation automatically denotes the physicality of that situation. That, of course, is untrue. You must travel the first inch before walking down the street, but that first inch is largely irrelevant to walking in general, given most steps are about 12 times that, and a walk down the street actually begins with a step, not an inch.These sort of ideas, though, actually led us down the path of hyperbolic geometry, which does not abide by Euclidean Laws (all lines converge into a point known as an angle), opening the potential for us to explore form without boundaries, such as which the Universe itself seems to express (as it is a hypersphere which, from the perspective of an observer, always rests at the center by perspective, although it has a margin).Here on earth, though, all lines converge to a point and abide by spatial laws. In a 200 meter race, Achilles would have overtaken the tortoise.The Fool: you didnt get mine???Lol, I saw it, but had nothing to say, because you're right. ^_^I've been crying about being able to edit our own posts forever, but I would settle for +1/-1 buttons.The Fool: you mean to edit anytimem, that would be cool I NEED that.http://www.debate.org...do you want to see a R.I.P arguement against Creationalism"The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 5,316 Add as FriendChallenge to a DebateSend a Message 7/22/2012 6:33:57 PMPosted: 5 years agoAt 7/21/2012 2:49:56 PM, Kinesis wrote:It's resolvable with modern mathematics. A convergent infinite series is a series which tends towards a finite amount.Consider: 1 + 1/2 + 1/4 + 1/8 ...It's provable that this sum is equivalent to 2. Denote the infinite series by X. Now, divide X by 2. We have 1/2 + 1/4 + 1/8 + 1/16 ...X - X/2 = 1 (because every number apart from the 1 will cancel out). So X = 2.Therefore, Zeno's infinite series converges on a finite number, and can be identified with a finite number.Mathematically, this is superior to the Fool, because he is using mathematical induction in a specific case then not extrapolating any rule, whilst this actually starts from a rule. The idea though is to be able to plot a line without knowing the gradient of the line, but knowing another line's gradient is less and has a 100m headstart.Let Achilles = RHS, Tortoise = LHS.At t=t, RHS = ax ; LHS = bx+100, where b [is directly proportional to] a.At t=0, RHS = 0 ; LHS = 100.These are the facts we know from the basics already told to us. From here we could plot a complex [meaning difficult] line graph on an area graph, but that is pointless. So:b [is directly proportional to] x.Therefore, b = ka, where k is a constant.RHS = ax ; LHS = kax+100.kax < ax.Thus, RHS is not parallel with LHS.Thus, the two lines will certainly cross when plotted on a graph.To work out the exact point based on the variables (other than x):k must be the y1/c (where c is the y-intercept, and y1 is the distance the tortoise travels when Achilles = y1 from starting point. For example, if the tortoise travels half as fast as Achilles and starts at 100m, it travels to 150m forwards, and so it has travelled 50m). This must be known.either a or b must be known to work out the proportions.c must be known, and y1 must be known if b is not known.I'm doing this all in my head, so I may be wrong with these "must be knowns", but the rest looks right to me.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...
 Posts: 5,316 Add as FriendChallenge to a DebateSend a Message 7/22/2012 6:35:13 PMPosted: 5 years agoAnd the points of the "must known variables" is because with these excluded, the problem is impossible: indeterminacies means it is not solvable. Which is where calculus helps us make up variables.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...
 Posts: 169 Add as FriendChallenge to a DebateSend a Message 7/22/2012 7:32:09 PMPosted: 5 years agoAt 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?I always thought it was because there was no constant to measure things against. Both are measured in comparison to one another and are relative and this is why it's a paradox.If you were to measure them both by the same point and not against each other it would be easier to solve right? Hare travels X distance over time T and tortoise travels Y distance over time T.Is that wrong?Things that make me happy! : At 6/22/2012 1:46:11 PM, Kinesis wrote: : Also, as an Englishman I'm obligated to be prejudiced against gingers and the French. : At 8/27/2012 10:00:07 PM, FREEDO wrote: : Every self-respecting philosopher needs to smoke a pipe.
 Posts: 5,316 Add as FriendChallenge to a DebateSend a Message 7/23/2012 12:33:12 PMPosted: 5 years agoAt 7/22/2012 7:32:09 PM, jedipengiun wrote:At 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?I always thought it was because there was no constant to measure things against. Both are measured in comparison to one another and are relative and this is why it's a paradox.If you were to measure them both by the same point and not against each other it would be easier to solve right? Hare travels X distance over time T and tortoise travels Y distance over time T.Is that wrong?The problem was, at the time, we didn't have calculus to solve the problem: this is why Pythagoras essentially laid the foundations solved it.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...
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/23/2012 3:59:15 PMPosted: 5 years agoAt 7/23/2012 12:33:12 PM, Stephen_Hawkins wrote:At 7/22/2012 7:32:09 PM, jedipengiun wrote:At 7/21/2012 2:17:36 PM, Lordknukle wrote:"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"As far as I can tell, the solution to this is that Achilles reaches each spot marginally faster than the hare, and therefore over an infinite amount of spots, will eventually overtake the hare.Thoughts?I always thought it was because there was no constant to measure things against. Both are measured in comparison to one another and are relative and this is why it's a paradox.If you were to measure them both by the same point and not against each other it would be easier to solve right? Hare travels X distance over time T and tortoise travels Y distance over time T.Is that wrong?The problem was, at the time, we didn't have calculus to solve the problem: this is why Pythagoras essentially laid the foundations solved it.The Fool: That is where calculas principles come from. Calculas doesn't solve the problem. Its much more simple then you think. It because Achilles progressive step are define in less then a whole step. Aka less then a unit. But the distance. Lets say a line is a whole unit. So every step is less then a complete unit of measurement . Aka A 0.5*1 unit. each step is half the last one and then again half that last one.... Etc Getting smaller and smaller. but never completing. the whole of the Race. Because each step is less then 1."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 7,102 Add as FriendChallenge to a DebateSend a Message 7/23/2012 4:08:14 PMPosted: 5 years agoAt 7/23/2012 12:33:12 PM, Stephen_Hawkins wrote:The problem was, at the time, we didn't have calculus to solve the problem: this is why Pythagoras essentially laid the foundations solved it.Pythagoras laid the foundation for trig, not alg/calc.a^2 + b^2 = c^2
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 7/23/2012 4:30:37 PMPosted: 5 years agoAt 7/23/2012 4:08:14 PM, Ren wrote:At 7/23/2012 12:33:12 PM, Stephen_Hawkins wrote:The problem was, at the time, we didn't have calculus to solve the problem: this is why Pythagoras essentially laid the foundations solved it.Pythagoras laid the foundation for trig, not alg/calc.a^2 + b^2 = c^2<(XD) I know."The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL
 Posts: 42 Add as FriendChallenge to a DebateSend a Message 7/24/2012 6:09:58 PMPosted: 5 years agoIt takes Achilles less time to cross each smaller subinterval, so the actual time it takes him to reach the tortoise is a finite number. For instance, if he has to cross 50 meters and it takes him 1 min to cross 25 meters, then it takes him 30 seconds to cross an additional 12.5 meters, 15 seconds to cross an additional 6.25 meters, and so on. He'll cross the entire distance in 2 minutes. Just take lim k -> inf SUM (1/2)^n for n = 0 to n = k, which is 2.
 Posts: 5,316 Add as FriendChallenge to a DebateSend a Message 7/24/2012 6:47:36 PMPosted: 5 years agoAt 7/23/2012 4:08:14 PM, Ren wrote:At 7/23/2012 12:33:12 PM, Stephen_Hawkins wrote:The problem was, at the time, we didn't have calculus to solve the problem: this is why Pythagoras essentially laid the foundations solved it.Pythagoras laid the foundation for trig, not alg/calc.a^2 + b^2 = c^2Bleh, I meant variables in mathematics i.e. nomials. I think. I might just say mathematics instead.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...
 Posts: 61 Add as FriendChallenge to a DebateSend a Message 7/26/2012 3:16:36 PMPosted: 5 years agoThe obvious answer to this paradox is that the scenario could never occur because it is fundamentally impossible to run a race.Consider this: to run a race it is necessary to get from point A to point B. To get from point A to point B it is necessary to run past the midway point. It is also necessary to run past the the midway point between point A and the midway point and again between point A and the midway point of the midway point, etc. In fact, there is an infinite number of midway points, and yet it is impossible to run past an infinite number of points. It is impossible to run a race!!"I would die at the stake rather than change a semi-colon!"