I stumbled across this Paradox today.
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The Tortoise challenged the great warrior Achilles to a 100 meter foot race, claiming that he would win as long as Achilles granted him a little headstart. Achilles laughed, for he was a mighty warrior swift of foot, whereas the Tortoise was heavy and slow.
"How long of a head start do you need?" asked Achilles, smiling.
"Ten meters," said the Tortoise.
Achilles laughs. "OK, you will most definitely lose, but we can race if you really want."
"Actually, I will most definitely win, and I can prove it to you with a simple argument," said the Tortoise.
"Go on then," Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
"Suppose," began the Tortoise, "that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?"
"Very quickly," Achilles affirmed.
"And in that time, how far should I have gone, do you think?"
"Perhaps a meter - no more," said Achilles after a moment's thought.
"Very well," replied the Tortoise, "so now there is a meter between us. And you would catch up that distance very quickly?"
"Very quickly indeed!"
"And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?"
"Ye-es," said Achilles slowly.
"And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance," the Tortoise continued smoothly.
Achilles said nothing.
"And so you see, in each moment you must be catching up the distance between us, and yet I - at the same time - will be adding a new distance, however small, for you to catch up again."
"Indeed, it must be so," said Achilles wearily.
"And so you can never catch up," the Tortoise concluded sympathetically.
"You are right, as always," said Achilles sadly - and conceded the race.
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The Tortoise challenged the great warrior Achilles to a 100 meter foot race, claiming that he would win as long as Achilles granted him a little headstart. Achilles laughed, for he was a mighty warrior swift of foot, whereas the Tortoise was heavy and slow.
"How long of a head start do you need?" asked Achilles, smiling.
"Ten meters," said the Tortoise.
Achilles laughs. "OK, you will most definitely lose, but we can race if you really want."
"Actually, I will most definitely win, and I can prove it to you with a simple argument," said the Tortoise.
"Go on then," Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
"Suppose," began the Tortoise, "that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?"
"Very quickly," Achilles affirmed.
"And in that time, how far should I have gone, do you think?"
"Perhaps a meter - no more," said Achilles after a moment's thought.
"Very well," replied the Tortoise, "so now there is a meter between us. And you would catch up that distance very quickly?"
"Very quickly indeed!"
"And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?"
"Ye-es," said Achilles slowly.
"And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance," the Tortoise continued smoothly.
Achilles said nothing.
"And so you see, in each moment you must be catching up the distance between us, and yet I - at the same time - will be adding a new distance, however small, for you to catch up again."
"Indeed, it must be so," said Achilles wearily.
"And so you can never catch up," the Tortoise concluded sympathetically.
"You are right, as always," said Achilles sadly - and conceded the race.
Was it really impossible for Achilles to win the race? Explain.
Note: Story adapted from Douglas Hofstaeder's awesome book, Godel, Escher, Bach.
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Let me try to offer an explanation here. Here are the race settings
Tortoise - Speed 10 m/hr
Achilles - Speed 50 m/hr
And assume that the race ground is a 100 m one.
Which means, the Tortoise can complete it in 10 hours.
Achilles can complete the race in 2 hours.
Then, the tortoise was never right..How can this be termed as a paradox?
What is the point in the original listing above makes this a paradox? According to my knowledge, a paradox is something that is very hard to reason out...
??????
I think the tortoise thought that Achilles will be trying to catch it rather than looking at the race end pole.. :)
Achilles can complete the race in 2 hours.
Then, the tortoise was never right..How can this be termed as a paradox?
What is the point in the original listing above makes this a paradox? According to my knowledge, a paradox is something that is very hard to reason out...
??????
I think the tortoise thought that Achilles will be trying to catch it rather than looking at the race end pole.. :)
hi, i think the paradox is supposed to run like this (if i remember right)
ReplyDelete- the tortoise has a headstart
- in X seconds, achilles can cover half the distance between him and the tortoise
so the paradox is that achilles, even though he can move at a faster rate than the tortoise, can never overtake the tortoise because he only moves half the distance between them on any given turn/snapshot of time.
of course, this doesn't hold true; at some point, achilles is close enough to the tortoise that the distance that the tortoise moves in X seconds is so little that achilles' movement speed is slower than the tortoise's since he only covers half the distance between him and the tortoise.
so the example would be, if the tortoise had a 10m headstart and moved at 1m/s, initially achilles would move at 5m/s, with his speed dropping exponentially.
if they both started at the same point, though, achilles would move at 0.5m/s initially. slower than the tortoise.
the paradox hinges on the idea that achilles' speed is variable based on the distance between the two, while the tortoise's remains constant.
hope this helped identify the paradox.
cheers,
dharma
another way to think of it is like this:
ReplyDelete- assume the tortoise moves at 1m/s and achilles moves at 10m/s
- assume the tortoise has a 20m headstart.
it takes achilles 2 seconds to move 20m, but in these 2 seconds, the tortoise has moved 2m ahead
it takes achilles 1/5s to move 2m, but in 1/5s the tortoise has moved 20cm ahead
and so on, ad infinitum. therefore there is always some distance ahead of achilles that the tortoise has moved by the time achilles catches up to where the tortoise was previously.
again, this is untrue in the real world because in the paradox, achilles' position is relative to the tortoise's timeframe, whereas in the real world, both of them are in positions relative to the same timeframe (the external world)
:) neatly described..
ReplyDeleteIts a very famous Zeno's Paradox.
ReplyDeleteTheoratically we can prove anything. Let us say I have to go from point A to point B.
In some time I'll reach half the way. Then I will cover half the way remaining. So everytime I move I'll be going half the way remaining. Does that mean I can never reach point B???
Think about it...