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## Question

I can't belive I didn't find this one here:

Achilles and the Tortoise is one of Zeno’s paradoxes. It appears to demonstrate that even the fastest of runners, Achilles, could not catch the slowest of creatures, the tortoise, if the tortoise were given a head start.

To get from one place to another takes time. If the distance between the two is very small, or the speed of travel is very fast, then it may take a very short amount of time, but it will take some time nevertheless. It is impossible to move from one point to another instantaneously.

To catch someone, you need to cross the distance between you and them, you need to move from where you are to where they are. If there is any distance between you at all, then this will take time.

Suppose that the person that you are trying to catch is moving away from you, then in the time that it takes you to get from where you are to where they are, they will have moved on. If you begin at point A, and they begin at point B, then by the time you reach point B they will be at point C. The person that you are trying to catch will no longer be at the point that you have reached.

To catch them, then, you will need to reach the point that they are now at, you will have to get from point B to point C. Doing so, though, will again take time. Point C is at a distance from point B, and so by the time you have reached point C, your target will have reached point D. This process can be repeated ad infinitum, without you ever catching your target.

Zeno illustrated this with the example of Achilles and the tortoise. If, in a race, the tortoise, who moves slowly, is given a head-start on Achilles, then no matter how quickly Achilles runs he will never catch the tortoise.

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Although the inherent flaw in this is that you are using a moving object as your basis for distance and time.

If you used a stationary object, such as a finish line, it is possible.

If Achilles is at 0.00, the tortoise is at 5.00

The next time stamp the tortoise is at 10.00

Where is Achilles with your logic of half the distance?

Is he halfway to 5.00 or halfway to 10.00?

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zeno considered there to be four possiblities:

space is discrete; time is discrete (discrete meaning there is a smallest unit which cannot be divided further.)

space is continuous, time is continuous (no smallest unit, infinitely divisible- this is his famous tortoise achilles paradox.)

space is discrete, time is continuous.

space is continuous, time is discrete.

he came up with a seeming paradox for each scenario.

personally, i don't really see the paradox. if you cut the amount of time traveled in half, of course the distance traveled will be halved.

but if you let the amount of time traveled be constant, the way we normally view motion, achilles will most certianly pass the tortoise.

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zeno considered there to be four possiblities:

space is discrete; time is discrete (discrete meaning there is a smallest unit which cannot be divided further.)

space is continuous, time is continuous (no smallest unit, infinitely divisible- this is his famous tortoise achilles paradox.)

space is discrete, time is continuous.

space is continuous, time is discrete.

he came up with a seeming paradox for each scenario.

http://www.suitcaseo...ox_Achilles.htm gives a brief glimpse into the depth of the paradox.

personally, i don't really see the paradox. if you cut the amount of time traveled in half, of course the distance traveled will be halved.

but if you let the amount of time traveled be constant, the way we normally view motion, achilles will most certianly pass the tortoise.

Sometimes Greek philosophers get too much credit? But in this case if you think there's nothing in it then you are just reading the paradox without knowing what its subject is (if you thought about your own first paragraph?).

As the link points out clearly, the discussion is not about whether Achilles reaches the tortoise (of course he does so in a finite amount of time, any statement saying "therefore he never catches up" is just for dramatic effect). I think the relativity argument is a bit over the top (plus I don't buy it).

To be honest I don't know what Zeno thought about space/time discreteness/continuity and I would like to know more.

Edited by voider
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no. It is like trying to add up parts of a reoccuring decimal. 1/3 = 0.3333333333333333333333333333333333..... or you could say 0.3 reoccururing. This paradox is just like adding all of the '3's indevisually. In fact if the man ran 3 times faster than the tortoise, it would be EXACTLY that.

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The paradox is also that in X time you close 1/2 the distance.

Therefore you never reach your target because you always only get half as close.

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