Zeno's Paradoxes

[amishrockstar]

Why --logically -- are the conclusions of Zeno's paradoxes of motion wrong?

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 feet. 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 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. 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.

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

Thanks

 

[toddpedlar]

It's wrong because it takes a finite amount of time to cover an infinite number of points. The main issue is that the Greeks had no clue when it comes to actually physically describing motion. Zeno's attempts fall flat when one actually understands how objects move from one place to another.

In the case of the tortoise and the hare, eventually as the hare starts to get closer to the tortoise, his speed enables him to cover the distance between his current location and the tortoise's future location in less time than the tortoise can cover the distance between HIS current location and his future location. It's a simple matter from the point of view of physics, and because of the fact that, again, finite distances contain an infinite number of locations, the Greek take on the issue fails. They are treating these infinity of locations AS THOUGH there was a finite distance between each - and I think therein lies the rub.

 

[amishrockstar]

That helps.
Thanks for the post.
; )

 

[SRoper]

Zeno's Paradoxes are more like Zeno's Puzzles--cute problems for high school math students to solve.