Achilles and the Tortoise: Exploring Zeno's Paradox

[gilboy64]

Could some one please explain the paradox written by Zeno, 'Achilles and the tortoise'? How can this be possable?

 

[cen2y]

Hastly, search google for a more throughly one:
Achilles and a tortoise raises. the tortoise starts ahead of achilles, let's call the distance d. In order for achilles to pass the turtle, he must pass the point the turtle is at now. When achilles has reached the point d, the tortoise has moved another small distance, d2. When achilles has reached d+d2, the tortoise has moved the distance d3, and so on, whenever achilles has come closer to the tortoise, it has moved another small distance and draws the conclusion, achilles will never reacht he trutle.

 

[gilboy64]

I was thinking in the terms of a bullet fired at a wall. You could say that if the bullet has to cover a distants d to reach the wall, that after a certain time T the distance between the wall and the bullet is 1/2d. After another certain time T the distance will be halved again 1/4d, if we asume the speed of the bullet is constant. If you carry on in this way the bullet will never hit the wall because although it is travals at the same speed it started off as, it is travaling through infinatly smaller distances??

Maby I am mad!

 

[Doctordick]

The fundamental issue of Zeno's paradox is very simple and is not quite the issue expressed above. What Zeno is pointing out is that continuity of space is a paradoxical concept. The central issue of infinity is the very fact that (by virtue of the definition of infinity) no matter how many times you step an infinite procedure, you are not finished. Zeno has given a valid procedure for delineating a specific set of points along the path of the race (valid if the path is continuous). His paradox is the fact that the tortoise cannot ever pass through the defined collection of points as they constitute an infinite set (no matter how many it has passed through, it's not finished by definition so how can it possibly finish).

Zeno is not claiming the hare can never pass the tortoise (he was not an idiot); he is merely pointing out a paradox in the mental concept of a continuous path. Furthermore, it is a well known physical fact that one cannot specify the exact position of any real object: to do so would be a direct violation of the uncertainty principal. Essentially, modern physics arrived at exactly the same conclusion (in a slightly different form). What is actually quite astounding is that Zeno perceived the existence of such a problem so long ago.

In modern physics, there cannot possibly exist a proof that any given object actually existed at every point along its path. To perform an examination of such an issue would require an infinite number of measurements and, as Zeno has so clearly pointed out, such an examination cannot be performed. The fact that an object exists along its path (when not being examined) is no more than an assumption convenient to our mathematical view of its behavior.

 

[Iacchus32]

I was thinking in the terms of a bullet fired at a wall. You could say that if the bullet has to cover a distants d to reach the wall, that after a certain time T the distance between the wall and the bullet is 1/2d. After another certain time T the distance will be halved again 1/4d, if we asume the speed of the bullet is constant. If you carry on in this way the bullet will never hit the wall because although it is travals at the same speed it started off as, it is travaling through infinatly smaller distances??

Maby I am mad!

Sounds to me like the only reason why it's a paradox, is because the've neglected to take into account one very important variable, it's velocity which, is not going to become halved everytime it gets half-way closer to the mark. So, does that make it a paradox or, just an incomplete assessment?