A philisophical Heisenberg and Zeno question

[wolf921]

Hello this is my first post.
I am a 15 year old 10th grade student in a philosophy program and now we are studying the famous Greek philosophers.

I have a problem that was asked by a teacher that I would like to answer. And although I have tried I have not come up with an answer on my own so I could use some help.

The question:
Can you disprove the Philosophers Parmenides and Zeno in your choice of either a logical reasoning way or a mathematical way?

If you don't know Parmenides said that all movement and change is merely an illusion. Zeno fortified his explanation with a few paradoxes. Since I am more interested in math I want to disprove Zeno mathematically.

Zeno's Paradox is this an athlete is attempting to run from the starting line to the finish line. In order to accomplish this, he must first run to the half-way point of the course. And once there, he must run to the 3 quarters point of the race. And after that he must somehow arrive successively at the 7/8ths, 15/16ths,31/32nd... point in succession. Indeed, this poor athlete must traverse an infinity of points just to finish the race. No matter how close our oily friend is to the finish line, he still has to run half the distance remaining before he can finish the race. Big problem--an infinity of points to traverse. But turning the problem around leads to the difficulty that our racer can't even start the race. In order to get to the 50 percent mark, he must first get to the 25 percent. And to get there he must manage to crawl to the 12.5 percent mark. And so on. To move the first angstrom, he must somehow be able to move the first half an angstrom.

I don't know how to disprove this. I could take the easy way and logically disprove them but this is interesting to me.

My first thought was to find if there is any length that is indivisible. Well not theoreticaly. But may be in the real world. Can anyone help me out?

The del X of Heisenberg's uncertainty principle:
After doing some research I found this theory (that I have never herd of) which after trying to learn about just confuses me. I am not stupid (well maybe compared to the people on the web site) I understand how atoms and sub atomic particles work, or i thought i did until I read this...

If some one could explain to me how to disprove Zeno it would be a great help. I would like if at all possible to understand the Heisenberg theory if it would be helpful to disprove Zeno. If there are any other more obvious ways to do it please tell me. The thing is that Zeno makes sense, but we all know he is quite wrong.
Thanks in adavance!

 

[Galileo]

Zeno's Paradox is this an athlete is attempting to run from the starting line to the finish line. In order to accomplish this, he must first run to the half-way point of the course. And once there, he must run to the 3 quarters point of the race. And after that he must somehow arrive successively at the 7/8ths, 15/16ths,31/32nd... point in succession. Indeed, this poor athlete must traverse an infinity of points just to finish the race. No matter how close our oily friend is to the finish line, he still has to run half the distance remaining before he can finish the race. Big problem--an infinity of points to traverse. But turning the problem around leads to the difficulty that our racer can't even start the race. In order to get to the 50 percent mark, he must first get to the 25 percent. And to get there he must manage to crawl to the 12.5 percent mark. And so on. To move the first angstrom, he must somehow be able to move the first half an angstrom.

It's really not a problem to traverse an infinite number of points. Actually, having an infinite number of points between any two given (not equal) points fits nicely in our sense of continuity. (We view space as continuous).
However, if I recall correctly, Zeno's problem was that it would take an infinite amount of time to get from A to B, because you have to reach an infinite number of other points in the process. That's where the error lies, taking a sum of an infinite number of terms does not necessarily yield an infinite answer.

I won't get too mathematical here (although it's not that difficult), but suppose Achilles wants to reach a wall 10 meters ahead of him. He starts walking with a constant velocity of 1 m/s. Common sense tells us this should take 10 seconds.
He has to traverse half the distance first (5 m), then half of that (2,5 m) and so on. The time he takes for a given distance is distance/velocity.
So the time he takes is:
$$t=\sum_{n=1}^{\infty}10\left(\frac{1}{2}\right)^n\frac{1}{1}=10\sum_{n=1}^{\infty}\frac{1}{2^n}=10(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...)$$
You may notice that the more terms you add the longer t gets, but that it will never be greater than 10 s.
As a matter of fact, you can get as close to 10 s as you like by adding enough terms.
In any case t cannot be infinite.

 

[Hurkyl]

Also, when you're dealing with an infinite sequence of events, it is no longer necessary for there to be a "first" or a "last" event.

 

[HallsofIvy]

Zeno has the unstated assumption that you cannot pass an infinite number of points in a finite time.

(How did Heisenberg get into the title?)