Math Pages Blog

God used beautiful mathematics in creating the world.

Wednesday, February 13, 2008

Counting the points

In the previous post, I wrote about an ancient paradox. The author of this paradox, Zeno of Elea, presented 7 paradoxes. In this post I will focus on another of his paradoxes.

This paradox arises from the following situation. Suppose you need to go to a shop to buy some milk. You live very near to the shop so you decide to take a walk. Lets suppose the shop is 100 meters away.

Photo by lime*monkey

If you read my previous post, you probably already guessed that Zeno claimed that it is impossible for you to get to the shop...

Where is the problem? Lets look on it in the following way - in order to get to the shop you need firstly to pass half of the distance to the shop. But even before this you have to pass a quarter of the distance. And before this an eighth of the distance... And so on.
It turns out that in order to get to the shop you need to pass an infinite number of such mid points. Zeno reasoned that this is impossible, because you would need an infinite amount of time for this. Therefore you cannot move at all. No matter how small the distance you move we can divide it in the some way and get an infinite number of mid points again. Conclusion - movement is only an illusion.

There is a simple solution to this paradox. Firstly, if we will sum all the time intervals the sum will be a finite number. This is the accepted solution to the paradox.
But this is not a full solution. In order to sum the time intervals we need to know two things - distance and speed. We know the distance, it is given in the problem. But we don't know the speed. We can decide that the man in the problem has a speed of ten meters per minute. If we would do so, it is trivial to find how much time it would take him to get to the shop:

$%5Cfrac%7B100%7D%7B10%7D=10%20minutes$

And for all the mid points we would get the following infinite series:

5+2.5+1.25+ .... =$%5Cfrac%7B5%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D=10$

As you can see the sum of all the intervals is indeed a finite number. But can we assume that speed exists at all? The whole point of the paradox is to show that motion is only an illusion. By assuming that something called speed exists we assume that the paradox is wrong from the beginning.

What is interesting in this paradox is that it doesn't describe our reality as we know it. There is a hidden assumption in the paradox that the distance can always be divided. But this is not true from a physical prospective. Our world is discrete - there is a basic unit of length called Planck length. Below this length distance is simply meaningless. So in our world we have only a finite number of points between us and the milk shop, so there is no problem to get there in a finite time.

Posted by Anatoly

Labels: math, paradoxes
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