## Friday, February 22, 2008

### Where the movement occurs?

Continuing the discussion on Zeno paradoxes, lets examine the moving arrow paradox. In the case you missed the two previous posts about Zeno, you can read about another two of his paradox here and here.

In this paradox the main problem is to explain when things move. It seems a strange thing to ask, but it is hard to give any answer to this one. We don't usually see any problem with our perception of time and space. But what will happen if we will start to analyze the world around us from a logical position? The results are often surprising.

Photo by Odalaigh

The paradox arises from the following line of thought: Lets take an arrow that is flying, and look on it in an infinitesimal time. More precisely, lets look on it in a point in time. In such a short time "period" (not even a period in fact, but a mere point in time), the arrow clearly doesn't move at all. The time is made from such points. Now, since the arrow doesn't move in anyone of these points, how can it be that it moves at all? It must remain at rest - and thus time and motion are illusions.

The explanation to this is very simple. In order for this paradox to work time must be infinitely divisible. But this is not true. There exists a basic unit, called Planck time. I already wrote about this in the first post about Zeno paradoxes, so I am not going to repeat myself here. What is interesting about this answer however is that it shows that Zeno paradoxes touch the very structure of our world. As such they are valuable, because they show us were we live.

The most interesting property of these paradoxes is that they show how easily we can deduct amazing facts about our world from pure logic. Zeno lived a long time ago and yet managed to create paradoxes that were only fully resolved scientifically by showing that our world is discrete. This was done only very recently. I believe that this is a great lesson both about logic and about science.