Friday, January 27, 2012

Friday Paradox:

Zeno put forward a number of paradoxes. The upshot of many of them is that motion is impossible. This is a pretty startling conclusion, but his reasons seem fairly plausible.

Suppose I want to go out to eat and I decide to walk to the restaurant. Let A be my house and B be the restaurant. In order for me to get from A to B, I must cover half the distance between them. So I must get to point C. But in order for me to get from A to C, I must cover half of that distance. So I must get to point D. But in order for me to get from A to D, I must cover half of that distance. So I must get to point E... you get the idea. The issue is that there are an infinite number of "halves" that I have to traverse and so I'll never be able to get the full distance. How can one ever traverse an infinite number of points? So I can never actually get to the restaurant. Of course, we can generalize this and so it looks like all movement is impossible. So, contrary to the way things appear, nothing is really moving!

This wonderful clip explains another of Zeno's paradoxes (sometimes called "Achilles and the Tortoise") quite well. It has quite a bit in common with the one I describe above and the visual representation of the issue might be helpful.

Now, of course, most of think motion is possible. So the trick is identifying where Zeno's argument went awry.

Good luck and enjoy the weekend!


Tyler Crozier said...

It is an interesting concept but I feel like it is almost just trying to argue with something that we deal with on a daily basis. I can say that we all have the ability to breathe water because we can inhale it but if you try it you'll obviously choke when it gets inhaled. He probably found that his theory doesn't quite work out when he made it to the bathroom on time.

Jesse Steinberg said...

We do indeed make it to the bathroom. But how is this possible given that we have to traverse an infinite number of points? How can one cover that territory? We seemingly do it, but it's not clear how. Zeno's point is that it might actually be an illusion. If we think about movement along this mathematical model it turns out movement is impossible, then maybe we should conclude that our beliefs about movement are actually mistaken. This sounds crazy. There has to be a counter-argument. But what is it?

S. Vetterly said...

Even though there may be an infinite number of points, there's only a finite distance. Zeno's paradox seems to leave out the concept of time and distance. First of all, the infinite number of distances will eventually add up to a finite number: the total distance. Therefore, his idea was wrong that the subject will never reach his destination. The same goes for the concept of time. As with the example of Achilles and the tortoise, as the length of each segment decreases, so does the amount of time it takes Achilles to travel it. It is physically impossible for an object travelling at a certain speed to travel a small distance in the same amount of time as it does to travel a large distance, all things kept constant. Because the segments of distance add up to a finite number, the time it takes to traverse that distance is also a finite number. As a result, Achilles will catch up to the tortoise, and you will get to the restaurant.