Zeno's Paradox

[Gonthrax]

Being half computer guy half physisist I like this sort of stuff so I just thought I'd put this out there for anyone who hadn't heard of it.

It is called Zeno's Paradox, Zeno was a famous mathmatition from Elea, a Greek city on the Italian coast. Zeno was well known for posing puzzling paradoxes that seemed impossible to resolve. Zeno's Paradox theorized that movement was impossable. He used the example of Achellies and the hare. Suppose you have a race between Achilles and a tortoise. Now suppose that Achilles runs 10 times as fast as the tortoise and that the tortoise has a 10 meter head start at the beginning of the race. Zeno argued that in such a situation, it would take Achilles an infinite amount of time to catch the tortoise. His argument went as follows:

By the time Achilles runs the 10 meters to the point where the tortoise began, the tortoise will have traveled one meter and will therefore still be one meter ahead of Achilles. Then, by the time Achilles covers a distance ofone meter, the tortoise will have traveled one tenth of a meter and is
still ahead of Achilles. After Achilles travels one tenth of a meter, the tortoise will have traveled 1/100th of a meter. Each time Achilles reaches the previous position of the tortoise, the tortoise has reached another position ahead of Achilles. As long as it takes Achilles some amount of time to traverse the distance between the point where he is and the point where the tortoise is, the tortoise will have time to move slightly beyond that point. No matter how long the race goes on, Achilles will have to move through every point where the tortoise has been before he can pass him. Each time Achilles reaches such a point, the tortoise is at another point. Therefore, Achilles will have to pass through an infinite number of points in order to catch up with the tortoise. If it takes him some time to pass through each one of these points, it will take hiim forever to catch up.

[Gonthrax]

Well I was going to post a PDF showing the work I wrote up disproving it but I can't attatch PDF files, so here is my simplified explination. Oh, and it has been disproven by other people too, so it's not like this is ground breaking or anything
The faulty logic in Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong. For example, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... is equal to 2. This type of series is known as a geometric series. A geometric series is a series that begins with one and then each successive term is found by multiplying the previous term by some fixed amount, say x. For the above series, x is equal to 1/2. Infinite geometric series' are known to converge (sum to a finite number) when the multiplicative factor x is less than one. Both the distance that Achilles travels and the time that elapses before he reaches the tortoise can be expressed as an infinite geometric series with x less than one. So, Achilles traverses an infinite number of "distance intervals" before catching the tortoise, but because the "distance intervals" are decreasing geometrically, the total distance that he traverses before catching the tortoise is not infinite. Similarly, it takes an infinite number of time intervals for Achilles to catch the tortoise, but the sum of these time intervals is a finite amount of time.

[V.S.]

it is a rather poor paradox. First of all, as gonthrax pointed out, an infinite amount of numbers don't have to add up to infinity. Secondly, neither achilles nor the tortoise move at constant speed. At some point achilles would reach the tortoise before the tortoise was able to move

[Gonthrax]

Originally posted by V.S.
it is a rather poor paradox. First of all, as gonthrax pointed out, an infinite amount of numbers don't have to add up to infinity. Secondly, neither achilles nor the tortoise move at constant speed. At some point achilles would reach the tortoise before the tortoise was able to move

Yea, it is pretty faulty, but for some reason it stood for several hundred years :huh:

[Oz]

Makes sense. Man, wanna come and do 2U maths for me? *shudder*. I still have problems integrating.

[Moppie]

http://www.automotiveforums.com/vbulletin/t13962.html

[Gonthrax]

Originally posted by Moppie
http://www.automotiveforums.com/vbulletin/t13962.html

Ahh damnit, you've beaten me to the punch Moppie

[fritz_269]

Originally posted by V.S.
it is a rather poor paradox. First of all, as gonthrax pointed out, an infinite amount of numbers don't have to add up to infinity. Secondly, neither achilles nor the tortoise move at constant speed. At some point achilles would reach the tortoise before the tortoise was able to move

Actually, I think it's a great paradox.

I challenge you to prove that a geometric series converges without using limit theory or calculus (which were invented 2000 years after Zeno). It can be done with just algebra (invented a mere 700 years after Zeno).

The concept of infinity is pretty darn tricky.

[Gonthrax]

Originally posted by fritz_269

Actually, I think it's a great paradox.

I challenge you to prove that a geometric series converges without using limit theory or calculus (which were invented 2000 years after Zeno). It can be done with just algebra (invented a mere 700 years after Zeno).

The concept of infinity is pretty darn tricky.

You make a good point, Zeno didn't really have the mathmatic skills to disprove this paradox (They hadn't been thought of yet ) And sense I beleave that the human mind is not capable of comprehending Infinity then I guess he has an excuse

[ChibiSF]

ahh! Now the concept of change. Is the world one of permanence or one of change? hmm....

[_W_]

It was Newton that resolved this paradox, by inventing a completely new branch of mathematics.