If I drop a ball from the height of 4 feet, before it hits the floor, it must travel half that distance--2 feet. And then it must travel half that distance--1 foot. Then half that distance--6 inches. Then 3 inches and so on and so on.
Technically, it must continue to travel half way on each and every measurement all the way down. From that explanation it must travel an infinite amount of space and should never hit the floor.
Now, I know it will hit the floor, so the flaw must reside in how the original explanation is laid out.
Cansomeone explain it (in simple non-mathematical terms) on where the flaw is? I never could grasp how to explain away the "infinite amount of halfing all the way down."
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
This paradox assumes that space is continuous, so you can keep halving the distance ad infinitum. Current quantum theory suggests that the fabric of space itself may actually consist of indivisible, subatomic distances, so you can only half the distance so many times, thus resolving the paradox. This was also discussed in another mathematics thread, Zeno the Stoic.
Posts: 265 | Location: Denver, Colorado, USA | Registered: 06-04-02
I respect the math involved. But can't one simply say that the space between any two (finite) points contains an infinite amount of points? Thus, you can still get from point A to point B even if you can halve forever.
Minnesota's explanation of the classic Zeno's Paradox is right on the money.
It wasn't until the modern era of mathematics that a rigorous treatment of limits was developed, to deal with (in this case) an infinite series with a finite sum. Such a concept was quite unknown to the ancient thinkers.
Posts: 2055 | Location: U.S. | Registered: 06-03-02
I assume in a real-world situation, gravity would take over enough for the bouncing to stop. Also, think about it this way. Let's assume each foot it travels takes 1/4 second. If you drop it from 4 feet, then it would take one second. The 2 feet back would take 1/2 second. So, this "unit" where it returns to a motionless position in midair is 1 1/2 seconds long. Then, it falls for 1/2 second and comes back up 1 foot for 1/4 seconds. This totals 3/4 seconds, half of 1 1/2. Using Zeno's Postulate/Theorem (?), that means the time unit will get smaller and smaller, but once 3 seconds is up, it would be stopped, because it is equal to/greater than the total of the units of time necessary for the ball to stop in midair.
Posts: 1363 | Location: Lowell, MA, USA | Registered: 06-03-02
Minnesota's answer should, in fact, have ended the whole thing. It was, as the Professor says, right on the money! I'd also like to commend Minnesota for the very neat answer to the question about walking in sand in the physics forum. Again, it was definitive.
I'm a little bit upset with you, SE, for your answer dragging in quantum theory. Analysis ala Cauchy was quite competent to handle the problem, before anyone even heard of Planck or Heisenberg or Born. We don't need lumpy space to account for Achilles overtaking the tortoise. (I'm inclined to think we'd be better off without lumpy space altogether, if we could only figure out how to iron it out).
Posts: 2612 | Location: Upper U.S. | Registered: 06-11-02
You know, me, maiku, I love that twilight zone physics. One thing I really enjoy about this forum is we can present a variety of relevant theories for any question, even those that are somewhat anti-intuitive, and haven't been proven yet. Besides, I feel the question of whether space is ultimately discrete or discontinuous is highly relevant to a discussion of the ancient Greek philosophers, who were really wrestling with some of the same issues about the nature of reality. While the simple mathematical argument does very effectively address the paradox, it doesn't necessarily capture the philosophical nature of Zeno's paradox and other inquiries of the ancient Greeks, who were as much philosophers as mathematicians (back then they were almost inseparable fields).
Posts: 265 | Location: Denver, Colorado, USA | Registered: 06-04-02
