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Topic: Zeno's paradox (Read 8400 times) |
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Kozo Morimoto
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Can the paradox be explained by the fact that the observations are looking at a constantly shorter time step. So the ultimately, the time between the previous observation and the current one becomes zero so no one seems to be moving, so the turtle does not seem to be overtaken. But if the observations were made at a certain constant time step, this no longer becomes a paradox?
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-D-
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Correct. what the greeks never figured out was Calculus. There is a concept of change over time such that each time unit is a constant amount which you can make infitesimally small. The other flaw that makes the paradox really fail is that you're adding up all the time units starting with 1, 1/2, 1/4, etc or even smaller than that. Such series don't add up to the sum of all time (ie infinite time) instead they converge on a number of finite amount of time. So what really happens is that for a certain amount of time, Achilles can not beat the tortoise which is certainly true if it had a head start.
I heard from a physics teacher who may as well have been pulling our legs that the Greeks pretty much proved to themselves that motion was an illusion. Take for example a javelin being thrown. At every instant of time the Javelin can not move. And since time is just an accumulation of all the instances, how could the Javelin move if it cannot move in any instant of time?
And then while they were standing around trying to figure out motion... the romans walked over them and built their own empire.
-D-
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Ryan
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The javelin thing is actually another of Zeno's paradoxes, but with an arrow.
Depending on which philosophers you read and if they are pre or post-Socratic, you'll get varying ideas. Plato held that all sensory experiences are illusions and not reality. Aristotle was the opposite. It's very interesting stuff (to me at least)...but I guess this ain't exactly a Greek phil board.
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Alex
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The problem with that system of reasoning is that a particular "instance" of time holds no velocity or force information. It is like a still picture in which both are excluded from consideration. This would be similar to saying since x=y, z does not exist.
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Chris
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on Jul 29th, 2002, 9:21pm, -D- wrote:
I heard from a physics teacher who may as well have been pulling our legs that the Greeks pretty much proved to themselves that motion was an illusion...
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One of my physics teachers explained the difference between engineers and physicists using a similar arguement.
A boy and a girl stand at the opposing goal lines of a football field. They advance towards each other, together closing exactly half the distance between them and then stopping. They continue advancing towards each other in this manner.
The physicist notes that, using this model of movement, the boy and the girl will theoretically never meet.
The engineer notes that while this is theoretically true, they will get close enough to each other for all practical purposes!
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Icarus
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Re: Zeno's paradox
« Reply #5 on: Oct 26th, 2002, 8:35am » |
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But the engineer is wrong! As long as this "stopping after halfway" approach continues (assuming a relatively fixed length of time per stop), they will never reach each other for "practical purposes" either! The truth is, after they get too close, they can no longer discern when "halfway" has been reached, and the movement model breaks down. In fact, the very notion of "position" becomes fuzzy when one looks on small enough of a scale!
Another take on the difference between Engineers, Physicists, and Mathematicians, which I heard in college:
An engineer, physicist, and mathematician are sitting in a house in front of a fire. At one point a spark from the fire sets the rug aflame. The engineer runs to the kitchen, gets a bucket from under the sink, fills it with water, and tosses the water on the rug to put it out. He then refills the bucket and sets it by the fireplace, to be prepared. Later he leaves, and as the other two watch the fire another spark leaps out, setting another portion of the rug aflame. The physicist grabs the bucket, runs to the kitchen, dumps the water down the sink, and puts the bucket away. Having reduced the problem to the previous case, he pulls the bucket back out, and proceeds as the engineer had. Afterwards he too leaves. Awhile later, another spark sets the rug on fire again. The mathematician declares that he has seen two solutions to this problem, so there is not need for him to solve it again, and leaves the house to burn.
Okay, so it's not great humor, but as someone who has been all three of these, I say that there is great deal of truth in this joke, including the failure of all three to put a grating over the fireplace!
One other thing: the solution given in this thread makes implicit assumptions about the nature of time and space. These assumptions are common to standard physics reasoning, but have never been tested (we have no way of testing them -- as yet). In other words, don't be so sure that you are smarter than those ancient greeks! 
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| « Last Edit: Oct 26th, 2002, 9:19pm by Icarus » |
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Taesong
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Re: Zeno's paradox
« Reply #6 on: Oct 29th, 2002, 6:57pm » |
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Actually, over time I think I have finally begun to understand why this ever was a paradox in the first place (or at least why the greeks thought it was a paradox). I think it has mostly to do with their rejection of Zero and Infinity as numbers (an idea I am sympathetic to if don't wholely agree with) in fact I suspect Zeno would believe that either neither are numbers or both are. I am sure Zeno is quite aware that you can outrun a turtle, rather I think this example is merely supposed to be showing a problem with classic geometry applied to the real world.
Basically, I think the problem he is trying to point out is, if you really move through every "point" in space between your start point and end point then you must spend no time at any point! In other words you end up with weird infinity/0 equations that don't work. Now of course we have come a long way and I doubt Zeno would reject calculus, in fact he may think it proved him right!!!
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Squander Two
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Some physicists claim (and I agree with them) that quantum mechanics proved Zeno right. Zeno was showing that something was up with the idea of infinite divisibility when applied to the physical world. Quantum physics showed that, in fact, you can't move an arbitrarily small distance: there is a minimum distance, although it's so small that movement looks continuous to us. When I move from A to B, I move a very large but finite number of very small distances inbetween, not an infinite number of infinitely small distances. This fact resolves Zeno's paradox.
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towr
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Re: Zeno's paradox
« Reply #8 on: Feb 26th, 2003, 8:10am » |
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hmm.. I never heard of distance-quanta..
Would be weird to have quanticized space, moving in one direction would go faster than in another (or be impossible)..
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Icarus
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Re: Zeno's paradox
« Reply #9 on: Feb 26th, 2003, 7:15pm » |
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Quantized distance and quantized time have both been suggested, but there exists no real evidence for either. Presumably the quantum of distance would be the Planck length, which is ~1.6x10-35 meters. The Planck time is the corresponding time interval (conversion factor = speed of light) is ~5x10-44 seconds. The Planck length is given by:
(The planck length is also defined by some using h instead of h-bar. In this case it would be ~4x10-35. The reason there are two definitions is that the length is not theoretically predicted by anything. It's just what you can come up with by playing around with the fundamental constants of nature. Since there are two versions of Planck's constant, there are two definitions for the Planck length.)
But quantized distance and time are not necessary to handle Zero's paradox. In fact, the paradox only exists when you quantize one without quantizing the other! This is exactly what Zero did - his paradox treats time in a quantized fashion without treating distance the same. It is this differing treatment that leads to the conclusion that nothing can move.
If you quantize both, or treat both as continuous, there is no paradox.
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| « Last Edit: Feb 26th, 2003, 7:22pm by Icarus » |
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