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Zeno's Paradox
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Draginvry
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PostPosted: Sun Jan 29, 2006 7:23 am    Post subject: Zeno's Paradox Reply with quote

This one's for you, Jack. A non-religious topic.

Today we will be discussing The Tortoise and Achilles, one of Zeno's classic paradoxi.

wikipedia wrote:
In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.


As you can see, holding to a mindet of infinite measurement means that Tortoise MUST win the race against Achilles, according to Zeno. If we are assuming infinitely divisible distance, then Tortoise will ALWAYS have a headstart against Achilles. Thus, Achilles can never defeat Tortoise in the race, no matter how fast Achilles is.

What does this mean for us? Is there truly an indivisible distance? A distance so small that you couldn't possibly move any smaller distance? A distance of say: a fraction of an electron. Surely if this were the case, then Achilles would win, because he would be covering more distance.

How do you reconcile Zeno's Paradox in a universe with infinitely divisible measurements?

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The Nameless One
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PostPosted: Sun Jan 29, 2006 9:31 pm    Post subject: Reply with quote

Since stuff can actually move from point A to point B, I don't think it's true. The same reason for which you can count without taking an eternity. Infinite parts? I skip them.

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StEvE21
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PostPosted: Sun Jan 29, 2006 9:47 pm    Post subject: Reply with quote

I agree with TNO.
This paradox just doesnt seem to make any sense at all to me.
Ex. 0 is undefined. if you have a distance of 0.0001 metres, that is a small distance, but compared to 0.000001 it is pretty big. you can never reach zero, because everytime the distance gets smaller, you can just keep adding a 0 and moving the decimal over every time. In this paradox, achilles is 100 feet from the tortoise, and then he is one foot away. The distance was divided by 100. However, that does not mean achilles will be 0.01 feet behind the tortoise in the same amount of time it took him to catch up to one foot. Get it?

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Draginvry
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PostPosted: Mon Jan 30, 2006 1:24 am    Post subject: Reply with quote

I think you missed the point. The numbers are irrelevant.

The paradox states that Tortoise gets a head start. Now, in order for Achilles to beat Tortoise, Achilles must FIRST catch up to where Tortoise was. But when Achilles catches up to where Tortoise was, Tortoise will have moved some small distance in the time it took for Achilles to get there. Achilles must now catch up THAT distance, which will take a small amount of time, in which the Tortoise will AGAIN have moved further. Achilles must now make up THAT distance. No matter how many times Achilles catches up that distance, Tortoise will always have moved some minute distance which Achilles must again catch up.

If you break this down infinitely (meaning that reality isn't digital) then Achilles can never truly catch Tortoise, unless Tortoise is sitting completely still.

In conclusion, reality must be composed of finite units.

Again, common sense says that if Achilles ran 100 feet, then he could simply run another 100 feet and end up 98 feet ahead of Tortoise. The paradox states that in order for this to happen, Achilles must first PASS Tortoise, which is where the paradox comes in.

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PostPosted: Mon Jan 30, 2006 2:11 am    Post subject: Reply with quote

Quote:
In conclusion, reality must be composed of finite units.


Exactly. But prepare for a lot of opposition, as I can almost never get away with stating the above quote without being met with endless debate. "Nuh-uh! There are an infinite number of steps between any two points!"

Quite simply, I don't see how there can be. Things wouldn't move at all, unless they were actually teleporting, as it would otherwise take an infinite amount of time for anything to move even the smallest possible distance (which is also infinitely small, in this case). There has to be a "minimum distance" between neighboring locations in three-dimensional space, or things just wouldn't work. At least, to my way of thinking.

Achilles can overtake the turtle, because they will eventually reach a point where the turtle hasn't moved at all in the incredibly small amount of time that it has taken Achilles to catch up from the last point at which they were even. In a universe where infinity exists, I don't see how objects could even move at different speeds in the first place.

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Lebowsk1
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PostPosted: Mon Jan 30, 2006 8:17 am    Post subject: Reply with quote

The problem is, if there are smallest units of space, then (if my understanding of this is correct) another paradox involving movement results, shown by another Zeno thought experiment:

A man fires an arrow at a tree. Now, if we were to comment on the arrow at any time we would observe that it is occupying a certain area of space. In the next moment the arrow is occupying another area of space. And on to the next moment, until the arrow hits the tree.

If this is the case... at which point is the arrow actually moving? At every point it is just occupying different points in space.

Have I understood this properly? I remember being really freaked out by Zeno when I first heard this stuff, but the arrow thought experiment always had more of an effect on me than the tortoise race.

Now everyone knows how the tortoise beat the hare. Wasnt to do with pacing yourself wisely at all, the hare just had an existential crisis and couldnt overtake.

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Atheist
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PostPosted: Mon Jan 30, 2006 8:27 am    Post subject: Reply with quote

The arrow is always moving, it's just at a different place at any particular moment, because a single moment doesn't include the element of time. This indicates that time must be limited in the same way that space appears to be, with a minimum unit of time separating each moment from the last. Otherwise, pretending for a second that there are an infinite number of steps between each moment and the next, how can the arrow exist at a point that's half-way between the two? What's half of infinity? How far has the arrow moved after 200 moments in time, if it takes an infinite number of them to move an infinitely small distance?

I suppose you can't really make assumptions about the nature of space-time, especially not assumptions based on the way a monitor displays pixels. The universe exists and functions in a way that's so very, very far outside our ability to understand, that it's truely is pointless to even consider it. At least, that's my position on the matter. You can try to wrap your head around it all you like, but it won't amount to anything. smile

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Lebowsk1
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PostPosted: Mon Jan 30, 2006 8:37 am    Post subject: Reply with quote

Well we've got to function in the world so I don't think it's a waste of time to try to at least get some kind of grasp of what it actually is.

I think the pixels comparison is just about the best we have to go on. You're right, it is insufficient, but it seems pretty good to me for a human way of comprehending something like this. I'm still thinking in terms of the arrow occupying a sequence of spaces, with "movement" being the illusion that is created by this process.

But still, whether 'movement' is real or illusory, the end result is one arrow in tree. ^^

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Draginvry
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PostPosted: Mon Jan 30, 2006 1:45 pm    Post subject: Reply with quote

The arrow paradox shows that time is finite.

I don't put much stock in that paradox, because I believe it is irrelevant. Time and space are the same thing. Thus, if space is finite, then time MUST be.

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Shaper
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PostPosted: Tue Jan 31, 2006 12:13 am    Post subject: Reply with quote

The example of the tortoise and Achilles seems to show (at least to me) that time and space must be finite, because if it weren't then this paradox would actually work in real life......which it doesn't, because obviously Achilles would eventually pass the tortoise.
Zeno has another example of a paradox which is kind of neat; if I were to shoot that arrow at the tree, it would be crossing a finite amount of space, yet I could divide that space in half, then divide those two spaces in half, and so on. I could divide those spaces infinitely, thus the arrow crosses an infinite number of points accross the finite amount of space...now there's a paradox! 8D

So I think this paradox shows us (at least philosophically) that space and time are finite like VeryGnawty said.

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StEvE21
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PostPosted: Tue Jan 31, 2006 7:01 am    Post subject: Reply with quote

*StEvE21's brain is beginning to hurt*

For the 1st paradox, are the speeds of the tortoise and Achilles considered at all?

What did einstein have to say about this subject?

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Draginvry
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PostPosted: Tue Jan 31, 2006 8:36 am    Post subject: Reply with quote

StEvE21 wrote:


For the 1st paradox, are the speeds of the tortoise and Achilles considered at all?


Like I said, the speeds are irrelevant. Achilles could be moving at the speed of light, and the paradox still holds, as long as Tortoise isn't sitting completely still.

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lispozord
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PostPosted: Tue Jan 31, 2006 10:23 am    Post subject: Reply with quote

With our current knowledge it isn't a paradox at all. It is possible to "add" an infinite amount of numbers and get a finite result.

So, we have Achilles running to point occupied by the turtle (which takes him some finite amount of time) and the turtle moves, so Achilles must run another distance to catch up. If we add up the amounts of time Achilles spend covering each consecutive distance we'll get a converging series, which sum is the total amount of time Achilles spent chasing the turtle. So, the whole paradox is just a weird way to look at things smile

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archdreamer
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PostPosted: Wed Feb 01, 2006 4:27 pm    Post subject: Reply with quote

lispozord hit it right on the 'noggin:

Let Achilles Begin at point 0, and the turtle at point 0.5

Let the speed of Achillies be twice that of the turtle.

So, as Achilles runs to point 0.5, the turtle advances to point 0.75, then as Achilles runs to point 0.75, the turtle advances to point 0.875. As Achilles runs to point 0.875, the turtle advances to point 0.9375.

If you carry this out an infinite number of times, they will eventually meet at point 1, as the series carried out by Achilles is a infinite convergent series that converges at 1, and the one carried out simultaneously by the turtle is also an infinite convergent series that converges at 1. Thus they meet at 1, then Achilles proceeds to overtake the turtle.

EDIT: ARGH, i put 2 instead of one.. sorry everyone. (i corrected it.)

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Last edited by archdreamer on Thu Feb 02, 2006 3:00 am; edited 1 time in total
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Lebowsk1
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PostPosted: Wed Feb 01, 2006 5:31 pm    Post subject: Reply with quote

Does Zeno's arrow thought experiment refer to a different kind of problem than the Achiles vs Tortoise one? I'm thinking along the lines of the arrow addressing the actual nature of movement itself while the race one tries to address the nature of relative movement (between different moving bodies)?

With regard the tortoise one, the above explanation seems good to me.

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