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Originally posted by selfAdjoint
Right.
I think Zeno is assuming space is infinitely divisible here, and time is atomic. So the length of time it takes to do something is the sum of the time-atoms consumed. But since space is continuous, you can form the segments 1/2, 1/4, 1/8, ... which form an infinite set, and if crossing each segment takes one irreducible atom of time, there will be an infinite number of them, which is absurd.
One unit of space per one unit of time is also the slowest speed that one can go (without stopping), a paradox in itself.Originally posted by quartodeciman
space discrete; time discrete ->
stadion paradox: A, B and C are blocks of space, like train carriages. Block A moves rightward one unit of space per unit of time with respect to B. Block C moves leftward one unit of space per unit of time with respect to B. The speeds of A and C with respect to B are maximal values, since nothing can move faster than one unit of space per unit of time (otherwise, a minimum unit of time can be subdivided). But A moves rightward two units of space per unit time with respect to C. C moves leftward two units of space per unit time with respect to A. That is paradoxical.
Don't agree with this. It assumes the existence of 'moments,' which contradicts the premise. I feel the arrow paradox is in the 'discrete space and discrete time' category. They have to be same at all times, either both discrete or neither, since they are so inextricably bound up in each other. (If there are no 'moments' in time then there are no precise positions in space. Therefore to assert otherwise is illogical rather than paradoxical).space discrete; time continuous ->
arrow paradox: An arrow goes from position A to position B. Therefore it has a discrete set of space positions between A and B. For every moment of time that the arrow is in transit, the arrow is at one position. So at every moment of time the arrow is stationary. But the arrow moves from A to B. That is paradoxical.
But there is no point between A and B. As above this is illogical rather than paradoxical (as written).space continuous; time discrete ->
dichotomy paradox: To move from point A to point B it is necessary to reach the halfway point between A and B at some time. To move from position A to the halfway point to B it is necessary to reach the quarterway point between A and B at some time. Each arrival requirement has an additional arrival requirement of half the previous distance. But there are only a discrete sequence of times. So, a mover must be at two or more positions at some minimal unit of time. That is paradoxical.
This is the paradox, but it is not a paradox because it assumes space and time are continuous. If you think about it assumes that they are not, and that's why it's a paradox. (If spacetime is continuous then there are no precise positions or moments and there is no paradox). The Achilles paradox is really a reduction ad absurdamof the argument for quantised spacetime.space continuous; time continuous ->
Achilles paradox: A tortoise runs a race with Achilles, who can run ten times as fast as the tortoise. But the tortoise gets a handicap of some distance ahead of Achilles for the starting point. Both runners start at the same moment of time. When Achilles arrives at the starting point of the tortoise, the tortoise has arrived at a point one tenth of that distance. When Achilles has run that distance, then the tortoise has arrived one tenth of the one tenth distance ahead of Achilles. No matter how long Achilles overtakes the former positions of the tortoise, the tortoise is still ahead of him. So faster Achilles never does overtake the tortoise. That is paradoxical.
I'm not sure I agree with that, since we assume that logical contradictions are impossible in physical reality, but no matter.Note that "paradox" doesn't necessarily mean "impossible". "Paradox" means hard to accept. [/B]
I don't think it matters what you call them. (If you do a search on Peter Lynds you'll find a couple of physics papers in which he argues 'instants' are an impossibility).Originally posted by quartodeciman
arrow paradox:
Canute: It assumes the existence of 'moments,' which contradicts the premise.
If 'moment' means a duration, then I will reject the use of that word. Maybe "instant" would be acceptable.
I see your point. But a paradox is a self-contradiction. If it models reality then unless reality itself is self-contradictory then the situation it describes is impossible, and there must be something wrong with the premisses underlying the paradox.Note that "paradox" doesn't necessarily mean "impossible". "Paradox" means hard to accept.
I like this, because "paradox" is antonymous to "orthodox". The latter means "straight opinion", so the former means "un-straight opinion". If needed, one can use "antinomy", which means "counter rule". [/B]
I can't think of any exceptions to this. Don't all paradoxes arise from faulty assumptions?Originally posted by Hurkyl
A lot of times, a paradox is merely someone making an unwarranted assumption which leads to a contradiction.
Originally posted by quartodeciman
It is best to consider together four famous paradoxes by Zeno of Elia. Someone later discovered that each one generates a different puzzle, depending on whether space is taken as discrete or continuous and time is taken as discrete or continuous.
space discrete; time discrete ->
stadion paradox: A, B and C are blocks of space, like train carriages. Block A moves rightward one unit of space per unit of time with respect to B. Block C moves leftward one unit of space per unit of time with respect to B. The speeds of A and C with respect to B are maximal values, since nothing can move faster than one unit of space per unit of time (otherwise, a minimum unit of time can be subdivided). But A moves rightward two units of space per unit time with respect to C. C moves leftward two units of space per unit time with respect to A. That is paradoxical.This assumes that time and space are granular (not just discrete), and that motion is continuous, a combination that only works if time consists of one instant, and space consists of one point.
space discrete; time continuous ->
arrow paradox: An arrow goes from position A to position B. Therefore it has a discrete set of space positions between A and B. For every moment of time that the arrow is in transit, the arrow is at one position. So at every moment of time the arrow is stationary. But the arrow moves from A to B. That is paradoxical.
First off an arrow has positive size, so there are many positions that it can occupy simultaneously, so it's not entirely clear that the position of the arrow is well defined.
Let's say, that we have an idealized arrow, and that the arrow's position is indeed always well-defined, and that space is granular and that time is continuous. Then you do have a series of intervals in which the arrow is at rest. However, the velocity of the arrow could still be a hidden variable that is not apparent during that interval. This leads to a qm-like situation where we know where it is, but we only have bounds on it's velocity.
space continuous; time discrete ->
dichotomy paradox: To move from point A to point B it is necessary to reach the halfway point between A and B at some time. To move from position A to the halfway point to B it is necessary to reach the quarterway point between A and B at some time. Each arrival requirement has an additional arrival requirement of half the previous distance. But there are only a discrete sequence of times. So, a mover must be at two or more positions at some minimal unit of time. That is paradoxical.
Once again, this assumes that time is granular so that there is a minimum time interval. This particular paradox assumes that the location of a moving object can be well-defined in a granular time system.
space continuous; time continuous ->
Achilles paradox: A tortoise runs a race weith Achilles, who can run ten times as fast as the tortoise. But the tortoise gets a handicap of some distance ahead of Achilles for the starting point. Both runners start at the same moment of time. When Achilles arrives at the starting point of the tortoise, the tortoise has arrived at a point one tenth of that distance. When Achilles has run that distance, then the tortoise has arrived one tenth of the one tenth distance ahead of Achilles. No matter how long Achilles overtakes the former positions of the tortoise, the tortoise is still ahead of him. So faster Achilles never does overtake the tortoise. That is paradoxical.
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Note that "paradox" doesn't necessarily mean "impossible". "Paradox" means hard to accept.
This last one is a red herring. "Never" refers to the inability to complete the sum, not to the amount of time being infinite.
Originally posted by Canute
I think that Zeno was well aware that the infinite series 1/2 + 1/4 + 1/8 ... = 1. It doesn't solve the paradox because it would take an infinite time to do the addition.
What do you think those difficult principles are?I suspect the paradoxes of Zeno are resolvable by acceptance of difficult principles, rather than by their rejection as being nothing but antinomies.
Originally posted by Canute
Ah, but that's cheating. Zeno is arguing that if spacetime is quantised it is logically impossible for us to travel at one metre per second. You've just assumed that we can, and thus avoided the problem.
I agree.(edit - I mean I agree that's a good way of putting it. You have a knack of putting it straightforwardly).Originally posted by quartodeciman
Canute:
I reckon that a postulation of a minimum spatial quantum means that all one can say is that something is AT some quantum position, so it is true that it stays AT one place until it DOESN'T(!)
Still the same subject though.I'm afraid I have stretched this topic thread too far from its original point, and I apologize for that. [/B]