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Zeno's paradoxes are easily solved if you substitute pixel-like points for geometric points... All you do is say that instead of being an infinite number of sizeless points in the universe there is a finite number of points with finite size... I've got a few theories based on this that also integrate general and special relativity, but I needn't go into detail on this stuff right now. Heh, I didn't read the whole article but isn't that essentially what it says? That all movement occurs as "jumps"? :)
There is no paradox, Zeno was simply wrong. The only "paradox" is that there are those who don't know where Zeno went wrong.
(I.e. that the sum of the infinite series 1/2 + 1/4 + 1/8 ... = 1)
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.
His paradox stands (imho) unless one assumes that spacetime is a continuum, in which case there is no paradox.
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.
As I've posted before, I think Zeno was "running the cases" of time and space separately being discrete or continuous. His program is obscure to us because our only source for his paradoxes is Aristotle, who scrambled them some.
But if spacetime is continuous there is no 'irreducible atom of time'.
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.
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.
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.
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.
Note that "paradox" doesn't necessarily mean "impossible". "Paradox" means hard to accept.
Nice post. A few comments.
One unit of space per one unit of time is also the slowest speed that one can go (without stopping), a paradox in itself.
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).
But there is no point between A and B. As above this is illogical rather than paradoxical (as written).
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.
I'm not sure I agree with that, since we assume that logical contradictions are impossible in physical reality, but no matter.
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.
"For every instant of time that the arrow is in transit, the arrow is at one position. So at every instant of time the arrow is stationary."
Canute: I feel the arrow paradox is in the 'discrete space and discrete time' category.
The paradox is more interesting in the discrete-space/continuous-time setting. Otherwise, no definite statement about the motion within one single minimum duration of time will be possible. But if there are two instances of time within one space position, that is prima facie a stationary state.
It takes the more complicated stadion situation to handle the discrete-space/discrete-time case.
Canute: They have to be same at all times, either both discrete or neither, since they are so inextricably bound up in each other.
I can't say I understand why (,or what follows).
Canute: But there is no point between A and B.
With space supposed continuous, there are plenty of points between A and B. A geometrical construction can locate the midpoint of any line segment, however small.
I would have expected a complaint that my conclusion to dichotomy is not the usual conclusion, and I would have to agree. The usual conclusion of dichotomy is that the mover can't get started. This would require the assumption of continuous-time. I changed it to match the discrete-time assumption. I consider this case the weakest.
Canute: If you think about it assumes that they are not [continuous], and that's why it's a paradox.
Reckoning the time interval for Achilles to attain the last point of the tortoise requires continuous-space, and itself requires a continuous-time value to keep the recurrence going.
The real answer to Achilles is that it deliberately considers only time-instances and space-points in which Achilles is behind the tortoise, and never considers any time-instances or space-points when Achilles isn't behind the tortoise. It is able to trap us in that range because of continuity.
quartodeciman: 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".
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).
It seems to me that to say that time is continuous is to say that there are no instants, that is you cannot 'reify' instants. I've argued this elsewhere with little success, however it seems obvious. Perhaps I'm missing something.
If time is continuous then all 'instants' are of arbitrary duration and position in time. If space is continuous then all positions are arbitrary, and reduce to mathematical points. Is there any argument against this? I can't find one.
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.
A lot of times, a paradox is merely someone making an unwarranted assumption which leads to a contradiction.
I can't think of any exceptions to this. Don't all paradoxes arise from faulty assumptions?
There are only two points in a race. From point A to point B. You could skew the race by saying there is a halfway point a runner must reach before point B is arrived at. That is the equivalent of running a completely different race, but still no different than running from point A to point B. The point here is that there is no halfway point in any given distance you so choose to move in. If the distance between point A and B is infinitely devisible - No halfway point is possible witout changing to to a finite system. Hence Zenos paradox gets lost in an infinte sea through contradiction ,,,,, over and over again. By placing a finite system over an infinitely devisible one - We contradict the whole premise of point A to point B.
All that is needed here is to allow the race to take place by which point (AB) becomes the whole point of the race. Point A is the runner with a destiny for point B. They meet to complete the (AB) point of the race. Don't excuse the puns - They are intended.
Even if this is incorrect, they are certainly interesting things to discuss.
I use "paradox" in the sense of
Bolzano, Bernhard. Paradoxes of the Infinite.
from the early nineteenth century. The word was used earler than that by Galileo in discussing infinite collections. For example, a small circle and a large circle with a common center have the same number of points on them, though one is obviously longer than the other. The sense that "paradox" means exactly a logically-contradictory or incoherent situation is a more recent usage. I suspect the paradoxes of Zeno are resolvable by acceptance of difficult principles, rather than by their rejection as being nothing but antinomies.
No one has to accept this particular rundown of the Zeno paradoxes. It was a later assessment. the paradoxes stand on their own and it is up to us what lessons can be learned from them. Most of us will not accept the eliatic conclusion intended by Zeno and his master, Parmenides, namely: all is one and cannot be rationally separated into parts, neither spatially nor temporally, nor any other way.
It is a matter of fact that if there are two points A & B separated by one meter, and I travel at 1 meter per second, I can traverse that distance in as little as one second. Not an infinite amount of time by any standard.
Zeno was wrong, anyone can prove him wrong. d = v * t is the controlling equation. Anyone who knows that the infinite series 1/2 + 1/4 +... = 1 (as you postulate Zeno did) would understand this: Regardless of how you split the series, v is constant. Thus t = d/v and the projected time of arrival is easily determined and is finite, in accordance with everyday experience.
Only someone who is distracted by the "sleight of hand" of the infinite series concept would see a paradox. It matters not that space is assumed to be continuous or discrete, that too is a distraction.
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.
What do you think those difficult principles are?
First, I don't agree that spacetime quantized has anything to do with being able to travel at 1 meter/second. A quantized space is traversable too. Logically, the series does not proceed to infinity.
Second, even if I did concede that, I would logically conclude that space is therefore NOT quantized as a result of experiment.
Either way, no problem. But I question whether Zeno was aware of these considerations.
"I suspect the paradoxes of Zeno are resolvable by acceptance of difficult principles, rather than by their rejection as being nothing but antinomies."
Canute, I was hoping you wouldn't ask about this. I will just watch for a while and see what (if anything) dawns on me.
Maybe I should have said:
"I suspect the paradoxes of Zeno are resolvable by acceptance of deeper principles, rather than by their rejection as being nothing but antinomies."
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