An effort to solve Zeno's motion Paradoxes

[altan]

It is downloadable at:


www.yapibozum.net/papers/achilles.doc (http://www.yapibozum.net/papers/achilles.doc)

Regards,

Altan TANRIVERDI

[Sikz]

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"? :)

[DrChinese]

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)

[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.

His paradox stands (imho) unless one assumes that spacetime is a continuum, in which case there is no paradox.

[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.

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.

[Canute]

Originally posted by selfAdjoint
[B]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.
But if spacetime is continuous there is no 'irreducible atom of time'.

[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.

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.

[Canute]

Nice post. A few comments.

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.
One unit of space per one unit of time is also the slowest speed that one can go (without stopping), a paradox in itself.


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.
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 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.
But there is no point between A and B. As above this is illogical rather than paradoxical (as written).

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.
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.

Note that "paradox" doesn't necessarily mean "impossible". "Paradox" means hard to accept. [/B]
I'm not sure I agree with that, since we assume that logical contradictions are impossible in physical reality, but no matter.

[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.

So:
"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).

dichotomy paradox:

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.

Achilles paradox:

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".

[Canute]

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 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.

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 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.

[Hurkyl]

A lot of times, a paradox is merely someone making an unwarranted assumption which leads to a contradiction.

[Canute]

Originally posted by Hurkyl
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?

[UltraPi1]

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.

[NateTG]

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.

Even if this is incorrect, they are certainly interesting things to discuss.


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.[QUOTE]

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.

[QUOTE]
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.

------

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.

[quartodeciman]

Forgive me.

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.

[DrChinese]

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.

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.

[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.

[Canute]

quartodeciman

I suspect the paradoxes of Zeno are resolvable by acceptance of difficult principles, rather than by their rejection as being nothing but antinomies.
What do you think those difficult principles are?

[DrChinese]

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.

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.

[quartodeciman]

"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."

[Canute]

Fair enough. I feel that 'difficult principle' is that spacetime is not quantised. I admit that it's quite hard to make sense of the idea, since it has metaphysical consequences as well as scientific ones. But in its favour that's just what we would expect if it were really true.

[Canute]

Here's an attempted modernisation of Zeno.

Zeno Restated.

Let us assume that spacetime is quantised.

Continuous motion implies that a fundamental quanta of matter is able to move from one point in spacetime to an adjacent one.

Take fundamental particle A at point P1. Let us say that A is an athlete amongst particles. How fast can it go?

Let’s separate time and space. In space A is a fundamental particle at a precise point in space. If it is to move to another position non-instantaneously then it must take time to do so. But how little time?

Let’s say that there are three positions, P1, P2, P3 in a straight line in space, three points on a piece of paper. These are three adjacent fundamental quanta of space (points).

A is at P1. If it wishes to get as fast as possible to P3 then it must take at least two instants. This is because for A to get from P1 to P3 in one instant requires instantaneous travel. If instantaneous travel is possible at the quantum level then all bets are off, we might as well put P1 and P3 on opposite sides of the universe. This means that A can move no faster then one change of point per instant, and only to a spatially adjacent one.

How slow can A go?. If it is to reach P2 at all then it must do so in no less than an instant. If it went slower than this it would have to stay at P1 forever. This means that A cannot go slower then one change of point per instant, and always to an adjacent one.

Thus in a world in which spacetime is quantised everything must move at the same speed, and athletes cannot catch tortoises.

What's wrong here?

[olde drunk]

what if the A exists at all positions P1 through Px and our observation, at Px, is a matter of perception. It has always beem there. Potentially or in reality? i dunno. it may simply appear where it is needed. to be seen? acted upon? collapsed?

TIME does not exist; it is a human necessity to percieve our physical framework and understand the experience. once i remove the limitations of time-space i can understand quantum theory.

IMHO, accepting that all probabilities are valid, i can then focus on one instant and/or an entire thread of probablities. they are linked so that my consciousness can absorb the experience and expand its awareness; the universe expands. again, we (humans) can only observe this instant (present), our total being may be in an infinite number of worlds absorbing their wonders. much like(but on a grander scale) our bodies having one foot in warm water, one foot on ice and our genitals--- errrr, you get the message. needless to say i haven't had time to polish this analogy.

peace,

[quartodeciman]

Travel in a quantized space-time system proceeds by jumps, with 0 duration for each jump. This would naturally be graphed by a step function with vertical lift.
So, your A could stay at P1 for 1, 2, 3 or more time units, then jump to the next space unit and stay there for the same number of time unit counts, and so on. The overall rate of progress from P1 to P2 and on can be made sequentially smaller, yet remain greater than 0.

This also offers a clue, I think, to resolving the arrow paradox in the continuous-time, discrete-space system. The positions sampled at different time instances within one space unit are equal, but that is exactly what we expect for a step function. Samples within an interval of the independent variable do not even try to approximate the jump that actually occurs at the end of the interval. So they are not representatives of the overall motion of an arrow (of length one space unit, if you want) at all.

(* is just for spacing)

^
|
s ************ ---|
p ********* ---|
a ****** ---|
c *** ---|
e ---|

time ->

On the other hand, there is no subdivision of a quantized time unit, so a maximum speed is automatically defined for anything that is confined to occupying a single space unit at a time. Does this perhaps define the limiting speed c? I read that Loop Quantum Gravity theory attempts to structure space and time units, using the Planck length and Planck time as units. It appears that the ratio of PL to PT is c. I don't know whether that theory exploits this.

http://scienceworld.wolfram.com/physics/PlanckLength.html
http://scienceworld.wolfram.com/physics/PlanckTime.html

[Canute]

Olde drunk

That seems to be a possibility. It does at least solve the problem.

Quartodeciman

I specified continuous motion above, so your solution doesn't work. However maybe motion isn't continuous, as you suggest. It seems an insane idea to me but then so do quite a number of current theories of motion. Do you really think that particles proceed in stops and starts? Doesn't it contradict the known laws of momentum and energy conservation?

I didn't quite understand your example.

I agree that c is connected with all this. It's interesting that by the the other view allowed by Galilean relativity photons are are actually at rest, (since time does not pass for a photon). Perhaps c is a limit because at c one is stationary, and you can't go slower than that, if you see what I mean.

[quartodeciman]

Canute:

I'm completely stuck over continuous motion in a discrete space (and/or time) system. Continuity demands that every intermediate value be assumed at some time within. But we otherwise tend to think motion in terms of FROM and TO times. So I stick with discontinuous motion (at the jump instances) and just shut my trap about it.

My example using Planck length and Planck time just sets smallest discrete intervals of distance and duration to values whose ratio just happens to be c. This presumes that jumps more than one space unit are forbidden. Then the maximum average speed looks like c. If jumps of magnitude greater than to an adjoining space unit are allowed, then a higher speed would be attainable. I repeat that I don't know whether this is theoretically significant.

About photons being at rest:
Everybody is at rest in Special Relativity as long as they are inertial and use their own proper measuring standards for reference. So an observer that trips the light fantastic (rides constantly within a photon stream) still has a proper space and time. What is missing is the ability to exchange light signals between this observer and a homie observer (that stays behind) after departure and separation. That is because any light signal must not exceed c. But the speed of the other observer for either observer is c, so no catchup can occur. This renders the relationship of the Lorentz Transformation impossible to apply.

[Canute]

Am I right that in your example (using P-length and P-time) for an object to go at any average speed slower than c entails that it pauses between positions?

I've had a few conversations about Zeno and quite a few people seem to be happy with the notion of intermittent motion. I can't see how it makes sense, but then the world is strange place.

I think I get your last para. Doesn't SR imply that there is no absolute reference for motion, and that therefore there is no absolute backdrop for it, i.e. that spacetime is a concept and not a thing?

[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(!)

More on SR:
It interests me that Einstein's two immediate predecessors in the development of the theory are Lorentz, a true physical theorist, and Poincaré, a true mathematical theorist. For Lorentz, it was physical dynamics that determined the Lorentz transformations and for Poincaré it was the underlying group structure that predominated. Einstein switched to kinematics and a special postulate, lightspeed invariance. As such, one might consider Einstein as presenting an intermediate position on the Lorentz transformation question. But he remained a pure physical theorist for several years, and there was much confusion about any distinction between his position and that of Lorentz. People talked about a Lorentz-Einstein theory back then. Einstein only changed his framework when he got hard to work on a generalization that would include gravitation. Then local Lorentz and general invariances took over his approach to a new theory.

Let's roll back to SR. For pre-relativity theory, it is rest (zero) speed that is uniquely central and neutral; for relativity theory, it is speed c that is uniquely central and neutral. Rest speed is just a particular choice of reference frame and not unique at all.

------

I'm afraid I have stretched this topic thread too far from its original point, and I apologize for that.

[Canute]

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(!)
I agree.(edit - I mean I agree that's a good way of putting it. You have a knack of putting it straightforwardly).

I'm afraid I have stretched this topic thread too far from its original point, and I apologize for that. [/B]
Still the same subject though.

[Canute]

Quarto

In case you're interested try running Zeno's race with two fundamental particles hopping across quanta of spacetime like stepping stones.

Before you know it you get indeterminate positions and differences in relative lengths due to relative velocities. Strangely familiar stuff.

[quartodeciman]

Canute,

I will think about it. Up front, it seems like a reduction to the stadion paradox.

The truth is, I tend to visualize discrete space as a kind of pinball display light sequence, where the sequence is actually of spatial states rather than motion as we normally think of it. In case you concluded that I favor discrete space, I declare my allegiance to continua for both space and time, in spite of quantum theories. But I favor considerations of both pictures. I think the Pythagoreans would not have uncovered irrational relationships if they had not held to a starting philosophy of numerical discreteness and fixed whole number ratios of sides to begin with.

Regards,

[Canute]

Interesting. If you are saying that our phenomenal world must be understood (at an everyday level) in terms of quanta but that ultimately it is a continuum then I agree.

But this leaves a metaphysical question. How do you get around the fact that if spacetime is a continuum then it reduces to nothing at the limit, and cannot be said to have any ultimate existence.

Canute

[quartodeciman]

I am not overwhelmed by questions of existence. I think of all theoretical thought as attempted representations. What there is to be got out of it, payoffs with predicted phenomena of sorts, is what matters. I am not much of a philosophical realist. Ultimate reality may have to remain shrouded forever.

[Canute]

Fair enough. Thanks for the discussion this far anyway.

[quartodeciman]

And my thanks to you.

[olde drunk]

How do you get around the fact that if spacetime is a continuum then it reduces to nothing at the limit, and cannot be said to have any ultimate existence.

good point, perhaps we do not really exist as we have come to define existence? maybe this world is only an experience that we believe to be real. all is energy! within this(our) framework the energy is experienced as physical/real. next step, we want the particle at either point P1 or at Px, as energy, it is there, instantly.

next question, what the hell is energy? lol

peace,

[Canute]

Originally posted by olde drunk
good point,
Thanks. I really didn't expect anyone to think so.

perhaps we do not really exist as we have come to define existence?
I'd say it's a lot more certain than just perhaps.

maybe this world is only an experience that we believe to be real.
But you can't deny that the experience does exist. As you say, the defintion of 'exists' is the big issue.

all is energy!
I can't agree with that. Energy exists in the usual scientific sense. Therefore it cannot be the thing that 'exists' outside our normal definition of existence.

next question, what the hell is energy? lol
I think it's possible that science is right and energy really is simply the ability to do work, rather than being a 'thing', in other words it's a property rather than an essence. It seems to make sense. If it was a thing then how could the net energy of the physical universe be zero?

[Sikz]

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.

Why can nothing move faster than one unit of space per unit of time? Why can't something move two units of space in one unit of time? There is no need to draw from that idea the concept of it moving 1 unit of space in 0.5 units of time... It simply skips the 1 unit marker.

Here's what I propose: Space and time are like a computer monitor- based on a "pixel" (atomic-like) structure. However, using the computer monitor analogy, think of this: Graphics displayed on the monitor do not require to be a whole number of pixels. Something can move at a speed of 1.3 pixels a milisecond (the 3 is actually repeating). Here's how that movement would look:

Actual Displayed
0.0 0
1.3 1
2.6 2
4.0 4
5.3 5
6.6 6
8.0 8

Displayed position on a monitor is equal to actual position in the universe. It doesn't matter if a particle's "true" (imaginary) position is 1.3; it interacts with other particles and exerts a gravitational field as if its location was 1. The only difference between imaginary positions and actual positions is in velocities- you can see from the example the imaginary velocity of 1.3 per 1 results in the skipping over of points 3, 7, etc. Therefore over a large distance imaginary velocities become obvious, but over short distances their effects are minimal.

That seems to resolve Zeno's paradoxes quite well without creating new paradoxes, doesn't it? Or does someone have a counterargument?

[Canute]

Your theory implies that motion can be instantaneous (more than one quanta travelled in one instant) or that physical quanta in motion have no precise location and are smeared across spacetime (partly at one location and partly at another).

This is inevitable in any theory of motion based on quantised spacetime, as QM illustrates.

[selfAdjoint]

Originally posted by Canute
Your theory implies that motion can be instantaneous (more than one quanta travelled in one instant) or that physical quanta in motion have no precise location and are smeared across spacetime (partly at one location and partly at another).

This is inevitable in any theory of motion based on quantised spacetime, as QM illustrates.

QM sensa strictu does not quantize spacetime, RQFT quantizes fields within spacetime, but continues to posit continuous (Minkowski) spacetime.

[Canute]

Originally posted by selfAdjoint
QM sensa strictu does not quantize spacetime, RQFT quantizes fields within spacetime, but continues to posit continuous (Minkowski) spacetime.
Can you unpack that a bit. I know cosmologist like to see the early universe as an ideal condensate, but I've never quite understood how this squares with Planck lengths and the like.

[Mr. Robin Parsons]

Zeno's paradoxes are based upon "addition in a series", yet Zeno's examples always(?) include division, usually by 2 (a halving) and we know (from Computer Science) that division is simply a (more complex?) form of subtraction, hence we can see that to 'add a series' while including the subversive subtraction, causes the appearance of a paradox to arise.

[Canute]

I've no idea whether that's true or not but it doesn't matter. What matters is that Zeno raised a paradoxical issue, whether or not he chose the best way of illustrating it.

[Mr. Robin Parsons]

Originally posted by Canute
I've no idea whether that's true or not but it doesn't matter. What matters is that Zeno raised a paradoxical issue, whether or not he chose the best way of illustrating it. Matters to me cause it is simply a bit of a math trick and I have this thing about people, people who fool people (start singing...HUH?)

[Canute]

Originally posted by Mr. Robin Parsons
Matters to me cause it is simply a bit of a math trick and I have this thing about people, people who fool people (start singing...HUH?) What's a trick? Zeno's paradox? You might like to read back through the posts - that criticism was dealt with a while back.

[Mr. Robin Parsons]

Originally posted by Canute
What's a trick? Zeno's paradox? You might like to read back through the posts - that criticism was dealt with a while back. In simple truth, in mathematics, there are only two (2) operations that can be performed, Addition and Subtraction, 'multiplication' and 'Division' are simply "Tabled" manners of doing addition and subtraction...NOW to claim that you are adding a series and to be constantly SUBTRACTING (Division by 2) is to fool yourself!

[Canute]

Fair enough.

[Mr. Robin Parsons]

It lies simply in the idea of there being an infinite number of numbers between zero and one, normally we do not attempt to count to one by counting all of the possible numbers in between as we know we would never get there...this approximates what Zeno's methodology does/accomplishes, gets you to try counting to a specified point and preconditions the math such that the opportunity to reach that outcome is precluded....a bit like the "Tree falling in the woods.." thingy as the very next statement precludes any kind of realistic responce, "..with nothing there to hear/record it.." then "what does it sound like?" a question that precludes proper respociveness by insertion of a conditional statement that 'precludes' the, then, following question...

Tell me, My Moral Rights, will they be respected?

[Mr. Robin Parsons]

Scuza..(but)...
for Quoting myself...
Tell me, My Moral Rights, will they be respected?
...This time...?

[Canute]

Is that to me? I'm afraid I don't underastand what you're talking about.

[Mr. Robin Parsons]

Originally posted by Canute
Is that to me? No! I'm afraid I don't underastand what you're talking about. Sorry...

[scott]

My simple understanding is this - the paradox says that becuse you have to cross an infinite number of "halfway" points on your journey from point A to point B, you'll never reach point B. This fails to consider that because the distance from point A to point B is fixed, and speed is fixed, the successive subdivisions must necessarily become smaller and smaller, requiring a smaller and smaller amount of time to traverse each one. Because the subdivisions become so small, I can traverse billions of them in a second. At some point, the subdivisions become infinitely small.

So in traveling from point A to point B, I traverse an infinite amount of subdivisions in an infinitely small amount of time, which, when all added together equals the amount of time to travel the distance from point A to point B at the given speed. This is demonstrated mathematically as pointed out somewhere above where the sum of a diminishing infinite series is equal to one.

[Canute]

Originally posted by scott
So in traveling from point A to point B, I traverse an infinite amount of subdivisions in an infinitely small amount of time,
Why does it take any time to get there then?

[UltraPi1]

An infinity of points between two chosen points is the same as saying there are no points at all other than the points specified. The time it takes between two chosen points is known by the completion or passage of the two points. Any subdivisions between the chosen points must be carried out by seperate points unrelated to the chosen ones.

[Mr. Robin Parsons]

Originally posted by scott
My simple understanding is this - the paradox says that becuse you have to cross an infinite number of "halfway" points on your journey from point A to point B, you'll never reach point B. This fails to consider that because the distance from point A to point B is fixed, and speed is fixed, the successive subdivisions must necessarily become smaller and smaller, requiring a smaller and smaller amount of time to traverse each one. Because the subdivisions become so small, I can traverse billions of them in a second. At some point, the subdivisions become infinitely small.
So in traveling from point A to point B, I traverse an infinite amount of subdivisions in an infinitely small amount of time, which, when all added together equals the amount of time to travel the distance from point A to point B at the given speed. This is demonstrated mathematically as pointed out somewhere above where the sum of a diminishing infinite series is equal to one. Nice insertion, attempting to use 'time' to factor out what the Halving does, it successfully assures and certifies that you will never reach the 'end' point....Insert arguement, ad infinitum, upon 0.99999999999to infinity as being equal to 1.0 (One)...absolute truth: "they never are!", rationalizing: "Yes! they are!" (at some point/level/stretch of the imagination...)

[Mr. Robin Parsons]

Originally posted by Mr. Robin Parsons
(SNIP) gets you to try counting to a specified point and preconditions the math such that the opportunity to reach that outcome is precluded....a bit like the "Tree falling in the woods.." thingy as the very next statement precludes any kind of realistic responce, "..with nothing there to hear/record it.." then "what does it sound like?" a question that precludes proper respociveness by insertion of a conditional statement that 'precludes' the, then, following question... (SNoP) Kinda funny, to me, as the 'principal' in use, the Idea of a precluded question, something that I have, and had, introduced to people, (the first? Don't know for certain...maybe...) over time, in the last ten + years, and yet, still, no one else figured it out from knowing that....Neat eh?!!

[Canute]

Originally posted by UltraPi1
An infinity of points between two chosen points is the same as saying there are no points at all other than the points specified. The time it takes between two chosen points is known by the completion or passage of the two points. Any subdivisions between the chosen points must be carried out by seperate points unrelated to the chosen ones. [/B]
But any division of spactime into points creates a paradox of motion. It doesn't matter you hypothesise an infinty or just two.

Robin - What you say, if I understand you right, is true. It is the whole point of the question.

[Mr. Robin Parsons]

Originally posted by Canute
(SNIP) Robin - What you say, if I understand you right, is true. It is the whole point of the question. (SNoP) Answering a precluded question is a redundant exercise, as it is posited in a manner as to ensure it's un-answerablity...

Change the wording of the 'tree' one, to: "If a tree falls in the woods, is there any sound, if there is no one there to record, or to hear it?" makes for a much more sensible approach, as it allows the person, being questioned, to see/know that there is an attempt to preclude the answer, a pre-condition that clearly applies to the ability to repsond to it, similar in Xeno's question, and I would suggest that if you re-arranged the question, brought in the "..and now half the distance travelled" at the end, rather then the middle, more people would see the "Common sense" knowledge that tells us it will remain 'unsolvable' as it will never achieve the responce point that has been indicated as desired....

[Canute]

But Zeno's paradoxes are paradoxes. They are supposed to be unanswerable. They are unanswerable because of the assumptions that they are based on, as you rightly point out. The purpose of such paradoxes is to act as 'reductio ad absurdam' arguments to show that the intial assumptions are false.

So the point of answering then is to find whether and in what way the intial assumptions are false.

[quartodeciman]

A slight improvement, if you will allow it.

The point of answering a paradox is whether and in what way at least one of the initial assumptions is inappropriate with respect to the other assumptions.

[Mr. Robin Parsons]

Originally posted by Canute
But Zeno's paradoxes are paradoxes. They are supposed to be unanswerable. They are unanswerable because of the assumptions that they are based on, as you rightly point out. The purpose of such paradoxes is to act as 'reductio ad absurdam' arguments to show that the intial assumptions are false.

So the point of answering then is to find whether and in what way the intial assumptions are false. Firstly, they are not "un-answerable" they simply appear as an "unanswerable" question cause the answer is simply that the permissions of mathematical theory allow you to develop towards infinity, and this example that Xeno offers tells of the differentiation between 'theory' and "Reality" inasmuch as, in reality you cannot accomplish what Xeno does, the appearance of "never being able to touch the end point", in reality the changing heat in the room will (would probably) cause the valence shell electrons to expand, (slightly) and then the exchange(s) of photonic energies begins, as the two atoms "make contact to resist contact"....when you get close enough to 0.99999Inf nature will fill in the space for you, ( = 1 ) cause it will not allow you to go anywhere nears as small as the numbers, which will continueously bring you to smaller, and smaller, 'points'...but never touching...

It is math theory applied erroneously to reality, cause in reality there is NOT an infinite distance (nor 'number of points') between two points, and Xeno's will attempt to prove to you that there is...that is math theory!

[Canute]

Originally posted by quartodeciman
A slight improvement, if you will allow it.

The point of answering a paradox is whether and in what way at least one of the initial assumptions is inappropriate with respect to the other assumptions. [/B]
Ok. But I take inconsistencies between assumptions to indicate the falsity of one or more of them.

[Canute]

Originally posted by Mr. Robin Parsons
in reality the changing heat in the room will (would probably) cause the valence shell electrons to expand, (slightly) and then the exchange(s) of photonic energies begins, as the two atoms "make contact to resist contact"....when you get close enough to 0.99999Inf nature will fill in the space for you, ( = 1 ) cause it will not allow you to go anywhere nears as small as the numbers, which will continueously bring you to smaller, and smaller, 'points'...but never touching...
What does nature fill in space with?

It is math theory applied erroneously to reality, cause in reality there is NOT an infinite distance (nor 'number of points') between two points, and Xeno's will attempt to prove to you that there is...that is math theory! [/B]
Zeno was not arguing that there are infinite distance between points. He was arhuing that this is the kind of problem that comes up if you assume that space consists of points and time consists of instants.

[Mr. Robin Parsons]

Originally posted by Canute
What does nature fill in space with? Didn't I metion energy, "photonic exchange of energies"...

Zeno was not arguing that there are infinite distance between points. He was arhuing that this is the kind of problem that comes up if you assume that space consists of points and time consists of instants. This problem comes up when 'math theory' is inappropriately applied to existent reality...little else...

Time is clearly a flow, and it doesn't "exist" sooooooo...

[Canute]

Just came across this if anyone is still interested. It's from http://members.aol.com/kiekeben/zeno.html

A brief analysis of the motion paradoxes

...The Racetrack and the Achilles are more difficult. (These are discussed together, for they are essentially the same paradox — that is, they generate the same basic difficulty.)

Nowadays, the standard solution to these paradoxes relies on the claim that (contrary to Zeno's assumption) an infinite series can in fact be completed. Thanks to advances in mathematics, we now know that the infinite series of fractions involved in e.g., the Racetrack, has a finite sum: (1/2 + 1/4 + 1/8 + ...) = 1. Hence one will of course reach the end of the track.

While I agree that the solution must depend in some way on this fact, I'm not so sure that no problems remain. One can imagine Zeno replying to the proposed solution as follows:

"Of course half the length, plus one fourth, plus one eighth, and so on, add up to the whole length. And that's just the point. The whole length contains an infinite number of finite parts. In order to traverse it, therefore, a runner would have to complete an infinite number of tasks. But how can such a thing to be possible?"

Some modern philosophers have argued that there are indeed serious problems with the notion of completing an infinite number of tasks. The best-known example of a current-day Zeno type paradox is the Thomson Lamp, named after James F. Thomson.

The Thomson Lamp
Suppose you have a lamp with a simple on/off switch. Press the switch when it is off and the lamp will be turned on, press it again and it will be turned off. Now suppose you run the following experiment. You turn the lamp on at the start of a minute. Thirty seconds later, you turn it off. In another fifteen seconds, you turn it back on, then 7 1/2 seconds later back off again, and so on throughout the midpoints of whatever time remains. Now the question is this. At the end of the minute, will the lamp be on or off?
Since the lamp has been turned on and off an infinite number of times, for every time it has been turned on, it has been turned off, and vice versa. At the end of the minute, therefore, it can be neither on nor off. But it must be one or the other.

Attempts to find fault in this paradox often attack some irrelevant aspect of the argument. Thus one sometimes hears the criticism that this situation is physically impossible, since no mechanism could operate indefinitely fast. The on/off switch would not be able to keep up. As a counter argument to this type of criticism, I offer the following simplified version of the Thomson Lamp:

Kiekeben's Odd/Even Paradox
Suppose a point P is moving between points A and B (just like in the original Racetrack). And suppose also that we stipulate that P is in the state "even" for the first half of the journey, "odd" for the next 1/4, "even" for the next 1/8, and so on. That is, we simply decide to classify P based on where along the journey it is, such that it alternates between what we call an "even" and an "odd" state. We can in addition stipulate that once it is in one state it remains in that state unless it gets switched according to the above rule.
What state will P be in at B? Just as with Thomson's lamp, it cannot be in either, yet it must be in one or the other. The only solution to this paradox, it seems, is to claim that there is something wrong with the way it is set up. The stipulated conditions simply cannot form a consistent set. But why not?

[Mr. Robin Parsons]

Originally posted by Canute
While I agree that the solution must depend in some way on this fact, I'm not so sure that no problems remain. One can imagine Zeno replying to the proposed solution as follows:

"Of course half the length, plus one fourth, plus one eighth, and so on, add up to the whole length. And that's just the point. The whole length contains an infinite number of finite parts. In order to traverse it, therefore, a runner would have to complete an infinite number of tasks. But how can such a thing to be possible?" There is your logical fallacy, right in the emboldened and the underlined.......The whole length contains an Infinite number of numbers, not parts, that is what Delineates reality, FINITE space...even though you can mathematize (count) it infinitely.....it has a finite number of measurable PARTS.

[Canute]

It's not my fallacy, it's a quote. Try the link.

Also it's not a fallacy. How can a length be made of numbers? I think that you're rather missing the point. If the issue was as simple as you say nobody would ever have taken an interest in the paradox, Zeno included. Did you not read the equivalent cases that were given?

[Mr. Robin Parsons]

Originally posted by Canute
It's not my fallacy, it's a quote. Try the link. O.K. Not yours, But a fallacy, none the less...
Also it's not a fallacy. How can a length be made of numbers? Not made, measured, but that is the point "infinitely numerable" (countable)...reality is that there are NOT an infinite number of 'pieces' or 'parts' between two points...get it? I think that you're rather missing the point. If the issue was as simple as you say nobody would ever have taken an interest in the paradox, Zeno included. Did you not read the equivalent cases that were given? No, No need to....already know the answer, God's Grace!

[Canute]

Originally posted by Mr. Robin Parsons
No, No need to....already know the answer, God's Grace!
Lol.

So you agree that spacetime is continuum then, infinitely divisible by measurement but not so in reality. Exactly Zeno's point.

[Mr. Robin Parsons]

Originally posted by Canute
So you agree that spacetime is continuum then, infinitely divisible by measurement but not so in reality. Exactly Zeno's point. Again you miss it, it is impossible to have an infinity within a finite space....mathematically you can count to the appearance of the infinite (fooled yourself if you believe that!) but reality is a finite perception, thing, event, and time doesn't exist, so why the heck would go for a continueum of Spacetime...

Your statement, or the one that is postulated as to represent what Xeno might have said spoke of an "infinite number of finite parts"...but that is inside a finite thing (the reason why you can invoke either, time, or the notion that you "know it adds to one" {eventually}...ha ha ha another 'self-fooly') to begin with, and you cannot have an infinity within a finite thing...except in math THEORY!!!!!!!

[Canute]

Who said anything about infinties in finite space?

[Mr. Robin Parsons]

Originally posted by Canute
Who said anything about infinties in finite space? YOU posted it! I've already quoted it from YOUR post..."The whole length contains an infinite number of finite parts." that is a finite length containing an infinity (Of finite parts! too!!) according to what YOU posted!


Do you Understand that??? [8)]

[Canute]

Originally posted by Mr. Robin Parsons
YOU posted it! I've already quoted it from YOUR post..."The whole length contains an infinite number of finite parts." that is a finite length containing an infinity (Of finite parts! too!!) according to what YOU posted!

Do you Understand that??? [8)]
Oh yes. Your change of words confused me.

But you have completely missed the point. It is utterly absurd to think that a finite length can have an infinite number of finite parts. That's why there is a paradox, as the quote I posted quite clearly points out. Why do you then argue that it's impossible? Of course it's impossible. Everyone agrees about that.

[selfAdjoint]

Originally posted by Canute
Oh yes. Your change of words confused me.

But you have completely missed the point. It is utterly absurd to think that a finite length can have an infinite number of finite parts. That's why there is a paradox, as the quote I posted quite clearly points out. Why do you then argue that it's impossible? Of course it's impossible. Everyone agrees about that.

What absolute bilge. Nobody but a few cranks believes that. The real line between 0 and one, a finite length, contains points in one to one correspondence with the decimal fractions 0.a1a2a3.... and there are infinitely many of those.

Or you can do the dichotomy. Divide this interval in two at its midpoint. Do you say that one of these halves contains NO points? If not divide each of the halves again. Where do you stop? What evidence have you, beyond your own prejudice, that we need ever stop?

[Canute]

You are confusing mathematics with reality. Of course it's possible to hypothesise that there are are infinite finite points between every two points. All one need do is assume that spacetime is infinitely divisible into infinitessimal but finite quanta. The point is that this assumption is incoherent and leads to paradoxes.

You yourself said "...ha ha ha another 'self-fooly') to begin with, and you cannot have an infinity within a finite thing...except in math THEORY!!!!!!!".

So call it bilge when I agree with you?

[Mr. Robin Parsons]

Originally posted by canute
All one need do is assume that spacetime is infinitely divisible into infinitessimal but finite quanta. Yes assume, but reality tells us very clearly about a Planck length, ergo, unless you can prove that wrong, your assumtion remains simply that, and it isn't me that it makes look like an ....assuming that is theoretical, not reality, and/so there is no 'paradox' to it, simply a self deception upon the idea of being able to number an infinity, you can, not!

The idea of (n + 1) as being representative of counting to the infinite is a self deception...no matter how large you got, measure the 'other' side, (means make it "times two" (n + 1) x 2 ...simple) none the less, you cannot 'number' an infinity, it is uncountable.....so there isn't even really a paradox, just some confusing (ill)logic...

(n + 1) represents an 'activity' not a 'thing' infinity would be a 'thing'

[selfAdjoint]

Originally posted by Canute
You are confusing mathematics with reality. Of course it's possible to hypothesise that there are are infinite finite points between every two points. All one need do is assume that spacetime is infinitely divisible into infinitessimal but finite quanta. The point is that this assumption is incoherent and leads to paradoxes.

You yourself said "...ha ha ha another 'self-fooly') to begin with, and you cannot have an infinity within a finite thing...except in math THEORY!!!!!!!".

So call it bilge when I agree with you?

I have scrolled through this entire thread and can't find any post by me that fits your characterization. I posted twice before, once with my conjecture about Zeno's intent, and once to correct the belief that quantum mechanics forces space or time to be discrete. Even LQG doesn't reduce to chunks of space: they have area quantized, so when you observe it you find one of a set (which might be continuous) of eigenvalues. There is no reason in experiment or theory to assume space is discrete.

[Mr. Robin Parsons]

Originally posted by selfAdjoint
I have scrolled through this entire thread and can't find any post by me that fits your characterization. I posted twice before, once with my conjecture about Zeno's intent, and once to correct the belief that quantum mechanics forces space or time to be discrete. Even LQG doesn't reduce to chunks of space: they have area quantized, so when you observe it you find one of a set (which might be continuous) of eigenvalues. There is no reason in experiment or theory to assume space is discrete. Thats probably because part of what he quoted is from me, this part...

Originally posted by MRP
You yourself said "...ha ha ha another 'self-fooly') to begin with, and you cannot have an infinity within a finite thing...except in math THEORY!!!!!!!". canute apparently ascribed it to you though, sooooo....it is mine...Ooops it isn't "ascribed" it is just posted my error, sorry! still it should have been as to avoid this kind of confu'i's'on, right?

[Canute]

Yeah sorry - something very strange is going on here which I can't figure out. Posts keep disappearing and swapping pages.

Self-Adjoint - I don't know why you thought my post was to you but it wasn't. from what you say I agree with you.

[Visitor]

Wheres the paradox? When you define the parameters to come to a certain result there is no paradox. Besides, brownian motion would eventually be greater than the distances traveled and the races would be over. Zeno never said anything about his thought experiments taking place at absolute zero.

Foolish humans.

[Mr. Robin Parsons]

Yikes!! [o)] one last time, you cannot use a finite tool, (math) to measure an infinite "thing", or "an infinity"...to think that you can, is to indulge yourself in an absurdity.....

"Ad Infinitum / Ad Absurdum"

[theophoretos]

Zeno was not wrong with his paradoxes of motion. These paradoxes are meant to point up the logical flaws inherent in our everyday conception of space, time, and motion which has been quantitatively codified in the classical mechanics. Since this logical flaws have been definitively shown in relativity and quantum mechanics, it'd be productive to solve Zeno's paradoxes, not with mathematics, but simply with the principles of new physics. I think that the paradoxes up to the last one simply imply the quantized notion of space and time and the uncertainty principle (this has been vaguely stated by Peter Lynd's article), while the last, most important of statium shows that in order for space and time to be quantized, a maximum cap on velocity has to be posited, the speed of light. In other words, there is a relation of logical mutual implication between quantum mechanics and special relativity. I wrestled with this problem here:
http://www.geocities.com/therapeuter/capra7777.html

I hope no one trivalizes Zeno by positing that he was not smart enough to even understand some basic notion or that he was playing logical tricks; and the philosophical background for the paradoxes is equally important to take account of.

[Canute]

Good point about not trivialising Zeno. But I don't agree that his paradoxes show c to be an upper limit to velocity. How do you get that result?

[selfAdjoint]

It's in his discussion of the fourth case, where both space and time are discrete and objects (represented by rows of "atoms") pass each other. A paradox arises in which a speeed is seen to be twice as big as itself. And he concludes that the formula for addition of velocities must be modified. Theophoretos exhibits the familar relativistic law, but I didn't see where he proved that is the one, rather than some other variation on v_1 + v_2.

[Mr. Robin Parsons]

How many 'spaces' can you count in an empty space??...and to apply Xeno to it, when you arrive at any answer, divide by 2, and keep going....


Can you figure it out from there?

[UltraPi1]

How many 'spaces' can you count in an empty space??...and to apply Xeno to it, when you arrive at any answer, divide by 2, and keep going....Assuming that the empty space is infinite in scale - There is an infinity of spaces possible. Whether the spaces are the size of plank, or the size of a breadbox ... Pick your poison - Then stick with it. Now if you assume that these spaces don't appear all at once - It will take forever to complete the task. In the mean time - Think of those existing spaces as moving around with laws regarding their movement of course. Therein lies our universe of discrete spaces that move in a world without halfway points to the infinite degree.

[Mr. Robin Parsons]

Originally posted by UltraPi1
Assuming that the empty space is infinite in scale - There is an infinity of spaces possible. Whether the spaces are the size of plank, or the size of a breadbox ... Pick your poison - Then stick with it. Now if you assume that these spaces don't appear all at once - It will take forever to complete the task. In the mean time - Think of those existing spaces as moving around with laws regarding their movement of course. Therein lies our universe of discrete spaces that move in a world without halfway points to the infinite degree. Assume that an empty space is defined by the FACT of the NON empty space around it and, wake up!

[UltraPi1]

Assume that an empty space is defined by the FACT of the NON empty space around it and, wake up!Not sure I follow. I'm curious as to where this empty space is - Certainly not in our neck of the woods. That means you have to leave the forest to the land of nada where there is no such animal as non-empty surrounding empty. I understand the point you were making, but your point is impossible to conceive by your parameters last I heard. My last post was simply conforming to the laws of reality, or non-existence if you will, where your's did not. If you want to have an empty space .. I can guarantee it won't have non-empties surrounding it. At least not by your accounts.

I replied to your post because the universe is the definition of an infinite empty space by conceptual means. It is defined with discete entities on an ongoing basis. The point I was making is that the universe will never be halved out, because you can never reach the a count of spaces within the infinite space. Not that the discrete spaces can't be halved out, but that they never will be. You choose to do so. The universe says otherwise.

Zeno is saying - If space is infinitely divisible you can never go from point A to B because first you must cross halfway point C to get to B, and so forth and so on, blah, blah, blah. One key point to be made is that there are no points in an infinitely divisible space between two points. Any attempt to do so leaves you with a finite structure by which you could never reach a conclusion as to the viability of a motion from point A to B where the space is infinitely divisible. In priciple - No proof is possible for an infinitely divisible space. Either space is infinitely divisible, discrete, or both. I say it is discrete entities that are potentially infinitely divisible.

[Mr. Robin Parsons]

What Xeno accomplished is to use a language, math, to get you to fool yourself into thinking that you could count all of the empty spaces in an empty space...Math is a language, and a tool, and is subject to abuse, and misuse, and mis-interpretations, just like everything else, in languages.

Usually when you count, you count something, (Items, existant items) but here, in the case with Xeno's application, he has gotten you to count the "empty spaces" in an empty space, AKA "Infinity" as math attempts to reveal it...it can't, BTW....

Much past that and you are just fooling yourself....

[UltraPi1]

Usually when you count, you count something, (Items, existant items) but here, in the case with Xeno's application, he has gotten you to count the "empty spaces" in an empty space, AKA "Infinity" as math attempts to reveal it...it can't, BTW....You might find this a little nutso, but I think the whole purpose of the universe is to count (create) the finite empty spaces of an infinite space. This process is ongoing and will take forever to complete. Keep in mind that I consider existence to be purely conceptual. There are no physical entities. They only come across to you as being that way.
So your existence is a hodge podge of conceptually discrete empty spaces, and each space acts in accordance with our known physical (conceptual) laws.

[Canute]

Hmm, do you know the 'The Jewel Net of Indra'?

[Mr. Robin Parsons]

Originally posted by UltraPi1
You might find this a little nutso, but I think the whole purpose of the universe is to count (create) the finite empty spaces of an infinite space. This process is ongoing and will take forever to complete. Keep in mind that I consider existence to be purely conceptual. There are no physical entities. They only come across to you as being that way.
So your existence is a hodge podge of conceptually discrete empty spaces, and each space acts in accordance with our known physical (conceptual) laws. Nothing, well written, in a philosophy forum, should be considered "nutso", especailly not the expression, simply, of idea/concepts/and/or/thoughts...so don't worry bout that..

It begs metaphysical, inasmuch as, resolving the Idea of 'Infinite', with existence, requires understanding(s) like; "In an infinity, the center can be everywhere!" as 'material existence' and 'Infinite' seem incompatable otherwise...but Xenos is a play 'tween Theory of math (language of math) and perception(s) of reality...

In a singular 'empty space' you could count to x ? Infinity? (absurdity) the number of "empty spaces" (non-existant) you could find, in there, then, A'la Xeno, when you think you have gotten them all, halve the space that that amount of numbers 'occupies', and keep counting!

(has the absurdity become clear to you, in that example?)