An effort to solve Zeno's motion Paradoxes

[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 traveled 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 traveled 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 separate 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 0.99999999999^{to 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 separate 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 0.99999^Inf 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 0.99999^Inf 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!