An effort to solve Zeno's motion Paradoxes

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SUMMARY

Zeno's motion paradoxes can be resolved by substituting finite-sized points for geometric points, eliminating the notion of infinite divisibility in space. The discussion highlights four key paradoxes: the stadion paradox, arrow paradox, dichotomy paradox, and Achilles paradox, each demonstrating the complexities of discrete versus continuous interpretations of space and time. Participants agree that Zeno's assumptions about the atomic nature of time and the continuous nature of space lead to contradictions, ultimately asserting that if spacetime is continuous, the paradoxes dissolve. The conclusion emphasizes that "paradox" signifies a challenging concept rather than an impossibility.

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  • Understanding of Zeno's paradoxes and their implications.
  • Familiarity with concepts of discrete versus continuous space and time.
  • Basic knowledge of mathematical series, particularly infinite series.
  • Awareness of philosophical implications in physics, particularly in relation to spacetime theories.
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  • Explore the mathematical foundations of infinite series and convergence.
  • Research the implications of discrete versus continuous models in physics.
  • Study the historical context and interpretations of Zeno's paradoxes in philosophy.
  • Investigate modern theories of spacetime, including quantum mechanics and relativity.
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Philosophers, physicists, mathematicians, and students interested in the foundations of motion, the nature of time and space, and the implications of Zeno's paradoxes in contemporary physics.

  • #61
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!
 
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  • #62
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.
 
  • #63
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.
 
  • #64
Originally posted by Canute
What does nature fill in space with? Didn't I metion energy, "photonic exchange of energies"...[/color]

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...
 
  • #65
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?
 
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  • #66
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.
 
  • #67
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?
 
  • #68
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...[/color]
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?[/color] 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!
 
  • #69
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.
 
  • #70
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!
 
  • #71
Who said anything about infinties in finite space?
 
  • #72
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?
 
  • #73
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?
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.
 
  • #74
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?
 
  • #75
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?
 
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  • #76
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'
 
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  • #77
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.
 
  • #78
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![/color] still it should have been as to avoid this kind of confu'i's'on, right?
 
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  • #79
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.
 
  • #80
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.
 
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  • #81
Yikes! 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"
 
  • #82
zeno was not wrong

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.
 
  • #83
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?
 
  • #84
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.
 
  • #85
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?
 
  • #86
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.
 
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  • #87
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!
 
  • #88
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 principle - 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.
 
  • #89
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, existent 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...
 
  • #90
Usually when you count, you count something, (Items, existent 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.
 

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