Zeno's Paradox

Discussion in 'Physics & Math' started by Xenu, Jun 22, 2002.

  1. drnihili Registered Senior Member

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    191
    No, time is not the issue.

    The problem is completion of a neverending series.

    Look at it this way. If I have a finite number of tasks, I can finish them because with each one I have fewer left to do. At some point I have 5 left, then 4, then 3, then 2, then 1, and when I get that task done then I've finished the series. I complete the series as a whole by completing it's final element. But an infinite series cannot be completed this way. I can't complete it by getting to the end of it. But if I can't get to the end of it, in what sense can I complete it? Remember, I have to do the tasks one at a time. And no matter how many tasks I complete, there are still infinitely many left to do.

    I understand that the total time spent working on the series is finite because each task takes less time than the previous. I understand how calculus can determine a least upper bound to the time spent. But calulus cannot explain how it is we reach the end of a series that doesn't have one. And that is the real puzzle.
     
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  3. yayacatfight Registered Senior Member

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    then zeno's paradox about crossing a room arises because zeno himself dictates how the person must cross the room, by going half the remainder. he will never cross the room, but only because zeno has restricted his motion.
     
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  5. drnihili Registered Senior Member

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    Zeno hasn't restricted his motion, merely described it. Yes there are other descriptions. But why should we think that Zeno's description is inaccurate?
     
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  7. yayacatfight Registered Senior Member

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    by describing the motion he has restricted it. zeno might as well have said count from 10 back to 0 but you may only go half the remainder. using other forms of motion you can certainly get there, but not in the way he has described
     
  8. Xgen Registered Senior Member

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    I don't understand why space can not be discreet. This save us from Zeno paradox. Do you agree that in 1-dimensional case the quatumized space limits infinity series to a finite irreducible quantity?
    Of cource if space and time are discreet other problems arise.
    But these problems come from wrong understanding of discretization. The real physical objects and times are too big and their movement appear smooth. However I don't see why space can not be discreet at scales below 10^-17m.
     
  9. drnihili Registered Senior Member

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    191
    The problem is that the macro scale is generated from the micro-scale. So if geometry fails in the micro scale, then it also fails at the macro scale. And if space is quantized then geometry fails at the micro level.

    You can solve this by completely redoing geometry so as to get rid of all irrationals (not an easy feat). An easier method might be to hold that space is continuous but motion is discreet. Of course that still leaves the arrow paradox....
     
  10. everneo Re-searcher Registered Senior Member

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    Space and time are, though might be continuous, limitted to planck scales (distance in the order of 10^-35 m and time in the order of 10^-43 s) for all practical purpose in our reality. Its something like a football can roll over a net lengthwise, breadthwise and diagonally. but a grain filtered down the net and lost for ever.
     
  11. drnihili Registered Senior Member

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    191
    I'm not sure what you're saying. Are you saying that Planck scale provides a lower bound or that everything is in integral motions of Planck scale? Also, does this scale apply to space? matter? motion? measurement?

    The net analogy suggest that Planck scale is merely a lower bound for matter. I.e. if you don't have a certain critical size, then you don't have matter. This is, however, a far cry from saying that space is discrete. Of course analogies can only be pushed so far.

    I'd really be interested to hear how you see this going. I can sort of understand how motion or matter can be discreet, but not space. And of course saying I can sort of understand soemthing doesn't mean that I don't think there are pretty severe problems with it. What makes most sense to me is that Planck scale provides a lower bound for measurement, or perhaps that measurement is discrete.

    Enlighten me?
     
  12. everneo Re-searcher Registered Senior Member

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    Planck distance (4.05*10^-35m) is the lower bound for observable distance. its difficult to determine any point within this distance. any matter of this extremely small size should be treated in QM way, if they exist. but when saying light travelling this distance in 1.35*10^-43 secs.(Planck time) this distance seems to be continuous. anyway i cannot assert space is not quantized.
     
  13. drnihili Registered Senior Member

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    191
    Well, if it's just a lower bound on measurement, that doesn't create much problem. Of course it does nothing to help resolve Zeno's paradox either.
     
  14. everneo Re-searcher Registered Senior Member

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    When i say i have 2 legs and can walk 10 km per day i am talking about reality. I don't bother about what zeno thinks about how space should be. i dont bother whether my leg-length is 1 m or 1.01354987535.......... m or the distance i covered on a day is 10 km or 10.00983875........km. For our existence and actions and observations, applying limits in natural way is suffcient. but i rembember you saying space is neither continuous nor discreet. would you mind elaborate further..?
     
  15. drnihili Registered Senior Member

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    191
    I don't think I said it wasn't continuous, that was someone else.

    Zeno's paradoxes are often taken to show that space cannot be continuous, but I don't think that's what they do really. So far as I can see (and I'm *not* a physicist) continuous space is the only option with much going for it. I'd be interested in seeing someone develop a theory of discrete motion within continuous space, but so far I haven't seen it

    The problems with discrete space is that that gemoetry on a small scale must mirror geometry on the large scale with respect to ratios. But it can't do that if space is discreet.

    Hmmm, it occurs to me that some versions of string theory have several dimensions that are tiny and looped, perhaps they could be use to reconcile the apparent inconsistencies, but I'd have to hear how.
     
  16. everneo Re-searcher Registered Senior Member

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    One of your earlier posts. Color emphasize is mine.

    However i agree with your point that geometric ratios must be the same irrespective of scale. euclidian space is continuous. honestly i am not sure about non-euclidian space. recent evidence points out that space might be continuous.
     
  17. wesmorris Nerd Overlord - we(s):1 of N Valued Senior Member

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    I don't think either system can fully capture "space" in terms of objective reality. At least I think it's safe to assert that any coordinate system to date is merely an approximation of "objective space". As such, this vein of the conversation is moot eh? I mean it's somewhat interesting sure, but it cannot reveal further truth to dispute existing coordinate systems in terms of applicability to Zeno's Paradox can it?
     
  18. everneo Re-searcher Registered Senior Member

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    Relativity prohibits any objective space in absolute sense. Quantized space can dissolve zeno's paradox. applying limits to infinte series was also able to explain zeno's paradox in continuous space. either space is continuous or quantized or both or neither is yet to be asserted, i think.
     
  19. drnihili Registered Senior Member

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    191
    Ok, I was basically taking a standard line on Zeno. I think we'll solve Zeno's paradox someday, we just haven't yet.

    Space is not continuous.

    Suppose that space is continuous. Thus Zeno's description of motion is accurate, traversing a unit distance involves traversing an infinite series of ever smaller distances. The series in infinite in that it is unbounded. Note that it is the series that is unbounded, not the distance. So in order to to traverse the distance, Achilles must complete each step of an unbounded series. But unbounded series cannot be completed as they have no final member. Thus space cannot be continuous.

    Remember, Achilles only does one member of the series at a time. He has to complete the series, and he has to do it one piece at a time. So suppose he gets to the other side. What was the last piece of the series he completed? Answer: none. For any member of the sequence, he completed infinitely many members after it. So long as Achilles is still finishing members of the sequence, he always has infinitely many more to go. How then does he manage to complete the sequence? No specific action of Achilles' can completes the sequence. But that's the just to say that he doesn't complete it at all. And yet we all know he gets across the road. So one of our assumptions must be wrong, and the most likely candidate seems to be the assumption of continuity.

    Calculus merely tells us the earliest point by which Achilles will have completed the series *IF* he completes it. It allows us to understand how a unit distance can be seen as being comprised of an infinite number of smaller segments. It does not tell us how those segments can be counted through one at a time.

    A solution to Zeno's paradox would have to tell us how to reach the end of an endless sequence. It would have to make sense of an infinity built one thing at a time. I'm not at all sure how to do that.
     
  20. wesmorris Nerd Overlord - we(s):1 of N Valued Senior Member

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    Not as a hypothetical. I think the true hypothetical cannot be solved. I think it's a testament to subjectivity. Sure, I get to the other side of the room in reality... but if I don't hold "real" rules and merely talk of numbering systems....

    Wait...

    I think that the problem with Zeno's paradox is lack of definition. The problem is not well defined.. this lets subjectivity lead to argument. If you choose the perspective of the guy crossing the room... you happily arrive on the other side. If you choose the perspective of the guy doing division.. you have to debate the applicability of math, then coordinate systems.. then applicability to reality.. then.. you'll stay unresolved unless you choose a lower limit.

    In the original problem presented.. it's just wrong. When you cross a room you use a stride of similar length for each step. As such when the sum of the distance travelled by each stride adds up to the distance required to cross the room, you're there. The division is superfluous and non-descriptive of the events happening. It's merely mental masturbation. Sure, it feels good.. but you're not gonna get anyone pregnant that way.

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    hehe.

    Maybe I just misunderstand the problem.

    Blah.

    Regardless, what's the difference if I'm standing at my destination? Zeno can piss off, I'll cross the room and his dead ass can't stop me damnit!
     
  21. yayacatfight Registered Senior Member

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    i don't see this as a contuinity of space problem. i see it as a description of movement problem. take a line segment from 0 to 10. it has an infinite amount of points, yet mathematically or physically you can get from 0 to 10 in a number of ways. if you restrict the form of movement to half the remainder, you cannot, physically or mathematically get to 10.
     
  22. ryans Come to see me about a dog hey Registered Senior Member

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    995
    It is definately a contrained motion problem. Obviously I am able to cross a room.
     
  23. Xgen Registered Senior Member

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    315
    Please can you explain further what you mean. As I understand the problem is that distance between two points in a discreet space will not be the same if their orientation changes. But do you understand that it depends in what units we measure distance. In macro scale the fhrase - "distance between A and B is 1 km" is sensible, but at micro scale distance have no sense if you do not say in what units you measure it. The concept "angle" also have no sense at micro scale, the triangles:

    ...................................................
    A......................A........................
    C.. B...............C..B......................
    ...................................................

    are the same and can not be distinguished in any way if AB contains equal number of elementary space quants. The non-discreet concept of the distance as fixed and independent from scale is wrong.
     
    Last edited: Jun 26, 2003

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