Zeno and Algorithms

Discussion in 'Computer Science & Culture' started by Canute, Oct 30, 2003.

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  1. Nasor Valued Senior Member

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    Let me make sure we're on the same page here. Xeno's paradox says that you can never reach a point because in order to reach it you would first have to cross an infinite number of discrete intervals, right?

    I don't see the paradox. Xeno is simply wrong. There is only a paradox if you believe that logic dictates it will take an infinite amount of time to pass through an infinite number of discrete intervals. In fact, this is faulty logic. There is no disconnect between logic and the real world here.
     
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  3. Canute Registered Senior Member

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    BBH

    Yes, I agree that R's paradox is relevant. Imo the cosmos is the set of all sets and subject to the same paradoxes. This is the 'non-dual' view of existence, a regression that ends at the notional limit in emptiness, a set that contains nothing but itself. However I have a lot trouble making that sound coherent whenever I try.

    Again I don't agree that there is a difference between the real world and a logical representation of it (unless the world is illogical). I would argue that quanta of spacetime are not logical, since if it is a continuum then those quanta are mathematical points and cannot be reified.
     
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  5. Canute Registered Senior Member

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    I don't think that's quite right. He said that if you relate the movement of Achilles and the tortoise algorithmically, according to some rule of calculation, and base that calculation on quanta of space and time then motion doesn't make sense. He made the same point in his paradox of the moving arrow.

    If you try the maths I think you'll find he's right.
     
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  7. BigBlueHead Great Tealnoggin! Registered Senior Member

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    Nasor: See my above post about infinite subdivision of time; Xeno is not "wrong", he is couching the situation in a frame of reference that is superficially reasonable but ultimately proves to rely on an apparently intuitive, but incomplete description of the situation.

    Wittgenstein talked about something similar with diagrams; he gave some examples of diagrams (in the TLP I believe) which demonstrated mechanical arrangements.

    The diagrams were made such that when you looked at them they appeared to "make sense", and thus demonstrate a mechanical principle, but when examined closely the diagram would turn out to be an incorrect representation.

    Similarly, Xeno's paradox frustrates the concept of time as an infinite connected series of instants by introducing a frame of reference where each block of time is less than half of the previous one, such that an infinite number of steps will not carry us up to a certain point in time.

    It's a trick, yeah, but if you don't spot the fraction problem then it sounds like it makes good sense.
     
  8. Nasor Valued Senior Member

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    I think you're wrong about this. Zeno's paradox relied on space and time not being quantized. In order for the paradox to work, you have to be able to continuously halve the distance between you and your destination. If space were quantized, it would simply be a matter of counting the number of quanta between you and your goal; you wouldn't be able to carry out the infinite series of divisions that is necessary for the paradox to exist.

    They didn't have any concept of space being quantized back then, so Zeno naturally believed that any distance could be divided into two smaller distances. This allows you to establish an infinite number of discrete distances between you and your destination. Zeno believed it must take an infinitely long time to cross an infinite set of distances, which is where the paradox arose.

    Either way, the paradox does not exist. Either space is quantized – in which case the paradox collapses because you can't keep taking half of the remaining distance – or space in not quantized, in which case the entire paradox is based around Zeno's faulty assumption about traveling over an infinite set of distances.
     
  9. Nasor Valued Senior Member

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    Don't be so sure. Zeno was part of the Eleatic school of philosophy, which believed that motion did not exist and that all apparent motion was merely an illusion. It was part of a rather bizarre set of beliefs about the universe being 'singular, eternal, and unchanging.' Most scholars agree that Zeno was trying to use logic to prove that the idea of motion was fundamentally irrational.
     
  10. Canute Registered Senior Member

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    I agree. Stll. whatever Zeno's motives his paradox works. It shows that if you calculate the race based on fundamental indivisible quanta the tortoise wins.

    If you postulate quanta of space and time, any size you like, then you get a paradoxical situation when calculating motion.

    Take two fundamental particles having a race. Suppose, for ease of measurement, each particle is one fundamental quanta in length.

    Make particle 1 travel at one quanta per instant. Then particle 2 must travel at least two quanta per instant to catch up. This means that particle 2 is smeared over two quanta in each instant and has no precise position.

    It makes no sense to say that particle 2 is in one position in the first half of the instant and a different position in the second half, since there is no such thing as half an instant by definition.
     
  11. Nasor Valued Senior Member

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    If space is quantized it would mean that the particles would move one quanta of distance instantly, wait a while, instantly move another quanta, etc. The particle that moves twice as fast is merely 'sitting around' for half as long between its jumps. Yes, the idea of a particle moving from one place to another instantly is very counterintuitive, but no more counterintuitive than the idea that space is quantized. For that matter, it's a lot less counterintuitive than many things in quantum physics.
     
    Last edited: Nov 26, 2003
  12. Canute Registered Senior Member

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    How long do particles wait around between leaps, and why, and what force starts them moving again? I can't see your hypothesis standing up to scrutiny for long.

    Anyway, if you factor these pauses into the mathematics exactly the same problem arises, it doesn't change anything. Either spacetime is not quantised or motion is in some sense an illusion, as Zeno worked out.
     
  13. Nasor Valued Senior Member

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    If space is quantized, then by definition things would have to move in 'leaps' because an object could never travel in-between two points with a continuous motion. Saying that space is quantized is implicitly saying that things travel in leaps.
    I don't know, why would space be quantized in the first place? I'm simply showing that it can work, not trying to build up a complete model of physics in a quantized space universe.
    We have two people, A and B. A is moving twice as fast as B. Lets say that A is moving at 2 meters/second while B is moving at 1 meter/sec. Let's also say that the meter is the basic quantized unit of distance. Person B starts 10 meters ahead of person B.

    Time----Distance from A to B
    0--------10m
    1-------- 9m
    2-------- 8m

    9---------1m
    10--------0m
    11-------1m

    What's the problem here?

    By the way, Canute, I suppose you would be distressed to learn that electrons can travel from one place to another without passing though the space in-between?
     
  14. Canute Registered Senior Member

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    Agreed.

    As leaps are illogical (imo) I would say that you're showing it cannot work.

    The problem here is that at t=10, which is a precise instant in time, person A is smeared over two meters and cannot be said to be any particular distance from person B, who is themself smeared over one meter. (Don't forget that here the quanta of measurement are assumed to be indivisible).

    You can get around this by increasing the resolution (using smaller and smaller units of time and distance) but you cannot ever get rid of the problem. If space and time are quantised then a moving object cannot be said to ever have a precise position in any instant of time.

    Not at all, even if turns out to be more than a theory.
     
    Last edited: Nov 27, 2003
  15. thefountainhed Fully Realized Valued Senior Member

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    We don't really know what Zeno based his premise that it is impossible for X to make infinitely many t- runs. The first premise being that to traverse A and B, x must make infinitely many t-runs.

    He must rely on a continous space and time if one assumes that the premise that it is impossible to make t infinite runs is provable because

    1. The distance between A nd B would be infinitely far(broken down into infitinitely many ts)

    2. To traverse A and B, x would take infinite time.

    The 3rd support, which he could have realied on would simply be that it is logically impossible to traverse the distance. This would require neither a continuous sace and time or a discrete space and time. It is simply paradoxical via contradiction to assume that x can traverse the infinite.

    Either way, the premise that one must traverse the distance between two ends to reach one end is false.
     
  16. Nasor Valued Senior Member

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    I don't see how this is any more illogical than energy being quantized, which we know to be the case.
    No, at t=10 person A is 0 meters away. This idea of 'smearing out' over a distance is entirely your own invention and there's no reason to suppose that it's necessary. Perhaps the jumps happen instantaneously.
     
    Last edited: Nov 29, 2003
  17. Canute Registered Senior Member

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    Instantaneously means in an instant. That means the particle A moves one, waits one, moves, waits one etc. Does this really seem likely to you?

    At T=0 the position of particle B is not known. It is somewhere within the ten positions that it traverses in that instant. It cannot be somewhere at T-1/4 instant, and somewhere else at T+1/4 instant. An instant is indivisible, that's how it is defined.
     
  18. Nasor Valued Senior Member

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    It seems no more unlikely than the original idea of space being quantized. It shouldn't be surprising that bizarre ideas have bizarre consequences.
     
  19. Canute Registered Senior Member

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    I think we're having a misunderstanding. I'm not disagreeing with you. We agree that if space is quantised then motion must be discontinuous. We agree that this is a bizarre idea, and we therefore agree that Zeno was right. Motion is illogical if spacetime is quantised.
     
  20. Nasor Valued Senior Member

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    I think you're correct. I thought you were saying that motion in quantized space could not be modeled; in fact it can be modeled, the model just turns out to be very strange.

    Until recently I would probably have agreed with you that this makes quantized space illogical. Now, having recently taken some classes that dealt with quantum physics, I'm convinced that our intuitive view of the universe is of little value when predicting what really goes on in nature. So I'm no longer willing to say that space can't be quantized simply because it leads to conclusions that seem bizarre.
     
  21. Canute Registered Senior Member

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    Fair enough. I go the other way and think that QM gives a false view of reality, and thus gives rise to bizarre conclusions.
     
  22. Nasor Valued Senior Member

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    I don’t see how you can doubt quantum mechanics in the face of its overwhelming amount of experimental evidence and ability to predict the natural world perfectly, but I guess you can believe what you like.
     
  23. Canute Registered Senior Member

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    I don't doubt that it has the ability to predict a lot very successfully, as do chemistry and biology. I just doubt that it'll ever turn into a fundamental theory.
     
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