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Zeno's Paradoxes

  1. Apr 27, 2003 #1
    There are a couple of Zeno's paradoxes that I just don't agree with. There may be something to them that I haven't understood properly, but I just don't agree with them as I've understood them.

    One is the paradox of motion. He posits that you can never get anywhere because you have to cover half of the remaining distance everytime. Since there are infinite halves, you can't ever get there. I disagree because you don't have to continue to cover half of each new distance, you can cover more than that, if you so desire. Also, once you have come into physical contact with something, I think it can be reasonably concluded that you have "gotten there".

    I would very much like that someone should explain where my misunderstanding of this paradox is, before I go on to the next one please.

    Any help would be appreciated.
     
  2. jcsd
  3. Apr 27, 2003 #2
  4. Apr 27, 2003 #3
    Ok, then that particular paradox is not considered authentic?
     
  5. Apr 27, 2003 #4
    I hope you don't mind me hogging your posts tonight. I like supporting Zeno, when given a chance.
    That's not the point Mentat. A 'half' is used because it's the most readily-understandable fraction. Even if you were to traverse three-quarters of the distance every time, Zeno's ultimate conclusion would still apply: that if you travel 3/4 of a length, and then 4/7's of the remaining-length, and then 7/9's of the remaining-length - ad-infinitum - then you will never reach the end of that length.
    I want to say more about this. But one of my favourite films is just starting...
     
  6. Apr 27, 2003 #5
    Re: Re: Zeno's Paradoxes

    Which film is that?

    I understand that it could be any fraction, and that a point particle, with no real determined size, couldn't get to the end. However, I have mass, and so I would eventually be touching whatever was at the "end of the line".
     
  7. Apr 27, 2003 #6
    Nahhh.... its only valid according to fundamentalist mathematicians in their ivory towers. Quantum Mechanics has its own versions of the paradox:

    http://www.tcd.ie/Physics/Schools/what/atoms/quantum/uncertainty.html

    No doubt it is easy, expedient, and comforting to tell little kids mom and dad know what life is all about and have a firm grasp on the situation, but the reality between adults is different. That's also why all these teachers and professors still teach outdated fundamentalist stuff first, and then get down and dirty with a more honest admission of the extent of their ignorance.
     
  8. Apr 27, 2003 #7

    Tom Mattson

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    The fraction used really doesn't matter. The reason Zeno's paradox fails is that, in his argument, he assumes that the infinite series of powers of 1/2 diverges to infinity, when in fact it converges to 1.

    It goes like this:

    Assume you are traveling at speed v and must cross a distance L in time T.

    Before you can get to L, you must get to L/2. Once you have gotten to L/2, you must then get to 3L/4, etc. This continuous halving of distances can be represented by the series:

    Distance=L/2+L/4+L/8+...

    Note that it takes an infinite number of steps to travel the distance. Since it is not possible to perform an infinite number of steps in a finite time, it is not possible to cross any distance L in time T at speed V.


    Zeno's mistake is in assuming that an infinite number of steps cannot be done in a finite amount of time. In other words, he tacitly assumes that it would require an infinite amount of time, and so could never be done.

    Is that true? Let's see.

    T=L/v=(1/v)Σn=0oo(1/2)n

    The series on the right is a geometric series. In order for Zeno's argument to hold up, it would have to blow up to infinity--but it actually converges to 1.

    That suggests the conclusion is wrong, but it doesn't say why it is wrong. The flaw in Zeno's argument is in assuming the divergence of a series that actually converges.
     
  9. Apr 27, 2003 #8

    ahrkron

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    I think it is here's where you miss the point of the paradox. You cannot "cover more than that" without covering the menitoned half.

    i.e., In order to cover 1m, you have to cover half a meter first, no matter how you do it, you will pass through the middle point. Similarly, on your way to cover the first half, you necessarily passed trhough the 25cm point. No matter what you are thinking during your movement (what you are "aiming for"), the matter is one of principle: you need to pass through an infinity of "middle-points".

    Since all those partial movements, the paradox goes, must take some time, and you need an infinity of them, the final destination cannot be reached.

    Does that help?
     
  10. Apr 27, 2003 #9
    Yes, if you've ever studied Calculus, it would be more apparent to you.

    The summation of a constanct divided by anything greater than -1 and less than 1 to the nth power is always a finite number, even when n goes to infinity.

    [sum]1/xn= c, when -1 < x < 1, regardless of bounds

    This can be verified by comparing the integration of such a term from a finite number to infinity, because the integratin can be used as an upper bound for the summation.
    for example,

    [inte]1/2x = -1/((ln2)2x

    so the integration of that from 1 to infinity is

    0 - (-1/((ln2)2) = 1/((ln2)2)
     
    Last edited: Apr 27, 2003
  11. Apr 27, 2003 #10
    Yeah, yeah, calculus suggests its not really a paradox. Philosophy suggests otherwise. I tried to keep it simple, but you asked for it. Zeno's paradox of motion is related to the Sorites Heap paradox.

    http://plato.stanford.edu/entries/sorites-paradox/

    This paradox and the Cretan Liar's paradox are both considered the most insoluable known. A central issue in the sorites paradox is vagueness.

    http://plato.stanford.edu/entries/vagueness/#5

    Something is vague if there are cases which do not fall clearly inside or outside the range of applicability of the term. That isn't to say vague words have no meaning. The word "heap" definitely has meaning, but its meaning is vague. Likewise the meaning of infinity is vague and it is debatable whether or not infinity is real.
     
  12. Apr 27, 2003 #11

    Tom Mattson

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    OK, you're going to have to explain the connection between Sorites and Zeno. For one thing, I don't see the vagueness in Zeno's paradox. In fact, everything seems to be quite well defined.

    As for the Cretan paradox, I don't see the vagueness there either, nor do I see the connection to Zeno. The Cretan paradox is an illustration of the breakdown of logic as a formal system, whereas Zeno attempted to prove the impossibility of motion. The former pertains to a mental construc (logic), while the latter pertains to something in the real world (motion).
     
  13. Apr 27, 2003 #12
    Here's another way of telling the story that takes into account Mentat's point of view:

    A mathematician and an engineer were both standing 20 ft away from this pretty girl when they were asked by another Zeno guy that if they could only walk half the distance to the girl each time what would they do?
    The mathematician exclaimed "Zeno, I will stand right here because I can conclusively prove that I'll never reach her", whereas the engineer said, "I agree but I can get close enough for practical purposes".
     
  14. Apr 27, 2003 #13
    Actually, the Cretan Liar's paradox is not widely considered vague. I mentioned it here just to put both paradoxes in perspective. That's the irony of it, the two most respected paradoxes cover both the vague and the well defined.

    Zeno's paradox is vague because the concept of infinity is vague just as the concept of a heap is vague. The concept of time isn't that well defined either.
     
  15. Apr 27, 2003 #14
    This ain't no paradox really!!!

    if they both go with a constant speeds (V1 and V2) then:
    dX1/V1=dt=dX2/V2 so if it was X1>X2 to become X2>X1 it has to be dX2>dX1 => dX2/dX1=V2/V1>1 => V2>V1 (it is enough).in fact it only seems like a paradox but it ain't cause as you shrenk dX you shrenk dt. at a infinite shrenking step you end up with 0/0=V=const.

    just mantein your speed Achilles and you'll win.
    wisdom is the might of the physically weaker.
     
  16. Apr 27, 2003 #15
    Re: This ain't no paradox really!!!

    He'll win at the end of eternity, what a paradox.
     
  17. Apr 27, 2003 #16
    The thing to remember in Zeno's paradox is that if he continues at a constant speed, as the distance travelled halves, so does the time. So, with continual halving, the distance approaches 0, and so does the elapsed time. But time keeps on a'going...
     
  18. Apr 27, 2003 #17
    Pardox of motion

    The essential problem, highlighted by Zeno’s paradoxes, is the inability of formal logic to grasp movement. Zeno’s paradox of the Arrow takes as an example of movement the parabola traced by an arrow in flight. At any given point in this trajectory, the arrow is considered to be still. But since, by definition, a line consists of a series of points, at each of which the arrow is still, movement is an illusion. The answer to this paradox was given by Hegel.

    The notion of movement necessarily involves a contradiction. Consider the movement of a body, Zeno’s arrow for example, from one point to another. When it starts to move, it is no longer at point A. At the same time, it is not yet at point B. Where is it, then? To say that it is "in the middle" conveys nothing, for then it would still be at a point, and therefore at rest. "But," says Hegel, "movement means to be in this place and not to be in it, and thus to be in both alike; and this is the continuity of space and time which first makes motion possible." (Hegel, op. cit., Vol. 1, p. 273.) As Aristotle shrewdly observed, "It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow." But what is this "now"? If we say the arrow is "here," "now," it has already gone.

    Engels writes:

    "Motion itself is a contradiction: even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." (Engels, Anti-Duhring, p. 152.)
     
  19. Apr 27, 2003 #18
    Re: Re: This ain't no paradox really!!!

    No... he will win at the end of time/change. Which doesn't signify the end of existence.
     
  20. Apr 27, 2003 #19
    Re: Re: Re: This ain't no paradox really!!!

    Which in other words is infinity, but anyway that is not the answer, since the paradox is only a trick of the mind, and not a real paradox (since movement does occur).
     
  21. Apr 27, 2003 #20

    drag

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    Re: Re: Re: Re: This ain't no paradox really!!!

    O.K. I believe that one is covered already. :wink:
    What's next Mentat ?

    Live long and prosper.
     
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