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The Paradox of Zeno

  1. Jul 16, 2006 #1
    "The Paradox of Zeno"

    "The Paradox of Zeno"

    The Paradox of Zeno is 2000 years old and its apparent ability to prove
    that all motion is impossible was not resolved until the mathematical
    techniques of Calculus became available, even though that technique is not
    required. One form of the paradox describes the flight of an arrow which has
    been shot at a target. The arrow is shot at a constant velocity, V, to a
    target at a distance, L, and the time of flight is divided into intervals.
    In the first interval, the arrow covers half of the distance to the target
    and, in each succeeding interval of time, it covers half of the remaining
    distance. Under the line of reasoning presented, the arrow never reaches the
    target because, after each successive interval of time, one half of the
    distance to the target that existed at the beginning of the interval
    remains.

    The author finds it incredible that this paradox has been taken
    seriously by intelligent men for over two millennia and has not been
    recognized as a form of trickery. If one accepts that in each successive
    interval of time the arrow traveled half of the remaining distance to the
    target, he must also accept that each of those successive intervals of time
    is half of the duration of the interval which preceded it. As a result,
    under the Paradox of Zeno, not only does the arrow never reach the target,
    the elapsed time of its flight never reaches the time, T, when the arrow
    would reach the target.

    There is no paradox in the Paradox of the Zeno. As long as one allows a
    cheap trick to fool him into considering only those events which occur prior
    to the arrival of the arrow at the target, he most certainly will believe
    that the arrow never reaches the target. The reality is that THE PASSAGE OF
    TIME DOES NOT SLOW AS THE ARROW APPROACHES THE TARGET AND THE ARROW REACHES
    THE TARGET WHEN IT SHOULD.

    To find the Paradox of Zeno be seriously considered as a topic which
    requires a level of mathematics beyond simple arithmetic for its resolution
    should lead a reasonable man to have serious doubts as to the mental
    capacity of the individuals who are engaged in the teaching process.

    The source material for this posting may be found in
    http://einsteinhoax.com/hoax.htm/ (1997);
    http://einsteinhoax.com/gravity.htm (1987); and
    http://einsteinhoax.com/relcor.htm (1997). EVERYTHING WHICH WE ACCEPT AS
    TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE WE HAVE ACCEPTED AS TRUE, IT
    MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND IT MUST BE MATHEMATICALLY
    VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS REQUIREMENT. THE WORLD IS
    ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP FROM THOSE IT HAS GRANTED WORLD
    CLASS STATUS.

    All of the Newsposts made by this site may be viewed at
    http://einsteinhoax.com/postinglog.htm.

    Please make any response via E-mail as Newsgroups are not monitored on
    a regular basis. Objective responses will be treated with the same courtesy
    as they are presented. To prevent the wastage of time on both of our parts,
    please do not raise objections that are not related to material that you
    have read at the Website. This posting is merely a summary.

    E-mail:- einsteinhoax@isp.com. If you wish a reply, be sure that your
    mail reception is not blocked.

    The material at the Website has been posted continuously for over 8
    years. In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE
    MATERIAL PRESENTED. There have only been hand waving arguments by
    individuals who have mindlessly accepted the prevailing wisdom without
    questioning it. If anyone provides a significant rebuttal that cannot be
    objectively answered, the material at the Website will be withdrawn.
    Challenges to date have revealed only the responder's inadequacy with one
    exception for which a correction was provided.


    ..
     
  2. jcsd
  3. Jul 16, 2006 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    > There is no paradox in the Paradox of the Zeno. As long
    > as one allows a cheap trick to fool him into considering
    > only those events which occur prior to the arrival of the
    > arrow at the target

    I agree that it's a pseudoparadox, but, IMHO, you vastly underestimate the force of the argument. It isn't a "cheap trick".

    In the pseudoparadox, we have this infinite, never-ending sequence of events that must happen:

    (1) The arrow reaches the L/2 mark.
    (2) The arrow reaches the 3L/4 mark.
    (3) The arrow reaches the 7L/8 mark.
    ...

    Now, in order to add "The arrow reaches the target" into this sequence of events, where does it go? Well, it goes at the end.

    Let me restate that -- "The arrow reaches the target" goes at the end of this never-ending sequence.

    Hopefully, now you see why it's not such a trivial problem. But it gets worse -- one can't simply naively believe that you can have things "after infinity"... because then one will want to start talking about things like 1 - 0.999... = 0.000...1. (That's a 1 after infinitely many zeroes)

    So, to resolve this pseudoparadox without introducing other problems, one needs a careful understanding of the infinite.
     
  4. Jul 29, 2006 #3
    Re: "The Paradox of Zeno"

    Aristotle solved the problem of motion along with the paradox of Zeno,
    using his own philosophical concepts. Zeno was a student of Parmenides,
    Parmenides believed that motion is impossible because it would imply
    that "the entity", the thing in itself aside from all sensory
    phenomena, would have to change. The only problem is that the only
    quality that the entity can have is "being" so any type of change would
    necessarily imply non-being of the entity, which is contradictory since
    the only defining quality of the entity is being or existence. Also the
    entity is always present, the sensory perception can be near or far or
    not at all, but the entity is always present, another problem with
    motion.

    So according to Parmenides motion is impossible on two fronts, a
    traveling arrow going from point A to point B would have to cease to
    exist at point A and begin to exist at point B, otherwise if it exists
    at every point on its path we run into Zeno's paradox. So, we have the
    problem of existence and the problem of sensory perception.

    Aristotle solved the problem by making motion "a thing in itself",
    stating that motion cannot be broken down into parts, movement must be
    seen as a whole. Aristotle's solution is a little more complicated than
    that, but that is the essence of it. All calculus did was give us the
    tools to break movement down in a special way as to allow mathematical
    analysis and avoid Zeno's paradox.

    John G.

    Egap wrote:
    > "The Paradox of Zeno"
    >
    > The Paradox of Zeno is 2000 years old and its apparent ability to prove
    > that all motion is impossible was not resolved until the mathematical
    > techniques of Calculus became available, even though that technique is not
    > required. One form of the paradox describes the flight of an arrow which has
    > been shot at a target. The arrow is shot at a constant velocity, V, to a
    > target at a distance, L, and the time of flight is divided into intervals.
    > In the first interval, the arrow covers half of the distance to the target
    > and, in each succeeding interval of time, it covers half of the remaining
    > distance. Under the line of reasoning presented, the arrow never reaches the
    > target because, after each successive interval of time, one half of the
    > distance to the target that existed at the beginning of the interval
    > remains.
    >
    > The author finds it incredible that this paradox has been taken
    > seriously by intelligent men for over two millennia and has not been
    > recognized as a form of trickery. If one accepts that in each successive
    > interval of time the arrow traveled half of the remaining distance to the
    > target, he must also accept that each of those successive intervals of time
    > is half of the duration of the interval which preceded it. As a result,
    > under the Paradox of Zeno, not only does the arrow never reach the target,
    > the elapsed time of its flight never reaches the time, T, when the arrow
    > would reach the target.
    >
    > There is no paradox in the Paradox of the Zeno. As long as one allows a
    > cheap trick to fool him into considering only those events which occur prior
    > to the arrival of the arrow at the target, he most certainly will believe
    > that the arrow never reaches the target. The reality is that THE PASSAGE OF
    > TIME DOES NOT SLOW AS THE ARROW APPROACHES THE TARGET AND THE ARROW REACHES
    > THE TARGET WHEN IT SHOULD.
    >
    > To find the Paradox of Zeno be seriously considered as a topic which
    > requires a level of mathematics beyond simple arithmetic for its resolution
    > should lead a reasonable man to have serious doubts as to the mental
    > capacity of the individuals who are engaged in the teaching process.
    >
    > The source material for this posting may be found in
    > http://einsteinhoax.com/hoax.htm/ (1997);
    > http://einsteinhoax.com/gravity.htm (1987); and
    > http://einsteinhoax.com/relcor.htm (1997). EVERYTHING WHICH WE ACCEPT AS
    > TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE WE HAVE ACCEPTED AS TRUE, IT
    > MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND IT MUST BE MATHEMATICALLY
    > VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS REQUIREMENT. THE WORLD IS
    > ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP FROM THOSE IT HAS GRANTED WORLD
    > CLASS STATUS.
    >
    > All of the Newsposts made by this site may be viewed at
    > http://einsteinhoax.com/postinglog.htm.
    >
    > Please make any response via E-mail as Newsgroups are not monitored on
    > a regular basis. Objective responses will be treated with the same courtesy
    > as they are presented. To prevent the wastage of time on both of our parts,
    > please do not raise objections that are not related to material that you
    > have read at the Website. This posting is merely a summary.
    >
    > E-mail:- einsteinhoax@isp.com. If you wish a reply, be sure that your
    > mail reception is not blocked.
    >
    > The material at the Website has been posted continuously for over 8
    > years. In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE
    > MATERIAL PRESENTED. There have only been hand waving arguments by
    > individuals who have mindlessly accepted the prevailing wisdom without
    > questioning it. If anyone provides a significant rebuttal that cannot be
    > objectively answered, the material at the Website will be withdrawn.
    > Challenges to date have revealed only the responder's inadequacy with one
    > exception for which a correction was provided.
    >
    >
    > .
     
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