Zeno's paradox

Discussion in 'Physics & Math' started by Roman, Mar 30, 2009.

  1. Roman Banned Banned

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    How is it a paradox?
    He just seems to be parsing his model of reality incorrectly. It doesn't seem so much a genuine paradox, more like something you may find on this page.
     
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  3. Tnerb Banned Banned

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    Paradox. I've always wondered about paradoxes

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  5. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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    Yeah, Xeno is pretty weak as far as paradoxes are concerned. Most people who are comfortable with limits regard it as a non-issue, but there are certain philosophers that have a problem with any kind of infinite divisibility, regardless of whether there is a finite limit associated with it. Which is fine, I suppose: either all motion is an illusion, or infinite divisibility is not inherently problematic. Seems like an easy choice to me...
     
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  7. Roman Banned Banned

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    The tortoise & Achilles one is solved with two sets of equations. Just graph that shit, and you find Achilles catches up with the tortoise at 10/9 units of time.
     
  8. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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    Ah, but Xeno never finishes graphing it, because he first has to graph the positions at time 5/9, but before that he must graph the positions at time 5/18, and so on. And since Xeno requires some finite time to graph any position, he never manages to complete the graph :]
     
  9. Roman Banned Banned

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    lol
    But he also requires less time to graph each position. Graphing to 5/9 only takes 1/2 the time, 5/18 a quarter, and so on. Or I could integrate.

    Infinite subdivision of a finite amount doesn't strike me as a paradox- it strikes me as dumb.
     
  10. prometheus viva voce! Moderator

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    Zeno's paradox is only a paradox if you assume (I guess intuitively) that you can't add up and infinite number of terms in the sum. What is called a paradox is not really a paradox at all because you can do precisely that, as you correctly point out.

    The terminology comes from opponents of a particular theory dreaming up "paradoxes" to disprove it. Then proponents of the theory can come back and argue against them, as was successfully done in the case of the twin paradox in special relativity for example.
     
  11. Absane Rocket Surgeon Valued Senior Member

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    If I remember correctly, the problem that Zeno had was that he could not comprehend how one could sum an infinite sequence to arrive at a finite sum.

    For example, to travel a distance of 1 unit you must travel 1/2 that, then 1/4 of the remaining, etc. In theory (an incorrect one, at that), you will never make it to 1 because of the infinite subdivisions. And, if you assume a constant speed of 1 unit per hour, it will take 1/2 an hour to get half way, another 1/4 an hour of to get another 1/4 the distance. Since you "never make it," it will take an infinite amount of time.

    But as we know, it should take 1 hour.
     
  12. Nasor Valued Senior Member

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    Yeah, I was about to say this and you beat me to it. Zeno's problem was his inability to believe that you can add an infinite number of non-zero values together and get a finite result. Once you accept that you can add an infinite string of non-zero numbers together and get a finite answer, the paradox goes away.
     
  13. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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    Well, to be fair, the objection is a little deeper than that. I'm not sure exactly what Zeno's thinking was, but the type of people who persist in this paradox tend to complain not about infinite sums giving finite answers, but about infinite sums being conceptually unacceptable in the first place. I.e., they don't like the idea that you do an infinite number of (sub)tasks, regardless of whether it takes finite time or not.

    Still not a very compelling objection, at least phrased this way, but there is a bit more to it than simple unfamiliarity with infinite sums.
     
  14. Absane Rocket Surgeon Valued Senior Member

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    Maybe the problem is whether one should be allowed to subdivide a finite length into an infinite number of subdivisions.

    But this is turning into a philosophical debate. I have no problem with Zeno's "paradox."
     
  15. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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    That is actually how some people propose to resolve it: you invoke quantum physics, and argue that at some (finite) point, it becomes meaningless to talk about finer divisions.

    Indeed, it's pretty hard to find anyone who is actually concerned about it these days. But we should bear in mind that its formulation preceded stuff like limits and calculus by centuries, and so it's not a simple matter of misunderstanding infinite sums... people nowadays don't tend to have nearly the intuitive difficulties with infinity.
     

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