Simultaneously

A paradox is something that, when it is, it isn't, paradoxically.

I'm fond of Zeno's paradox which states that Achilles, even though he was the greatest athlete in the land, could not beat a tortoise in a footrace.  Sound ridiculous, doesn't it?  And yet, using logic which at the time was considered state-of-the-art, he managed to convince his fellow philosophers.  This demonstrates that, if nothing else, there are limits to the practical application of reason.

ElLagarto ElLagarto
56-60, M
7 Responses Jun 23, 2007

I agree with your observation and admire your methodology. The logic is sound, it's the leap to the conclusion that is flawed. Bravo.

Oh, I can see that this is old, but I remember this one. :)<br />
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I'd argue that reason explains anything that is within reason's grasp to explain; That logic can produce a conclusion that is as correct as its axioms.<br />
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I didn't mean to argue that Zeno's logic must be flawed *because* it is self evident that the conclusion is wrong. That was probably Zeno's point - illustrating that impossibility. What I mean is, his conclusion doesn't actually logically follow from his reasoning. (And that's not just our "current concept of logic", there's an actual element in the statement that is dropped!) It just sounds similar, but dropped the important logical point of "time". It's a bit subtle, though, so many miss the actual point where his logic fails, even though they understand THAT something wrong must be going on. Anyway, thus, he doesn't really manage to illustrate a point, it's just a schoolboy's error. (Forgive me for the metaphor, but it COULD be.)<br />
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"Each time Achilles had run half the previous distance." Basically, if we look away from the conclusion, it's a very thorough and tedious mathematical explanation of the point that Achilles wouldn't catch up *before* having run two times the initial distance between him and the turtle.<br />
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It's entirely valid up until the unexplained jump to the conclusion, which is easy to see if we look at it mathematically.

No.

"Reason cannot be used to understand, much less prove, anything of real consequence in this life." This means the consequence is what proves the reasoning and brings with it understanding? <br />
Is that what should be understood from your statement?

A marvelous illustration of humor's fragility. Actually I would maintain that one of mankind's greatest flaws is the belief that reason explains anything. We are capable of reason, consequently we believe it separates us from the lower beasts. But it is wildly overrated. Think of it as an elaborate parlor trick. Xeno believed that space and time were infinitely divisible - even he didn't actually believe the tortoise would win the race - it was an example to illustrate a point - something pedantic minds were doing even then. Reason cannot be used to understand, much less prove, anything of real consequence in this life.

Warning: Boring rant. :P Don't read on. Please.<br />
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There are limits to the practical application of reason, since not everything can be accurately broken down into simple concepts and thus represented adequately in the 'world of theory' - but logic *itself* doesn't fail in Zeno's paradox. Actually, that example (as with any theoretical thought experiment where all values are pre-defined) is entirely possible to look at, understand and 'solve' through logic alone - and it's not much of a brain tickler for logicians of today's standards.<br />
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It seems they had a very hard time wrapping their head around *that specific kind of* abstract concepts back then, if several of the top notch logicians didn't understand what Zeno was doing mathematically in his reasoning.<br />
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For those who don't know - Zeno said that if the turtle gets to start 10 m ahead of Achilles, and that the turtle would run 5 meters in the time that Achilles ran 10 (could also be 5 cm, as it doesn't really have to do with the logic), the turtle would, according to logic, always remain ahead, since each time Achilles would halve the distance between them, the turtle would have moved further.<br />
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Ie:<br />
Achilles runs 5m in the same time the turtle runs 2.5m.<br />
Achilles runs 3.75m (half the new distance between them) in the same time the turtle runs 1.375m<br />
Achilles runs half the new distance between them in the same time the turtle runs half that distance<br />
Etc. etc.<br />
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The conclusion here being that this would 'continue infinitely', and thus that Achilles would 'at no point' pass the tortoise. Blatant mathematical error here, of course, in the conclusion.<br />
And not only that - guess what? In the ultimate consequence of this, Zeno found that Movement™ is an illusion, since it would take an infinite amount of time to move *anywhere* if one broke it down like this. Horror!! XD<br />
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To do the old folks justice - Aristotle did see the fallacy and commented it. Still, I doubt anyone would get the kind of recognition today that Zeno did back then from such a proposal. He's still famous, but only because he received recognition for his ideas *back then*.

Scientific paradoxes are interesting. According to me though, the old Greek paradoxes are inconsistent as they divide time/space "weirdly". [For example in Zeno's paradox; why does the time converge?]