Zenon's paradox

[Elysian]

The only time I pick up an old dusty book from my shelf is when I'm sick at home from work. And today, after getting tired with crappy tv sitcoms or surfin the net, I found a book about philosophy. In one of the chapters I found an interesting, mind-puzzling story called Zenon's paradox.

 

The story is about the greek hero Achilles, the fastest man in town, who will race with an old turtle. Since Achilles is a fast runner and the turtle is slow, Achilles gives the turtle some advantage by giving him a head start. The simple question is then who will win the race? The obvious answer is Achilles, but according to the old greek philosopher Zenon, if one uses strict logic, Achilles can never win this race... why?

 

Well the book says that once Achilles has reached where the turtle started the turtle has bridged some distance - not much but still - which Achilles now has to run in order to catch up. And in this time again the turtle has gone for some distance and Achilles is still in behind of the turtle. This process continues forever and Achilles will never be able to pass the turtle.

 

I'm not sure I understood it right, but I guess that this really tiny tiny distance the turtle covers each time is divided into smaller and smaller distances that Achilles need to catch up, and because the distance is divided in ad infinitum, forever, Achilles will never win... does it make sense? But then again we know for sure that Achilles will win, so what is it that is wrong with Zenon's paradox? Is it so that the Zenon's logic doesn't take into account that the speed is different for Achilles and the turtle, or is it something else?

[Johnny B]

Zenon just wanted to show that numbers are infinite, and i'm sure that you've no problems with that.

Your Hero Achile ofcourse wins the RACE if that was what you doubted , but numerically he'll always stay behind that lazy turtle.

[Cara.]

I don't get it. Sure, Achilles will cover the same distance, but he'll do it in less time, so eventually he'll catch up and pass the turtle. This puzzle assumes Achilles and the turtle have the same speed, but we're already told Achilles is faster.

 

The version of Zenon's paradox I'm familiar with involves a frog... This is dim memory from high school: Imagine a frog on a lily pad at one end of a pond. The frog needs to get to the other end of the pond by hopping from one lily pad to another. If the lily pads are spaced such that he covers half the distance each time, will he ever get to the end of the pond?

 

The issue here I think was that there is no number such that half of it is zero, or something like that...

[Caano Geel]

its the notion of mathematical infinity

-

it says you can never reach infinity because it is not a rational number and so cannot be counted.

 

the trick here is that it demonstrates the requirement to count as fundamental property. And the clever bit is that it shows that countable numbers can be expressed as ratios of other countable number - [this the def. for rational numbers]. The realy really clever bit is that it does this on the small scale rather than the grand scale by using an infinite precision.

 

The simplest way to see this is to imagine that we have a ruler that measures the distance between archy and the turtle and gives us a number, call it 'A'. if we know 'A' then we know what half of A is. Now imagine our rules is slightly more fine grained and gives us an extra precision of 'e', so instead of a measurement of 'A' we can instead get a measurement of 'A+e'. Again because we know 'e' we know what half of 'A+e' is. and so on.

 

Now imagine our ruler is kind of magical and the closer we look at it, the better precision we get from it. So instead of having a constant extra precision of 'e', 'e' now depends on how closely we are looking, the closer we get, the better it is at judging the real/accurate distance. In other words, the more precise we get, the smaller 'e' gets. And as a consequence the only thing that determines the value of 'e' now is our own perception - i.e. how closely we can look, not the values that 'e' can hold.

 

In this way the magical ruler is a analogous to a measure of a number, where the precession is determined by the increment we use. i.e.

2+0.01 or 2+0.001 or 2+0.0001 we can make that extra precision as small we like because there is no limit to the number of '0''s we may use. Such numbers are now called 'real numbers' their characteristic is that there is no limit to their decimal representation or precision. So as a consequence, they are uncountable - the only way to count them is to fix the precision to a single number, ( for example setting our precision to 0.01 would give us a 100 points between any two whole number, i.e. 0-1 or 1-2 ..) -- in the same way as the number unique measurements you can get from our magic ruler is determined by the maximal level of precision we ourselves can observe.

[Castro]

^^^^ You had to go out and ruin the thread for everybody. Must you always show off your high school calculus? :rolleyes:

[Caano Geel]

look, just 'cos your too dump to follow, dont attempt to take every one else down with you :rolleyes:

[Castro]

^^^^ The really dumb person here is Achilles. Allegedly the fastest man in town and yet goes out to have a race with an old tortoise that he can't even beat.

 

Do you think Achilles had psychological issues?

[Cara.]

People didn't have psychological issues back then. They had character.

 

He had trouble with one of his heels though.

[Castro]

^^^^ I told you he had issues. The man wore heels and raced old tortoises for crying out loud.

[Cara.]

You can't judge him by today's standards!

[Elysian]

JB and CG... You guys make it sound so easy! Anyway, CG's magic ruler was a good illustration of the problem, and I understood that part. But what bothers me is that although Zenon's reasoning seems correct, the conclusion is for sure wrong. So there has to be something incorrect with Zenon's paradox... right?

 

Castro lol, who said Achilles wore heels

[Castro]

^^^^ The paradox ignores time as a variable. That's what's incorrect.

 

And Achilles did wear heels, at least two of them. He was a drag queen you know.

[Caano Geel]

Originally posted by Elysian:

JB and CG... You guys make it sound so easy! Anyway, CG's magic ruler was a good illustration of the problem, and I understood that part. But what bothers me is that although Zenon's reasoning seems correct, the conclusion is for sure wrong. So there has to be something incorrect with Zenon's paradox... right?

Maybe not, at the most pedantic level, we never actually reach anywhere or touch anything.. since at an atomic level the electron shells around atoms act to repel each each other, so there is always a gap between any 2 objects and at a quantum level everything is a probability density so there is no certainty about anything being anywhere, just a likelihood

 

.. i think u're better of just ignoring 'ol fidel there, much like his other major organs his brain has long vacated the space previously occupied by thought in favor of the wonders of heels

[Johnny B]

Originally posted by Elysian:

JB and CG... You guys make it sound so easy! .... what bothers me is that although Zenon's reasoning seems correct, the conclusion is for sure wrong. So there has to be something incorrect with Zenon's paradox... right?

 

Castro lol, who said Achilles wore heels

shhhhhhhhhh, don't get SB here with your Paradox now that you already got Cara, CG, and Fidel here, that is if you want to keep your Paradox alive.

 

So you got problem with the conclusion but just can't put finger into exactly where?

hang on here becouse i'm gonna take you for a rough ride to bedrock, since you know that pedants are boring

 

16/64 = 1/4 that fraction is correct right? good.

now if Zenon came to that conclusion by canceling a 6 from top and a 6 from bottom and concludes that the fraction 16/64 equals one fourth reasoning, that he canceled equal number from both the numerator and the denominator , and anybody who claims that i'm wrong is welcome to work it out and come to a different conclusion.

 

Now, what would you say about his reasoning and conclusion?

 

Heyyy, Achiles did wear heels !

[Cara.]

^You would too if you had to wear a dress.