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#1 2007-01-02 08:50:53

Zeroface
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Registered: 2006-12-29
Posts: 11

Strange things about Xeno's Paradox

According to what Xeno said about measuring and motion. Something can't travel to somewhere without first travelling half-way. So this means that a measurement of 0 to 1 would be 0.5. This is not logical and this post is to explain what a quack Xeno was.

P.S For those who don't know, Xeno was a greek person who
liked measuring things. cool


0 can be nothing and something.
                        0

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#2 2007-01-03 02:43:17

Patrick
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Registered: 2006-02-24
Posts: 1,005

Re: Strange things about Xeno's Paradox

Wikipedia wrote:

What the paradox says is that if you are walking 5 feet, you must first move 2.5, then 1.25, then 0.625, etc, dividing by 2, and never reaching 0 because division doesn't reach 0. But humans don't walk via division of distance. We walk via subtraction of step length from distance. So if we take 1-foot steps, our objective is 5 feet away, then 4, then 3, and eventually we're there.

What you have to remember, is that Zeno was trained in argueing, not math(primarily anyway).

Last edited by Patrick (2007-01-03 02:44:18)


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#3 2007-01-03 11:38:45

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

According to what Xeno said about measuring and motion. Something can't travel to somewhere without first travelling half-way. So this means that a measurement of 0 to 1 would be 0.5. This is not logical and this post is to explain what a quack Xeno was.

I'm sorry, but I don't see where you explained how Xeno is a quack.

To me, this is a very important critique of our view of space as being continuous.  The problem is that we don't know what happens at the Planck length.  My opinion on this is that it is a scientific concept (not mathematical), and thus, I accept space as being continuous on the fact that the statement has yet to be falsified.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2007-01-03 23:56:19

Patrick
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Registered: 2006-02-24
Posts: 1,005

Re: Strange things about Xeno's Paradox

Well, as I see it, Zeno of Elea(it's not Xeno) got this all wrong. Instead of saying before you can move one unit, you have to move a half unit, you should be saying when you move one unit you also move a half unit.


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#5 2007-01-04 07:34:08

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

Instead of saying before you can move one unit, you have to move a half unit...

But is that first part wrong?  If so, how?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2007-01-04 07:59:38

Patrick
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Registered: 2006-02-24
Posts: 1,005

Re: Strange things about Xeno's Paradox

Because it's implying that moving a half unit is a seperate action from moving the whole unit. An action that has to be done on it's own and before the other.


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#7 2007-01-04 10:10:33

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

Patrick wrote:

Because it's implying that moving a half unit is a seperate action from moving the whole unit. An action that has to be done on it's own and before the other.

Are you saying that to move a half unit, one must also move a full unit?  Because if that isn't the case, then they are separate actions.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#8 2007-01-04 15:02:06

Toast
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Registered: 2006-10-08
Posts: 1,321

Re: Strange things about Xeno's Paradox

Well I agree that you have to move half a unit before you move one... but that wouldn't make a measurement from 0 to 1 = 0.5.

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#9 2007-01-04 16:29:13

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

...but that wouldn't make a measurement from 0 to 1 = 0.5.

You lost me there.  Is the measurement equal to 0.5?  Or is whatever your measuring from 0 to 1 equal to whats at its median?

I'm not even sure why you're talking about measurements being equal to something.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#10 2007-01-04 19:04:20

Toast
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Registered: 2006-10-08
Posts: 1,321

Re: Strange things about Xeno's Paradox

Zeroface wrote:

According to what Xeno said about measuring and motion. Something can't travel to somewhere without first travelling half-way. So this means that a measurement of 0 to 1 would be 0.5. This is not logical and this post is to explain what a quack Xeno was.

P.S For those who don't know, Xeno was a greek person who
liked measuring things. cool

That's what I was responding to

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#11 2007-01-04 19:18:45

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

I guess I really don't understand that either.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#12 2007-01-04 19:20:11

mikau
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Registered: 2005-08-22
Posts: 1,504

Re: Strange things about Xeno's Paradox

To get from point a to point b you need to move half the distance there first. Then from there to b you need to move half the remaining distance and so on. So its a "Half the distance to the goal" thing that keeps repeating. The limit will be the goal point but it will never actually get there. Which I'm guessing is why Zeroface is calling him a quack. (Because we all know you can get there and move beyond it.) But his wording is still unclear.

Last edited by mikau (2007-01-04 19:29:00)


A logarithm is just a misspelled algorithm.

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#13 2007-01-04 19:47:03

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

But the argument is against continuous space, not whether or not you can travel.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#14 2007-01-04 20:16:49

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Strange things about Xeno's Paradox

aye, but i think he may have been using one to prove/disprove the other. I don't know. It was the first thing that ran through my head.


A logarithm is just a misspelled algorithm.

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#15 2007-01-05 06:59:35

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Re: Strange things about Xeno's Paradox

Ricky wrote:
Patrick wrote:

Because it's implying that moving a half unit is a seperate action from moving the whole unit. An action that has to be done on it's own and before the other.

Are you saying that to move a half unit, one must also move a full unit?  Because if that isn't the case, then they are separate actions.

Okay, you got me wrong there. I thought you would understand that all I was saying, was about the case where the goal was to move one unit. In that process, the half unit movement is a part of the full unit movement, not a seperate process.


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#16 2007-01-05 09:15:31

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Strange things about Xeno's Paradox

I think Zeno is more clever than you're giving him credit for. He obviously didn't actually believe that his assertions were true - he just made them and then used the real world to contradict them. Hence it being called Zeno's paradox instead of Zeno's theorem.

Zeno liked to make up things that used apparently sound mathematics and yet that were clearly false, to annoy all the mathematicians of the time. A more famous example is of the tortoise and the athlete, although the reasoning used is kind of similar.


Why did the vector cross the road?
It wanted to be normal.

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#17 2007-01-05 10:18:15

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Re: Strange things about Xeno's Paradox

mathsyperson wrote:

I think Zeno is more clever than you're giving him credit for. He obviously didn't actually believe that his assertions were true - he just made them and then used the real world to contradict them. Hence it being called Zeno's paradox instead of Zeno's theorem.

Zeno liked to make up things that used apparently sound mathematics and yet that were clearly false, to annoy all the mathematicians of the time. A more famous example is of the tortoise and the athlete, although the reasoning used is kind of similar.

Well, that's exactly my point! It doesn't really contradict the real world, or the world as we experience it anyway, because the premises are wrong. This is one of the things that can qualify a statement as a paradox, so Zeno succeded in making a paradox I guess. Though, I don't think it's of any importance to mathematics, only to the study of rethoric.


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#18 2007-01-05 10:35:21

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

I'm sorry, I'm still not understanding.  Which premise is the one which you are saying is wrong?

And I don't see this as a paradox of math.  Just of a continuous space universe.  If space is not continuous, then the paradox breaks down as there will be a point where you can't split a distance in half.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#19 2007-01-05 13:21:44

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,555

Re: Strange things about Xeno's Paradox

It would help if we knew what space and length is.

I defined Space as "The region in which objects exist.", and Length as "Distance. How far from end to end." in the math dictionary.

(If anyone else would like to offer concise definitions better than those, please do.)


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#20 2007-01-05 14:26:56

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

I don't think the definitions are the problem.  Just let space and length be as we naively think about them.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#21 2007-03-19 15:53:30

elliptic=modular
Member
Registered: 2007-03-19
Posts: 5

Re: Strange things about Xeno's Paradox

hmm...i think that what zeno was referring to is that all change can be reduced to .0000...on to infinity...1. since the limit of this is zero, we can never move, hence the paradox. the problem with this is that all change cant in fact be reduced to .0000...1. which of course, leads us to the nature of the continuity of change. this leads us on to continuous functions on the complex plane, i think...or maybe im just a freshman who knows nothing.

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#22 2007-03-19 16:08:57

elliptic=modular
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Registered: 2007-03-19
Posts: 5

Re: Strange things about Xeno's Paradox

o wait, i totally got it.
the universe cant be continuous, or else we run into this paradox, where any division of distance can be reduced to .000...1. obviously a paradox. therefore, space must be discrete, which is the only other logical logical possibility. a sort of proof by contradiction of the discreteness of the universe. as mentioned before, the plank length is the discrete unit of space, just like the quanta is the discrete unit of energy...which was discovered by max planck. this in itself leads to interesting physical conclusions, but also mathematical ones. yay!

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#23 2007-03-19 17:43:24

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

the universe cant be continuous, or else we run into this paradox, where any division of distance can be reduced to .000...1. obviously a paradox. therefore, space must be discrete, which is the only other logical logical possibility. a sort of proof by contradiction of the discreteness of the universe. as mentioned before, the plank length is the discrete unit of space, just like the quanta is the discrete unit of energy...which was discovered by max planck. this in itself leads to interesting physical conclusions, but also mathematical ones. yay!

The bolded part is a logic fallacy known as a False Dilemma.  You are forcing me to choose between X and Y, when in reality other choices such as Z exist.

A more accurate statement is "which is the only other possibility, that I can think of".

Perhaps movement is discrete and space is continuous.  Perhaps space is continuous but acts with very weird properties on the small scale.  Actually, we already know the 2nd is true.

The list goes on...


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#24 2007-03-20 16:10:00

elliptic=modular
Member
Registered: 2007-03-19
Posts: 5

Re: Strange things about Xeno's Paradox

Ricky wrote:

the universe cant be continuous, or else we run into this paradox, where any division of distance can be reduced to .000...1. obviously a paradox. therefore, space must be discrete, which is the only other logical logical possibility. a sort of proof by contradiction of the discreteness of the universe. as mentioned before, the plank length is the discrete unit of space, just like the quanta is the discrete unit of energy...which was discovered by max planck. this in itself leads to interesting physical conclusions, but also mathematical ones. yay!

The bolded part is a logic fallacy known as a False Dilemma.  You are forcing me to choose between X and Y, when in reality other choices such as Z exist.

A more accurate statement is "which is the only other possibility, that I can think of".

Perhaps movement is discrete and space is continuous.  Perhaps space is continuous but acts with very weird properties on the small scale.  Actually, we already know the 2nd is true.

The list goes on...

ah...you have got me thinking again. in fact, when i wrote that i almost put in the "that i can think of" part. tongue of course, "those strange properties" that is, quantum physics (i think thats what you mean) dont mean the universe is continuous do they? i mean, the discrete space theory of the plank length is accepted, right?

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#25 2007-03-20 16:22:13

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Strange things about Xeno's Paradox

i mean, the discrete space theory of the plank length is accepted, right?

I've never even heard of a non-philosophical based discrete space theory.  That is, one which comes out of actual physics instead of mind-experiments.  The plank length is how much one has to "zoom in" until something known as quantum foam is observed.  This is where space gets really weird and freaky and general relativity, which requires space to be nice and smooth, breaks down.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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