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Re: Is movement impossible?
[Re: PHeMoX]
#154119
09/15/07 18:48
09/15/07 18:48
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Joined: Oct 2006
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AlbertoT
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The point is that you consider " infinite " same as a " potential " number rather than an " actual " number
This was the ancient Greeks' idea of infinite
In the modern math infinite is considered as an ordinary number which can be manipulated same as any other number
You must not consider the series : 1/2 + 1/4 + ...
same as some thing unfinished
You must consider it same as any other normal math expression which supplies one and only one result
It is somewhat similar to the irrational number issue
The diagonal of a square having side 1 is : square(2)
Try to write this number
You can not , the number of figures after the decimal point being infinite
Does it mean that the diagonal of a square is infinite ?
Dont students use irrational numbers along with integer numbers ?
This is the logic explanation
As far as the Physical explanation is concerned, well actually it would not deserve even an explanation being evident that movement does exist regardless of the phylosophers' nonsenses
However
You claim that you have reached your goal when the ditance is : 0
But you must also measure such distance so you have to do with the measuring errors due to measuring istruments
Halving and halving your distance , sooner or later your super sophisticated istruments will tell you : hey you have reached your goal
Last edited by AlbertoT; 09/15/07 19:08.
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Re: Is movement impossible?
[Re: AlbertoT]
#154120
09/15/07 19:52
09/15/07 19:52
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Joined: Mar 2003
Posts: 5,377
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fastlane69
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Quote:
Nope , I dont agree
It will not take an infinite amount of time
If you shorten your step each time , the number of steps tend to "infinite " that's true , but the lenght to "zero" thus also the interval of time to cover each step tends to zero
Keyword: "tends to". Tends to in math means a limit or in this case, as n->inf....
...but how long does it take to get to infinity? If you start counting the infinite 1/2 numbers, whether this is time or distance, it will still take you an infinite amount of time. Don't forget that the nature of Zeno's paradox is the disconnect between mathematical models and physical reality. And that is what I'm saying: that it "tends to" but doesn't ever "acheive" is the basic definition of asymptotically approaching.
Using a series limiting process to "solve" the paradox does nothing of the sort for it assumes you can actually physically carry out that infinite sum... which is exactly what causes the problem to be paradoxical in the first place!!!
Or in the words of Wikipedia:
Quote:
Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned.
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Re: Is movement impossible?
[Re: fastlane69]
#154121
09/15/07 20:17
09/15/07 20:17
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Joined: Oct 2006
Posts: 1,245
AlbertoT
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Quote:
....
...but how long does it take to get to infinity? If you start counting the infinite 1/2 numbers, whether this is time or distance, it will still take you an infinite amount of time. Don't forget that the nature of Zeno's paradox is the disconnect between mathematical models and physical reality.
No sorry, it is not like that
You must disconnect between the mathematical models and the physical reality
Of course it will take a long time if you try to implement he Zeno's paradox in practice but the Zeno's paradox is a logic paradox and it must be solved in the domain of logic
You can not introduce in a logic discussion partical elements which have nothing to do with the logic
The Zeno's paradox is nothing else that one version of the ancient greeks' nightmare for the " infinite ", the most famous being the irrational numbers
The great Pitagora became almost crazy being not able to explain why a so evident definite entity such as the diagonal of a square can not be measured by an integer number
take the number ;
square(2) = 1.41421.....
This number has an infinite sequence of figures after the decimal point , you will never reach the end of it, nevertheless it is the measure of a well defined segment
This is the "key point" of the Zeno's paradox
The solution has been found two thousand years later thanks to the modern analysis and the theory of numbers
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Re: Is movement impossible?
[Re: Matt_Aufderheide]
#154123
09/16/07 17:25
09/16/07 17:25
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Joined: Oct 2006
Posts: 1,245
AlbertoT
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I quote fro Wikipedia " Status of the paradoxes today Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Most philosophers, and certainly scientists, generally agree with the mathematical results. Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this, he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's." As you can see the explanation which I provided in my previous post is normally accepted by all scientits "Zeno's paradoxes are still hotly debated by philosophers in academic circles" Well actaully I have not so much respect of Phylosophy 
Last edited by AlbertoT; 09/16/07 17:27.
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Re: Is movement impossible?
[Re: AlbertoT]
#154125
09/18/07 01:34
09/18/07 01:34
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Joined: Sep 2002
Posts: 8,177
Netherlands
PHeMoX
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Quote:
asymptotically reach it which means it will take an infinte amount of time to cover the final infinitely small distance
Which in practice means you'll never reach your goal.
Quote:
Nope , I dont agree
It will not take an infinite amount of time
If you shorten your step each time , the number of steps tend to "infinite " that's true , but the lenght to "zero" thus also the interval of time to cover each step tends to zero
This is the tricky part of the Zeno's paradox
Uhm, regardless, when each next step becomes half the step size of the previous then how on earth can you EVER reach your goal, especially when you go on for infinity dividing your next step by 2? It's impossible.
By the way, there's no such thing as a 'final' step when something really is infinite, even when the difference between infinite and (infinite+1 step more) becomes 'irrelevant' in terms of usage of our language.
Anyways, as fastlane said, you really can only reach your goal this way when your very first step practically gets you there (for example when the whole journey only takes 2 steps in total), since your next step will be half of your last in distance.
Cheers
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Re: Is movement impossible?
[Re: PHeMoX]
#154127
09/21/07 20:49
09/21/07 20:49
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Joined: Oct 2006
Posts: 1,245
AlbertoT
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Quote:
you really can only reach your goal this way when your very first step practically gets you there (for example when the whole journey only takes 2 steps in total), since your next step will be half of your last in distance.
Cheers
You are simply playing with words, same as many phylosopher normally do
What does it mean ?
If you claim : It is not possible to move from A to B , you must be aware, first of all, that B does actually exist
How do you know it ?
You should at least receive a signal from B, maybe just some photons of light
So something must be able to move , anyway
Not only
The paradox claims that you will never reach the goal
Ok , suppose that such claim is true
Howeve it admit that you can get closer to your goal
Isnt' a movement ?
As you see if you get out of the domain of math and logic , Zeno's paradox is simply a nonsense
Last edited by AlbertoT; 09/21/07 21:00.
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Re: Is movement impossible?
[Re: AlbertoT]
#154128
09/21/07 21:20
09/21/07 21:20
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Joined: Sep 2002
Posts: 8,177
Netherlands
PHeMoX
Senior Expert
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Posts: 8,177
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There is movement, but it's not possible to reach your goal if each next step is half of your last step ánd when the first step doesn't cover more than half of the total distance you want to move. If you keep dividing a cake by two forever, you'll only get smaller and smaller crumbles, but never the full cake. There will always be a next crumble (infinitely small and beyond) that gets divided by two. I tends to zero, but never reaches it. I do understand that in practice you should not think in terms of distance dividing by two, however íf you would do so... basically the 'formula' you would use to move is one of slowing down for infinity dividing your distance by two. One thing that I don't quite get though is how you can mathematically go from infinitely small, to zero, to negative numbers. In a logical way this doesn't quite make much sense. Perhaps that's your problem with this "paradox"? Quote:
You are simply playing with words, same as many phylosopher normally do
What does it mean ?
If you claim : It is not possible to move from A to B , you must be aware, first of all, that B does actually exist
Actually, I didn't say that we aren't aware of A and B, nor did I say that it's impossible to move from A to B. It ís however impossible to reach your goal when you use the method of movement as Zeno said. His conclusion of 'ow then there can't be movement' is ridiculous though,
Cheers
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