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Re: Is movement impossible?
[Re: Helghast]
#154109
09/14/07 22:40
09/14/07 22:40
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Joined: Jan 2007
Posts: 1,565
innsbruck, austria
achaziel
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Serious User
Joined: Jan 2007
Posts: 1,565
innsbruck, austria
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Quote:
It's a strange claim by the way, because everytime we reach the door we have proven to be able to cover all these infinite halfway points all the way up to the door, regardless of the scale you choose before you start.
/signed.
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Re: Is movement impossible?
[Re: achaziel]
#154110
09/14/07 23:41
09/14/07 23:41
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Joined: Mar 2003
Posts: 5,377
USofA
fastlane69
Senior Expert
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Senior Expert
Joined: Mar 2003
Posts: 5,377
USofA
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Consider that you take steps that are 1/4 the length of the room. In two steps you've reached the halfway mark... so far tracking with Zeno. In three steps you've reached half of what was left in two steps... Zeno is feeling awefully smug! In four steps you've reached the destination... Zeno is quite confused at this point.  The easier resolution lies in realizing that just because you cross 1/2 the distance you are not restricted to only that or put another way the discreteness of steps resolves the paradox. By this I mean that Zeno's paradox is completely true: if you took a step that was EACH TIME half the distance to the end of the room, (in other words if you shortened your step each time) you would never reach it (or rather asymptotically reach it which means it will take an infinte amount of time to cover the final infinitely small distance). But this doesn't mean that ALL movement is impossible. When the paradox states that you have to move half the distance, that is true... but at the point where you reach the 1/2 way mark you don't stop... you actually go further than half way. So if you take steps that are a third the distance to the end, after the first step you have covered the 1/2 way mark (Zeno Smiles), after the second step you have covered half the remaining distance (Zeno Laughs), but after the third step you are at the end (Zeno Crys). So the paradox is no paradox at all: if you follow it exactly, shortening your step by half, then it's true; if you take same sized steps the whole time, then it isn't.
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Re: Is movement impossible?
[Re: fastlane69]
#154112
09/15/07 10:26
09/15/07 10:26
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Joined: Oct 2006
Posts: 1,245
AlbertoT
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Posts: 1,245
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Quote:
if you shortened your step each time you would never reach it or rather asymptotically reach it which means it will take an infinte amount of time to cover the final infinitely small distance.
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
We focus on the term "infinite" , ignoring the term " infinitesim" but both terms must be considered simultanuosly
If so you get a finite result
Zero*infinite = finite ( in some cases)
If the distance if "d" and the speed is "v" the time "t" to reach the target is, according the common sense
t = d / v
A finite value
But you can get the same result also via the Zeno's paradox
Cosider the time t as sum of the partial time : t1,t2,...tn
Where
tn= 1/ 2^n d/v
for n = 1,2,3.....infinite
you have t = d/v( 1/2 + 1/4 + 1/8 + 1/16 + ....)
The number of addendum between the brakets is infinite, nevertheless their sum is : 1
so, again, you get t = d / v
same as common sense would suggest
Obviously I ma talking in principle ,in practice if you try to cover a certain distance using the Zeno's method you will actually take a long time , if not an infinite time, but this has nothing to do with the paradox itself
Last edited by AlbertoT; 09/15/07 10:29.
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Re: Is movement impossible?
[Re: AlbertoT]
#154114
09/15/07 11:48
09/15/07 11:48
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Joined: Sep 2002
Posts: 8,177
Netherlands
PHeMoX
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Senior Expert
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Posts: 8,177
Netherlands
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Quote:
Obviously I ma talking in principle ,in practice if you try to cover a certain distance using the Zeno's method you will actually take a long time , if not an infinite time, but this has nothing to do with the paradox itself
This is where you lost me. That's exactly the prove I had in mind about why these particular infinites are not finite when all added together.
On the other hand I now do understand why in theory it must be finite, after all you're talking about áll numbers in between two numbers which all together make up the difference between those numbers, not less and not more. The numbers in between are infinite in amount, so if added up altogether they must be finite. BUT in this paradox each subsequent number is half of the last, so even when infinite, it can never consist of every number inbetween and thus can not become a finite number.
Code:
you have t = d/v( 1/2 + 1/4 + 1/8 + 1/16 + ....)
The number of addendum between the brakets is infinite, nevertheless their sum is : 1
You're actually missing a lot of numbers in between each step because of the step size (half of each old step size), so isn't it not infinite and therefore not every number between those two points? The finite number 1 in this case simply means 0 to 1 and the infinite amount of numbers in between, but what if you're skipping a lot of those infinite numbers? Isn't it smaller than the 'other' infinite?
Quote:
Is universe a reality or just a dream ?
Ever had a dream inside a dream waking up in your dream but not in reality just yet? Or ever had a serious déjà vu?
Pure theoretically the question is about another possibility.
Problems with all this, and that's where most people stop thinking about this, are our definitions of 'real', 'reality' and 'dream'. It's the good all Matrix movie question "how do you know something is real?". Well, actually we don't know, but by definition it must be... I think there's plenty of room for philosophers to seriously think about this, question reality and so on. The thing about philosophy is that some parts of it are rather hard to prove once you've made up your mind about something like 'reality', you'll either need to prove it somehow or convince people...
Cheers
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Re: Is movement impossible?
[Re: PHeMoX]
#154115
09/15/07 12:38
09/15/07 12:38
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Joined: Jan 2003
Posts: 4,305
Damocles
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Joined: Jan 2003
Posts: 4,305
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There is no such thing as an infinitly small distance. At the atomic level, there is still the law of quantum uncertainty. So there is at a certain point no senseful determination of a smaller distance. So there is a limited amount of steps to reach the goal, and not an infinit. ---- We can state that the worl we see is real: -> the world is so complex, we can interact with it, and the reactions are even more complex. Simulating such a complex world would be almost imposible. And even if: there is no economical reason for someone to make such a complex simulation, just to observe our primitive (for them) reactions. Would be cool though to be able to change the simulation. I would assign me free money, and pretty girls and a big boat.  And call me: "Big Daddy", or "Luder Horst".
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Re: Is movement impossible?
[Re: Damocles]
#154116
09/15/07 13:33
09/15/07 13:33
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Joined: Sep 2002
Posts: 8,177
Netherlands
PHeMoX
Senior Expert
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Senior Expert
Joined: Sep 2002
Posts: 8,177
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Quote:
-> the world is so complex, we can interact with it, and the
reactions are even more complex.
Simulating such a complex world would be almost imposible.
And even if: there is no economical reason for someone to
make such a complex simulation, just to observe our primitive (for them) reactions.
Perhaps from where we are standing and in our perspective this is true, but perhaps the reason behind the simulation was something other than just personal gain, scientific insight, economic benefit or other such very human things. We can't quite tell. 
Perhaps the world ís a dream, only created by giving the real us some chemicals that causes us to dream? In such a case it wouldn't necessarily be about any benefits except that we're not moving in the real reality.
Personally I don't think complexity is a boundary, why would it be? If you look at us 3000 years ago and now there's a massive and back then unimaginable difference in level of technology. Why would super complex things be impossible?
Cheers
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Re: Is movement impossible?
[Re: AlbertoT]
#154118
09/15/07 14:59
09/15/07 14:59
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Joined: Sep 2002
Posts: 8,177
Netherlands
PHeMoX
Senior Expert
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Senior Expert
Joined: Sep 2002
Posts: 8,177
Netherlands
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Quote:
The number of addendum between the bracket is infinite it is not just a long number, while the result is 1 , it is not just something between 0 and 1 , it is 1 (one)
Modern maths treats infinite same as an ordinary number
That's exactly what I've said actually, but perhaps I simply drew the wrong conclusion. I do understand that if all those numbers are added that it will come close to 1.
Anyways, here an image to show what I mean:
Still it bothers me that it wíll reach the full 100% when added, that doesn't make much sense if the next number, eventhough infinite, will always be half of the last one. Even when an infinite sequence, the numbers will always be divided by two each time. Why would it still reach the full 100%?
There are plenty of formulas that come close to 0 with a nice curve but will never reach it, even if it goes on forever simply because of the nature of the formula (divides the next number by two for example). Honestly I don't see why this Zeno problem is any different from that,
Cheers
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