| A twist on Zenos infinity paradox
Posted: 9/14/2008 11:50:26 AM | I just walked into my living room from my bedroom. I had to go half the distance an infinite amount of times to get here. I did arrive. Thus the math problem is solved. The rabbit will reach the hole because I reached my living room.
You say I did not go half way all the time? But I argue I did. I can not cover a distance without first covering half the distance. I am in my living room so I covered all the halves. I guess I found the end of infinity.
No, I think you found a different problem with a different set of rules to solve. You didn't solve the problem at hand within the rules set up by the problem itself. BTW, I think you found a wonderful solution to your problem - so maybe if you put your mind to the task of solving this problem, we might have a discussion.
Although that's spot on, I was hoping it would give rise to philosophy as well.
Fair enough. What's the philosophical question? If Zenos' infinity paradox accurately describes infinity? How could it not? | |
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| A twist on Zenos infinity paradox
Posted: 9/14/2008 11:57:56 AM |
Fair enough. What's the philosophical question? If Zenos' infinity paradox accurately describes infinity? How could it not?
See? I figured if left in a thread about infinity with a riddle that the questions would arise naturally.
How could it not? By being misunderstood. | |
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| A twist on Zenos infinity paradox
Posted: 9/14/2008 12:11:17 PM |
See? I figured if left in a thread about infinity with a riddle that the questions would arise naturally.
How could it not? By being misunderstood.
But... but... isn't that how ALL philosophical questions work?   | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 4:33:00 AM | What a crazy tread.
And as far as rabbit goes in reality (in our 3 plus 1 dimensional world) he will get to that hole just fine.
In Zeno’s paradox he will newer reach it –or should I say he would have to hop infinitely.
It is one of the ancient Greek philosopher, Zeno of Elea, more famous paradox of motion and it’s actually called the dichotomy paradox:
‘’That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ‘’—Aristotle, Physics VI:9, 239b10
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 5:37:14 AM | ‘’That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ‘’
While this is true, it's only a paradox as far as the actual breaking down into fractions goes.
There's no law of motion that says we can't pass over the half way point mid-stride. We still hit it before the goal.
I'm more interested in the part where you can break down anything infinitely.
Next month we may see the smallest get smaller... Again. | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 7:22:48 AM |
Although that's spot on, I was hoping it would give rise to philosophy as well. Consider anything that moves. Say a tennis bal launched from a tennis launcher. How does it move? In a single second, say it travels a metre.
But then, it needs to cross the first 1/2 metre before it can reach the second 1/2 metre, in the first half-second.
But then it has to cross the first 1/4 metre in the first 1/4 second, before it can reach the second 1/4 metre, in the first 1/4-second.
But then it has to cross the first 1/8 metre in the first 1/8 second, before it can reach the second 1/8 metre, in the first 1/8-second.
And so on, and so on.
By using Mathematical Induction, we can divide by 2 forever. But then we'll get zero.
Eventually, we get:
But then it has to cross the first 0-th metre in the first 0-th second, before it can reach the second 0-th metre, in the first 0-th-second.
So what is the velocity of the ball? Well, 0 divided by 0 is indetermindate. It can be anything and we don't know what it is. So we don't have any clue about the speed of the ball. Nor can we prove it moves at this point.
That is the problem with ALL movement. Everything can only be observed moving macroscopically. But when we start looking microscopically, we find that eventually, we won't be able to tell how fast it is moving, and the only way that is possible is if we really can't see it properly.
Xeno's problem was that to measure speed more accurately, you need to take more samples, by splitting your sample of distance and time in half. But to Xeno, it seemed that you would get to a point at which the thing didn't seem to move. So if you added all those non-movements up, you would get a speed of 0, which would contradict the way things look when you just take a much less accurate method of measurement.
Fancy that, the less accurate result makes sense, and the more accurate result doesn't. Who would have thunk it?
once you know that speed is truly indeterminate, what happens when you add it all back | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 8:52:01 AM | I was a math student for many years. There are many aspects of math that I still enjoy, but they tend to be the aspects that can be related back to the real world. The subleties and paradoxes arising from the study of the infinite (on both the large and small scales) stopped interesting me long ago because they are pure speculation.
Everything in the universe is finite. For example: (age of the universe)X(speed of light)=(maximum size of the universe).
Note: Let K=1/2+1/4+1/8+... then 2K=1+1/2+1/4+1/8+... therefore K=2K-K=1. | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 5:58:25 PM | ^^^ Geometric series: 1+ a + a^2 + ... + a^(n-1) = (a^n - 1) / (a - 1)
So, taking n to infinity, we get 1+ a + a^2 + ... = (a^infinity - 1) / (a - 1)
So, taking a < 1, we get 1+ a + a^2 + ... = (a^infinity - 1) / (a - 1) = (0 - 1) / (a - 1) = 1 / (1 - a)
So 1+1/2+1/4+1/8+... = 1/ (1-1/2) = 1 / (1/2) = 2
So 2K = 1+1/2+1/4+1/8+... = 2
So K = 1
It's a direct result of geometric series. | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 6:41:40 PM | | Frogg0 was close - 3 hops. He noticed he only made it a foot in the first hop and a half foot in the second hop, knew about Zeno, recognized the problem so in the third hop he landed on his head and rolled straight down the hole. | |
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| A twist on Zenos infinity paradox
Posted: 9/16/2008 7:41:56 PM | | This reminds me of Dangerous Knowledge (a BBC Documentary I think). It's about how Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing's minds drove them all to suicide trying to prove/figure out the impossible, but changed how we view reality. Very interesting. | |
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| A twist on Zenos infinity paradox
Posted: 9/18/2008 11:35:31 PM | There is no last jump. The rabbit stays still because it would take an infinate amount of energy to make the last jump and that would make him infinately massive. Notice the statement says "...he is getting increassingly tired. " Meaning, the energy required for each successive jump is double the sum of all of the energy needed in the previous jumps (it's exponential). The Rabbit does not have enough energy because there is no end to space but the rabbit has energy limits. He comes to a point where he's approached the speed of light and all motion stops. I saw Zeno's rabbit jump into the Black Hole in Disney's Alice in Wonderland. He also did a series of Energizer batteries commercials.
Allan Turing committed suicide not form trying to solve difficult problems. But,because he was imprisoned and tortured by the UK government for being homosexual. He bit an apple injected with Cyanide. Apple Computers has the symbol with an apple with a bite in it. Turing helped the UK government crack the German Enigma cypher code used by their U-Boats aka Underwaser boten to communicate in code. | |
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| A twist on Zenos infinity paradox
Posted: 9/18/2008 11:58:51 PM | | This rabbit would inevitably have to divide by zero...then he would be going in another direction. Which direction...who knows? | |
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