For every step I take forward before I do so I must make half of that distance. I am now at the half point between making a step and making a step. Before I can make the step I must go half way between the half way point and the finish. Before I can make the next step I must go half way between the 3/4 step and the finish.
Therefore motion is impossible since we are infinitely approaching it but never making it.
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Not the Same Thing16% Voted for by Mephitic ID Synergy, beauty-of-neptune, WhiteShadow, TeChNoWC.
I don’t believe this is the way motion works, but in the case of the Tortoise and Achilles in the original Zeno’s Paradox, the math makes sense. In that case, no, we only get 99.999999999% there, but in your last statement, you said motion is impossible because we never get there. True, we never reach the goal, but we have moved 99.9999999% of the way there. Maybe not 100%, but it’s still motion.
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It doesn't actually work that way.12% Voted for by Mephitic ID Synergy, WhiteShadow, eightball.
But you aren’t really dividing distance, you are dividing time. Distance = rate time. The rate, I assume, stays constant. If you can continually slow the pace of your step, to the point where the degree of motion would be inperceptable on the macrocosmic scale in which we operate, then you are some sort of demigod – but certainly not a human.
It’s not that you’ll never get to the end of your step. What you’re suggesting is that if you had the ability to analyse infinite points along this continuum then you would be able to infinitely divide the progress of the step by two.
One way to look at what you have presented is to use the distance = rate time formula that I presented earlier. As I stated, rate will stay the same throughout. If we divide distance by two – going half the way – then we must do the same to the other side of the equation. Thus, time is halved for every time that distance is halved. So no matter how many times we halve distance, nothing changes. Rate remains the same. Given the total distance of the step, the pedestrian completes the step in a given amount of time, determined by the rate of his or her step.
The situation that you presented assumes that the rate changes, though nominally it says time changes. While this may be possible in one’s imagination, it is not possible, nor is it correct according to the formula for distance and time, in reality.
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If one thing is infinite, everything must be infinite8% Voted for by Just Paper, Muriel.
It wouldnt matter anyways. There is nowhere to stop anyways. We have no barriers. Motion is very real, but movement is but an illusion. We move on top of the Earth, just as the Earth moves under our feet.
Always, Sam
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Pffffft8% Voted for by WhiteShadow, crazimyke.
The entire theory is ridiculous. If one is able to take half of a step, why can they not take another half? If they take two half-steps, then they have taken a full step. Eventually your foot will reach its destination relative to Earth. If one can take 1/2 or 3/4 or 7/8 or 15/16 of a step, why would they not be able to finish the whole thing? This is a fault in human imagination, not in movement itself.
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Er8% Voted for by WhiteShadow, justonewish.
But motion is possible because we do not just take half-steps. It’s that simple – motion does not work as described in the paradox.
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The Eleatic Paradox and LogicVoted for by GaryCGibson.
Reading logic such as is used in writing computer programs (Boolean and etc.) isn't quite the same as classical and ordinary symbolic logic, yet each could perhaps resolve the problem of infinities within finite paradigms.
Zeno's paradox does perhaps have a inadequate premises and provides a false conclusion. In ordinary lo0gic the premises must be valid for-themselves.Yet one can consider the issue perhaps just as a matter of placing infinities within finite parameters; One can subdivide any infine area with infinite measures, yet not use a standard measure unit and place an infinite number of thoise within.One can find some interesting correlations in economics with Mises' criticism of Wicksell's premises in the proportionality theorem. Mises also wrote a frequency theory with Riechenbach, I believe. -
zeno's paradoxVoted for by Fallingleaves.
but first you need to define the refference frame, AFTER you have taken a step, only then can you cut it in half. And even in theory, we step in addition af halves. I take a half a step, and my foot continues to move throug the other half. I do not infinately take half steps, because then motion would be impossable, but we just don’t work that way. .. Or you could look at it differently. If we did walk like that, we would have to train ourselves to choose to walk to a different spot, stepping halfway and ending up at the origionaly desired location
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What?Voted for by Michael L.
If this is meant to be some sort of mathematical asemtote, I don’t understand.
Please clarify and/or give an example.
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Mathematical trickVoted for by Obvious Child.
The time taken for you to complete half steps also halfs each time. this is a trick of maths. Say each step takes one second, it takes half a second to get half way and so on. You never make a full step, but then you never go through a whole second.
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No such thing as indefinite divisibilityVoted for by ennoia.
Quantum physics holds that you cannot divide space indefinitely. Eventually you come to a point where you can no longer chop up.
This is called Planck's length, after clever Professor Planck hypothesised and proved. It's exactly 1.6 x 10-35 metres.
Similarly time is not indefinitely divisible - it too seems to have quantum property. The smallest amount of time is Planck's time - the time it takes a proton travelling at the speed of light to cross Planck's Length.
So when we step a metre, we are not travelling an infinitely divisible metre, we are travelling through a metre constituted of little finite Planck lengths.
"What??" you scream in annoyance, and start throwing things at me. "How ridiculous? But how does one cross these Planck lengths themselves?"
Well, when we cross a Planck length our atoms exist in it for an instance, before crossing into the next Planck length.
"But that's even more ridiculous! If we can’t exist in half a Planck length as we cross into the next one - then are you saying we cease to exist as we move from Planck length to Planck?"
Yes, actually. That's the whole idea. That we move in frames, so to speak, through these Planck lengths. Even though it seems strange to suggest we exist in one frame and then in the next instant, we exist in the next frame, it has been observed that is exactly what happens at sub-atomic level.
I suppose, if Planck’s length was quite a bit larger life would appear like those very old films with only a few frames per second, with an observable distance between each picture. But then it is worth bearing in mind that we would not be conscious between each frame to observe the transition between frames – and life would appear just as it does now.
Even so, its such a small distance, its not worth worrying about “splinching” between frames, but it does eliminate the possibility of infinite divisibility.
Zeno’s paradox seems to be solved by science, but still it is an interesting concept.
I suppose once again the human mind tries to out smart the universe, but once againt he universe got there first…
All sounds a bit 42, doesn’t it?
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The Limits of ConceptionVoted for by Raven King.
Assume the Achilles runs at 10 m/s while the tortoise runs at 1 m/s, and that Achilles has granted the tortoise a 10 meter head start. After one second, Achilles will have reached the tortoise's starting point, and the tortoise would now be 1 meter ahead of Achilles. After two seconds, Achilles would now be 8 meters ahead of the tortoise, having run a total of 20 meters. The tortoise, conversely, would now only be 2 meters ahead of its starting point.
Achilles would have passed the tortoise during the second second, specifically during the second decisecond when both Achilles and the tortoise would have been between the 11th and 12th meter marks (I am counting the meter marks from Achilles' starting point). Zeno's paradox applies when I try to calculate the exact moment when Achilles and the tortoise are even. At the end of the second decisecond, Achilles would be 0.8 meters ahead of the tortoise, so let's consider what happened before the second decisecond. Before the second decisecond, the tortoise was 0.1 meters ahead of Achilles. Now if we let another decisecond pass, then Achilles will have passed the tortoise, so let's reduce the unit of measurement. After a centisecond, the tortoise will now be only 0.01 meters ahead of Achilles. But if we let another centisecond pass, then Achilles will have passed the tortoise, so we have to reduce the units again. But this only leads to an infinite regress, as we would never reach zero but only longer strings of zeroes with a one at the end.
But Zeno's paradox does not really disprove motion so much as it demonstrates the impossibility of knowing the exact moment when Achilles and the tortoise are even. Achilles will in fact pass the tortoise, but we can never know exactly when Achilles passed the tortoise. Thus we cannot know the moment at which Achilles passed the tortoise, but only the moment during which Achilles passed the tortoise. The paradox also illuminates the fact that an infinite set of numbers can add up to a finite sum, but it does not disprove the existence of motion.
Zeno's paradox is more an example of the unusual properties of numbers and the limits of human conceptual ability than a disproof of motion. Our minds can only approximate the truth because our conceptual abilities are limited. That's why quantum physics appears to be contradictory, it lies outside our ability to properly conceive. Describing the quantum realm would be analogous to a society of color-blind people who have evidence that color exists trying to explain what color is. They could not properly conceive color and it would most likely appear largely contradictory to them. Numbers are a part of our conceptual framework, and while useful, they are only an approximation of reality, albeit an often close one.
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I am a MadmanVoted for by WhiteShadow.
You can't have a half of a hole can you? No, its just one hole. SO how can you have a half of a step?
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Try ItVoted for by creason.
If you think this "Paradox" exists the have someone shoot a bullet at your face. Using this logic you won't die because it has to keep going half way. Try it for yourself and you will see this has no real life application.
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Bird train paradoxVoted for by TeChNoWC.
Yes, this paradox is a bit like the bird and the two trains. If a bird sat on the top of a train, and another train was approaching that train (as in they would collide) and the bird flew from one train to another and back again, you can use the same formula to prove that the bird would make infinte trips to and from one train to the other. This is because as the trains get really close, the bird just makes really quick trips. Although, this does only work mathematically, not practically.
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Let's rephrase it the way it was originallyYou're trying to go across a room. Before you get there, you have to get 1/2 the way there. Before you get half the way, you have to get 1/4... and so on. Since distances are infinitely small, that means we have to do an infinite amount of things in a finite amount of time in order to move...Voted for by peripheral visionary.
(plato couldn't solve it, btw, and those of you who thought you did weren't answering the write question, sorry) -
Zeno Came before CalculusAnd Calculus lets you add up an infinite amount of numbers into a finite number, which really helps untangle the knot, yes you can break movement into an infinite amount of steps, but it's easy to add them back together again and prove what intuition already tells us, movement is easy and logical.Voted for by Makessenseright.
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April 5, 2005
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clarafication on Zeno's beHALF
I think what Zeno was saying was that to make one step you have to travel half of the distance first and then half the remaining and so on but never make one entire step. But suppose the step you were taking was half the size of the step mentioned earlier. By the same reasoning above, you would never be able to complete that half-sezed step. Similarily, you could never complete a step half of that size, or one fourth of that size or even a step .00000000000000000......1 of the original size. That is Zeno's reasoning for saying motion is impossible. 99.999999.....% of an infinitely small step is still no motion at all.Xelgaroth
August 10, 2007
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