Posted on 02/23/2006 3:58:12 AM PST by S0122017
And ofcourse meant analog not analogue :)
But english aint my first language so..
Ah, Zeno ...
The so-called 'Zeno' paradoxes originate in the deficiencies of mathematical thought/expression amongst the ancient Greeks. Having no form of the calculus or the ability to express, let alone 'solve' series summations, etc., they viewed these conundra first with disdain and, ultimately, alarm.
If an arrow cannot arrive at it's target ... If Achilles cannot overtake the hare ... then, whatever allows a dropped rock to reach the earth ? While it is perfectly obvious to our imperfect senses that the arrow DOES indeed hit the target and Achilles inevitably WILL leave br'r Rabbit in a cloud of dust, the hapless Greeks had nought but their deductive logic and descriptive geometry to attack and 'solve' Zeno's evil little puzzles.
(Like 'the wonder of the dancing bear ...', I've always thought it was simply 'brilliant' of Zeno to design and put forward these puzzles. Especially considering the miserable tools he and his peers had to work 'em with. Did you ever LOOK at their 'notation'? No powers, no Zero, NO DECIMAL POINT! Over 2000 years would elapse 'til these puzzles were ultimately tamed.)
Back to the paradoxes themselves ...
The motion of a physical body happens 'relative' to that of some convenient frame of reference. In the instance of the arrow, it's motion begins independent of the target and proceeds independent of the target until it's position and that of the target coincide. However else you choose to describe this -- chopping up the motion of the arrow into segments or particles of 'time' (duration) or quanta of space-time, or whatever -- doesn't change the inevitability of the moving body traversing its course and, conveniently, arriving at a point in space-time coincident with the point occupied by the stationary target.
Achilles' pursuit of the hare is a bit more complex, in that the frame of reference includes two bodies in concurrent and independent motion. However else you choose to describe it, the two 'relative' motion vectors will eventually sum up to any of several 'solutions' in which Achilles (always assuming he stays on the course, of course) WILL overtake and pass the hare.
Of course, we simplify somewhat outrageously in these expressions of the paradoxes. The motion of the arrow is 'ballistic' and proceeds without any addition of energy or guidance, once begun. The motion of the target is zero. Another simpler way to express it is that ONLY the arrow is moving in this frame of reference.
We further ASSUME that Achilles and the hare will remain with some constraining 'course'. The paradox still 'works' even if Achilles is pursuing the hare across the fields, up hill and down dale. However, this doesn't add anything but complication to our intent in sponsoring this 'race' in the first place.
There ARE no paradoxes here. Only clumsy and/or inadequate descriptions of what's going on. Note that these so-called paradoxes are most useful as a framework in which to teach reasoning and to demonstrate the utility of mathematic tools which enable more 'useful' descriptions of the scenarios Zeno proposed.
I boggled at some of the descriptions offered here to process and resolve the so-called paradoxes.
E.g. [paraphrasing]: "The arrow approaches the target in finite steps expressed as quanta of time, reducing the distance by half, then half again, ad infinitum, until it reduces the distance to the minutest time quantum possible. It then 'quantum teleports' to complete it journey." (Just how, pray tell, does it KNOW when it's time to teleport? Oh, well. Details, details ... )
E.g. [paraphrasing]: "Achilles and the hare approach each other sufficiently close to invoke a 'quantum entanglement' that trips up the hare (but NOT, of course, our Hero) and enables Achilles to win the race." (All this without any actual 'contact' one with the other that would seem to produce either squashed hare or a tumbling Hero. Oh, well. Details, details ... )
Did poor William of Occam live in vain ?
Thanks for the analysis. Being a person who is reasonably strong in logic but somewhat weak in math, this paradox has always fascinated me. It is ridiculous of course, but appeals to the weaknesses in my own reasoning process.
One thing that really fascinates me (something I've read which I suppose might be false) is that supposedly, Zeno and his followers utilized these paradoxes as an excuse to basically lie around doing nothing---since they had proved that it was impossible to accomplish anything. This tickles me to no end, but perhaps it is just another urban myth (or I should say Greek myth).
I am pretty confident that you realize that my entanglement explanation was purely a joke.
Mrs. Centurion will assure you I have absolutely NO sense of humor. Actually, I DID notice and thought it rather clever.
However, after multiple boggles, I felt a need to attempt to unwind this discussion into something less obscure or weird.
Not to worry. I have confidence this whole rant will not modify the perceptions or world-view of any of the participants one iota.
"Not to worry. I have confidence this whole rant will not modify the perceptions or world-view of any of the participants one iota."
indeed. a most enjoyable thread.
"I have always thought that this is a surprisingly silly paradox. Motion is inexorably related to time, and clearly so. The fact that the paradox depends upon the restraint of "any single instant" exposes it as just a ridiculous play on words. By definition, nothing moves "during any single instant"!
In fact, the use of the word "during" in that phrase is contradictory in and of itself. If no time passes, there is no "during"."
Agreed that in some ways one leaves the common sense approach to logic behind when entering quantum mechanics. The question you pose goes to the essence of "the calculus". Motion is dx/dt, and in integral calculus dt --> zero as a limit.
If dt it were, in fact, zero then there would be no dx, or displacement.
Perhaps you don't understand the question? This requires Deep Thought...
If you understand the question, maybe you (big thinker that you are) can help me out?
Actually, though, I wasn't talking about the "question", I was talking about the method and the result.
what if God is a Quantum Particle Entity?
I think the article may have benefited from a description of:
a. What the program was supposed to do?
b. What answer they got by not actually using the program?
c. A better explanation of how the photon "flirts" with the program components.
Don't know about you, but there appeared to be gaping holes in this article. I got a few posts deriding my IQ 'cause I dared to question what was being reported. As for the "somehow it works" part, I've seen enough junk science to be skeptical when anyone says: "and a miracle occurs here".
This is just an article for laymen.
If you want the real science article you can search for it, but it isnt going to be easy reading.
Maybe they could have tried to explain some details, but would it have mattered? They dont claim to have made a supercomputer, just found what might be an alternative way it can work.
With quantum mechanics articles, these days I just assume the reporter doesn't know what he's talking about so the article may be full of holes. But, knowing the rudimentary basics of quantum mechanics, this does seem possible.
Well put.
"A stopped clock is right twice a day." -- old saying.
:')
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