The paradox: Before an object can travel a distance 'd', it must travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 etc. Since the sequence goes on for ever, it therefore appears that the distance 'd' cannot be travelled. But we do travel a finite distance 'd' within a finite TIME. If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel any SPECIFIED distance within a finite (calculable) time. Then what is wrong with Zeno's paradox? What Zeno did is, he introduced the idea of "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add infinitum, is same and finite, in sussession. It really means that we are moving at continuously (?) dcreasing average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER BECOMES NEGATIVE we cannot imagine how it is possible - it is a black hole! In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here the number of periods of time (of same duration) is same as the number of terms in the series therefore is endless. If we move at a constant linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of time and in time 2T+, we would cross double that distance. "An arrow in flight is identical to arrow at rest." This is wrong according to physics. A moving arrow is longer than an arrow at rest. In fact even in Newton's idea of acceleration (L/T^2) we use idea of 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
> The paradox: Before an object can travel a distance 'd', it must > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > etc. Since the sequence goes on for ever, it therefore appears that > the distance 'd' cannot be travelled. But we do travel a finite > distance 'd' within a finite TIME. > If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel > any SPECIFIED distance within a finite (calculable) time. Then what is > wrong with Zeno's paradox? What Zeno did is, he introduced the idea of > "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF > TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add > infinitum, is same and finite, in sussession. It really means that we > are moving at continuously (?) dcreasing > average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER > BECOMES NEGATIVE we cannot imagine how it is possible - it is a black > hole! > In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously > and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here > the number of periods of time (of same duration) is same as the number > of terms in the series therefore is endless. If we move at a constant > linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of > time and in time 2T+, we would cross double that distance. > "An arrow in flight is identical to arrow at rest." > This is wrong according to physics. A moving arrow is longer than an > arrow at rest. > In fact even in Newton's idea of acceleration (L/T^2) we use idea of > 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in > velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
Surely, this is a troll.
When we subdivide the interval, there is *no change* in velocity.
velocity = distance/time
We cut the time in half. We cut the distance in half. Velocity remains the same. Try it yourself...
An arrow is moving at 64 meters per second from x = 0 to x = 64. This would take one second. But instead, we will do the motion by successive halves. The arrow moves at 64 meters per second from x = 0 to x = 32. This takes 1/2 of a second. Then the arrow moves at 64 meters per second from x = 32 to 32+(64-32)/2 = 48. This takes an additional 1/4 of a second. Then the arrow moves at 64 meters per second from x = 48 to 48+(64-48)/2 = 56. This takes an additional 1/8 of a second. etc. The intervals become shorter and shorter by a factor of 1/2 but so does the time. The only thing that puzzled Zeno is he realized that there are an infinite number of such halvings. Hence, the only ambiguity is due to Zeno's lack of calculus. He could not know that the sum from one to infinity of 1/2^n has a limit. He lacked the necessary tools to prove it. -- C-FAQ: http://www.eskimo.com/~scs/C-faq/top.html "The C-FAQ Book" ISBN 0-201-84519-9 C.A.P. FAQ: ftp://cap.connx.com/pub/Chess%20Analysis%20Project%20FAQ.htm
On 24 Sep 2002 12:02:34 -0700, vgopa...@rediffmail.com (V.Gopal) wrote:
> The paradox, etc. etc...
Sorry, what was the MESSAGE again?
F.
P.S. In the subject line you asked: "Zeno's paradoxes - What is wrong?" ??? Don't understand that question. WHAT *should* be wrong? Nothing's _wrong_ there.
It seems to me that Zeno's basic assumption was that one can not do an infinite number of things in a finite time. This is simply false -- one can travel half the distance, then half the remaining distance, etc. So in some cases one can do infinitely many things in a finite time.
Incidentally, I have read that all we know about Zeno is via Aristotle, who presented Zeno's arguments specifically to rebut them. Conceivably, Zeno's original writings were more subtle....
-- # Paul R. Chernoff chern...@math.berkeley.edu # # Department of Mathematics # 3840 # # University of California "Against stupidity, the gods themselves # # Berkeley, CA 94720-3840 struggle in vain." -- Schiller #
vgopa...@rediffmail.com (V.Gopal) wrote in message <news:38af3945.0209241102.5fa7260e@posting.google.com>... > The paradox: Before an object can travel a distance 'd', it must > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > etc. Since the sequence goes on for ever, it therefore appears that > the distance 'd' cannot be travelled. But we do travel a finite > distance 'd' within a finite TIME.
<deleted>
Thanks for the post as this is one of my favorite subjects!
I'd like to elevate the discussion by providing the following link, which talks about discrete steps:
<chern...@math.berkeley.edu> wrote: > It seems to me that Zeno's basic assumption was that one can > not do an infinite number of things in a finite time. > This is simply false -- one can travel half the distance, > then half the remaining distance, etc. So in some cases one > can do infinitely many things in a finite time.
> Incidentally, I have read that all we know about Zeno is > via Aristotle, who presented Zeno's arguments specifically > to rebut them. Conceivably, Zeno's original writings were > more subtle....
> It seems to me that Zeno's basic assumption was that one can > not do an infinite number of things in a finite time. > This is simply false -- one can travel half the distance, > then half the remaining distance, etc. So in some cases one > can do infinitely many things in a finite time.
> Incidentally, I have read that all we know about Zeno is > via Aristotle, who presented Zeno's arguments specifically > to rebut them. Conceivably, Zeno's original writings were > more subtle....
> -- > # Paul R. Chernoff chern...@math.berkeley.edu # > # Department of Mathematics # 3840 # > # University of California "Against stupidity, the gods themselves # > # Berkeley, CA 94720-3840 struggle in vain." -- Schiller
On 24 Sep 2002 12:02:34 -0700, vgopa...@rediffmail.com (V.Gopal) wrote:
>The paradox: Before an object can travel a distance 'd', it must >travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 >etc. Since the sequence goes on for ever, it therefore appears that >the distance 'd' cannot be travelled. But we do travel a finite >distance 'd' within a finite TIME. >If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel >any SPECIFIED distance within a finite (calculable) time. Then what is >wrong with Zeno's paradox? What Zeno did is, he introduced the idea of >"MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF >TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add >infinitum, is same and finite, in sussession. It really means that we >are moving at continuously (?) dcreasing >average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER >BECOMES NEGATIVE we cannot imagine how it is possible - it is a black >hole!
(Do you know what a black hole is apart from knowing that "it sucks matter and light"?)
>In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously >and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here >the number of periods of time (of same duration) is same as the number >of terms in the series therefore is endless. If we move at a constant >linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of >time and in time 2T+, we would cross double that distance. >"An arrow in flight is identical to arrow at rest." >This is wrong according to physics. A moving arrow is longer than an >arrow at rest.
According to classical physics movement is continous. The graph of a point particle moving with constant positive velocity on the real line starting at the origin surely crosses the space-time-points (d/2^n, t/2^n), (d>0, t>0). These points given, little information about the whole trajectory is provided without knowing the equation of motion. Or frankly speaken: You could imagine the particle doing anything between these points. It could go from the origin to the moon and to d/2 in time. After crossing d it could boldly go where no point particle has gone before. According to quantum mechanics trajectories (discrete or continous) are no way to describe physics. So what's wrong with physics or zenos paradox?
>In fact even in Newton's idea of acceleration (L/T^2) we use idea of >'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in >velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
What is L/T^2? If discrete motion's your notion of motion without time it's definitely NOT what Newton had in mind. He "invented" calculus to solve the kepler problem. The twin paradox really is an effect of general relativity though it is always mentioned in the context of special relativity. (In this context it IS a paradox.) It is discussed without infinite accelerations if that is what you mean by "change in velocity without time". (Do you know what the twin paradox is despite of knowing that "one grows older because he stays at home"?)
---------------------------------------------------- Some people have got a mental horizon of radius zero and call it their point of view - David Hilbert
> "V.Gopal" <vgopa...@rediffmail.com> wrote in message > news:38af3945.0209241102.5fa7260e@posting.google.com... > <delurk> > <snip> > > "An arrow in flight is identical to arrow at rest." > > This is wrong according to physics. A moving arrow is longer than an > > arrow at rest.
> Actually it is slightly shorter (by about -1.25 * 10^(-11) percent assuming > a very fast-moving arrow of about 540km/h) :)
> <snip> > </delurk>
The exact length depends on your frame of reference. It varies anywhere from L=length to 0, depending upon where you are. Even at that, if the velocity is constant, any observer who is not accelerating in respect to the arrow will see the same effect (though they may not see the same speed). One observer might see velocity V and another 1e6*V. But both will see the arrow travel from 0 to 1/2 D in 1/2 of the time as it takes to go from 0 to D.
Therefore, the frame of reference is irrelevant towards the argument unless the observer or the arrow is accelerating.
Of course, there will be some accelerations on any arrow not at infinite distance from any mass or which is undergoing any forces. (Pull of gravity, force of air friction, buoyancy of the air, etc). But this is posted to sci.math and not sci.physics, so I think we can dispense with the trivial forces. -- C-FAQ: http://www.eskimo.com/~scs/C-faq/top.html "The C-FAQ Book" ISBN 0-201-84519-9 C.A.P. FAQ: ftp://cap.connx.com/pub/Chess%20Analysis%20Project%20FAQ.htm
In <38af3945.0209241102.5fa72...@posting.google.com>, on 09/24/2002 at 12:02 PM, vgopa...@rediffmail.com (V.Gopal) said:
>The paradox: Before an object can travel a distance 'd', it must >travel a distance d/2. In order to travel d/2, it must travel d/4, >d/8 etc. Since the sequence goes on for ever, it therefore appears >that the distance 'd' cannot be travelled. But we do travel a finite >distance 'd' within a finite TIME.
That hasn't been a problem since Achilles beat the snot out of Zeno and ate the tortoise. That's when Zeno started calling him a kill ease.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamt...@library.lspace.org
> "V.Gopal" <vgopa...@rediffmail.com> wrote in message > news:38af3945.0209241102.5fa7260e@posting.google.com... > <delurk> > <snip> > > "An arrow in flight is identical to arrow at rest." > > This is wrong according to physics. A moving arrow is longer than an > > arrow at rest.
> Actually it is slightly shorter (by about -1.25 * 10^(-11) percent assuming > a very fast-moving arrow of about 540km/h) :)
From the arrow's point of view, it's the rest of the universe that's a little shorter ;).
> From the arrow's point of view, it's the rest of the universe that's a > little shorter ;).
Accordingly then, from a photon's point of view, the entire universe would look like a point. :*)
> Cheers - Chas
-- Ioannis http://users.forthnet.gr/ath/jgal/ ____________________________________________ "You cannot go against Nature, because going against Nature is part of Nature too".
"Dann Corbit" <dcor...@connx.com> wrote in message <news:amqfn602ac4@enews3.newsguy.com>... > "V.Gopal" <vgopa...@rediffmail.com> wrote in message > news:38af3945.0209241102.5fa7260e@posting.google.com... > > The paradox: Before an object can travel a distance 'd', it must > > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > > etc. Since the sequence goes on for ever, it therefore appears that > > the distance 'd' cannot be travelled. But we do travel a finite > > distance 'd' within a finite TIME. > > If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel > > any SPECIFIED distance within a finite (calculable) time. Then what is > > wrong with Zeno's paradox? What Zeno did is, he introduced the idea of > > "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF > > TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add > > infinitum, is same and finite, in sussession. It really means that we > > are moving at continuously (?) dcreasing > > average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER > > BECOMES NEGATIVE we cannot imagine how it is possible - it is a black > > hole! > > In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously > > and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here > > the number of periods of time (of same duration) is same as the number > > of terms in the series therefore is endless. If we move at a constant > > linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of > > time and in time 2T+, we would cross double that distance. > > "An arrow in flight is identical to arrow at rest." > > This is wrong according to physics. A moving arrow is longer than an > > arrow at rest. > > In fact even in Newton's idea of acceleration (L/T^2) we use idea of > > 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in > > velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
> Surely, this is a troll.
> When we subdivide the interval, there is *no change* in velocity.
> velocity = distance/time
> We cut the time in half. We cut the distance in half. Velocity remains the > same. Try it yourself...
> An arrow is moving at 64 meters per second from x = 0 to x = 64. This would > take one second. But instead, we will do the motion by successive halves. > The arrow moves at 64 meters per second from x = 0 to x = 32. This takes > 1/2 of a second. > Then the arrow moves at 64 meters per second from x = 32 to 32+(64-32)/2 = > 48. This takes an additional 1/4 of a second. > Then the arrow moves at 64 meters per second from x = 48 to 48+(64-48)/2 = > 56. This takes an additional 1/8 of a second. > etc. > The intervals become shorter and shorter by a factor of 1/2 but so does the > time. The only thing that puzzled Zeno is he realized that there are an > infinite number of such halvings. Hence, the only ambiguity is due to > Zeno's lack of calculus. He could not know that the sum from one to > infinity of 1/2^n has a limit. He lacked the necessary tools to prove it.
"The intervals become shorter and shorter by a factor of 1/2 but so does the time. The only thing that puzzled Zeno is he realized that there are an infinite number of such halvings. Hence, the only ambiguity is due to Zeno's lack of calculus. He could not know that the sum from one to infinity of 1/2^n has a limit. He lacked the necessary tools to prove it." The paradox is engendered not because Zeno did not know that the sum of the convergent series of infinite number of terms is finite but because he realised that infinite number of 'activities' cannot be performed within a finite time. If each activity (say that of measuring or expressing) takes a finite and a constant period of time (that can be communicated, 'once and for all') then infinite number of activities would definitely take infinite time. If the distance 'd' is constituted of 'N'UNITS OF DISTANCE and if PERIOD of time required to travel N/2 UNITS OF DISTANCE is 't', and, if 't' is the period of time requird to travel N/2^2, N/2^3, N/2^4 ----UNITS OF DISTANCE then we can never cover the complete distance 'd' because although the distance 'd' is finite, 'd' being continuous, NUMBER OF UNITS OF LENGTH WITHIN 'd' IS INFINITE (number of units of length within 'd' is not a natural constant.) Here I have described the process of natural radioactive decay, except that I have simply replaced 'mass' by distance 'd'. There is nothing subtle here, the paradox is based on wrong fundamentals - it simply does not exist if we move with uniform velocity.
> It seems to me that Zeno's basic assumption was that one can > not do an infinite number of things in a finite time. > This is simply false -- one can travel half the distance, > then half the remaining distance, etc. So in some cases one > can do infinitely many things in a finite time.
Zeno came from the school of Eleatic philosophy that believed the universe never changes. Zeno came up with his paradoxes to defend the philosophy of Parmenides, and he seeks to show that what we perceive as change is an illusion.
Zeno's arrow paradox "proves" that motion is impossible be arguing that the arrow can not occupy the same space it occupied before it moved.
Calculus does not refute most of Zeno's arguments. For example, showing that an infinite sum can have a finite limit does not change the arrow paradox. In fact, assuming that time is infinitely divisable just makes the arrow paradox harder to refute.
> Incidentally, I have read that all we know about Zeno is > via Aristotle, who presented Zeno's arguments specifically > to rebut them. Conceivably, Zeno's original writings were > more subtle....
> Zeno's problem (in regards to his dichotomy paradox) is that he had no > concept of the limit of an infinite series.
I always thought Zeno presented the paradox with a sarcastic bent. That is he was showing a logical fallacy by proving an impossibility. I don't think he believed his false proof but was meaning it to be educatory.
Actually, Zeno discovered quantum mechanics long before Planck wrestled with the paradox of infinite energy being irradiated by a black body at a finite temperature because of the assumed continuity of the frequencies of the radiation. Zeno's paradox disappears when time is quantized. Perhaps it is like the individual frames on a film strip. The motion is definitely discretized, but the illusion, when played, is continuity. What exists between the individual frames? Nothing! How much time passes between each quantum event that eventually builds up to what we perceive as our continuous existence? None! (Ok, I am guessing here because you may ask, if time doesn't pass between events, does it pass at all? I would say yes, but that each event is actually a "tick" that leads to our perception of the passage of time"). So, in each frame if a pursuer is farther along than the pursued, at some point the pursuer, in one quantum instant, will jump ahead of the pursued. In our macroscopic world, continuity is an excellent approximation to the discrete events that are the true basis of our existence. But there are examples that, if analyzed rigorously, will disprove the notion of continuity.
"V.Gopal" wrote: > The paradox: Before an object can travel a distance 'd', it must > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > etc. Since the sequence goes on for ever, it therefore appears that > the distance 'd' cannot be travelled. But we do travel a finite > distance 'd' within a finite TIME. > If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel > any SPECIFIED distance within a finite (calculable) time. Then what is > wrong with Zeno's paradox? What Zeno did is, he introduced the idea of > "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF > TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add > infinitum, is same and finite, in sussession. It really means that we > are moving at continuously (?) dcreasing > average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER > BECOMES NEGATIVE we cannot imagine how it is possible - it is a black > hole! > In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously > and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here > the number of periods of time (of same duration) is same as the number > of terms in the series therefore is endless. If we move at a constant > linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of > time and in time 2T+, we would cross double that distance. > "An arrow in flight is identical to arrow at rest." > This is wrong according to physics. A moving arrow is longer than an > arrow at rest. > In fact even in Newton's idea of acceleration (L/T^2) we use idea of > 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in > velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
> > "V.Gopal" <vgopa...@rediffmail.com> wrote in message > > news:38af3945.0209241102.5fa7260e@posting.google.com... > > > The paradox: Before an object can travel a distance 'd', it must > > > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > > > etc. Since the sequence goes on for ever, it therefore appears that > > > the distance 'd' cannot be travelled. But we do travel a finite > > > distance 'd' within a finite TIME. > > > If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel > > > any SPECIFIED distance within a finite (calculable) time. Then what is > > > wrong with Zeno's paradox? What Zeno did is, he introduced the idea of > > > "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF > > > TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add > > > infinitum, is same and finite, in sussession. It really means that we > > > are moving at continuously (?) dcreasing > > > average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER > > > BECOMES NEGATIVE we cannot imagine how it is possible - it is a black > > > hole! > > > In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously > > > and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here > > > the number of periods of time (of same duration) is same as the number > > > of terms in the series therefore is endless. If we move at a constant > > > linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of > > > time and in time 2T+, we would cross double that distance. > > > "An arrow in flight is identical to arrow at rest." > > > This is wrong according to physics. A moving arrow is longer than an > > > arrow at rest. > > > In fact even in Newton's idea of acceleration (L/T^2) we use idea of > > > 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in > > > velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
> > Surely, this is a troll.
> > When we subdivide the interval, there is *no change* in velocity.
> > velocity = distance/time
> > We cut the time in half. We cut the distance in half. Velocity remains the > > same. Try it yourself...
> > An arrow is moving at 64 meters per second from x = 0 to x = 64. This would > > take one second. But instead, we will do the motion by successive halves. > > The arrow moves at 64 meters per second from x = 0 to x = 32. This takes > > 1/2 of a second. > > Then the arrow moves at 64 meters per second from x = 32 to 32+(64-32)/2 = > > 48. This takes an additional 1/4 of a second. > > Then the arrow moves at 64 meters per second from x = 48 to 48+(64-48)/2 = > > 56. This takes an additional 1/8 of a second. > > etc. > > The intervals become shorter and shorter by a factor of 1/2 but so does the > > time. The only thing that puzzled Zeno is he realized that there are an > > infinite number of such halvings. Hence, the only ambiguity is due to > > Zeno's lack of calculus. He could not know that the sum from one to > > infinity of 1/2^n has a limit. He lacked the necessary tools to prove it.
> "The intervals become shorter and shorter by a factor of 1/2 but so > does the > time. The only thing that puzzled Zeno is he realized that there are > an > infinite number of such halvings. Hence, the only ambiguity is due to > Zeno's lack of calculus. He could not know that the sum from one to > infinity of 1/2^n has a limit. He lacked the necessary tools to prove > it." > The paradox is engendered not because Zeno did not know that the sum > of the convergent series of infinite number of terms is finite but > because he realised that infinite number of 'activities' cannot be > performed within a finite time. > If each activity (say that of measuring or expressing) takes a finite > and a constant period of time (that can be communicated, 'once and for > all') then infinite number of activities would definitely take > infinite time. If the distance 'd' is constituted of 'N'UNITS OF > DISTANCE and if PERIOD of time required to travel N/2 UNITS OF > DISTANCE is 't', and, if 't' is the period of time requird to travel > N/2^2, N/2^3, N/2^4 ----UNITS OF DISTANCE then we can never cover the > complete distance 'd' because although the distance 'd' is finite, 'd' > being continuous, NUMBER OF UNITS OF LENGTH WITHIN 'd' IS INFINITE > (number of units of length within 'd' is not a natural constant.) Here > I have described the process of natural radioactive decay, except that > I have simply replaced 'mass' by distance 'd'. > There is nothing subtle here, the paradox is based on wrong > fundamentals - it simply does not exist if we move with uniform > velocity.
Greg Fleischman <fleisch...@iit.edu> wrote in message <news:3D935E61.BAA16FFB@iit.edu>... > Actually, Zeno discovered quantum mechanics long before Planck wrestled > with the paradox of infinite energy being irradiated by a black body at a > finite temperature because of the assumed continuity of the frequencies of > the radiation. Zeno's paradox disappears when time is quantized. Perhaps > it is like the individual frames on a film strip. The motion is > definitely discretized, but the illusion, when played, is continuity. > What exists between the individual frames? Nothing! How much time passes > between each quantum event that eventually builds up to what we perceive > as our continuous existence? None! (Ok, I am guessing here because you > may ask, if time doesn't pass between events, does it pass at all? I > would say yes, but that each event is actually a "tick" that leads to our > perception of the passage of time"). So, in each frame if a pursuer is > farther along than the pursued, at some point the pursuer, in one quantum > instant, will jump ahead of the pursued. In our macroscopic world, > continuity is an excellent approximation to the discrete events that are > the true basis of our existence. But there are examples that, if analyzed > rigorously, will disprove the notion of continuity.
> "V.Gopal" wrote:
> > The paradox: Before an object can travel a distance 'd', it must > > travel a distance d/2. In order to travel d/2, it must travel d/4, d/8 > > etc. Since the sequence goes on for ever, it therefore appears that > > the distance 'd' cannot be travelled. But we do travel a finite > > distance 'd' within a finite TIME. > > If we CONTINUOUSLY move at a CONSTANT LINEAR VELOCITY we can travel > > any SPECIFIED distance within a finite (calculable) time. Then what is > > wrong with Zeno's paradox? What Zeno did is, he introduced the idea of > > "MOTION WITHOUT TIME." In Zeno's paradox we imagine that PERIOD OF > > TIME taken to move the distanceS d/2, d/4, d/8 --------d/2^100-----add > > infinitum, is same and finite, in sussession. It really means that we > > are moving at continuously (?) dcreasing > > average velocity. IF WE ASSMUNE THAT DECREASE INCREASES AND NEVER > > BECOMES NEGATIVE we cannot imagine how it is possible - it is a black > > hole! > > In case of Zeno's paradox DECREASE IN VELOCITY DECREASES continuously > > and RATE OF DECREASE APPROACHES ZERO but cannot become zero. But, here > > the number of periods of time (of same duration) is same as the number > > of terms in the series therefore is endless. If we move at a constant > > linear velocity 'v' we can cover the distance 'd' in T=(d/v) units of > > time and in time 2T+, we would cross double that distance. > > "An arrow in flight is identical to arrow at rest." > > This is wrong according to physics. A moving arrow is longer than an > > arrow at rest. > > In fact even in Newton's idea of acceleration (L/T^2) we use idea of > > 'accelerated MOTION WITHOUT TIME'. In fact the idea of 'change in > > velocity' WITHOUT TIME' engenders the famous "TWIN PARADOX'.
'Acceleration WITHOUT SPACE-TIME' compels us to introduce the idea of 'time corpuscles'. Within each 'time corpuscle' there is change in velocity. If the distance d/2 is travelled at velocity V; d/4 is travelled at velocity V/2, d/8 is travelled at velocity V/4, one would definitely take infinite time to cover the distance 'd'. But where is the 'duration' within which the velocity changes and become half each time? If motion is continuous we have to place acceleration within dimensionless space-time corpuscles that we have to assume to exist whenever velocity changes. Since velcity has to change infinite times, there are infinite number of 'space-time corpuscles' Here cahnge in velocity is quantized. This is acceptable IF we assume that existence of space-time corpuscles requires no proof or explannation.
> "PoorRichard" <poorrichar...@hotmail.com> wrote in message > news:gjbk9.5314$9V6.120@fe01... > > Zeno's problem (in regards to his dichotomy paradox) is that he had no > > concept of the limit of an infinite series.
> I always thought Zeno presented the paradox with a sarcastic bent. That is > he was showing a logical fallacy by proving an impossibility. I don't think > he believed his false proof but was meaning it to be educatory.
He believed it, since he was one of the original: "All is one, and one is all" moron philosophers.
> > "PoorRichard" <poorrichar...@hotmail.com> wrote in message > > news:gjbk9.5314$9V6.120@fe01... > > > Zeno's problem (in regards to his dichotomy paradox) is that he had no > > > concept of the limit of an infinite series.
> > I always thought Zeno presented the paradox with a sarcastic bent. That > is > > he was showing a logical fallacy by proving an impossibility. I don't > think > > he believed his false proof but was meaning it to be educatory.
> He believed it, since he was one of the original: > "All is one, and one is all" moron philosophers.
How could he believe you could not walk to the finish line when it is so easily proven false? Just do it.
> > > He believed it, since he was one of the original: > > > "All is one, and one is all" moron philosophers.
> > How could he believe you could not walk to the finish line when it is so > > easily proven false? Just do it.
> Zeno's philosophy had nothing to do with walking, > since it had nothing to do with *physics*.
?? I thought we were talking about the old paradox that you could never get to the finish line because after you walk halfway there you always have halfway to go. Were we not?
> " > > Zeno's philosophy had nothing to do with walking, > > since it had nothing to do with *physics*.
> ?? I thought we were talking about the old paradox that you could never get > to the finish line because after you walk halfway there you always have > halfway to go. Were we not?
We were, but the question stills comes up? What does that have do with physics? It apparently concerns Geometry.
The force never was with Zeno, and likewise since the Zeno disease is contagious, the force is also not with Einstonians.
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