Zeno's paradoxes - What is wrong?
V.Gopal
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'.
Dann Corbit
news:38af3945.02092...@posting.google.com...
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.
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Franz Fritsche
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.
Paul R. Chernoff
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 cher...@math.berkeley.edu #
# Department of Mathematics # 3840 #
# University of California "Against stupidity, the gods themselves #
# Berkeley, CA 94720-3840 struggle in vain." -- Schiller #
James Harris
> 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:
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf
James Harris
zeno
(Sorry, could not resist.)
In article <amqup4$2lg2$1...@agate.berkeley.edu>, Paul R. Chernoff
PoorRichard
concept of the limit of an infinite series.
"Paul R. Chernoff" <cher...@math.berkeley.edu> wrote in message
news:amqup4$2lg2$1...@agate.berkeley.edu...
Matthias Plaue
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
Michael Brown
<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>
--
Michael Brown
My inbox is always open (remove the obvious):
emb...@i4free.NOSPAM.co.nz
Dann Corbit
news:xRgk9.10246$Y3.19...@news.xtra.co.nz...
> "V.Gopal" <vgop...@rediffmail.com> wrote in message
> news:38af3945.02092...@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.
Shmuel (Seymour J.) Metz
at 12:02 PM, vgop...@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 spam...@library.lspace.org
Chas Brown
Michael Brown wrote:
>
> "V.Gopal" <vgop...@rediffmail.com> wrote in message
> news:38af3945.02092...@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 ;).
Cheers - Chas
Morpheus, Inc.
[snip]
> 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".
V.Gopal
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.
Russell Easterly
"Morpheus, Inc." <morp...@olympus.mons> wrote in message
news:3D9226...@olympus.mons...
> Chas Brown wrote:
> [snip]
> > 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. :*)
>
But the photon won't notice the point because zero time has passed.
Would that mean that, from the photon's point of view, the position of the
universe is infinitely uncertain?
> > Cheers - Chas
> --
> Ioannis
Russell
- Zeno was right. Motion is impossible.
Russell Easterly
"Paul R. Chernoff" <cher...@math.berkeley.edu> wrote in message
news:amqup4$2lg2$1...@agate.berkeley.edu...
> 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.
>
See:
http://www.utm.edu/research/iep/z/zenoelea.htm
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....
Russell
- the universe is one dimensional
Adam Russell
"PoorRichard" <poorri...@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.
Greg Fleischman
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.
Dann Corbit
Therein lies the rub. When we get to an infinite number of subdivisions,
the subdivisions are no longer finite, but (rather) infnitesimals. Surely
you realize that.
V.Gopal
'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.
James Hunter
"Adam Russell" <adamr...@directvinternet.com> wrote in message
news:amu476$99cqn$1...@ID-122512.news.dfncis.de...
He believed it, since he was one of the original:
"All is one, and one is all" moron philosophers.
Adam Russell
"James Hunter" <jim.h...@jhuapl.edu> wrote in message
news:an58na$nbo$1...@houston.jhuapl.edu...
How could he believe you could not walk to the finish line when it is so
easily proven false? Just do it.
James Hunter
>
> "James Hunter" <jim.h...@jhuapl.edu> wrote in message
> news:an58na$nbo$1...@houston.jhuapl.edu...
> >
> > 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*.
Adam Russell
?? 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?
James Hunter
>
> "
> > 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.