One wants to run a 100m sprint. It will take a finite time to do so.
Now to run 100m, one needs to cover 50m which also would take a finite
time. To run this 50m,one needs to complete 25m which itself takes a
finite amout of time. And so on....

And mathematics tells us that the sum of infinite finites is infinite.
Which means it will take one an infinite amount of time to reach 100m.

So is motion discontinuous rather impossible according to mathematics?
Or is this logic wrong?

Cheers,
Leo.

Who are you? Zeno, warrior philosopher? The sum of terms (1/2)^n, wnere
n runs from 0 to infinity is 2. Is that finite enough for you?

Zena could have done better than that had she taking off her creaking
leather garments and thought about it.

Bob Kolker

i was just thinking about the problem imagining myself in an
antiquated epoch when series and calculus were unknown.
Of course as it stands today the logic i presented is invalid.

"Robert J. Kolker" <bobkol...@attbi.com> wrote in message news:<3D7777D7.80
20...@attbi.com>...

leo  wrote :

.... of which there are only 2...

.... of which there are 4 ...

True, but in this case there are only a finite sum of finite lengths, which
is finite.

It would be a dreadful shame if mathematics failed us on such a point of
logic. We have found it so terribly useful.

The logic is wrong. Basic algebra is certainly reversible, but your
hypothesis is not.

There will only be an infinite number if they are infinitely small -
factual, intriguing, but pointless on it's own. Great for integral calculus
though.

No matter how many tiny fragments you chop a distance into, so long as the
fragments have finite dimensions there will only be a _finite_ number of
them - not infinite as you state near the start. Now you will find the
finite number of finite distances all just add back up.

Whew! For a moment there I thought I would lose integral calculus - without
which, all of physics becomes just so much pointless paperwork.  ;-)

Galen

  Killfile
    -- A-Team: Acar, Friedman, Malenor, Symmetry, R.Kolker, Helen
    -- B-Team: J Silva, Homer, j domani
  _______   If I'm silent, don't assume I agree with you.  _______

leo  wrote :

Lol, invalid as it has always been.

Incidentally, where n tends toward infinity:

(1/2)^1 + (1/2)^2 + (1/2)^3 + ... + (1/2)^n tends toward 1, not 2.

Proof:

Sn = (1/2)^1 + (1/2)^2 + (1/2)^3 + ... + (1/2)^n --(1)

Multiply by 1/2

Sn.1/2 = (1/2)^2 + (1/2)^3 + (1/2)^4 + ... + (1/2)^n + (1/2)^(n+1) --(2)

(1) minus (2) gives:

Sn.1/2 = 1/2 - (1/2)^(n+1)

therefore:

Sn = 1 - (1/2)^n

the second term is negligible, so:

Sn = 1

Galen

  Killfile
    -- A-Team: Acar, Friedman, Malenor, Symmetry, R.Kolker, Helen
    -- B-Team: J Silva, Homer, j domani
  _______   If I'm silent, don't assume I agree with you.  _______

Not true. An infinite series can converge to a finite number. I even
gave an example.

Mathematics did not fail. Zeno did.

Zeno, warrior philospher.

Bob Kolker

I said, sum from n = 0. That adds (1/2)^0 = 1 to the series.

Bob Kolker

Mathematics did not fail. Zeno did.

Incidentally Zeno did fail and he admitted it. Yes he is a true
warrior as u were able to percieve.

And he likes your sense of sarcasm.

Cheers,
Leo.

"Robert J. Kolker" <bobkol...@attbi.com> wrote in message news:<3D78E94F.20
60...@attbi.com>...

Ah, true, however (regaining context) I was speaking of the sum of
end-to-end lengths between two fixed points - the 100m race of the original
post.

But Bob knew that already.

Galen

  Killfile
    -- A-Team: Acar, Friedman, Malenor, Symmetry, R.Kolker, Helen
    -- B-Team: J Silva, Homer, j domani
  _______   If I'm silent, don't assume I agree with you.  _______

1ED6%news.270...@octothorn.com>...

You are consistently talking nonsense, as far as I can tell. However,
tell me, how is it that you are engaging in a debate, silly as it is,
with somebody who you proudly claim to be in your "killfile"?

8a...@posting.google.com>...
 Galen Rutledge <news.270...@octothorn.com> wrote in message news:<B9A055A6.
 1ED6%news.270...@octothorn.com>...

  you proudly claim to be in your "killfile"?

   Because he is a few pints short of a Galen?

3...@posting.google.com>...

Hi Leo,

Your logic is not wrong, but the foundation of math is in a sorry
state, mainly because it assumes that a line segment has an infinite
number of points.
     One ridiculous implication of this (shown by Cantor):  The
"infinite" number of points on a line (1 dimension) is equivalent to
the "infinite" number of points in, say, an infinite plane (2
dimensions.) In fact, the same logic would make 1 dim equivalent to n
dimensions, for arbitrary n.
     You and every are invited to visit my site
http://www.thinhtran.com where I suggest a solution that meets the
Occam's razor test. Click "The Resolution Limit Postulate as
Foundation of Physics."

.78617...@posting.google.com>...

There is absolutely nothing "ridiculous" about that fact.

Nonsense.

I would suggest you spare people the waste of time of going there.

-- Helen.

        I think you just don't understand what mathematicians mean by a "line
segment". They mean a particular abstraction, not the first ten feet of
your driveway.

        DS

DANGSTON TRAN(DT)
THE POINT ASSUMPTION OF PHYSICS AND ITS PROBLEM
Since I have an infinite number of points, each of size zero, the
mathematical relation for the length L1 is:

infinity * 0 = L1
For another line segment of length L2, where  L2 not equal to L1, the
formula would read:

infinity * 0 = L2

Since the left side of the two equations are exactly the same while L1
and L2 are different from each other, the point assumption leads to an
ambiguous description of physics.

LEO:
If the number of divisions tend to infinity(not infinite),thier size
tends to 0(not zero). Tending to infinity and zero means gets close
but not there. Hence left hand sides of the 2 equations are
mathematically as well as functionally incorrect.
Let L1 = 1.
We can have L1 = 1 * 1 = 2  * 1/2 = 100 * 1/100 = billion * 1/billion
= the biggest number u can think of * reciprocal of it. But not 0 and
infinity.

DT
THE RESOLUTION LIMIT POSTULATE
I will speculate that the point assumption is only an approximation of
how Nature operates. An approximation will have to break down at some
point. As the next logical step, I postulate the possible existence of
a classical length scale L0 and a classical time scale T0 that could
not be reduced any further.
My eyes also have limits in resolving time. A movie appears continuous
to me only because the time lapse between frames is smaller than the
resolution ability of my eyes. A card trickster can cheat me by
dealing the second card (instead of the top card) right before my eyes
because his action is too fast for my eyes to resolve; etc.
LEO:
Note again that 'your eyes' are different from mine and every1 else's
and also different from the 'eyes' of a camera or a robot or watever.
So which eyes physics ought to trust? Your point that L0 and T0 cant
be broken down anymore seems matematically and logically unconvincing
to me.

DT
Can I generalize this idea of space and time resolution to all physics
phenomena? I think I can because, strictly speaking, the word
"phenomena" means "phenomena as observed", and all observations
involved measurements.
LEO:
Precisely. Thats what i wanted to say. But phenomenon as observed by
whom? We have to take an objective view of physics which would allow
us to divide and keep dividing L0 and T0.
You yourself answer this point as follows.

DT
We must differentiate space and time as man made concepts and space
and time as they mean to physics. As man made concepts, space and time
could be either continuous or discontinuous, depending on personal
perception and/or inter-personal agreement. But in physics by "space"
we mean "space as measured" (e.g., with a 'standard unit length', a
'standard unit area', a 'standard unit volume'), by "time" we mean
"time as measured" (e.g., with a watch, a clock).

DT
v = dx/dt (limit of Dx/Dt when Dt tends 0)
In this definition I see the ghost of the point assumption because in
order for Dt to approach zero, the idea of a point of zero measure in
the time dimension must be permitted.
LEO:
No. This is precisely where i disagree with your whole theory. Tending
to zero does not necessarily allow existence of zero. A time tending
to zero would signify an instantaneous action like an impulsive force.
What will zero time signify? Nothing. What can you measure in zero
time and for that matter what is zero time? Physics concerns
measurables. You can always measure the smallest period of time
interval but how do you measure zero time? Hence i think,contrary to
your stated position, that zero time is not permitted in what you call
the point assumption.

Cheers,
Leo.

5...@posting.google.com>...

Anything that is expressed as the function of time is and must be
considered as continuous. If L=mT, then L is continuous and we can
calculate L for all values of T and verfiy the truth by experiments.
Here linear speed is given by the slope 'm' of the straight line L=mT.
We must understand that at T=0 'm' does not become indeterminate; at
T=0, 'm' has the same value that it has for all other finite of L and
T satisfying the given condition. Therefore 'm' indicates all
theoretically possible velocities between 'm'=0 and 'm'=infinity. Here
at all values of 'm' both T and L are continuous and L is continuous
because T is continuous. Time 'T' here is always 'time without change'
'Contiguity/continuity' becomes a problem whem we want to
express/communicate
continuous spatio-temporal change, like linear acceleration. If we
want express linear acceleration we have to find the formula for the
open curve: L vs T, which has to show the slope 'm' as the continuous
variable. During acceleration all the speeds or slopes=m=L/T must have
the common denominator T=0. There must be infinite number of speeds
within each unit of time. Therefore I do not think we can draw any
open curve (in which 'm' has to be a continuous variable in
space/time) in one direction in space/time. If we want to make an open
curve - line - move in one direction in time then we have to make use
of two ideas:(1) time without change and (2) change without time.
Change within time is inexpressible, but change without time is
illogical. If L/T is the continuous variable then unit of time (dT)
remaining constant dL (or dispalcement per unit time) must change -
elongate. This is the raeon why there must an upper limit to the
velocity that can be given to any physical object.

Wait a second. Something's wrong with this here argument. Are you
meaning to tell me that just because f(x)=(e^x-1)/x is undefined for 0
because f(0)=(e^0-1)/0=(1-1)/0=0/0=Undefined that we cannot find the
limit of f(x)=(e^x-1)/x as x->0? This cannot be so. I propose we
accept the system of L0 T0 but under one constraint: That L0 and T0 do
NOT be taken to be the smallest possible units. There actually must be
two orders of infinity, and those are the set of infinity to which a
given interval is a part, and the set of infinity which is a part of a
given interval.

I suggest you go to http://www.utm.edu/research/iep/ for reference.
That seems to be a good place to look if you want questions answered,
and also, answers questioned. ;P

Length (or in general measure) is a property of point sets. The simplest
measure theory is built on the algebra of semi closed intervals (so
called sigma algebras).

See any standard textbook of measure and integration.

As to tending to zero, this is the limit concept, which replaced the
infinitesimal, correctly criticized by Bishop Berkeley. All of the
operations of calculus are properly found on the limit concept. It took
nearly 200 years from the time Newton (and Leibnitz) invented calculus
to put the mathematics on a firm logical foundation.

Bob Kolker