Re: Calculus XOR Probability

Virgil said:
In article <MPG.1ec408b1bbb254a098acba@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1ec2b1ba7220538998acb0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:


Distances between points depends only on their locations, so that
is all one has to pay any attention to.

But, when two points have locations that cannot be distinguished, how
do you derive direction from the set of two points?

You don't, because you don't have two points unless you can distinguish
between them.

Very good. So, if the neighboring points cannot be distinguished, and change in
direction at that level is lost.



The length of a broken or polygonal line depends only on the
location of of the "joints" and not at all on the direction of the
segments between joints.

What is the slope of the staircase at x=0. It's infinite.

Why does TO seem to think that changing the direction between points has
any effect on the distance between them? He must never have come across
circles.


I ahve come across them and around them. Is the diameter equal to half the
circumference? It's shorter? COuld that have anythign to do with NOT changing
direction?


Does that
ever change as the staircase approaches the limit? No. That point at
x=0 always has slope infinite, as opposed to the slope of the
diagonal. The direction between 0,0 and 0,1/n never changes.

But that direction is irrelevant to the total length.

One can straighten out the staircase without changing its.

I assume you meant length. That's true, even in the limit.



So that something that only takes into account the locations of
those "joints" includes everything needed.

As long as those locations are ON the curve, yes.
But they need not be at any particular points of the curve.

No, but they need to be somewhere on the curve.


In particular to approximate the length of the staircse, one need not
take a point at any of the corner points of the staircase, and the LUB
of such approximations will still be the correct length.


No, the locations of the staircase in the limit are
indistinguishable from the locations of the points on the
diagonal.

Then, based on those LOCATIONS, they have the same lengths.

No, that depends on direction.

Not in mathematics, it doesn't. In TOmatics, anything may go, but
mathematics is much stricter, things must make sense.





I don't think you can ever get this.
When you reduce the difference in locations down to an infinitesimal
level, they appear to be the same object, but the difference in
direction does not also have a limit of 0, but is unchanged.

In the limit, the horizontal and vertical "directions" disappear, or get
averaged out to 45 degree directions (TO has long argued that 0*oo = 1,
but now when that appears to be the case, a zero slope combined with an
infinite slope gives a slope of 1), TO now rejects his own claim.

It appears that way, but they never get any more parallel to the diagonal. I
think yours is a false assumption.




* Do you mean that it fails to have some property which is
mathematically required before one is allowed to use the word
"limit"? If so, what property is that?

No, it fails to have a property that is required to use that
limit to measure the diagonal.

WRONG! Only the locations of the "joints" if any, are relevant, and
in the limit there are no longer any joints except the endpoints of
the entire diagonal.

And the locations on the joints are supposed to be point ON THE CURVE
to ensure parallel orientation and proper measure.

Only if directions were being measured, which they are not. When only
distances are being measured, only distances count.

So, you measure with your ruler at a 45 degree angle to what you're measuring?



In other words, each element must have a point parallel to the
curve in order for the measure to work.

That may be a TO requirement, but it is not a mathematical one. The
only mathematical requirement for a curve to have a length is the
the lengths of its polygonal approximations have a finite upper
bound, and then that upper bound is then defined to be the length.


If the endpoints are on the curve.

Actually all the points of the polygonal approximation are required to
be on the curve, and taking the LUB of lengths assures that the
endpoints of the curve are included in some of them.


So, where your type of limit does NOT measure direction, it
cannot be used as a measure of a curve. Surely you can understand
that.

Why should we understand what is not true? There is only one way,
mathematically, to measure the length of a curve, as has been
described above. In many, but not all, instances it reduces to
evaluating a definite integral

Yeah, and that works fine, when the points are on the curve.

One takes a finite sequence of points along the curve, finds the metric
distance between consecutive points of that sequence, and adds up those
distances. That is polygonal approximation to the length of the curve.
If the set of all such polygonal approximations is bounded, then its
least upper bound is, by definition, the length of that curve.

Any measure of the length of a curve in a metric space that does not
agree with this result is wrong.


Uh, if length is not a type of measure, then I don't know what
is.

Arc length is the upper limit, provided that limit exists, on the
lengths of polygonal approximations to the curve.

And that would be some real value associated with the curve, a
measure of it.

See above!



I mean, take the formula for the one, and perform algebraic
operations on it to transform it into the formula for the other.
Derive it formulaically.

Been there, done that.

No you haven't. Stop lying.

Let the one step staircase go from (0,0) up to (0,1) over to (1,1).

The n-step stair then goes from (0,0) to (0,1/n) to (1/n,1/n) to
(1/n,2/n) and so on to ((n-1)/n,(n-1)/n) to ((n-1)/n,1) to (1,1).

Each is of length exactly 2.

The n step stair lies between lines y = x and y = x + 1/n, though with
points on both lines.

As n -> oo, line y = x + 1/n has line y = x as its limit,
and any points caught in between get squeezed onto line y = x.

So that the limit of the steps is the diagonal.


Yes, I have seen that argument, but you did not symbolically state the formula
for the staircase, and then symbolically derive the formula for the diagonal
from it. That I would consider proof, as long as it's valid.

--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... Distances between points depends only on their locations, ... But they need not be at any particular points of the curve. ... the locations of the staircase in the limit are ... mathematics is much stricter, things must make sense. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Distances between points depends only on their locations, ... But they need not be at any particular points of the curve. ... the locations of the staircase in the limit are ... Actually all the points of the polygonal approximation are required ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Distances between points depends only on their locations, ... But they need not be at any particular points of the curve. ... the locations of the staircase in the limit are ... Actually all the points of the polygonal approximation are required ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the limit there are no longer any joints except the endpoints of the ... mathematical requirement for a curve to have a length is the the lengths ... of its polygonal approximations have a finite upper bound, ... we need only verify that the sum of the distances between ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... But they need not be at any particular points of the curve. ... mathematics is much stricter, things must make sense. ... distances are being measured, only distances count. ... If the set of all such polygonal approximations is bounded, ...
    (sci.math)