Re: Calculus XOR Probability

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

cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:

<snip intro regarding equivalenceof "y=x" and "2y=2x", versus
equivalence of the diagonal and the limit of the staircases>

As you didn't respond to the above, can I assume that you agree
with it? Or do you disagree that figures in the x/y plane are
adequately defined by sets of points (x,y) satisfying formulas
such as y=f(x), or f(x,y)=c for some constant c?

To be honest, my response was "blah blah blah". You can reiterate
that your limit doesn't take into account anything but location,
and I can reiterate that that's the problem with your limit,
forever and ever.

When you say there's a "problem" with my limit, what do you mean?

I mean that it doesn't take into account anything but location, and
you're using it to measure distance.

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?


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. 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.


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.



* Do you mean that my definition of limit is not well-defined
(i.e., you can't figure out what the set "lim {C_n}" is for some
sequence {C_n})? If so, what sequence {C_n} of curves and point p
do you find it not possible to decide whether "point p is in limit
{C_n}" is true or false?

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. 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.



* 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.


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.


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.


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.


This is beginning to sound Berkeleyesque. Are you sure you're feeling
okay?

It is TO who is creating all the flash and glitter, signifying nothing.
The mathematics involved is quite straightforward and pedestrian.

Yep, that was another tree falling in the woods. Timber!


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.



Secondly: I see no problem with a definition of a curve which lacks
"points with directions", since I can determine the direction of a
curve, /at/ a point, using the standard topology on R^2.

I have already shown you how I define the diagonal as a set of
points; and how that set of points is the same as that of the limit
of the staircases, as a set of points. I assume you agree at least
with this as a starting point: Lim(sequence of sets of points) =
(set of points).

Locationwise, the points on the staircase approach those on the
diagonal. Directionwise, they do not, but stay at a constant 45
degree angle to the diagonal, causing the error ration of sqrt(2).
I'm getting tired of repeating this.

As no single point has a direction, it takes at least a pair of distinct
points to determine a direction, TO is , as usual, off his rocker.

Yes, and in a curve, each point has two neighboring points in a sequence of
points, determining the direction at that point.


In the staircase, the slope between two arbitrary points becomes more
likely to be close to the slope of the diagonal the farther apart
they are. Any abritrarily close points are either in a norizontal of
vertical direction.

But only with reference to another point. No point in isolation has a
direction, only in reference to another point!

Agreed.



You appear to mean something else by "direction of a point".

That definition doesn't work on the staircase, where you have corners
with no discernible direction. It's somewhere between 0 and oo.


Can you tell me /how/ you determine the correct "direction" of the
point (0,0) in each of the following simple figures in the x/y
plane?

Remember, direction is necessary for measure.

Not to measure the radius of circles. In fact for any measure of length
it takes TWO points to establish either the direction or the distance.

No one point in isolation has either direction or distance.


Correct!



As a point in itself it is without direction or measure. It can be
thought of, in the context of the Cartesian plane, as an
infinitesimal square parallel to the x and y axes.

Or even better, as a circle of 0 radius, so it in effect has ALL
directions as all circles do.


That's what I said to Chas. It would have all, or none, depending on how you
thought about circles. But, in the context of an empty xy plane, I would
picture each point as an infinitesimal grid element.


The remainder of the concepts discussed in the previous post appear
unresolvable until the question of "what do we mean by the diagonal
line?" is resolved, which is in turn contigent on the question
"what do we mean by the point (0,0)?", so I will stop here.

<snip>

Until you define what you mean by statements like this, I
cannot express an opinion regarding whether such a statement is
true or false; instead it is meaningless.

Then you're being clueless. You should be able to parse that
statement. Or, did you read "probabilities" as "possibilities"
again? Remedial English is down the hall to the left.

This is Remedial Mathematics. The current topic is "what is a valid
mathematical argument?". Stay in your seat until the bell rings.

If that's your attitude, the discussion's over. Have a nice day.

There has been no discussion. TO has again been pontificating in areas
of his nearly perfect ignorance, and refusing to become informed.


Discussion's a two way street, Maestro.

--
Smiles,

Tony
.

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