Re: Calculus XOR Probability

In article <MPG.1ed52352581c4a8298ace5@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1ed3f9dc9cbbac5398ace1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1ed3a8683e8cf9d898acdb@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article
<MPG.1ed290581a4f392198acd4@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:

For the last time, no. If the limit of the staircase is
anything different from the diagonal, which it is, then
there is no contradiction.

There is no mathematically valid model in which the limit of
the sequence of staircase functions is anything but the
diagonal function.

If TO wished to claim otherwise, then he must create and
present to us the entire system in which he claims his
allegations hold, as they do not hold in any current system.

Okay. Here goes.

Rather than a set of points, let us define both the staircase and
the diagonal as sequences of segments defined as a pair of reals
which represent the x and y coordinate differences between
subsequent points.

If one is going to represent segments by numbers, it takes 4
numbers for each segment, not just 2. And if "coordinate
differences" are to be used at all, then they cannot both be zero
for any segment.


No, they cannot both be 0 for any segment of any length. However,
your first statement is incorrect. The direction and magnitude of
each segment is represented by the pair of reals which represents the
x and y differences between endpoints. The other two numbers you say
are required denot the location of one of the endpoints, for a full
specification of direction, magnitude AND location of the segment.
However, with one starting point for the sequence, the starting point
for each segment is given by the sum of the x coordinate of that
point and all x coordinate changes for the segments up to the segment
in question, and the sum of the y coordinate of that starting point
plus all the y changes for the segments up to that segment. So, the
second pair of numbers you say are required are redundant.


No, they are not redundant. They must be supplied. But can be, as here,
supplied indirectly provided a startmg point for the whole is provided
(which TO did not originally require be provided).

In fact, the coordinates of the first point are, like all the other pairs in
the line, an offset, this time from the origin. So, even that first pair is a
pair of differences.

Which , if omitted, causes the whole to fail.


Only the
starting point and the subsequent changes are required to specify the
line.

And the fact that each endpoint is the start of the next line segment.

Obviously.

Math is often the business of pointing out the obvious.


Actually, for a staircase, one can reconstruct the whole from much less:
staring point, direction and length of each segment, left or right at
first joint, and number of joints or number of segments. But to
reconstruct any single segment, one needs both endpoints or equivalent.

Yes, if you know it's a staircase, you can use that knowledge to define it
using fewer parameters, but that definition wouldn't lend itself to
comparison
with the diagonal line, as my approach did very well. You obviously don't
really have a sound objection to it, so why pretend? Can you ever admit it
when
I supply a valid argument or construction? Are the segments of the staircase
in
the limit, of form {0, 1/n} and {1/n, 0}, the same as the segments of the
diagonal, of the form {1/2n, 1/2n}? Not even in the limit.

Except the limit does not contain any 'segment' at an angle from any
other 'segment' as all 'segment endpoints' are now ON the diagonal.

Not unless the vertices of your polygonal approximation lie on the
curve being approximated.

They do!

Not in the staircase example. Only half do.

I don't know where TO's staircases wander off to, but mine are placed so
that all the points of my polygons lie on them.


For each n, there will be a polygon of length 2, so length = LUB >= 2.
In the limiting case, every point is ON the diagonal so that every
polygon will be of length <= sqrt(2)

In the limiting case the vertices become arbitrarily close to, but not
identical with, the diagonal.

A point which is "arbitrarily close" to a line is, in standard geometry,
ON the line. TO's illusionary geometry does not count anywhere except in
TOland. There is no basis for it in mathematics.

What definition of arc length is TO using? He very carefully does
not say. But according to what he seems to be using, one can
equally show that the circumference of a circle of radius 1 equals
8.


I don't know what you "seem" to be seeing, but what you're suggesting
does not follow from anything I've said.

The LUB of taxicab metric lengths around a circle is 8.

That's very nice. Are your taxis made out of newspaper? Watch out for the
cellophane flowers.

http://mathworld.wolfram.com/TaxicabMetric.html
http://planetmath.org/encyclopedia/CityBlockMetric.html\http://en.wikiped
ia.org/wiki/Unit_disc

No. As TO's measure of arc length is incompatible with the
mathematical definition of arc length, TO must also prove that the
mathematical definition is flawed, which he has not done.

Chas already did that for me. Thanks, Chas. :)

Wrong! All he showed is that standard length does not match taxicab
length.

He was supposed to be proving that infinite induction didn't work

Which he did. And which others have also done.

Well, obviously, now I have to define a measurable limit
definition for sets of "points", on top of everything else.
Fine, it's on my list.

Until it is done, TO is wrong.

You crack me up, Virgil. Why don't YOU invent something for a
change?


I invented a parametrization of the staircase which showed TO to be
wrong.

In your dreams.

In everyone's opinion who has an opinion except for TO.

Furthermore, since I accept standard mathematics, I am able to use
the inventions of all standard mathematicians, whereas TO, having
rejected essentially all of standard mathematics, must invent
everything for himself.

Poor TO.

TO's poverty is self-inflicted, so those of us who are less masochistic
have little sympathy for him.

Does it actually cause you pain to question the foundations of your
knowledge?

I question them all the time. Which is what gives me equal right to
question TO's. By comparison, I find TO's wanting.



So, you calculated the error in my insistence on infinite
induction to have a value of sqrt(2)? That's very interesting.
I'd like to see your calculations. Moron.



On the contrary, TO's ERROR is (2 - sqrt(2)).

TO claims a measure of 2 whereas we have several times in several
ways shown that the true value is sqrt(2), so that TO's error is
the difference. It appears that TO is incompetent even at
elementary arithmetic.


You're just flailing, now, Virgil.

Such responses indicate that even TO's arithmetic is failing him.


Yeah right, and infinitididdit is the pinnacle of Virgilogic.

At least I have some form of logic, whereas TO only has his misleading
intuitions.
.

Relevant Pages

  • Re: Cantor Confusion
    ... For any given n, the number of steps, the staircase is defined as the ... offsets from the start of the segment to its end. ... version of the infinite case obviously has a length of sqrt. ... infinite-case induction that doesn't rely on such differences, ...
    (sci.math)
  • Re: Cantor Confusion
    ... For any given n, the number of steps, the staircase is defined as the sequence of segment offset pairs: ... the pair of numbers denote the x and y offsets from the start of the segment to its end. ... His conclusion is that, since one can prove inductively that the length of the staircase is 2 for every value of n, that proof only applies in the finite case, since his version of the infinite case obviously has a length of sqrt. ... By the way, the only other counterexample to infinite-case induction suggested made obvious use of a discontinuity based on a buried difference with a limit of 0 in the infinite case, and thus violated the rules as I put forth. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the sequence of staircase functions is anything but the ... numbers for each segment, not just 2. ... And in standard mathematics there are no such distinguishing ... You crack me up, Virgil. ...
    (sci.math)
  • Re: Cantor Confusion
    ... So, the notion of a sequence derives really from an inductive definition such as Peano's, and not from the one primitive in set theory, membership, alone. ... For any given n, the number of steps, the staircase is defined as the sequence of segment offset pairs: ... No, that is no simpler, and does not capture the direction or magnitude of any segment in a single pair. ... If this is a valid formulation of the two objects, and an explanation for Chas' counterexample to infinite-case induction, where does this fit with set theory? ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... of thing as the members of the sequence. ... infinite number of infinitesimal stairs, the length IS 2, because that's ... Then you have no business talking about the identity between the staircase ... because the segment definition preserves the notion of direction IN THE ...
    (sci.math)
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