Re: Calculus XOR Probability

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.



Let us compare the two thus in a segment-wise manner,
maintaining the same number of segments in each, and see if the
segments which describe the staircase approach those that describe
the diagonal. Where n=1, we have two segments to the staircase, {0,1}
and {1,0}, for a total change of {1,1}. Dividing the diagonal into
two segments we have {1/2,1/2} and {1/2,1/2}, also for a total change
of {1,1}. Now, as n increases we have {1,1}=sum(x=1->n:
{1/n,0}+{0,1/n}) for the diagonal, and sum {1,1}=(x=1->n:
{1/2n,1/2n}+{1/2n,1/2n}) for the diagonal. While the locations of the
points in each segment become arbitrarily close, the vectors defining
the segments of which the lines are made never become close, but are
always at a 45 degree angle to their corresponding segments in the
other line.

WE are using the mathematical definition of arc length based on the
Cartesian distance between points and the LUB of finite polygonal
approximations.

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.


When you look at the distance traveled, you sum all the x components
of the vectors in each line and sum all the y components, and you get
{1,1} in both cases, and the distance is sqrt(2).

When you look at the lengths of each, you sum the length of each
vector in the line. For the staircase we have sum(x=1->n: 1/n+1/n)=2.
For the diagonal we have sum(x=1->n: 1/sqrt(2)+1/sqrt(2))=sqrt(2).
Because of the difference in vector direction, even at the
infinitesimal scale, the staircase is longer than the diagonal.

Is that an "entire" enough "system" for you?

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.


Well, do you agree that /if/ the limit of the staircase is
/not/ "anything different from" the diagonal, then there /is/ a
contradiction?

Of course. If there were not distinguishing characteristics
between the diagonal and the staircase in the limit, or some
other explanation for the discrepancy, then I would have to admit
that you may have a real counterexample to refute the validity of
infinite induction.

And in standard mathematics there are no such distinguishing
characteristics. I have, in fact, given a specific and concrete
example of the staircases as parametric functions of the diagonal
distance whose limit is the diagonal itself.

No, you made a leap in saying that the points become the same set in
the limit, just because they are arbitrarily close. You didn't derive
one formula from the other algebraically, as asked.

On the contrary, I , and others, have done precisely that.


And, none of you
have explained why the error is sqrt(2).

The error is TO.




However, the differences I pointed out not only serve as a
probable cause of the discrepenancy, but lead to an exact
quantitifcation of what the dsicrepancy is.

Except that they provably do not hold in any standard mathematics,
and TO has not produced any other system in which they do hold.

But I have /defined/ the limit of the staircases and the
diagonal as /sets/ which are identical.

Sets of points, which do not lend themselves to additive measure.

They are sets which have well defined arc lengths in the only sense
that any set of points is allowed to have an arc length in standard
mathematics. Where is TO's definition of the 'arc length of a set
of points' which is self-consistent.


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.

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.




And that's what I mean when I say, your knowledge of what
constitutes a mathematical argument is sorely lacking;
particularly your knowledge of what a mathematical definition
is.

Uh huh. When I point out exactly why the measure fails, including
how the error is calculated, I don't know anything about making a
mathemtical argument. But, when you use a definition of limit
which doesn't lend itself to linear measure, and then blame the
fact that the measure of the limit isn't correct on infinity, for
vague reasons, that's a mathematical argument? Where do YOU think
the error ratio of sqrt(2) comes from? Infinity?

Everyone following these posts except TO knows where it comes from.
The error comes directly from TO's insistence on his false
"principle of infinite induction".


Thanks, I needed that.

TO needs something that he clearly lacks.

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.
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... 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 ... may have a real counterexample to refute the validity of infinite ... Except that they provably do to hold in any standard mathematics, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of staircase functions is anything but the diagonal function. ... as sequences of segments defined as a pair of reals which represent the x ... When you look at the distance traveled, you sum all the x components of ... Mathematics is very picky about semantics. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... According to the definitions of arc length in calculus, ... parallel to the curve they're approximating. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Rather than a set of points, let us define both the staircase and the ... If one is going to represent segments by numbers, ... What definition of arc length is TO using? ... Except that they provably do not hold in any standard mathematics, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the locations of the staircase in the limit are indistinguishable ... mathematical requirement for a curve to have a length is the the lengths ... it cannot be used as a measure of a curve. ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
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