Re: Calculus XOR Probability

In article <MPG.1ed52ce0d87f8a298acea@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Matt Gutting said:
Tony Orlow 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. 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.

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? :D

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.

First of all, it's not that "the points become the same set in the limit".
It's
that the limits of the two sets of points are identical (the same set of
points). Nothing "becomes anything in the limit".

Your objection is semantic? Take it to alt.picky.english.

Mathematics is very picky about semantics. That is one reason why it is
so picky about definitions. Ambiguity in mathematics is almost always a
bad thing, so it is to be avoided.

But TO revels in ambiguity.

Second, the statement Virgil is making is not a leap; it's a consequence of
the definition of limit. Unless you have a different definition.

I just offered one that explains the discrepancy.

Except that it is not a mathematically adequate definition. TO's
definitions never are.
.

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
    ... 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
    ... According to TO's infinite induction principle it would necessarily be ... The length of the staircase in the limit IS 2, ... The segments have directions, which alternate between vertical and horizontal, ... We're not defining the curve by the endpoints any more. ...
    (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
    ... Rather than a set of points, let us define both the staircase and the diagonal ... as sequences of segments .... ... of the sequence of curves, which clearly distinguishes between the two. ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
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