Re: Calculus XOR Probability

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. You didn't derive one formula from the
other algebraically, as asked. And, none of you have explained why the error is
sqrt(2). It's obviously the inverse of the angle between each segment of the
staircase and its corresponding segment in the diagonal. It's quite clear. :)



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


That sentence good not.




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? ha ha :D




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".


ha ha ha hee hee ho ho hummmm..... Thanks, I needed that.

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.

This insistence that "infinity did it" is no different, nor any more logical,
than the creationists' "Goddiddit" philosophy. In fact, it rather seems to me
that all the political haggling over this stuff in the 19th century had a lot
to do with maintaining a mystery about the infinite, rather than making any
sense of it, for religious and philosophical reasons. Thankfully, I am not
bound by such constraints, because my religion allows that we CAN understand
the infinite, and that this is an important goal for mankind. Demystifying the
world is not a bad thing. Superstition is dumb. Evolution happens every day.
Get used to it. :D


That's a very sound
mathematical argument, if ever I heard one.


Infinitididditology: The Cantorian Occult Philosophy

Have a nice day!
--
Smiles,

Tony
.

Relevant Pages

  • 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
    ... So, the diagonal line as a whole is the limit of the staircase as a whole, as ... are where the risers meet the treads, but then there are infinitesimal segments ... Besides - points don't have directions. ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... So, the diagonal line as a whole is the limit of the staircase as a whole, as ... the size of them shrunk accordingly, in the infinite limit of this process, the ... points persay, but line segments. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... staircases converges pointwise; you don't *have* a staircase as the limit. ... Uh, yeah, you do have an infinite sequence of infinitesimal steps, and the ... mathematical requirement for a curve to have a length is the the lengths ... Some point on the segments must be parallel to the ...
    (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)