Re: Calculus XOR Probability

Virgil said:
In article <MPG.1ec3d7695f98b59c98acb6@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx said:

Is lim n->oo {C_n} a real number, or is it a set of points?

It's a sequential set of points, that is a line of some sort, with a real
measure called length.

What does TO mean by a "sequential" set of points.

The usual meaning of a "sequence" is a set ordered in such a way that
all but possibly one member has an immediate successor, all but possibly
one has an immediate predecessor, and that any lessor member is linked
to any greater through a finite set of successors.

Since TO cannot impose this structure on the points of a line, he must
mean something else.

I mean everything above except for the limitation of finiteness on the number
of successors between any two points.




So why do you claim that I am using the limit to measure distance or
length? I am using the limit to take the limit of a sequence of curves,
nothing more.

To measure distance or length of a set of points in R^2 with the usual
metric, I use the usual approach, just like everyone else.

That's good, because this approach doesn't work.

It has worked perfectly for everyone for thousands of years. Why should
it stop working now?

When you prove the length in the limit is 2 instead of sqrt(2),
that's an indication that the object in the limit is NOT exactly the
same as the diagonal line


As he does not prove any such thing, TO is way off base.

That was the concluson Chas logically drew, erroneously or not.


What he does prove is that TO's version of arc length behaves that way,
but that is quite different from any actual version of arc length.


It wasn't MY version of arc length. That was supposed to be Chas' example of
how lim(x->oo: f(x))=f(lim(x->: x)) (or something similar) doesn't necessarily
hold. He says that's the problem with the proof, and I say it's because Chas is
using a version of "limit" that doesn't lend itself to accurate measure of arc
length. So, what are you talking about? I didn't create this example.


and the reason for the
discrepancy is easily seen to be due to the angle at which each of the
approximating segments intersects the diagonal it's supposedly measuring.


Angles are irrelevant. For each finite sequence of points along a piece
of an arc, one can find the point-to-point length without any attention
being paid to any angle, but only to the distances between successive
points. If the set of all such polygonal lengths has a least upper
bound, that LUB is the length of the curve.

Yeah, if the segments are parallel at some point to the curve.


I do not know what sort of measure of a curve TO is talking about which
does not fit that definition. And neither does TO.

There's a lot you don't know, despite your opinion of yourself.



Do you understand what I'm saying about the elements of the staircase needing
to be parallel to the diagonal in order for the "usual metric" to work?

We can easily understand what you are saying, but not why you are saying
it, because it isn't true.

If you say that, then you obviously don't understand what I'm saying.



No, you are misunderstanding me. I reject your example as demonstrating that
the concept of an inductive proof of equality holding in the infinite case is
invalid, because the error in your staircase example is easily explainable in
the way I've been describing.

But since that easy explanation is quite false, its ease does not excuse
its irrelevance.

Your declaration of falseness is false. That handwaving nonsense is pathetic,
Virgil. Really, you can do better than that. Why don't you go measure a sine
wave using vertical elements, and tell me how that works out? Show me one
example where the segments are not parallel to the curve, which actually works,
or admit you're full of crap.



I use the limit to find the limit of a sequence of curves. I don't
"measure" anything with it; nor have I claimed that you /should/ be
able to "measure" anything with it.

Fine so far.


I use the /usual method/ to "measure" the lengh of any curve, both
before and after taking the limit, based on sets of points in R^2 using
the usual metric.

What is the usual method for measuring the staircase in the limit? As far as
I'm concerned, you proved it had length 2.

As TO is totally unfamiliar with distinguishing between valid and
invalid proofs, his judgement of what constitutes a valid one is just
another of his irrelevancies.

Irrelevancies? Whatever. The proof is valid. The limit is not of a sort which
can be used to measure arc length of the diagonal, however. They are slightly
different objects, one with length 2, and one with length sqrt(2).



Finally, I use the principle of "infinite induction" to /deduce/ that
"length(limit)=limit(length)"; i.e.:

Yes, that's very nice, and I have no problem with that. You're measuring the
length of an infinite number of infinitesimal steps, not a straight line.


THE PRINCIPLE OF INFINITE INDUCTION is what I (erroneously) "use to
measure" the length of the limit of the curves;

Don't be so hard on yourself. You got the right answer. You did very well.


because the principle of infinite induction claims that I /can always/
"use" the limit of the lengths of the curves "to measure" the length of
the limit of the curves.

Yeah, and that worked out pretty good, didn't it?


It is this /third/ assertion (premise B, "infinite induction") which
causes a "problem", not the first two.

No

Yes! TO's alleged principle of "infinite induction" has been shown to be
invalid but impeccable logic several times, but TO's unreasoning faith
in it persists.

If you consider Chas' argument impeccable despite my far more exact argument
which explains the precise error, then you really have no clue about the nature
of analytical thought. That's too bad. Good luck with that. No wonder
Transylfinity manages to persist.


What do you mean by "the usual metric". Maybe this is the problem here. Is
the
metric usually wrong?

The usual metric for the Cartesian plane is that for points whose (x,y)
coordinates are (a1,b1) and (a2, b2), the distance between them is
sqrt( (a1-a2)^2 + (b1-b2)^2 ).

Similar formulae hold in Cartesian spaces of other dimensions.


Oh, well, that's just the Pythagorean distance formula in Cartesian
coordinates. According to that, the staircase has length 2. Since it never
travels in the diagonal direction, even in the limit, it makes sense that the
length does not become sqrt(2). The slope is always oo at 0. The diagonal, of
course, is sqrt(2), with slope everywhere equal to 1.




Yeah, well, as I understand it, the usual metric is generally used parallel
to
whatever one is measuring.

Parallelism is irrelevant, and need not even be defined in the space in
question, so long as it is a metric space.

TO seems in every mathematical question to have an unerring ability to
single out one irrelevant property as being the essential one in his
warped version of mathematics.

The distance between two points does not depend at all on the direction
between them. If it did, then rotating the space in which the points
occur would change the distance between them.


The sky is blue, Virgil, and love is good. (sigh) And puppies are cute.
--
Smiles,

Tony
.

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