Re: Calculus XOR Probability

In article <MPG.1ede89caba9a168398ad2b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1edd2d52db47c88e98ad1c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1edce444870c902a98ad09@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Then how does TO suggest that one measure the length of curves
which actually curve, such as a section of circular arc. One cannot
do it in the same way. Does TO wish to have a separate rule for
length for every curve?

Mathematics does a much better job of it, by having one definition
that fits for all curves for which "length" makes any sense at all.

Calculus does it quite well, using the notions I'm espousing.

Not so!

Indeed!!!

Calculus works quite well, but does not need, or use, any of TO's ideas
in order to do so.


Look
again at the page you posted on arc length. The segments all have
endpoints on the curve, thus some point on the curve between those
endpoints is parallel to the segment.

Not at all.

One can get the proper upper bound with no more than one segment of any
approximating polygon parallel to any segment of the staircase. And, of
course, as n increases, one segment becomes of insignificant effect.

That doesn't sound right to me.

TO's approval would foster my doubts but his doubt confirms my
certainty.

I'm not even referring to the locations of the points, but the
directions and lengths of the segments. that's all that's required.

Except that in the limit all such 'segments' degenerate into mere points.

No, they never fully degenerate. The infinitesimal lengths are sufficient for
maintaining direction.

Except that there are no such things as infinitesimals in the reals, nor
infinitesimal vectors in any real vector space.

What does not exist cannot maintain a direction.



It's also a series of diagonal segments as I orignally showed.

Then show me one limit segment where either end is not on the line
x = y.

The very first riser, from (0,0) to (1/n,0).

Then it extends beyond the limit set, and is not a part of that limit.


Is that second point of the form
(x,x)? Nope, not unless you claim that 1/n is absolutely equal to 0 for
infinite n,

It is quite enough to note that lim_{n increases unboundedly} 1/n = 0,
without any mention of "infinity" at all.




Only if one segment is a sequence. The limit set is {(x,x){0<=x<=1}

One segment is a sequence of two pairs, yes.

Of form {{(x,x) to (x,x)},{{(x,x) to (x,x)})

That's the limit of the point locations, not my limit.

It is the only limit that makes any mathematical sense in a real vector
space.


I can't believe this
notion of the curve as the limit of the sequence of segments isn't mainstream
somewhere. I feel like I'm already inventing half of mathematics. (sigh)

While we, who are more familiar with mathematics, note that whatever TO
is "inventing", none of it is mathematics by those standards that
mathematics requires of itself.



When the "two" endpoints of a "segment" coincide, one does not have
a segment, one only has a point.


When the endpoints are infinitesimally different

Then they are not possible in any standard geometry.

Until TO can produce a system of geometry in which infinitesimals exist,
and show us his limiting process in that geometry, he has nothing.


Yes, sir.

So where is you vaunted system, TO?




So, you think your line is a set of points. How do you know the
limit is also a set of points?

When a segment degenerates to having less than two endpoints it has
degenerated to a single point. Each of the risers and treads has
degenerated to a point in the limit.

What is the point of this question? What do you THINK the
sequence of segments becomes?

A set of points!

Well, I've already shown how the segments are different in the two
cases

Not in mathematics.

Call it what you like. The linear limit is valid, and you haven't shown
otherwise. Your big objection is "it's too original". (yawn)

Our objection is that there are definitions of length which TO ignores
in order to define lengths in a way which, among other things, conflicts
with mathematical lengths, and requires existence of things which do not
exist.


Lim_{x -> oo} x*sin(1/x) = 1, but oo*sin(1/oo) is meaningless. In
taking that limit as x -> oo, one NEVER allows x= oo.

Mathematical definition of lim_{x ->oo} f(x) = L:
for every positive real epsilon there is a positive real delta
such that | f(x) - L | < epsilon whenever | x | > delta

Mathematical definition of lim_{x -> +oo} f(x) = L:
for every positive real epsilon there is a positive real delta
such that | f(x) - L | < epsilon whenever x > delta

Mathematical definition of lim_{x -> -oo} f(x) = L:
for every positive real epsilon there is a positive real delta
such that | f(x) - L | < epsilon whenever x < -delta


That's fine, but none of it provides for linear measure.

TO has not defined his "linear measure" in any way compatible with
mathematics. Mathematics has a definition of the length of a finite
polygon in any Cartesian space, and a definition of the least upper
bound of any bounded above set of reals.

Yeah, that's essentially what I'm doing by defining the curve as a
sequence of segments in the limit.

On the contrary, if one applies the mathematically defined "arc length"
measure to the limit of the staircases, one can easily show that there
is no finite polygonal approximation of length greater than sqrt(2),
therefore the mathematical length cannot be any larger that sqrt(2).



The only legitimate measure of the length of a "curve" is the LUB of
lengths of finite polygonal approximations for that curve.

That works great. Just keep your vertices on the curve, and you can be sure
the
gons are paralle to the curve, for good measure.

Not at all necessary. The "curve" being approximated need not even have
any straight sections to which a polygonal segment can be parallel.

Note that in the mathematical definition, there need not be, and often
is not, any approximating polygon of length EQUAL to that of the curve
being measured, e.g., for a circular arc. It is only necessary that the
polygonal lengths be bounded, and then the LUB is the desired length.


And for the limit curve of the staircases, there is no finite polygonal
approximation with length greater than sqrt(2).

uh huh.



There is no reason to believe that infinitesimal differences cannot
exist on the Cartesian plane. It's just not normally done. So what?

There is even less reason to suppose that they do. So there!

Oh. Ouch. You got me.



In any case, you're talking about the limit
as n->oo, so what makes YOU think oo exists?

TO seems to think that in a limit process the limiting value of the
variable is actually achieved, but in standard mathematics, that
NEVER happens. See the above definitions, for example.

Of course, you asked a different question from last time, so I am
not sure you know WHAT you're asking. The limit of the staircase
is a staircase in the limit.

Not in standard mathematics as there are no actual segments of
zero
length. When a supposed segment is so degenerate as to have zero
length it becomes merely a point. Any sense of direction has
vanished along with its "length", and it is indistinguishable from
other points in that respect.


That's a mistake. Infinitesimal segments are exactly what calculus is
originally based on, and you know it.

But calculus got rid of them as soon as it could, and present day
calculus avoids them as being less than rigorous and leading to
occasional wrong answers.

Yeah, Cantor called them "bacilli cholera", but Cantor's work didn't make all
modern physics a dn calculus possible, now, did it? I know there has been
difficulty creating rigorous definitions for infinitesimals, but Robinson did
it, did he not?



Now you have a more "rigorous"
definition, but you should not forget the opriginal inspiration.

Nor the occasional wrong answers that it inspired.

As opposed to the wrong answers that your notion of limit inspires?

TO claims wrongnessof standard mathematics, but has no proofs.

We claim, and have proved, that in standard mathematics, TO's
assumptions are false. TO has produced no system in which they are not
false.

Not every indicated limit does exist, and the ones required by TO's
fantasies don't.


Oh?

Oh!

.

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