Re: Calculus XOR Probability

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

Matt Gutting said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:



What is missing is a statement of /exactly what you mean/ by
"the length of (the limit of the staircases) is {whatever you
propose}".

The limit of the staircases is the series Sum(n->oo:
{1/n,0},{0,1/n}). That's n repetitions of a step with length 2/n,
for a total length of 2.

You're assuming again that you can interchange the sum and the
limit process. The length of the limit of staircases need not equal
the limit of the length of staircases with the standard definition.

Do you mean to say that the limit of the staircases is a series?
That's how your sentence is phrased, but it doesn't seem to make
sense. You're apparently making a sequence of geometric figures
(staircases), then stating that the limit is an infinite series,
presumably evaluated in the same way that infinite series typically
are - although you need to be clearer about the meaning of "n->oo
(1/n,0),(0,1/n)". How does the limit of a sequence of geometric
figures get to be a sequence of real numbers? Or is that what you
meant?

I guess that wasn't exactly clear. I wasn't sure what notation I
should use. When I say "Sum(n->oo: {1/n,0},{0,1/n})", I should have
said "lim n->oo sum(x= 1->n: length({0,1/n})+length({1/n,0})".

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. 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.



I mean
that each stair consists of the two pairs, one denoting the tread and
the other the riser, the curvilinear length being the sum of their
lengths, which is 2/n. So, the overall length becomes lim n->oo
sum(x=1->n: 2/n)=2.

That presumes something not in evidence, that in the limit there will be
any points not on the diagonal. In fact several people, including me,
have shown that in standard mathematics all the limit points ARE on the
diagonal, so that TO's formula cannot work in that limit set.

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.


For each of these pairs of the form {1/n,0} and {0,1/n} are
corresponding segments in the diagonal of form {1/(2n), 1/(2n)}, each
with a length of sqrt (2)/(2n), for a total length of sqrt(2)/n,
compared to 2/n for each stair. The diagonal is lim n->oo sum(x=1->n:
sqrt(2)/n)=sqrt(2).

The diagonal itself is a set of points. The length of the diagonal is
sqrt(2).

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



In order for me to understand your answer, you must first state
/exactly what you mean/ by "the limit of the staircases"; which
you have not done in the above paragraphs. Is "the limit of the
staircases" a function? Is it a real number? A set of line
segments? A set of pairs of pairs in R^2 x R^2? A white
elephant?

I stated already it's a sequence of line segments.

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.



"defined as a pair of reals which represent the x and y
coordinate differences between subsequent points". Each of those
pairs represents a line segment.

So, each staircase is a sequence of line segments. How do you
decide that the limit is also a sequence of line segments?

Because that's the way the line is defined.

What "sequence of line segments"? Show us any one of those alleged limit
"segments" which has two distinct endpoints. Unless such "segments" can
be shown to exist, TO is doing his dream stuff again.

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, they are indistinguishable on
the finite scale, but when a segment is defined as {0,x}, and x is nonzero,
which infinitesimal is, then that's a vertical segment. Only when both reals in
the pair are absolute 0 do you have a point without direction.


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, the
first alternating between vertical and horizontal segments, and the second
consisting entirely of diagonal segments.




The closest you get is this cryptic comment: "Because of the
difference in vector direction, even at the infinitesimal scale,
the staircase is longer than the diagonal." But this doesn't
tell me what "the limit of the staircases" is; it simply
mentions several (undefined) properties you propose it to have.

It's a staircase with oo stairs, each 2/oo long, given riser and
tread. What is your question?

*My* question is, since you haven't actually defined oo, how can
you tell whether oo or 2/oo exist?

Because that's the LIMIT. You want to take the limit as n->oo? Well,
oo has to exist, doesn't it?

NO! The whole point of limits is that you approach but never arrive.

The the staircase never becomes the diagonal.


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. You are looking only
at the locations of points, which is the same as the values of real numbers.


You have a "taxicab" distance of 2? It
doesn't matter WHAT rectilinear approaching path you take, it'll
always be 2. So, if you think the limit of the staircase DOESN'T have
a length of 2, it's not a taxicab distance, and the object is no
longer a staircase. If it's still a staircase, with an infinite
number of infinitesimal stairs

Since in the standard Cartesian plane, where these staircases exist,
there are no infinitesimals, that cannot happen.


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


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


The difference between the diagonal and the
staircase cannot be distinguished by location alone. By defining the
curve as a sequence of segments, rather than a set of locations, the
difference is quite detectable, because the segment definition
preserves the notion of direction IN THE LIMIT.

Not in standard mathematics. If TO wants a system in which his dreams
hold, he must build it from scratch, since his dreams crash in any real
system.

You mean "crash INTO any standard system".




For example, presumably there is some point p = (a,b) in R^2
that is in the limit of the staircases. Does that point satisfy



How do you know that the limit of the segments exists, and that it
is a segment?

Because that's the way it's defined

Definition does not guarantee instanciation.


Oh?


, whether as a starting point and
a vector, or two endpoints. When the points or offsets are
infinitesimal, the locations may be indistinguishable, but the
direction is not.

Except that in standard geometry there is no infinitesimal, so any
directions vanish when the two endpoints become one.


I'm obviously not constrained, in mathemtics or anywhere else, by the
"standard" or "normal" treatment. Those who are ultimately make no difference.



Until then, you haven't defined what you mean by "the length of
(the limit of the staircases)"; all you have defined is "the
limit of (the length of the staircases)"; and at least in its
result, we are all in agreement: the limit of the length of the
staircases is 2, and the length of the diagonal is sqrt(2).

But you disagree that the limit of the staircases is anything
other than the diagonal, whereas I have demonstrated a form of
limit which shows clearly that there's a difference, and which
accounts precisely for the error.

I don't see a clear definition of limit. Can you fill in the blanks
here:

DEFINITION: The limit of a ____(insert name of mathematical object)
is a ___ (insert name of a mathematical object) satisfying the
following criteria: ______.

The limit of a curve is curve satisfying the following criteria:

A curve is defined as a series of pairs {x,y}, the first denoting the
x and y offset of the first point from the origin in R^2, and each
subsequent pair being the offset of the next point from the last.


According to this "definition", a section of circular arc is not a curve.

Sure it is. It's the limit of a section of a regular polygon, defined as a set
of segments, as the number of segments increases without bound.


The offsets are defined with a formulaic relation to the number n of
points defined, such that knowing n and the relation, one can specify
each offset which defines the curve.

According to this "definition", a section of circular arc is not a curve.

Well, if you're going to repeat this, I guess I'll just have to whip out the
concepts put forth by Bresenham in drawing this beast. At each point in the
circle with center at the origin, the y offset of each segment is proportional
to the x position of its starting point, and the x offset is negatively
proportional to the y position of its starting point, the proportion for the
two determining length of each segment. Where the proportion is infinitesimal,
the curve goes from a polygon to a circle.


The limit as n->oo is defined to be the infinite sequence of xy
offset pairs which are each the limit of the xy pairs as defined by
the relation for any n.

That does not define one limit except as another limit, which is,
itself, undefined.

And you define the limit of the curve as the set of all limit points, which is
absolutely no different.


I think this last part is missing a little something, but you'll
probably point that out.

One also has the problem of proving that the limits indicated actually
exist.

Define "exist". The sequence of pairs "exists", does it not?


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


Oh?
--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... By defining the curve as a sequence of segments, rather than a set of locations, the difference is quite detectable, because the segment definition preserves the notion of direction IN THE LIMIT. ... the 1/n's have a limit of 0 as n->oo, but what that essentially means is that, for any given actual infinite n, 1/n is infinitesimal, and larger than absolute 0. ... definition of "the limit of the staircases" is a mathematical object ... A curve is defined as a series of pairs, the first denoting the x and y offset of the first point from the origin in R^2, and each subsequent pair being the offset of the next point from the last. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... are where the risers meet the treads, but then there are infinitesimal segments ... the length of the limit of the set of staircases D is not ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... are where the risers meet the treads, but then there are infinitesimal segments ... the length of the limit of the set of staircases D is not ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... as sequences of segments .... ... staircases is 2; which no one is disagreeing with; and that the length ... I stated already it's a sequence of line segments. ... Definition 2 says that the endpoints of these line segments are real ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... The limit of the staircases is the series Sum(n->oo: ... number of steps, n, and the varying number of segments, m, in a ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
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