Re: Calculus XOR Probability

cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow 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 ....

In other words, rather than saying "there is no natural number x such
that 2*x = 3", let us say "there is no rational number x such that 2*x
= 3".

In other words, let's defined a type of limit that lends itself to linear
measure. You asked for a better definition of limit for this purpose. Well,
here it is. Your discrepancy of sqrt(2) is explained yet again.


But the domain of discussion is figures which are defined as sets of
points in R^2. That is what my argument (which you claim is fallacious)
is based on. So if you change the argument to one regarding points
which are /not/ in R^2, that is like taking an argument about the
naturals, and trying to refute it with an argument about the rationals.

If your "principle of infinite induction" does not apply in R^2, then
where does it apply? Your statement was extremeley generous in its
presumptions; I saw no mention of special conditions when it should
apply.

The fact is, your proof was correct, as I said from the time you presented it.
The length of the staircase in the limit IS 2, because it's not the same thing
as the diagonal. Locationwise, they are indictinguishable. Directionwise, they
are not. You offered a pointwise definition of limit which made the diagonal
appear to be the staircase in the limit, and blamed the discrepancy in lengths
on infinite induction, whereas I am presenting an alternative form of the limit
of the sequence of curves, which clearly distinguishes between the two. You
offered a solution that didn't work as a counterexample to my claim, and I
showed you what's wrong with it, and how to properly compare the curves so the
problem makes sense. If you can't grok that, oh well.



Does it surprise you that these two statements provide different
answers?

Does it surprise you that your directionless point-wise definition of the limit
gives an incorrect measure and mine does not? It shouldn't.


It doesn't particularly surpise me that two /different/ definitions of
limit give /different/ results when applied to the same sequences. What
makes your /argument/ incorrect is your lack of a mathematical
definition of the terms you propose.

What, like "pair", "x and y offsets", "origin", or what? Which terms do you not
understand? Does it surprise you that my definition of limit demonstrates
exactly the kind of difference between the curves that I try to explain to you?



... 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


No, because all that I see you have done is to note (using
unnecessarily obfuscatory language) that the limit of the length of the
staircases is 2; which no one is disagreeing with; and that the length
of the diagonal is sqrt(2), which also no one is disagreeing with.

And the legth of the limit of the staircases is 2, which you DO disagree with
because you fail to see the obvious *linear* difference between that and the
straight diagonal.


You assert that the limit of the staircases is a thing you have
defined, and that it's length can be determined to be 2, but you have
not defined yet those terms, mathematically.

Instead you give various, contradictory "definitions":


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


Definition 1:

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.

What is the series Sum() you are describing here? All I can assume is
that you mean something like:

Sum(n->oo 1/n) = 1/1 + 1/2 + 1/3+ ... + 1/n + ...

I believe I fixed that in another post. I am trying to keep up with a bunch of
people here. Do you read the other subthreads?

lim n->oo sum(x=1->n: {0,1/n}+{1/n,0}=2/n) = 2


which sum does not converge; so whatever you mean by adding {a,b},{c,d}
+ {e,g},{g,h}, where a,b,c,d,e,f,g and h are real numbers representing
segments (I assume), it seems at odds with anything I've ever seen
before.

At any rate, what is this sum? Is it the real number 2? How can the
limit of the staircases be the real number 2?

The length of the limit of the staircases is 2, just like all the finite case
staircases of which it is the limit. Don't you remeber what we're talking
about? Sorry about the bad "grammar" there, which I believe I corrected,
perhaps for you, but try to read a little between the lines.




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. See above, "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.

Definition 2 is not the same as Definition 1: definition 1 claims
something about a "series Sum" of a sequence of line segments, whereas
definition 2 talks about merely a sequence of line segments.

"Definition 1" was a calculation of the length of the staircase in the limit,
using the actual definition, which you call "Definition 2". I was summing the
lengths over the sequence to get a total length.


Definition 2 says that the endpoints of these line segments are real
numbers; but it gives no explanation of why some particular point p
might b the endpoint of some line segment in this limit.

Slow down. We're not defining the curve by the endpoints any more. The set of
pairs which defines the curve is not a set of point coordinates, but a set of
xy offsets. They are segments with direction and length, relative location
because they are contiguous, and absolute location of the beginning given by
the first pair. So, where these pairs can be calculated based on the number of
segments, we can take the formula for calculating these segment pairs, and
compare them. If they are the same formula in the infinite case, preserving 1/n
as a non-zero infinitesimal, then they are the same curve in the limit.


Is the point (1/2,1/2) an endpoint of /some/ segment in the limit, for
/every/ sequence of curves we might consider? I doubt it; but so far
your /definition/ has said nothing that allows me to answer this
question.

Again, forget endpoints. Think vectors, okay? The sequence defining the
staircase is ({0,1/n},{1/n,0}) n times. The corresponding sequence in the
diagonal is ({1/(2n),1/(2n)},{1/(2n),1/(2n)}) n times. In the standard
universe, the limit of all these are {0,0}, but if we consider 1/n to be a
nonzero infinitesimal, then there is a clear difference. The staircase is
vertical and horizontal, with segments of the form {0,x} and {x,0}, and the
diagonal is, well, diagonal, with segments of the form {x,x}.




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?


This is your third definition: what is a staircase with oo stairs? what
is oo? what is 2/oo? How does this correspond to a sequence of line
segments with endpoints in R^2? What is a line segment with endpoints
in R^2, whose endpoints are 2/oo apart?

Hopefully I already answered these questions, mostly by telling you to forget
about endpoints.



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 b = 1 - a, or does
it not?

The tread of one step meets the riser of the next at a point on the diagonal.
Where the riser meets its tread, that corner is NOT on the diagonal, even if it
may be only an infinitesimal difference away, and consider coincident with the
line according to standard finitist limits.

If each line segment has endpoints which are in the reals, then how can
such a point be anything other than a finite or 0 distance from the
diagonal?

It can be an infinitesimal distance, or an infinite distance, in
nonstandardland. If the distance is infinitesimal, it's considered zero int he
standard world, but that's not the whole universe.




Given that point p, what is the "vector direction, at the infinitesimal
scale" associated with it? Can we deduce it from the values of a and b?
For example, how do I determine the "vector direction, at the
infinitesimal scale" at the point (1/2,1/2) (which I presume is in the
"limit of the staircases")?

The point (1/2,1/2) is in every staircase for n>1, for sure.

Not by the way I defined it; this is true only if n is even.

Oh yes, you're right. For some reason I was thinking nested subdivisions.
Bleed-over from another subthread I suppose. Sorry 'bout that.

Since we're talking about segments, it doesn't make sense to talk about the
direction at a "point" on the curve. If that point is on the curve of the
staircase, then it has two directions, since the only points in the staircase
that are on the diagonal are the bottom corners, where tread meets riser. A
better question might be to ask whether that point is in the staircase in the
limit, which ultimately, as you point out, depends on whether your infinity is
even or not. I don't think this can be determined. On the infinite level,
finite differences are insignificant, and the difference between a remainder of
0 or 1 can be ignored for the most part. I think it's impossible to say in
general whether that point is in the staircase. Half the time it is. If you use
an even infinity it is. But, that's not always the case. The fact remains that
none of the segment pairs are of the same form, much less containing the same
values, in the two curves.


The direction of
the tread before it is horizontal, and the diretcion of the riser after that
point is vertical. Remember, directions are not defined for points, but for
segments. That point has not direction of its own, hence the need to look at
the limit, not of the points, but of the segments.


Fine: how do we determine those segments in the limit, whose endpoints
you claim are real numbers? Can you name one of them?

Again with the endpoints!! Enough with the endpoints!! Oy!! Why, why always
with the endpoints??? I already defined the form of every segment pair in each
curve. Must I bake ruggaleuch too?



Given two points p and q in R^2 which are in the limit, how do I
determine whether p and q have the same or different "vector
directions, at the infinitesimal scale"?

Points do not have directions, ultimately. The segment {1/2,0} is horizontal,
and {0,1/2} is vertical.

Is {1/2,0} a segment or a point?

Do you see where I said, "The segment {1/2,0} is horizontal"??? It's right
there, above your question.

Oh, now I think I see your confusion. You're wondering where the coordinates of
the other endpoint are, right? You didn't notice I was using squiggly brackets,
and forgot that these are locationless segments, whose relative location is
given by the fact that one begins where the last ended, and the absolute
location of the curve given by the first pair, which is the offset of the
starting point from the origin. Is that what you forgot, or is there something
else?




Once you have addressed these questions, we can suppose that your
definition of "the limit of the staircases" is a mathematical object
called "L". /Then/ I can evaluate a statement you might make of the
form "the length of L is {whatever you propose}".

Are you sure you won't ask the alreayd answered questions, again?


No, I'm sure I will, assuming I maintain an interest; because your
several definitions above don't answer this simple question: if the
limit of the staircases is a set of line segments, whose endpoints are
points in R^2; then can you name a /single line segment/ in your limit
by telling us its endpoints, both of which are points in R^2? Any one
at all?

The m-th stair (starting at 0) has riser from (m/n,m/n) to (m/n,(m+1)/n) and
tread from (m/n,(m+1)/n) to ((m+1)/n,(m+1)/n). There, that's ALL of them.
Happy?




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.

No, I don't claim what you imply.

I claim that, if by "limit of the staircases" you mean /my definition/
of limit, and by "length" you mean /my definition/ of length, then it
follows that the limit of the staircases is the diagonal, and it length
is sqrt(2); and not any other thing.

Okay, but that leaves open other possible explanations for the discrepancy
besides "infinite induction don't work", namely, the pointwise definition of
the limit being unsuitable for linear measure.


However, I don't disagree that one could define a limit function and a
length function that result in the conclusion that the length of the
limit is the limit of the lengths for every curve.

You don't? Then, I guess I am not sure what your objection is. Perhaps you can
stop thinking of the endpoints and look at the pairs as I've defined them at
this point. I'm not sure.


In fact it is trivial to make such a definition: for any sequence {C_n}
of curves define lim n->oo {C_n} as the line segment ((0,0),(1,1)); and
for any curve C, define length(C) = sqrt(2). Then length(lim {C_n}) =
lim {length(C_n)} for all sequences of curve {C_n}.

It's trivial, useless, and stupid; but at least it's a /mathematical/
statement one can analyze.

Uh, yeah that is trivial, usless and stupid. And I thought you were perhaps
beginning to be able to comprehend my notion. Huh! You had me fooled for a
second there. :)


On the other hand, you haven't been clear enough in your definitions of
"limit" and "length" for me to make sense of your claims,
mathematically speaking; so they are meaningless to me.

Well, if I call something a segment and you immediately ask if it's a segment
or a point, I really have little doubt as to why my statements might be
meaningless to you. I explained carefully that the curve is defined as a
sequence of pairs denoting xy offsets of each successive segment, but you
apparently can't understand that, or forget the beginning of the sentence by
the time you get to the end. I dunno what the issue is.




The remainder of your "definition" leaves me as desirous of a
definition as before: you have simply introduced new, undefined terms
to define a previously undefined term. This renders your definition no
more meaningful than it was before, mathematically speaking.

Your failure to understand what I've defined as a limit isn't my problem.


Your failure to understand how to clearly formulate and communicate
your ideas is your problem.

It's not my job to teach you to read.



<snip>


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.


No, his objection is that you imply that for each point in the limit,
there is some /unique/ continuous /curve/ of points which can be
identified as "becoming the point in the limit".

That is not a feature of the definition of "limit" I gave (nor is it,
in general, a feature of various other definitionsof "limit" I have
seen).

Your definition of limit is inappropriate for this measure, and mine, whether
you can understand the concept of defining a curve as the limit of a sequence
of segments or not, is the correct notion of limit for this purpose, and does
NOT give an erroneous answer.


It's not that "2" is an "erroneous answer", it's an unjustified one
unless you clearly say what you mean by "limit" and "length". Your
"definition" of these terms is changeable and imprecise; it practically
gives no "answer" at all.

If you say so.



It does not require a /unique/ sequence to be identified with each
point in the limit; simply that /at least one/ such sequence exists for
a point to be considered a limit point of the sequence of sets of pairs
in R^2.

Nor does it generate a continuous /curve/ of points which is associated
with a particular point in the limit; it provides a discrete /sequence/
of points which converges to a point in the limit.

Yes, and your directionless points are not adequate for measuring the curve.


They never claimed that feature; they claimed to be a function I called
"the limit"; and claimed that a certain function yielded a quantity I
called "length" equally applicable to both the limit and on the
elements of the sequence..

These happen to correspond to the standard definitions; but naturally I
gave the definitions which were most familiar to me.

This was in response to your claim that it is not the case that if one
defines a limit L and a function f, that it is possible that L {f(x_n)}
can be different than f(L{x_n}).

If that is /not/ your principle of infinite induction, then can you
restate it please?


For equalities:
(E m,n e R: n>m -> f(n)=g(n)) -> f(oo)=g(oo))

For inequalities:
(E m,n e R: n>m -> f(n)>g(n) ^ lim(n->oo: f(n)-g(n))>0) -> f(oo)>g(oo))




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. Did you read any of it? Is
this supposed to explain why my limit definition "doesn't make sense", as you
claimed in your next post to have shown? Nice hand waving.


Aside from the fact that you have not even provided a definition of
what kind of mathematical object "the limit of the staircases" is (a
set? a real number? an equation?), I don't see how your discussion of
limit above applies to anything that is not a collection of segments;
which is to say your definition is (at best) simply providing an
example, not providing a proper definition.

I said first off it was a sequence of segments.

Then you claimed it was a series Sum.

That was to get a measure on the curve.

Then you claimed it was segments with endpoints in R^2.

With the curve in R^2, of course the endpoints of the segment are in R^2, but
the curve is not defined using the endpoints, but using the segments.

Then you claimed it was segments with endpoints /not/ in R^2.

When did I claim THAT?

Later in this post you go on to say that it might be a
sequence of cycles, whatever they may be.

You mean in talking about a sine wave? You can't imagine what I might means by
"cycles"? (sigh)

Finally you claim that it is
a sequence of segments with "infinitesimal nonlinearities".

That's what I said to begin with, and it's compatible with what I'm saying now.


If you can't follow the very
first statement of the argument, then get some ritalin or something. Claiming I
haven't said what I started out saying is disingenuous.


It is not disingenous to note that you give at least 5 different,
loosely related, yet contradictory definitions in this very post; not
one proper definition.

That may be your perception....



For example, let C_n = {(a,b): b = sin(n*a)/n^2}. According to my
definition of limit, lim n->oo {C_n} = {(a,b): b = 0}, as you should be
able to see for yourself by applying my definition.

Yeah, and that'll get you another shitty measure, which you will continue to
blame on "infinitididdit". This creationist math is deplorable.


I'm not "blaming" anything on anything else. I'm asking you to fill in
the blank below, using your definition of limit:

lim n->oo {(a,b) : b = sin(a*n)/n} = ??????

By your limit method, it's the x axis. It's hard to apply the limit I defined
to this, since it is a continuously changing curve. I'd need to define the sine
curve itself as a limit of a set of segments, as the cycles got proprtionately
shorter with the amplitude, thus preserving the directions of corresponding
points in each cycle. Let's say the sine wave is
lim n->oo Seq(x=1->oo: Seq(z=1->n: {2*pi/n,cos(z*2*pi/n)})). That is, the sine
wave is a sequence of oo cycles, each of which is a sequence of segments with x
offset 2*pi/n (for n segments per cycle), and y offset cos(x*z), where z is the
position of each segment in the sequence.

Okay, so now we want to apply your formula to this limit somehow. Where you
have sin(a*n) you are dividing the x coordinate by n, and where you are
dividing this by n, you are dividing the y coordinate by n. So, our x offset
becomes 2*pi/n^2 and our y offset becomes cos(x*z)/n, or cos(z*2*pi/n)/n. Since
we divided both the x and y offsets by the same amount, our direction has not
changed, but the sine wave has been compressed both vertically and
horizontally, so that it is essentially like looking at the finite sine wave
from an infinite distance. The waves bocome infinitesimal but not flat. For
every z=0, at the beginning of each cycle, the slope is still 1. So, ultimately
it's not the same thing as the x axis.


Whether your "measure" is "shitty" or "not-so-shitty", I should think
you should be able to answer this question. Can you name a single line
segement ((a,b),(c,d)) which is in the sequence of line segments with
endpoints in R^2 making up this limit?

I just defined the offsets. You can sum them any number of times to get the
endpoints of any given segment.


For that matter, can you name a single line segment ((a,b),(c,d)) which
is a "part of" the function y = sin(x)?

If you can't, how on earth do you say you have defined it,
mathematically speaking?


Could you walk us through how your definition of limit applies to this
sequence? What do you claim you mean by "lim n->oo {C_n}" in this case?
Is it a set? Is it a function? Is it a real number?

It would be a sequence of cycles, as wavelength and amplitude approach 0.

Is "a sequence of cycles, as wavelength and amplitude approach 0" a
sequence of line segments with endpoints in R^2? If so, how? If not,
this is yet another definition of "limit" you are introducing.

See above, if that made any sense to you.


It
will also have infinitesimal nonlinearities to it

How can a line segment with endpoints in R^2 have an "infinitesimal
nonlinearity to it"? Is this yet another thing which you will tack onto
your definition, so that a limit is a sequence of elements from R^2 x
R^2 x R? What do you /mean/ by that?

Uhhhhh....what? The nonlinearities arise due to non-infinitesimal changes in
direction from one segment to the next. There is no nonlinearity within any
segment.


, and the limit will appear to
have a length with an error of pi. Big surprise.


It's not surprising that you claim it; what's surprising is that you
expect that it makes any sense whatsoever as a mathematical claim.

Surprise!


Cheers - Chas



--
Smiles,

Tony
.

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