Re: Calculus XOR Probability
- From: Tony Orlow <aeo6@xxxxxxxxxxx>
- Date: Mon, 5 Jun 2006 10:05:34 -0400
Virgil said:
In article <MPG.1edfa107c7d8ccfe98ad32@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
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.
On the contrary, it is a specific demonstration of the fact that
limiting processes do not always commute.
According to TO's infinite induction principle it would necessarily be
the case that
Lim_{x->+oo} Lim {y -> +oo} (x - y)/(x+y) = -1
must be equal to
Lim_{y->+oo} Lim {x -> +oo} (x - y)/(x+y) = 1
You have implied transfinite nonsense in there.
Lim {y -> +oo} (x - y)/(x+y) = (x-oo)/(x+oo), and you really just take that to
mean -oo/oo=-1, since x-oo=-1*(oo-x)=-1*oo and x+oo=oo. But, without
assumptions of oo-x=oo=oo+x, your argument is wrong. In both cases you end up
with (oo-oo)/(oo+oo)=0/(2*oo)=0. In my system, if oo is specific, we have a
specific infinitesimal value, 0/(2*Big'un) = Lil'un/2, and not absolute 0.
Neither of your answers are correct.
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.
What "directions" do the points have?
The segments have directions, which alternate between vertical and horizontal,
as in the finite case of the staircase.
You offered a
pointwise definition of limit which made the diagonal appear to be
the staircase in the limit
What mathematical rules or definitions did that limit violate?
None? Then TO's only objection must be that he does not like the result.
No, my objections have been clear as day, and include an exact explanation of
the value of the error.
Well there are many more who do not like his results, so they must be
wrong, too.
Or does TO claim infallability and the right to speak ex cathedra?
Yeah sure, Virgil. I'm god and you must bow down to my authority. I thought you
understood that. ;)
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?
TO cannot give any mathematically valid justification why our answer is
wrong and his is right, whereas we can give such reasons why his is
wrong and ours is right.
No, you can't, haven't, and never will. I have given perfectly valid reasons
why your non-parallel limiting case gives an incorrect measure of arc length,
and how the definition of the limit of a curve can be changed so as to avoid
this mistake. I get the feeling that pearl's just being munched up with the
rest of the Purina Swine Chow. Oh well, I have a whole bag of pearls.
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.
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?
TO does not have any adequate definition of limit for objects in the
standard Cartesian plane, and has no system of his own in which such an
inadequate definition works either.
What, because of your unfounded claim that infinitesimal distances cannot exist
on the Cartesian plane? You'll have to wave your hands faster if you expect
that to fly.
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.
Except that the limit as n increases unboundedly cannot contain any n's.
Uhhhh......whatever.
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
But {0,1/n}+{1/n,0} is not a distance. In fact it has no well defined
meaning at all. So such a a nonsense expression proves nothing.
Indeed the length of each segment, according to the "usual metric", is given as
the square root of the sum of the squares of the x and y offsets in each
segment. That makes the length of {0,1/n}+{1/n,0} equal to 1/n+1/n=2/n.
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.
That is not at issue. There are two "limit" processes here, the varying
number of steps, n, and the varying number of segments, m, in a
polygonal approximation to the length, and the limit processes here do
not commute.
That's because your "polygonal approximation" to the diagonal is nowhere
parallel, but everywhere at a 45 degree angle, to the diagonal being
approximated. That's been explained to you. That's where you get the error of
sqrt(2), as the inverse of the cosine of the angle between them.
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.
We have quite enough trouble reading the nonsense TO puts ON the lines
without having to read between them.
Yes, there seems to be trouble even just with basic reading at times.
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.
The limit cannot possibly be a"sequence" of line segments as any such
"sequence", distinctly having both a first and a last member, must be
finite.
Says you, without good reason.
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.
But TO again requires the existence of infinitesimals, which are not
required, nor even allowed, in standard plane geometry.
Too frickin' bad. I'm obviously not restricting myself to the safe world of the
standard mathemtician when it comes to number or set theory, and base my
approach largely on geometric considerations, so I don't why on Earth you would
think that the restrictions of standard geometry would make any difference
either.
So while there may be some never-never land off in TO's imagination
where his dreams hold, there is nowhere in mathematics where this
particular one holds.
Not yet. :)
Okay, but zero vectors, which is what one gets in the limit, have zero
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?
length.
With absolute oo as the limit, you may think so, but as I said above, when the
infinitesimal 1/n is preserved, rather than zeroes away, one gets infinitesimal
vectors which preserve direction in the form of the ration between x and y
offsets.
It's a staircase with oo stairs, each 2/oo long, given riser and
tread. What is your question?
Since lim-{ n -> oo} 2/oo = 0, why isn't the TO-distance equal to 0?
Because we are considering a specific infinity, in which case 2/n is a specific
nonzero infinitesimal, thus preserving the direction of the segment.
Hopefully I already answered these questions, mostly by telling you
to forget about endpoints.
Since without those endpoints one cannot even define the staircase at
all, why should we forget them?
I defined the staircase without them, as a sequence of pairs, denoting
segments' x and y offsets.
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.
It is the whole universe in the standard world of mathematics, and TO
has not provided us with any other standard.
Provide you with a standard? Anything outside of the current standard is
nonstandard by definition.
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
Continuous curves in the plane, as images of real intervals, including
the staircases, can be oriented, in which case every point DOES have a
direction. But using those directions gives precisely the results that
TO object to.
If Chas's example was what you consider a proper use of those directions in the
staircase vs. those in the diagonal, then of course I object. The whole point
of the example was that the results were wrong. It was supposed to be a
counterexample to infinite induction, remember?
, since the
only points in the staircase that are on the diagonal are the bottomFor any finite n, only n+1 points are actually on the diagonal, but the
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.
distance from any point to the diagonal is less than 1/n.
That makes the limiting distance equal to zero.
Yes, I understand that, and it's fine for some applications, but does not
guarantee a good measure of arc length, as Chas demonstrated. You agree that
what I describe pertains to all finite n, but I maintain that it applies
equally to infinite n. There are always points in the staircase not on the
diagonal, which means it's a different curve.
Again with the endpoints!! Enough with the endpoints!! Oy!! Why, why
always with the endpoints???
Because if one knows what happens for those endpoints, one also knows
what happens for all points between endpoints. Take heed of the
endpoints and you know everything.
Yeah, that's one of two ways to define the curve, and the only one you seem to
be able to wrap your head around. It's probably not very flexible anymore.
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.
TO's insistence on wiggle room so that he doesn't have to admit that
his "infinite induction principle is as bad as everyone else knows it is
does not persuade anyone.
Wiggle room? Is that what you call a precise quantitative explanation of the
error in Chas' example? Huh! That phrase sounds more like it applies to your
squishy "infinity did it" explanation. Maybe that's more like "flailing space".
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.
Our objection is that that is not standard mathematics, that standard
mathematics already has solved the problem of arc lengths, and that
allowing the throwing out any standard mathematics just so that TO can
maintain a bad principle would be a bad principle.
I can o0nly chuckle at that.
I'm not sure.
Doubt can be the beginning of wisdom.
Which is why you'll never achieve that goal.
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.
But it is TO's job to know how to write mathematically sound statements
if he wishes to influence those who write mathematics. It is a job he is
failing at.
Failing at a hopeless task may be a waste of time, but is no source of shame
unless one fails to recognize the hopelessness of it. So, why am I responding
to you? Because occasionally you raise valid points for me to consider. The
rest of the time you're just pissy and obstinate. Oh well. Have a nice day.
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.
Every curve is entirely defined by its points. Segments are totally
irrelevant, except possibly as shortcuts to specifying these points,
and are never necessary.
Specifying the curve by segments is no shorter a cut than specifying it by
points. Both require the same amount of information. It's just a better
definition of the curve, at least for the purposes of measuring arc length.
--
Smiles,
Tony
.
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