Re: Calculus XOR Probability
- From: Virgil <vmhjr2@xxxxxxxxxxx>
- Date: Mon, 05 Jun 2006 15:19:59 -0600
In article <MPG.1eee042025cb3ae198ad47@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
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.
On the contrary, it is TO's nonsense.
In standard mathematics, the order in which limit processes are applied
can effect the final results, as in the perfectly legitimate example
above.
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.
If one uses TO's nonsensical definitions of limiting processes,
anything may happen, but not if one uses mathematical definitions.
The mathematical definition for Lim {y -> +oo} f(x,y) =L says that for
each x
for each epsilon > 0
there is a y_0 large enough so that
whenever y > y_0, then also | f(x,y) = L | < epsilon
It does not say that either x or y ever assumes any infinite value at
all.
Note that in defining, say, Lim_{x -> 0} sin(x)/x = 1, the value x = 0
is specifically prohibited.
Similarly for Lim_{x -> +oo} f(x), the "value" x= +oo is specifically
prohibited in the limit process.
In my system
TO does not have a system.
Neither of your answers are correct.
Perhaps not in TO's non-system, but it is perfectly correct in standard
mathematics.
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.
WRONG AGAIN! TO's objections have been less clear than mud. There is a
certain result that TO wants, so he begs every question to try and
support his result.
That sort of reasoning leads to bridges that fall down.
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. ;)
How quaint, and how arrogant.
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.
While our reasons may never satisfy TO, assuredly his non-reasons will
never satisfy anyone else.
I have given perfectly valid reasons
Valid only in TO's mind, and nowhere else.
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.
There is no requirement in standard mathematics for any parallelism.
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.
I have a system of standard mathematics in which it works this way.
To has no system in which anything works.
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.
As long as TO does not have any working system, he cannot expect anyone
else to accept his castle in air approach.
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. :)
TO keeps promising, but never delivering. His "system" is pure vaporware!
Okay, but zero vectors, which is what one gets in the limit, have
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?
zero
length.
Because we are considering a specific infinity
Limits in real mathematics never actually use any infinities, specific
or otherwise.
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.
To remove the standards we have and leave a vacuum, does not improve
things.
But that is precisely what TO is trying to do.
, since the
only points in the staircase that are on the diagonal are theFor any finite n, only n+1 points are actually on the diagonal,
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.
but the
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.
Since TO does not have a system yet, he is still pushing his vaporware.
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?
TO's lack of any system allows him to wiggle every time he gets caught
in another self-contradiction.
Vaporware like TO's does not sell.
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.
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.
Beeing obstinate about mathematical truths is one of the things
mathematicians do.
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.
Not in mathematics. Mathematics already has a measure of arc length than
depends only on the points of the curve.
.
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