Re: Calculus XOR Probability

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

<snip>

Right now, we're working on Premise A: the limit of the sequence of
stairsteps curves is the set of points D, which is the diagonal line.

Well, what we are doing is debating whether premise A or premise B is at fault
for the faulty conclusion C.

If a point is a line segment in the limit at the length goes to 0, then a point
has direction...

Look at the definition of "limit" I gave. A point is not the limit of a
line segment. The /set/ of points, the diagonal, is the limit of the
sequence of curves "staircases".

So, the diagonal line as a whole is the limit of the staircase as a whole, as
the number of steps increases and the size of the steps decreases accordingly,
but the steps as they reach infinitesimal size in the limit are not the points
which make up the diagonal line?

You are still not using the definition of the limit I gave; you are
using something you imagine that I said. I never said anything about
staircase steps "becoming infinitely small" or "reaching infinitesimal
size".

I thought the example you gave was one where, as the number of steps grew, and
the size of them shrunk accordingly, in the infinite limit of this process, the
diagonal would be produced, but would not have the length expected as the limit
of the lengths in the finite case. Am I mistaken?


It's really very simple. I said, loosely, that the /points/ in the
diagonal are approached arbitrarily closely by the the /points/ in the
sequence of staircases. Nothing more than that.

Right, You're saying the points on the staircase approach the points on the
diagonal, in terms of location. But, the elements of the staircase aren't
points persay, but line segments. In the limit, they still retain their
direction, which directly accounts for the discrepancy of sqrt(2) in the
length.

I think we're both starting to feel like broken records on this one. (sigh)


Perhaps you feel that the points of the line
are where the risers meet the treads, but then there are infinitesimal segments
between them, at right angles to each other. In any case, the problem with your
concept of the diagonal line being the staircase in the limit is that in the
limit you lose the concept of the mutually orthogonal segments which constitute
the staircase. If you maintain the linearity so that the points are indeed the
limit of the segments then the answer is clear.

Yes, if I had claimed to do the things you say, there would indeed be a
problem. But that's not what "taking the limit" means.

Well, it is. You are postulating a staircase of n steps, with n risers and n
treads, as n approaches oo. You say the points become exactly those on the
diagonal, but at no point on the staircase is it parallel to the diagonal. The
derivatives of the two lines are different, the one being continuous and
constant, and the other being discontinuous.


When I say "the diagonal line is defined by the points (x,y) satisfying
x+y=1, x, y>= 0", do you see any mention of "except it must be
self-parallel", or "only if the points (x,y) have a particular
direction" or "only if the points are not actually infinitesimal line
segments"?

No, but I see a formula that defines y in terms of a continuous function on x,
such that y=1-x. Does that define the staircase? Is that really the limit of
the formula for the staircase? If you claim so, please derive that.


If you accept that "the set of points satisfying y = mx + b" is the
same as "the line with slope m and y intercept b", then any other
function which produces the /same set/ of points /really does/ produce
the line with slope m and y intercept b: they produce exactly the same
thing.

Sure, if the slope is equal to m, which it is not on the staircase at any
point.


Now, suppose we define L as the set of points on y = mx+b with 0<=x<=1.
This definition doesn't need to /also/ include "and define the length
of L as sqrt(1+m^2)". We /deduce/ the length of L from the definition
of L, and the fact that we are working in the x/y plane R^2.

When dealing with straight lines in Cartesian space we use the Pythagorean
theorem. Did you use that on the staircase?


The length of L is not /part/ of the definition of L; it is a
/consequence/ of the definition of L.

Right, as a line with slope m over the domain.


Likewise, the length of the limit of the set of staircases D is not
/part/ of the definition of the limit; it is a /consequence/ of the
definition of the limit.

Right, and the limit, as you aptly proved, is 2.




Besides - points don't have directions. /Curves/ can be said to have
directions /at/ points. I haven't claimed anything about maintaining
"directions at" points, anymore than I have said anything about
maintaining "lengths between" points. This is still just Premise A.

Premise A is that the diagonal line is the staircase in the limit, right? If
you mean the normal, self-parallel diagonal line, then this is not a correct
statement. Once the stairs are reduced to infintiesimal size and appear to you
as points...

I don't mean to sound like a broken record; but you are /still/ not
using the definition of limit I gave. Instead of talking about what you
imagine me to be saying, why not focus on what I'm actually saying?

I am tryign to ficus on the example given, and argue for the correct
interpretation of its failure.


What is /really/ meant by saying

"the sequence (1/2, 1/4, 1/8, ..., 1/2^n, ...) has limit 0"?

It means that as n grows without bound, 1/2^n shrinks without lower bound,
technically.


Do you agree that this is /the same number/ as

"the limit of the sequence (-1/3, 1/9, -1/27, ..., ((-1)^n)/3^n, ...)"?

It's not like in one case we get a "different" 0, because "the sequence
is different". Why? Because that is not how the limit is /defined/.

Well, in standard mathemtics, no distinction is made between these two zeroes,
but one can be made between the infinitesimal values produced at infinite n
using a variable n and formulaically compared values.


To clarify: Suppose (a_1, a_2, a_3,..., a_n, ...) is a sequence of
rational numbers.

Could you say what you think it means to claim "the limit of the
sequence (a_1, a_2, ..., a_n, ...) = pi"?

The standard practice is to say that for any finite x, there is a sufficiently
large finite n such that a_n-pi<x. Correct?


<snip>

Now, you blame premise B, that the inductively proven constant equality of L=2
is only true in the finite case, and not in the infinite case, and that is why
the proof fails. You postulate something magical happening in the infinite case
with no more justification than, "what do you expect from infinity?"

RIght. So premise B is false

(momentarily dumbfounded - goes out for a smoke and some sun...)


Thank god this isn't a non-smoking forum (sparks one up as well) :)

In my hasty reading, I missed your "redefinition" of Premise B above.

Premise B in the "Han-style" argument is the claim "always true for
finite, therefore /also/ true for the limit". That premise is (at
least) unfounded in this case; because it has not been derived directly
from the definition of the limit that I gave.

I am not sure it can be. The standard treatment of the limit was chosen for
it's non-reliance on any concept of actual infinity, since it defines the limit
purely in terms of the finite case. I don't have an argument with that
formulation, really. But, it doesn't address the issue of whether a constant
relation that holds for all finite cases can be said to hold in the infinite
case as well. Over the last year, a number of inductive proofs have been
offered that have really brought home this issue for me. My conclusion is that
inductively proven equalities between expressions on sufficiently large n hold
as well for infinite n, much like what Han is suggesting. I freely admit that
properties in general do not have this quality, and that inequalities in
particular may or may not, but maintain that equalities do indeed hold for
infinite n.


However, we /can/ derive a result using my definition ("the length of
the diagonal is sqrt(2)") which contradicts his premise: thus the
/conclusion/ "always true for finite; but /not/ true for the limit" is
correct in this case (which is what it now appears you were calling
"Premise B" above).

Yes, I know what the intent of your example was, to show that this proof
structure is invalid in general, but I have disagreed on that point since long
before this thread. It seems clear to me what the error in this example is: an
essential difference between the continuous diagonal and the discontinuous
micro-step diagonal.


RIght. Ummm....is that a valid justification? I haven't seen any reason why a
formulaic equality proven for all n>m doesn't hold in the infinite case...

But I just gave you a perfectly valid reason!

No, you gave an example of a fractal line that you assumed was the same as a
straight line, and derived a contradiction due to that assumption, then blamed
it on the general proof structure. I know you didn't make up this example, and
I'm not blaming you, but I'm not agreeing that an undifferentiable fractal line
is exactly the same as a continuous line with a constant derivative. And, I am
maintaining that, in absence of such unwarranted declarations of identity,
inductive arguments of equality hold for infinite n.


If the limit really /is/ the diagonal, and the length of the diagonal
really /is/ sqrt(2), then what other conclusion can there be besides
that Premise B really /is/ "at fault" here?

Those are big IFs. I think we can agree that the diagonal distance between the
corners of a square is sqrt(2) times the side of the square. My position is
that the diagonal line is different from the limit of the staircase, and that
the proof holds for this object in the infinite case. Your position is that
there is no difference between the diagonal and the limit of the staircase, so
this is obviously the result of claiming the proof holds for the infinite case.
But, if you truly think the diagonal is the limit of the staircase, then you
should eb able to drive the formula of the one from the formula of the other as
a limit, and you should be able to differentiate the staircase in the limit,
whch you really can't.


The "blame" here falls on the assumption that the limit of a sequence
is anything other than /what it is defined to be/. This is the
"essence" of Premise B ("the magic wand" property of limits):

* the vague /intuition/ of limits as something "becoming" something
else "at infinity",

as opposed to

* the precise /definition/ of limits as something which the sequence
"gets arbitrarily close to".

There is really nothing vague to saying that if an equation is true for all
sufficiently large n, then it is true for infinite n.


<snip several repetitions of this same conflation between "becoming"
and "approaching">

Right now, I'm just claiming Premise A:
the limit of the staircases is D, the diagonal line. You don't need to
appeal to anything else yet; just evaluate that claim.

It's been done. There is an inherent difference between the two lines, which is
accurately reflected in their different lengths, as amply shown to result from
the cosine of the angle between the diagonal and the steps.

No; the two different definitions produce identical sets of points,
which we can also express as "D, the set of points (x,y) such that x+y
= 1, x,y >= 0". Unless you are claiming that the diagonal is not D,
then the limit of the staircases is exactly the diagonal line.

That's what I'm claiming. Please state the formula for the finite staircase,
and derive the formula for the diagonal as the limit of that formula as the
number of steps approaches oo. If you can do this, then we have something to
discuss.


This is really no more unusual or bizzare than observing that the two
sequences of rationals I gave above both have the same limit: 0. Why?
Because that's just what the definition says it is, in both cases.

Many sequences have terms with a limit of 0, just like the many inverses of
those terms which have no limit, but diverge to oo.


There is no reason to "distinguish" between the 0 which is the limit of
one sequence, and the 0 which is the limit of the other sequence,
because in both cases, 0 is provably the unique real number which
/satisfies the definition/.

Yes, as a standard real, there is no difference between the one 0 and the
other, but distinctions can be made between the behavior of the two sequences.


In our case, the fact that the set of points D is a line, that it
therefore has length, etc., is not affected by "how we got" that set of
points. Instead, it is a /consequence/ of the fact that D is in R^2,
and the way we define length in R^2 tells us that the length of D is
sqrt(2), regardless of "where it came from" or"how we got it".

And yet, you proved that the staircase always has a length of 2, regardless of
the number of steps, and this doesn't change in the infinite case. It's a clear
indication that this fractal diagonal is a different animal from the one we
normally consider in the Pythagorean sense.


"sqrt(2)" not somehow "attached" to D by way of the definition that
generates D, anymore than there is somehow something different
"attached" to the limits of the two sequences with limit 0.

That sentence not read good, but the error of sqrt(2) is directly explained by
the angles between the infinitesimal elements and the diagonal.


Premise A only claims that a certain set of points is the limit of the
sequence of staircases, where the limit is defined as (roughly), "every
point which is approached arbitrarily closely"; and that this set is
identical to the set of points D, which is the diagonal line.

Right, the diagonal is the limit in terms of location, but when it comes to a
metric on the line, direction is important too. Do you measure something with a
ruler placed at odd angles to what you're measuring, or does the ruler only
measure accurately parallel to its own direction?


And that's /all/ it claims. Because that's all "the limit" means in
this case. Really.

So, in order to claim that Premise A is false, you must then show that,
given my definition, there is a point on the diagonal which is not in
the limit of the sequence of staircases; or that there is point which
is in the limit of the staircases which is not on the diagonal.

Or that the derivatives of the two lines are entirely different, or that the
formula for the one cannot be derived from the other.


You have demonstrated neither thing; so the only sensible conclusion is
"Premise B is false".

No, the sensible reaction is toe recall that non-parallel elements cannot be
used in accurate measurements.


Cheers - Chas



--
Smiles,

Tony
.

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