Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:

*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?
Yes, or writing it out without shorthand, I want to take the limit as
n increases without bound.

Well, oo has to
exist, doesn't it?
Not necessarily.

Oh. Then the symbol doesn't necessarily mean anything. Can you take a limit as
n approaches soemthing that doesn't exist?

No, and I'm not. It's not true that "n approaches infinity"; n increases without
bound. And one can certainly take a limit as n increases without bound.

Right, but you can't say what the curve IS in the limit without considering
having REACHED the limit.



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.
That's exactly it. There's no requirement that the limit object be the same sort
of thing as the members of the sequence.

You don't require it, perhaps, but then again, if you think they are the same,
then what happened to your arclength measure?

If it's still a staircase, with an
infinite number of infinitesimal stairs, the length IS 2, because that's the
nature of the staircase. In any case, you're talking about the limit as n->oo,
so what makes YOU think oo exists?
I'm not talking about the limit as n approaches anything, as you seem to imply
from the way you write "the limit as n->oo". I'm talking about the limit as n
increases without bound. I don't believe oo exists as a number.

Then you have no business talking about the identity between the staircase "in
the limit" and the diagonal. If oo doesn't exist, then they never are the same,
and the whole discussion goes out the window.

See my reply above.


The staircase approaches the diagonal, location-wise, but in the end, it's not
the same object, as I've demonstrated.


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.
Can you prove that assertion?

I have demonstrated a concept of limit that shows it. while the treads and
risers become infinitesimal, their direction never changes, and never
approaches the direction of the diagonal.

You haven't defined "infinitesimal" to anyone's satisfaction, certainly not
to mine.

Any finite divided by any infinite yields an infinitesimal. If you divide the
unit interval by n, as n->oo, the subintervals have lengths approaching 0. For
any specific, non-absolute infinity, you have an infinitesimal value. As the
number increases without bound, the subintervals shrink with a bound of 0, but
never ultimately reach that bound, as n never ultimately reaches absolute oo.
But, perhaps you just aren't partial to infinitesimals.



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.

See?

No; you're assuming that the limit is some construct involving infinitesimals.

I am assuming that when n is a specific infinity, 1/n is a specific
infinitesimal, and that's part of the problem we're discussing.



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 stard finitist limits.

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. 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.
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, 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.
If the locations are truly indistinguishable, then the endpoints are identical,
and the result is a point, not a segment.

Incorrect. Even an infinitesimal is larger than absolute 0, so n/n is not 0/0,
but 1, and n/0 is still infinite, even when n is infinitesimal. So, the risers
are still vertical, not diagonal. Infinitesimal differences are not equalities.

If an infinitesimal is larger than 0, one can distinguish between one endpoint
and the other one an infinitesimal distance away, no?

Yes, on the infinitesimal scale, not on the finite scale, as in standard
mathematics.



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.
Okay, so how about the infinitesimal scale?
{0,1/n} is still vertical, and {1/n,0} horizontal, even if n is infinite. Those
0's are absolute 0's. There is no horizontal change in any riser, or vertical
change in any tread. 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. Direction is maintained.

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?
I still have questions about your answers to the questions.
Just as long as they're not the same questions that I already answered, or
we're just going around in circles, which I suppose serves some purpose anyway,
but seems rather like a waste. Anyway, carry on....

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.

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.

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.

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

Both blanks have to be filled with terms which either are agreed upon
generally, or are defined in turn according to the template provided.

Once you can fill in those blanks, then we have something we can talk
about. Until then, your definition is not sufficiently well-formed to
be able to discuss anything related to it.

Your serve.
By your definition of "curve", the set of points {(0,0),(1,1)} is a curve.
Is that intentional?

By my definition, the segment between them is a curve. When defining a curve
simply as a sequence of points, a two point sequence is allowed, as two points
determine a line. For a continuous curve, one can apply Archimedean principle
to the sequence, but sparse curves are not unheard of. Look at a graph of the
stock market. It's a sparse curve, down to the minute, or hour, not the moment.

Generally, my understanding of curves is that they're defined almost everywhere.
I'll check up on this.

That's the common notion of a curve, but a general definition may be adopted
that distinguishes between continuous and discrete curves. What exactly do you
call a series of points separated by finite space? It can be considered a
discrete curve, and Archimedean principle may be applied to get the common
notion of curves as continuous.



It looks as if by your sentence about "formulaic relation" you mean something
like "the offsets (x_n,y_n) are determined by a function whose domain includes
n, and which doesn't change for any n". Is that a correct interpretation?
Does the function need to be specifiable by a formula, or can it be a list
of input-output values? If it can't, why not?

If you want to find the limit of the curve as n->oo, then you need to be able
to specify the segments using a formula of some sort, because you can't list
all of an infinite set of pairs. It must be parameterized with n, in order to
find a limit as n->oo. Do you see any other way?

When you say, "the offsets (x_n,y_n) are determined by a function whose domain
includes n, and which doesn't change for any n", that seems a little off. Let's
say A n e N A m e N m<=n -> E {x_mn, y_mn} such that x_mn=f_x(m,n) and y_mn=f_y
(m,n). In other words, where the curve includes n points there are n segments
(including initial offset form the origin) each defined as offsets {x_mn, y_mn}
which are calculated using f_x and f_y based on the position in the sequence,
m, and the length of the sequence, n. Does that clear things up a little?

Okay, that's better. I think.

Whew! ;)



By "the limit as n->oo", I assume that you mean "the limit *of a sequence
of curves C_n* as n->oo". I'm trying to figure out what you mean by the
sentence though. The n *could* refer to an indexed curve in the sequence
of curves, or to an indexed point on a specified curve. Or, I suppose,
it's possible it might refer to something else. Clarification?

My pleasure. The variable n here denotes the number of segments in the curve.
Each of those segments has a position in the sequence, m, from 1 through n
(including the initial offet). Each n, or number of segments, denotes a
different curve, and as n->oo and the number of segments increases without
bound, we have the "curve in the limit". What groups all these curves together
as one family is the pair of formulas that give the offsets in each segment of
the curve, f_x and f_y. Did that help clarify things? :)


To an extent. I was indexing curves by their position in the sequence, which
was coincidentally the number of segments; so that agrees with what you're
saying. But I'm still unclear about what you mean by "we have the 'curve
in the limit'." Is the "curve in the limit" one of the curves in your
sequence of curves?

Sure, if we consider n from 1 through Big'un, we can sonsider the value at
Big'un to be the value in the limit.

If so, at what position can it be found?

Big'un.

Are there curves beyond it in the sequence? What do they look like?

Well, where the staircase becomes infinitesimally close to the diagonal, there
is still distance between them at points. One CAN continue to apply the formula
regarding the staircase to greater values than Big'un, in which case one is
going into the subinfinitesimal range of differences. While this process can
continue, getting closer than even a first order infinitesimal distance, the
directions continue to be different between the two at every point in the
curve.

Alternatively,
is it produced from the curves of the sequence by some process or mechanism?
If so, how is it produced?

When the curve is defined by the pairs denoting the x and y offsets produced by
each segment of the curve, using a formula for the offsets based on the number
of segments and the position of each segment, then you can take that formula to
the limit as the number of segments approaches oo. If the limits of two curves
defined this way are equal, then they are the same, and will have the same
measure.


Matt


--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... and the object is no longer a staircase. ... segments, rather than a set of locations, the difference is quite detectable, ... but 1, and n/0 is still infinite, even when n is infinitesimal. ... The limit of a curve is curve satisfying the following criteria: ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... According to TO's infinite induction principle it would necessarily be ... The length of the staircase in the limit IS 2, ... The segments have directions, which alternate between vertical and horizontal, ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... curve, which is precisely what causes the problem here. ... infinite case, and that is why the proof fails. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... infinite case, and that is why the proof fails. ... The "length" of smooth plane or space curve is only defined on intervals ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... So, the diagonal line as a whole is the limit of the staircase as a whole, as ... are where the risers meet the treads, but then there are infinitesimal segments ... Besides - points don't have directions. ... You postulate something magical happening in the infinite case ...
    (sci.math)