Re: Calculus XOR Probability

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



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

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.

You're assuming again that you can interchange the sum and the limit process.
The length of the limit of staircases need not equal the limit of the length
of staircases with the standard definition.

Do you mean to say that the limit of the staircases is a series? That's how
your sentence is phrased, but it doesn't seem to make sense. You're apparently
making a sequence of geometric figures (staircases), then stating that the
limit is an infinite series, presumably evaluated in the same way that
infinite series typically are - although you need to be clearer about the
meaning of "n->oo (1/n,0),(0,1/n)". How does the limit of a sequence of
geometric figures get to be a sequence of real numbers? Or is that what you
meant?

I guess that wasn't exactly clear. I wasn't sure what notation I should use.
When I say "Sum(n->oo: {1/n,0},{0,1/n})", I should have said "lim n->oo sum(x=
1->n: length({0,1/n})+length({1/n,0})". I mean that each stair consists of the
two pairs, one denoting the tread and the other the riser, the curvilinear
length being the sum of their lengths, which is 2/n. So, the overall length
becomes lim n->oo sum(x=1->n: 2/n)=2.

For each of these pairs of the form {1/n,0} and {0,1/n} are corresponding
segments in the diagonal of form {1/(2n), 1/(2n)}, each with a length of sqrt
(2)/(2n), for a total length of sqrt(2)/n, compared to 2/n for each stair. The
diagonal is lim n->oo sum(x=1->n: sqrt(2)/n)=sqrt(2).



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.

So, each staircase is a sequence of line segments. How do you decide
that the limit is also a sequence of line segments?

Because that's the way the line is defined. So, you think your line is a set of
points. How do you know the limit is also a set of points? What is the point of
this question? What do you THINK the sequence of segments becomes? Ask Newton.



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?

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

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. 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.



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.



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.

--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... as sequences of segments .... ... staircases is 2; which no one is disagreeing with; and that the length ... I stated already it's a sequence of line segments. ... Definition 2 says that the endpoints of these line segments are real ...
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  • Re: Calculus XOR Probability
    ... the size of them shrunk accordingly, in the infinite limit of this process, the ... So for example, if the curves are the staircases, and p is the ... point of the sequence C. ...
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  • Re: Calculus XOR Probability
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    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... are where the risers meet the treads, but then there are infinitesimal segments ... the length of the limit of the set of staircases D is not ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the limit of the staircases is the diagonal; ... My definition of "limit of a sequence of sets of pairs in R^2", ... segment of a curve", dated 4 May, are available at: ... segments, it seems at odds with anything I've ever seen ...
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