Re: Calculus XOR Probability

Tony Orlow wrote:
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.

What is this about "reaching"? Is there an aspect of your definition of "limit"
which requires something to be "moving" in some sense? Because I don't see any
requirement of that sort. The whole point of a sequence converging to a limit
is that the limit doesn't have to be included in the sequence.


<snip>


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

Without a definition of limit, you haven't demonstrated this.


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.

If you could provide a *definition* of "infinite" (rather than just a
*description* of it, such as "a number greater than any finite") then it would
be easier to discuss infinitesimals.

If you want to define an infinite number as "a number greater than any finite",
then that's fine; but you have to show that such numbers exist before using them
in any statements.


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.


Again, I don't think there's a clear definition of infinitesimal. Can you
fill in the blanks here:

1) A finite number is a ___ (type of, e.g. real, natural, etc.) number which
satisfies the following criteria: ____.

2) An infinite number is a ___ number which satisfies the following criteria:
_____.

3) An infinitesimal number is a ____ number which satisfies the following
criteria: _____.

To prevent circularity, "infinite" may not use "infinite", "infinitely",
or related formations in its definition; "finite" may only use "infinite"
or related words in its definition if "infinite" does not include "finite"
or related words; and "infinitesimal" may use "infinite" and "finite" in
its definition only if those words do not use "infinitesimal" in their
definitions.

Once you have provided definitions in this form, the next thing to do
is show that there are indeed numbers which satisfy the criteria you've
specified.

Until you've done that, you don't have any kind of mathematical tools.
Only after you've done that can we start discussing things like "finite and
infinitesimal scale", and so on.


<snip>

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.

If you're going to change the definition of a curve ("adopting a general
definition", as you say), then you need to state that clearly; it would help
distinguish things if you adopted a new name for it ("generalized curve,"
perhaps?). As far as what I call a series of points separated by finite space,
I'd call it exactly that - or more likely, "a set of discrete points".


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.

Assuming that Big'un exists: see my request above for definitions.


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.

I'm not exactly sure what this means.


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.


I still don't get the idea of a number of segments "approaching oo". Your
terminology here is very confusing, especially since you've stated elsewhere
that oo isn't a specific number.

Matt

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