Re: Calculus XOR Probability

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

<snip intro regarding equivalenceof "y=x" and "2y=2x", versus
equivalence of the diagonal and the limit of the staircases>

As you didn't respond to the above, can I assume that you agree with
it? Or do you disagree that figures in the x/y plane are adequately
defined by sets of points (x,y) satisfying formulas such as y=f(x), or
f(x,y)=c for some constant c?

To be honest, my response was "blah blah blah". You can reiterate that your
limit doesn't take into account anything but location, and I can reiterate that
that's the problem with your limit, forever and ever.

When you say there's a "problem" with my limit, what do you mean?

I mean that it doesn't take into account anything but location, and you're
using it to measure distance.


* Do you mean that my definition of limit is not well-defined (i.e.,
you can't figure out what the set "lim {C_n}" is for some sequence
{C_n})? If so, what sequence {C_n} of curves and point p do you find it
not possible to decide whether "point p is in limit {C_n}" is true or
false?

No, the locations of the staircase in the limit are indistinguishable from the
locations of the points on the diagonal.


* Do you mean that it fails to have some property which is
mathematically required before one is allowed to use the word "limit"?
If so, what property is that?

No, it fails to have a property that is required to use that limit to measure
the diagonal. Where f(x)->g(x), f'(x)<>g'(x). Same location. Different
direction. Different distance.


* Do you mean some other thing that has /nothing to do/ with whether my
limit is a perfectly sensible function, which we write as "lim n->oo
{C_n}", and refer to loosely as "the limit of {C_n}"? In which case how
is that other thing a "problem" with my limit?

It's all about applying that notion of limit to a measure of the line. If you
want to approximate a curve as the limit of a set of straight line segments,
then the direction of the curve must be taken into account, which is generally
done by either having both endpoints of each segment ON the curve, or some
point within each segment touching the curve as a tangent. In other words, each
element must have a point parallel to the curve in order for the measure to
work. So, where your type of limit does NOT measure direction, it cannot be
used as a measure of a curve. Surely you can understand that.


You can view spatial
objects as sets of points, but that doesn't afford you the kind of measure you
assumed in your example.

So if you agree that the "problem" is /not/ that my definition of limit
is not a prefectly valid definition of a limit, /nor/ with the
assumption that spatial objects "can be viewed" as equivalent to sets
of points; then it must be a "problem" with /some other assumption/ in
the argument.

Perhaps it is the assumption that "length of a curve" is a "measure" as
you define it?

Uh, if length is not a type of measure, then I don't know what is. This is
beginning to sound Berkeleyesque. Are you sure you're feeling okay?


That requires some notion of how the points are
connected, that is, the direction of the curve at each point. When you can show
me that the linear formula is derivable from a description of the finite
staircase, then you'll have a case that it's the staircase in the limit.


Firstly: I have given explicit, easy to verify definitions and proofs
that the set of points satisfiying the conditions "y = 1 - x, x,y >= 0"
is exactly the set of points I defined as "lim n-> oo {C_n}", where C_n
is the nth staircase. If that doesn't "show" you that the linear
formula "y = 1-x" is derivable from the sequence of the staircases,
then I'd like to know what you mean by "derivable".

I additionally gave the somehwat different example of how one can
"derive" the equation "y = 0" as the limit of the sequence of equations
{y = sin(n*x)/sqrt(n)}: simply note that -1 <= sin(n*x) <= 1 for all
naturals n, reals x; and that 1/sqrt(n) approaches 0 as n->oo. Thus the
"limit of {sin(n*x)/sqrt(n)}" is the function y(x) = lim n->oo
sin(n*x)/sqrt(n). It should be obvious that y(x) = 0 for all x; we
usually write this as "y = 0".

What more are you looking for in terms of "derive the formula"? Where
is any mention of "directions of points" made, explicitly or implictly?

I mean, take the formula for the one, and perform algebraic operations on it to
transform it into the formula for the other. Derive it formulaically.


Secondly: I see no problem with a definition of a curve which lacks
"points with directions", since I can determine the direction of a
curve, /at/ a point, using the standard topology on R^2.

I have already shown you how I define the diagonal as a set of points;
and how that set of points is the same as that of the limit of the
staircases, as a set of points. I assume you agree at least with this
as a starting point: Lim(sequence of sets of points) = (set of points).

Locationwise, the points on the staircase approach those on the diagonal.
Directionwise, they do not, but stay at a constant 45 degree angle to the
diagonal, causing the error ration of sqrt(2). I'm getting tired of repeating
this.


By "direction" in the usual sense, I would think that one means
"direction towards arbitrarily nearby points also in the set". To
determine the "direction" of any of point p in a set C, we only need
look at the set of points which are "nearby" p - we don't need to
/also/ look at what method we used to generate these points.

In the staircase, the slope between two arbitrary points becomes more likely to
be close to the slope of the diagonal the farther apart they are. Any
abritrarily close points are either in a norizontal of vertical direction.


You appear to mean something else by "direction of a point".

That definition doesn't work on the staircase, where you have corners with no
discernible direction. It's somewhere between 0 and oo.


Can you tell me /how/ you determine the correct "direction" of the
point (0,0) in each of the following simple figures in the x/y plane?

Remember, direction is necessary for measure. Let's see what you got....


By limit in the following examples, I mean the previously defined
"location" limit. I assume it is clear that the point (0,0) uniquely
satisfies the requirements in every case below; at least to the extent
that the the point (0,0) is a point "without direction".

I would like your answer to allow me to say "in example (the example)
the set of points p with (x,y)=(0,0) and direction {insert your value}
uniquely defines (the example) because {insert justification here}".

* x = 0, y = 0

As a point in itself it is without direction or measure. It can be thought of,
in the context of the Cartesian plane, as an infinitesimal square parallel to
the x and y axes.


* lim n->oo {y = 0; -1/n <= x <= 1/n} (line segments parallel to the x
axis)

Horizontal, the same as all the objects of which it is the limit.


* lim n->oo {y = x; -1/n <= x <= 1/n} (ditto, to y=x)

Diagonal, just like all the elements of which it is the limit.


* lim n->oo {y = -x; -1/x <= x <= 1/n} (ditto to y = -x)

Diagonal in the negative direction, forthe same reason.


* lim n->oo {y = x/n; -1/n <= x <= 1/n}

Horizontal, which is the limit of y' as n->oo and y->x/oo=0. That is, in the
limit, the straight line not only shrinks to infinitesimal length, but rotates
to infinitesimal slope.


* x^2 + y^2 = 0

You list this separately from the next, so do you mean this not to be the limit
of circles around the origin as radius decreases? If so, I guess it's a
directionaless point, with no measure, so it doesn't matter. All this talk of
single point direction when there's no measure involved is a little wasteful.


* lim n->oo {x^2 + y^2 =1/n^2} (circles centered on the origin)

All and none, like all the circles of which it is the limit.


* lim n->oo {(x - 1/n)^2 + (y - 1/n)^2 = 1/n^2} (circles centered on y
= x, intersecting (0,0))
Same, but the location is infinitesimally different from the last.


* lim n->oo {(x-1/n)^2 + (y-1/n)^2 = 1/n^2}) (circles tangent to the x
and y axes)

Ditto


* x^10000 + y^2 = 0

Directionless point.

Pointless direction. What does this have to do with anything?


* lim n->oo {x^(2*n) + y^2 = 0}

* lim n->oo {x^(2*n) + y^2 = 1/n}

* lim n->oo {x^(2*n) + y^2 = (sin(n))^2/n}

* |x| + |y| = 0

* lim n->oo {|x| + |y| = 1/n} (shrinking diamonds)

* lim n->oo {|x|^n + |y|^n = 1/n}

* lim n->oo {|x|^n + |y|^n = sin(n)^2/n}

* C = {(x,y) : f(x,y) = c} where f(x,y) = c iff x = 0 and y = 0}.

The remainder of the concepts discussed in the previous post appear
unresolvable until the question of "what do we mean by the diagonal
line?" is resolved, which is in turn contigent on the question "what do
we mean by the point (0,0)?", so I will stop here.

<snip>

Until you define what you mean by statements like this, I cannot
express an opinion regarding whether such a statement is true or false;
instead it is meaningless.

Then you're being clueless. You should be able to parse that statement. Or, did
you read "probabilities" as "possibilities" again? Remedial English is down the
hall to the left.

This is Remedial Mathematics. The current topic is "what is a valid
mathematical argument?". Stay in your seat until the bell rings.

If that's your attitude, the discussion's over. Have a nice day.


Cheers - Chas



--
Smiles,

Tony
.

Relevant Pages

  • 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
    ... What is the slope of the staircase at x=0. ... infinite, as opposed to the slope of the diagonal. ... As long as those locations are ON the curve, ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the locations of the staircase in the limit are indistinguishable ... mathematical requirement for a curve to have a length is the the lengths ... it cannot be used as a measure of a curve. ... The mathematics involved is quite straightforward and pedestrian. ...
    (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. ... According to the definitions of arc length in calculus, ... parallel to the curve they're approximating. ...
    (sci.math)