On Thu, 2 Nov 2006 23:37:44 +0800, "riverman" wrote:


wrote in message
.. .
On 1 Nov 2006 16:27:51 -0800, "riverman" wrote:

Whether or not _you_ can measure to _your_ satisfaction that two points
on a 2-dimensional plane are _absolutely_ the same distance from an
initially-chosen point (in this case, a "target"_), those two points
certainly exist.

Yes, they do.

The random selection of a second point (the landing of
Dart A) "x" distance from the first point (the "target") creates a
radius from which a circumference may be scribed. The second dart (Dart
B) and its landing point have no relevance and can be ignored.

Not necessarily, it depends on what is being asked. "Conditional
probabilities" do exist. But in the case of what you are discussing (the
existance of arcs), I concur; we can ignore the second dart for now.

Well, I guess it's good that at least some of the time, you don't argue
with yourself...

A third
dart is thrown (Dart C). According to your theory, that dart can easily
and readily strike any point on the disk or any point outside of the
circumference created by the selection of the first and second points,
up to and including "un-measurably" close to the inside or the outside
of the circumference, but can never actually strike a point on the
circumference. IOW, the third point (Dart C) can only create a second
radius that must be less than or greater than the first radius.

Yes, that's correct also. There is a statement in calculus that asserts that
no matter what two numbers you choose on the number line, there is always
another number between them. No matter how close to the circumference you
get, you can always get closer. But you cannot get there unless you, well,
get there.

Oh, geez...if there's a statement and all...well, anyone thinking about
math better cut it out...just think of all the books that'll need to be
changed if someone ****s up and comes up with something new...

With
not being able to select a second point on the circumference, arcs, in
such a world, don't exist.

No one said you cannot select a second point. What is being said is that the
probability of another dart hitting that point, or any other point on that
circle, is zero. Thats because the point is infinitely small. The
probability of hitting something infinitely small is infinitely
small....zero, in fact.

Infinitely small is not "zero." One can choose to "round it off" and
just call it "zero," but it isn't, in fact, non-existent.

Here's another hint: consider the points in a tangent to point/Dart A
and the points in lines perpendicular to that tangent and...why, shoot,
sooner or later, one might account for all the points in the plane, and
then, uh-oh...

HTH,
R