Re: Cantor Confusion

On Sat, 25 Nov 2006 03:12:20 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:

In article <lcbem29apkc14rfr1m1jtqaqj2anh1e41l@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
On Fri, 24 Nov 2006 01:10:39 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:
...
Pray, tell me. I would say that a straight line in the Euclidean plane
is a line of the form ax + by = c. With the standard formulas for angles
it is easy enough to get rectangles. And I would say that a rectangle is
a square when the sides have equal length (this is the point where the
measure creeps in). So we have a rectangle enclosed by the lines:
x + y = 1
x - y = 1
- x + y = 1
- x - y = 1
Now define the Manhattan measure:
d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2||
and we see easily enough that the figure enclosed by the lines above is
a square with sides with length 2.

A circle is a figure where each point has the same distance to a common
centre, and it is also easy to show that the points on the boundary of
that square have the same distance to the origin: 0.

Well for one thing points equidistant from any point define a sphere
not a circle unless one assumes "on a plane" when the Euclidean plane
isn't defined to begin with. But my question was directed not at the
definition of a circle or square on a Euclidean plane but at the
definition of a square with the Manhattan measure.

You completely misunderstand what I wrote. I *start* with an Euclidean
plane without measure (i.e. distance function). With that we can at
most define a rectangle. Neither a square, nor a circle.

Actually I think I understand you very well. How is it exactly you
start with a Euclidean plane? The fact that you start without a
distance measure doesn't allow you to start with a plane Euclidean or
otherwise. If you begin by assuming this you wind up by assuming that
and pretty soon you find yourself assuming what you were supposed to
demonstrate in the first place, that square circles are conceivable.

In other words we have two different figures defined in the Euclidean
metric, one as a curve and one with straight lines. Now I'm not trying
to quibble over the modern math definition of either figure at the
moment, just trying to point out that you have certain characteristics
and properties defined in the Euclidean metric and then apparently
claim that if you use some other metric and don't use the Euclidean
metric the two figures are the same.

I do not use Euclidean metric at all. Where, above, do I use Euclidean
metric?

I suspect we're using the phrase "Euclidean metric" in different ways.
When I use the term I'm referring not just to measures of distance but
to all definitive characteristics which go into definitions of such
things as dimensionality and geometric figures in addition to distance
measures. For example I see no definition of yours for "plane" which I
think would be impossible to define without a Euclidean metric. On the
other hand if all you're describing are variable measures of distance
then you'd have to explain how you obtain those measures without an
underlying Euclidean metric and corresponding assumptions. You can't
just assume them as modern mathematikers are wont to do.

For example is it possible to define plane squares with non Euclidean
metrics?

Of course.

Well then let's see some definitions for planes, circles, and squares
which don't explicitly or implicitly rely on Euclidean assumptions.

And can we define right angles without the parallel postulate
people simply ellide when operating with non Euclidean geometries? No.

I think you can. But I used Euclidean geometry above.

But the problem here is what kind of definition for squares doesn't
rely on straight lines, right angles, planes, and so on?

When I
ask a question such as "why are square circles unimaginable" the
defining metric for "squares" and "circles" is Euclidean and not
Manhattan and I don't expect answers that if we look at Euclidean
figures through some other metric we'll find they are imaginable.

In that case you should use better formulations in your questions.

And you should use better formulations in your answers.

And
when I follow-up to point out that in the Manhattan measure all circles
are squares (but not the other way around) you should state that your
formulation was insufficient.

Except that your answer relies on non Euclidean assumptions that
circles are squares. If I ask why one sided quadrangles are
unimaginable and you reply that they aren't if you start counting from
four would you consider your answer responsive to the question asked?
All you're doing is answering a question that wasn't asked in terms
employed by the original question. I can make up private definitions
just like everyone else does but that doesn't make definitions true.

In fact my personal favorite private definition for distance metrics
is one I made up for the real number line which runs 1, 2, e, 3, pi,
4, 5, . . but I don't try to pretend that when I'm trying to analyze
real numbers that that is a true definition.

Besides as far as I can tell you still haven't answered my question as
to whether there are any curves at all with the Manhattan measure. In
fact I can't even find it reprinted above.

~v~~
.