Re: Cantor Confusion

On Fri, 24 Nov 2006 01:10:39 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:

In article <34i6m255sfv5upv6nenqm6menle2gp8l8b@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
On Tue, 21 Nov 2006 03:04:42 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:
In article <8m8pl2pj1icbeven2hq7rp4hq1rufqh1u2@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
...
Why are square circles unimaginable?

They are not. With the Manhattan measure of the plane, each circle is
a square.

Then what is a square?

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. It looks to me that
you've just defined a square with the Manhattan metric with the
properties of a circle in Euclidean plane metric. What's the point of
that if you don't define a square with different properties in the
Manhattan metric?

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 suppose I should ask instead are there any Euclidean curves in the
Manhattan metric? If not the issue is moot. But my primary concern is
that I see people all the time defining lines, figures, etc. with the
Euclidean metric then going on to do non Euclidean mathematics with
them. I mean Euclidean definitions require the Euclidean metric. And
if you want to use some other metric you need some other definition
using that metric. In which case my original question should be
rephrased "why are square circles unimaginable in the plane Euclidean
metric"?

For example is it possible to define plane squares with non Euclidean
metrics? And can we define right angles without the parallel postulate
people simply ellide when operating with non Euclidean geometries? No.

Then people complain that what they do gives the same answers. But
that's only because they're using the same words yet asking different
questions, with concepts defined in the Euclidean metric but operated
on in some non Euclidean metric but not defined in that metric. 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.

~v~~
.