Re: Cantor Confusion

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.

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?

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

Of course.

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.

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
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.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.