Re: Cantor Confusion

On Sun, 26 Nov 2006 03:32:30 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:

In article <962hm2tg9dqirfmerue74rasendvcggoll@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
On Sat, 25 Nov 2006 03:12:20 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:
...
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 plane as defined by Euclides. With all his axioms (or postulates).

The fact that you start without a
distance measure doesn't allow you to start with a plane Euclidean or
otherwise.

Why not? Euclides did not have a distance function either.

He didn't try to make an argument for square circles either. You do.

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.

Well the latter was not an assumption. There was an explicit question.

But no demonstrated answer. You just claimed square circles are
conceivable according to a metric you can't demonstrate in the same
Euclidean terms you claim to rely on.

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.

In that case you are using quite non-standard terminology.

Modern mathematikers use quite non standard terminology to refer to an
imaginary real number line. If it doesn't bother them no reason it
should bother me.

For example I see no definition of yours for "plane" which I
think would be impossible to define without a Euclidean metric.

See Euclides' postulates.

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.

You read more in the word "metric" than is in there.

Not at all. Even in the restricted sense of "distance measure" a term
like "metric" has to be self consistent with underlying assumptions of
measure and what is measured and how.

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.

Now you require that without assumptions rather than metrics... That
is something completely different. Let us recap the Euclidean postulates:
(1) Two points determine a line
(2) Any line segment can be extended in a straight line as far as desired,
in either direction
(3) Given any length and any point, a circle can be drawn having the length
as radius and that point as center
(4) All right angles are congruent
(5) The parallel postulate
Now, that is precisely what I did use (after translation to analytical
geometry).

The problem is you have a host of undefined terms for things like
points, lines, length, circles, right angles, etc.If Euclid understood
these things one way and you understand and employ them another
they're not Euclidean assumptions at all just revisionist private
definitions. To justify your claim that square circles are conceivable
you also claim a metric you don't demonstrate on the basis of
Euclidean assumptions you claim to rely on.

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?

See above, postulate (4).

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.

How could I give a better formulation than stating that in the Manhattan
measure it is possible?

Your stating it has nothing to do with whether it's true.

I did state that explicitly, while you did not
state that you did use Eucledian measure.

No reason I should if the words themselves rely on the same Euclidean
assumptions you claim to rely on.

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.

No. I show that with a particular measure all circles are squares. There
is no assumption.

So you say but so you don't demonstrate. What you say is nothing more
than a claim. You might just as well reply "square circles are
conceivable" and let it go at that. You haven't explained why square
circles are imaginable except something called the Manhattan metric
says so. Anyone could do as much just by saying "circles are squares"
without all the metric nonsense.

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?

You are using a few words that you did not define, and again using some
assumptions. Quadrangle I can understand. What is one-sided? Does it
mean that the top side is different from the bottom side? In that case
I can show you quadrangles on a Moebius strip that have only one side.
On the other hand, I think you are referring to the edges. In that case,
in topology a connected set (as a quadrangle is) has a single boundary.

So you wind up with a four sided figure with one side? I don't see
that you're taking your words or mine seriously. How is it you get
from one side to another regardless of topology and how do you get
past the points of intersection?

All you're doing is answering a question that wasn't asked in terms
employed by the original question.

The terms employed by the original question did not refer to Euclidean
metric. So my answer was valid.

Terms like "circle" and "square" don't refer to Euclidean metric? Okay
then. Obviously this discussion is going nowhere. We might just as
well move on to a discussion as to why points are lines and cats are
dogs since it doesn't appear you have any ability to take these things
seriously.

I can make up private definitions
just like everyone else does but that doesn't make definitions true.

A definition can not be false.

Another interesting and rather self serving claim from someone who
obviously prefers false definitions to true definitions.

It can be the case that there is nothing
(within the current context) that satisfies the definition, but it is
still not false.

So is it the case that definitions cannot be self contradictory or
that self contradiction is not false?

Hell I don't mind if modern mathematikers need to make up another word
for "false" so they can feel useful for a change. But I think we need
to be clear on what we're talking about. I mean if you think
definitions cannot be self contradictory what is it you imagine goes
on in the context of self contradictory predicates in definitions? Is
there a variable logic for different predicates, a private definition
for self contradiction that makes definitions exempt from being false?

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.

That is not enough for a definition.

Sure it is. I just said so and you just said that definitions cannot
be false.

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.

Pray define "curve".

Don't need to. For purposes of the present discussion it is enough to
understand that circles are curves and squares aren't. If you maintain
otherwise I'd like to know how and whether there are other curved
figures besides circles in the Manhattan metric which are squares.

Quite frankly I don't understand how it is you can pretend to say
anything of mathematical significance without understanding the
difference between curves and straight lines. Do you imagine you can
just babble on about sets, bijection, injection, infinities and so on
and pretend you've said something about a subject like square circles?

~v~~
.