On Thu, 15 Mar 2007 13:59:08 GMT, Sam Wormley <swormley1@mchsi.com>
wrote:

Bob Kolker wrote:

Sam Wormley wrote:

Give me something better, Bob, or are you arguing there isn't a better
definition (if you can call it that).

You are asking for a definition of an undefined term. There is nothing
better. If one finds a definition of point it will have to be based on
something undefined (eventually) otherwise there is circularity or
infinite regress. We can't have mathematics based on turtles all the way
down. There has to be starting point.

Here is my position. If an alleged definition is no where used in proofs
it should be eliminated or clear marked as an intuitive insight.

Bob Kolker

Fair enough--However, for conceptualizing "defining" a point
with coordinate systems suffices.

However it does not suffice for the definition of lines and arguments,
proofs, and justifications based on such assumptions. Defining points
is hardly essential to definition of lines based on such definitions.

~v~~

Sam Wormley wrote:

Fair enough--However, for conceptualizing "defining" a point
with coordinate systems suffices.

Yes indeed. Point is a tuple of elements from a ring. But even these
have be grounded upon undefined terms.

The fact that RxR with a metric satisfies the Hilbert Axioms for plane
geometry implies that points can be taken to be pairs of real numbers.

The fact that the Hilbert Axioms for the plane is a categorical system
makes me feel warm and fuzzy about identifying a line with a set of
points (number pairs) that satisfy a first degree equation in the
co-ordinate variables.

This is a point (sic!) that Lester Zick is genetically incapable of
grasping.

hahahahaha you are poor philosopher.

Obviously. That's why I became a mathematician.

You did? Gosh, congratulations!

Brian Chandler
http://imaginatorium.org
(just wanting to be part of this golden thread, this irridescent
braid, this)

SucMucPaProlij wrote:

Intersection of lines can define a point and we both know it just as we both
know that line is made of points.

If you don't think that line is made of points then how do you explain the
fact that two lines can have common point? If two lines are intersecting in a
point is this point one part of both lines or is it created during
intersectioning?

Maybe he thinks there are objects other than points on lines. If so, they are
not ever mentioned in any axiom system for Euclidean Geometry.

Likwise for planar curves. L.Z. rejects the usual definition of a cirlce as a
set of points on a plane a given distance (the radius) from a specified point
(the center). If a circle does not consist of its points, what else besides
points lie on the circle? If there are any such objects they are never
mentioned in the axioms.

Zick'w problem (among several problems he has) is that he simply does not
comprehend what an axiomatic system is. He cannot comprehend the notion of
undefined terms or objects whose only properties are given in the axioms. For
example, whatever a point is, given two distinct points there is one and only
one line (whatever a line is) containing them.

SucMucPaProlij wrote:

Intersection of lines can define a point and we both know it just as we both
know that line is made of points.

If you don't think that line is made of points then how do you explain the fact
that two lines can have common point? If two lines are intersecting in a point
is this point one part of both lines or is it created during intersectioning?

Maybe he thinks there are objects other than points on lines. If so,
they are not ever mentioned in any axiom system for Euclidean Geometry.

Likwise for planar curves. L.Z. rejects the usual definition of a cirlce
as a set of points on a plane a given distance (the radius) from a
specified point (the center).

If a circle does not consist of its
points, what else besides points lie on the circle?

If there are any
such objects they are never mentioned in the axioms.

Zick'w problem (among several problems he has) is that he simply does
not comprehend what an axiomatic system is.

He cannot comprehend the
notion of undefined terms or objects whose only properties are given in
the axioms.

For example, whatever a point is, given two distinct points
there is one and only one line (whatever a line is) containing them.

On Tue, 13 Mar 2007 20:48:34 +0100, "SucMucPaProlij"
<mrjohnpauldike2006@hotmail.com> wrote:

In the swansong of modern math lines are composed of points. But then
we must ask how points are defined? However I seem to recollect
intersections of lines determine points. But if so then we are left to
consider the rather peculiar proposition that lines are composed of
the intersection of lines. Now I don't claim the foregoing definitions
are circular. Only that the ratio of definitional logic to conclusions
is a transcendental somewhere in the neighborhood of 3.14159 . . .

point is coordinate in (any) space (real or imaginary).
For example (x,y,z) is a point where x,y and z are any numbers.

That's nice. And I'm sure we could give any number of other examples
of points. Very enlightening indeed. However the question at hand is
whether points constitute lines and whether or not circular lines of
reasoning support that contention.

line is collection of points and is defined with three functions
x = f(t)
y = g(t)
z = h(t)

where t is any real number and f,g and h are any continous functions.

Your definition is good for 10 years old boy to understand what is point and
what is line. (When I was a child, I thought like a child, I reasoned like a
child. When I became a man, I put away childish ways behind me.....)

Problem is you may have put away childish things such as lines and
points but you're still thinking like a child.

Are points and lines not still mathematical objects and are lines made
up of points just because you got to be eleven?

~v~~