Re: Is a line segment composed of points?

On Fri, 09 Nov 2007 09:58:44 -0800, vreddyp@xxxxxxxxx wrote:

On Nov 8, 11:24 pm, Lester Zick <dontbot...@xxxxxxxxxxx> wrote:
On Wed, 07 Nov 2007 21:39:15 -0800, vred...@xxxxxxxxx wrote:
On Nov 8, 5:55 am, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1194471215.470102.253...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
David R Tribble <da...@xxxxxxxxxxx> wrote:
venkat wrote:
A point seem to represent a "split" or absence of material and as well
as the content or existence of the material.

David R Tribble wrote:
Interesting analogy, which allows us to say that an uncountable
interval [a,b] consists of an uncountable set of points [a,c)
and another uncountable set of points (c,b] surrounding the
zero-width "crack" at point c. The distance between the two
intervals, i.e., the width of the crack, is exactly zero.

venkat wrote:
Yep, that's exactly what I'm saying. A location or point has no
magnitude or extent, and it is not even infinitesimally small, and
hence it can't be a building block for a finite extent line segment.
This "line is composed of points" concept was simply comforting to
people by relating spaces of different dimensions by seeing that an
higher space is "built" by the building blocks of lesser dimensional
spaces. I'm saying this understanding is incorrect. You can never
build a N dimensional object by using blocks of lesser dimension.

If, as you say, a line segment is not in fact composed of an
uncountable infinitude of zero-width points, then what /is/ it
composed of?

What is the thing where two intersecting lines intersect?

Maybe he has a new geometry in which lines can cross without
intersecting! Or maybe not.- Hide quoted text -

It doesn't need a different geometry. Since both lines and points are
imaginary objects of zero width, lines do intersect in points.
However, I don't think the fact that lines intersect in points is
enough to indicate that lines are composed of points. My problem was
the finite "length" of the line and its contents. Zero lengths simply
do not make up finite length whether it is adding up or union or
summing up - whatever you call. Infinitesimal lengths do build up a
finite length. So it is better to say a point has infinitesimally
small length, area, volume etc.

Points of zero length, lines of zero width, plane of zero thickness
simply do not exist, not even by imagination - because zero stands for
"nothing ", non-existence - or say don't exist.

In general I agree with you but you might try framing the analysis
this way:

Is a solid composed of surfaces or defined by surfaces?

Is a surface composed of lines or defined by lines?

Is a line composed of points or defined by points?


These questions make lot of sense to me. I would accept the second
part of each of these questions - that is, a line is defined by its
end points but not composed by points as its contents. I can see a
question to me here - can points exist as "end points"? According to
my earlier post, since a point has zero extent in all dimensions, it
should cease to exist. Then how can points exist as boundary of a
line? They can, because boundary is just a seperation of a region from
outside. A seperation or split doesn't have any extent, and points can
serve as boundary of a line segment.

As an example, if a measuring scale is cut into two pieces at 6 inch
mark, the mark now belongs to both pieces and it serves as boundary
point. So a point can be split into multiple points, but they can't
make up a fininte extent.


Well I can generally agree without getting into philosophy. If points
represent intersections of lines then we can alway take the definition
of line segments to correspond to intersections of lines with lines.

~v~~
.

Relevant Pages

  • Re: Is a line segment composed of points?
    ... interval consists of an uncountable set of points [a,c) ... uncountable infinitude of zero-width points, ... imaginary objects of zero width, ... my earlier post, since a point has zero extent in all dimensions, it ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... hence it can't be a building block for a finite extent line segment. ... imaginary objects of zero width, ... I don't think the fact that lines intersect in points is ... my earlier post, since a point has zero extent in all dimensions, it ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... interval consists of an uncountable set of points [a,c) ... people by relating spaces of different dimensions by seeing that an ... imaginary objects of zero width, ... I don't think the fact that lines intersect in points is ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... interval consists of an uncountable set of points [a,c) ... people by relating spaces of different dimensions by seeing that an ... uncountable infinitude of zero-width points, ... imaginary objects of zero width, ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... interval consists of an uncountable set of points [a,c) ... people by relating spaces of different dimensions by seeing that an ... uncountable infinitude of zero-width points, ... imaginary objects of zero width, ...
    (sci.math)
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