Re: Is a line segment composed of points?

On Nov 11, 4:08 am, "Timothy Golden BandTechnology.com"
<tttppp...@xxxxxxxxx> wrote:
From your OP I think we should accept that interpretation leaves both

sides with ground to defend.

Thanks for the nice post.

I don't think they have enough ground if they don't collapse
infinitesimals to zero, which should not be allowed by any pure
interpretation. When it comes to "multiplication, division" sense,
zero is not the opposite or counterpart for infinity, Simply because
zero is meaningful only in "addition, subtraction" sense. In a
reduction process, you get zero by subtracting a quantity from the
same quantity. But you can't get a negative infinity by subtracting a
quantity from any other quantity. The difficulty in reaching -oo by
subtraction is similar to reaching infinitesimal (not zero) by
division. The following picture should make it much clearer.

additive number line:
-oo ...........0 ...........+oo

multiplicative number line:
infinitesimal ...........1 ...........oo

Note that there is no zero or negative numbers in "multiplicative"
line. Both of them do not make sense here.


But I think that your own interpretation
is a nice entry into brane theory.

Thanks.

The people working in those spaces
are clearly regarding each dimensional level as its own critter. I
would say that your criterion is very close to their critter. Next
from lines come paths but both should be regarded as existent in some
dimensional medium shouldn't they? For instance a first line segment
can be built in 1D but then you can shift up to 2D, etc. Within the 2D
space your segment has some freedoms that can allow for curvature and
so on.

Curvature of extents of lesser dimensional objects into higher
dimensional spaces is again stemming from the interpretation of seeing
these non-existent objects (lines, surfaces) as standalone objects of
infinitesimally small extent (width, thickness), which is not allowed
in my interpretation. The boundary of an object is defined by the
extent of the higher dimensional object which it is enclosing. The
extent of the higher dimensional object might require its boundary to
be curved.

So, the boundaries of objects are curved, but objects themselves are
not curved.

- venkat


My own sense on continuum indicates that traditional real analysis is
excessive. We need only ponder distance as magnitude and come to a
sense of continuum which even an ape can understand. The breakdown of
such a simplistic entity is not worthwhile, as evidenced by the
troubling of real anal-ysts to construct both rational and irrational
values whereby if both aren't covered then there are holes in your
line. A calculus sense of continuum suffices in my opinion so that
these other properties are extraneous and can be dismissed as academic
overhead.

Mathematics is open to interpretation and you are doing a fine job.
Reality is a far more puzzling beast and in that mathematicians
divorce themselves from reality they have created a divergent form of
freedom that leaves little to be criticized without that criticism
being dismissed as merely interpretation. The deeper puzzles are
physical and I encourage you to keep a foot or at least a toe in that
water.

-Tim

On Nov 10, 11:53 am, vred...@xxxxxxxxx wrote:



On Nov 10, 7:15 pm, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Nov 10, 6:13 am, vred...@xxxxxxxxx wrote:

On Nov 10, 9:42 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Nov 9, 11:20 pm, vred...@xxxxxxxxx wrote:

On Nov 10, 8:37 am, Lester Zick <dontbot...@xxxxxxxxxxx> wrote:

On Fri, 09 Nov 2007 09:58:44 -0800, vred...@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.

Time to make some basic observations now. Not sure if these are
correct, but I can't see why they should be incorrect either. Sorry
for not using formal mathematical terminology, but I can always
clarify anyway.

In the abstract geometrical world, all that we have is regions.
Regions of lesser dimension serve as borders for regions of higher
dimension (objects) if they enclose the objects. However, a boundary
is not part of an object. Although a boundary has an extent in its own
dimensions, its extent is zero in the dimensions of the object which
it encloses,.

A line segment is a region of a single dimension object enclosed by
two points as boundary. The end points of the line segment simply
separate what we see as line segment from what it is not. So, the end
points are not part of the line segment.

The same logic holds good for points of intersection for lines - the
intersection points are borders for line segments and are not part of
any line.

This seems to indicate that neither the end points nor the
intersection points are part of the line segments. May be because
there is no "is-part-of" relation between objects of different
dimensions.

Each object can only be cut into objects of same dimension but never
into objects of lesser dimension. This holds good even for uncountable
number of cuts, assuming that you can't cut an object at its boundary
(or end point), since boundary is not part of the object.

Take a line
segment [0,1]. For every natural number
N, make a cut at 1/N. Note that there is no cut omega.
Note further that no cut is at the boundary. How long is
the piece that contains the endpoint 0?

The length of that piece is infinitesimally small, but not zero. By
the way, it doesn't "contain" the endpoint 0.

To get infinitessimal lengths you have to extend the real
numbers (which do not contain infinitessimals) to something
which contains an uncountable number of infinitessimals. How
do you decide which infinitessimal represents the length
of the piece which has zero as one endpoint?

Rational numbers should be enough to represent the infinitesimals,
because the number of cuts are represented by N. Am I missing
something here?

What are its endponts?

Its end points are 0 and 1/N where N is uncountably large and 1/N is
not zero.

The real numbers do not contain infinite or infinitessimal
numbers. So you have to extend the real numbers. When you do you
get an uncountable number of N where N is uncountable large.
How do you decide which one you are going to use?

This must be beyond my simple sense. I'm not sure if real numbers or
rational numbers can represent infinitesimals, but if a certain kind
of man-made numbers can't represent infinitesimals, it should not
bother or restrict the nature and its on-goings. Ideally the same
number system which represent- Hide quoted text -

- Show quoted text -...

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Relevant Pages

  • 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, ... my earlier post, since a point has zero extent in all dimensions, it ... Rational numbers should be enough to represent the infinitesimals, ...
    (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, ... my earlier post, since a point has zero extent in all dimensions, it ... Rational numbers should be enough to represent the infinitesimals, ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... For instance a first line segment ... imaginary objects of zero width, ... my earlier post, since a point has zero extent in all dimensions, it ... Rational numbers should be enough to represent the infinitesimals, ...
    (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, ... my earlier post, since a point has zero extent in all dimensions, it ... Rational numbers should be enough to represent the infinitesimals, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Infinitely many non-standard infinitesimals. ... Infinitesmals are not zero. ... David, there are non-zero infinitesimals which are all equal to zero at the ...
    (sci.math)
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