Re: Is a line segment composed of points?

On Nov 11, 7:37 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
Speaking of curvature again, I want to make my interpretation more
clearer. The lines, surfaces, spaces etc are not curved relative to
their own space, but their boundaries may be. For example, a 2D
surface is not curved relative to a 2D observation point, due to
limitation in defining straightness in 2D space, but its 1D boundary
may be curved in the view of the 2D observation point.

So whether an object is curved or not is relative to the observation
point.

- venkat

It is interesting that you introduce observation here. Should we be
considering an observation medium? For instance within the lower D
supposedly curved space no ability to get a twisted meter stick
exists. So injecting some of reality into the problem an entity which
plods along at a fixed interval of say one unit within the space may
track a path with some approximate accuracy, having produced a survey
of the path when he has traversed from one starting point and merely
measuring a series of angle in a 2D space. As you say this 2D system
is flat. Now though stepping out to a 3D rubber *** model of the
same system this 3D observer will not be shrunk and expanded as the 2D
observer was. Now as the 2D observer is observed by a 3D observer each
step can be measured by a 3D entity with a 3D meter stick. Angles can
be taken again though the information will be a bit more confusing.
The 3D survey will yield results consistent with the 2D survey though
more complex informationally.

If we were to allow the 2D observer to course out a straight path with
his single unit transitions and return to the same or approximately
the same starting point then he might trouble over it a bit. From this
special path he might opt to try again from some positions nearby. If
we have a continuous space then could the 2D entity continue surveying
and find a shorter path that returns? The method of measure will break
down as much as the space breaks down so this model seems pretty
troubling. To what degree do we even care if the system is realistic?
Are singularities realistic? In some regards they are e.g. quantized
charge but then we have flipped things around a bit from the concept
of space that we were placing objects into versus now considering
those objects to be singularities in the space.

Another criticism of the curvature study is that we have only granted
one additional dimension for this analysis. To what degree is this a
universally decisive system? It is only a first step.
Should we then consider general dimensional systems? What could be
gained by entering such an overwhelming level? If we were to work in n
dimensions and discover an n-1 dimensional relationship then we might
take some meaning that would then allow for a progressive domain. In
terms of this angle of criticism all of the study of fixed dimension
systems of branes or strings that is going on should gell into a
general dimensional format.

I am happy to be corrected anywhere that substantial criticism can be
waged. I feel quite unconvinced by much of the existing mathematics
though much of that may just be my own misunderstanding. In that this
mathematics should be convincing by virtue of its being mathematical
then I'll try to be open to an argument in its favor.

-Tim

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