By standard geometry a line segment is defined by at least a pair of points.
In that case a line segment depends on the existence of points, and not vice versa.
Says whom? A Point is just a geometrical abstraction, physically the Planck distance would be the lower limit of any spatial considerations. If you are arguing against the geometrical abstraction of points then you should remind yourself that you utilize that abstraction as well.
In other words, standard geometry uses locality as the exclusive building-block of Geometry.
By using this exclusive observation, you cannot get non-locality, and indeed you did not understand my post.
This is your fundamental confusion with standard geometry; a line is the nonlocal expression of points, just as a plane is the nonlocal expression of lines and a volume the nonlocal expression of planes and a hyper volume the nonlocal expression of volumes…ECT. Even as I mentioned before since any line segment can be considered to be comprised of smaller line segments a line segment is also the nonlocal expression of those smaller line segments. The same can be said of planes, volumes, hyper volumes and other aspects of geometry save the point which has no extents. If you bother to study geometry you would know this but you simply choose to make unfounded and irrelevant assertion to help you rationalize the time you have spent thinking you have been developing something new when you have simply been wasting your time.
A volume, a plane or a line can non-local w.r.t to each other, whare a point cannot be but local w.r.t to any other dimension.
It all depends on how one is geometrically defining ‘local’ in that application, another result of your misunderstanding of ‘local’ when applying it to geometry.
Since you get a volume, a plane of a line as things that are based on sub-things, you cannot get their non-local nature in addition to their local nature.
In other words, your obvervation is closed under a one and only one exclusive observation, which is locality.
Bravo !
And by define a line segment as collection of sub-elements, you are unable to get it as a non-local object. The reason: you are using again locality as exclusive observation.
As usual for you, the facts are precisely the opposite of what you say. Because of those subcomponents any local expression, a given line segment plane or volume, can also be a nonlocal expression of some combination of line segments, planes of volumes. You are the only one on this thread who “cannot get” it and your “observation is closed under” your own need to think you have come up with something new.
The meaning of "atom" is exactly "non-composed" or "indivisible". Actually an atomic state is the opposite concept of anything that is based on sub-elements (and please do not give me the physical atomic state as an example, because the name "atom" was originally given to it because people thought that it is non-composed or indivisible).
What you mean much the same way that you now think line segments, planes and volumes are indivisible?
We have been over this before on another thread (do you need me to refer you to that post and the link you continue to ignore?). The term “Atom” can also refer to something in its simplest form. A single line segment is its simplest form, this does not mean that said segment cannot be divided into smaller segments. 2 is the simplest form of 1 + 1 but it can still be replace or has a self identity with 1 + 1 it is just that 1 + 1 is a more complex form of 2. Another point that was addressed in that other thread in term rewiring, which you continue to ignore yet claim to use “=” as self identity, the foundation of term rewriting. With your assertion of “=” as self identity and “. = ___” that means anywhere we find “.” In your assertions we can rewrite it as “___”.
By this problem alone you cannot see beyond the exclusive observation of locality that is learned as the one and only one point of view of the mathematical science (geometry or not) for the past 3000 years.
By your sad devotion to your misinterpretation of basic geometry you have convinced yourself that you have found something new in what has been there all the time if you just bothered to look at it.
By learning non-locality in addition to non-locality, the exclusive local observation is going to lose its exclusivity.
Ironic, since your own assertions are based solely on what “local observation” you choose to use at that given time as exemplified by “From . point of view . = ___” while “From ___ point of view ___ < and > .”. You will see this only when you finally allow your selective conclusions from your exclusive points of view not to exclude you from seeing that non-locality has been a fundamental part of geometry from its inception.
Non-locality is a new thing and cannot be found (yet) in any already established school that teaches locality as the one and only one possible observation of Math.
As I have shown above non-locality is very easy to find in standard geometry, you would have to go out of your way not to find it and in fact it is geometry that lets us define what we consider local or nonlocal in some given consideration, but the fact that you miss such a fundamental aspect of geometry is understandable, as you have gone so far out of your way to try and claim you have found something new.
By using the Cyclops' local-only observation you cannot even lie to yourself, because you are under the illusion of one and only one observation.
I have no desire to lie to myself; you are the only one on this thread who seems to find the ability to lie to yourself as an admirable trait. As far as illusions go, that anyone considers (other than yourself) there is “one and only one observation” is an illusion you have created yourself that you ascribe to others yet depend on so intently in your own assertions that “From . point of view . = ___” while “From ___ point of view ___ < and > .” extolling the differences “From . point of view” as “only one observation” and “From ___ point of view” as “only one observation” and exclusive for the other “.”. You remark in this post about your distain for breaking things down in to sub components and have remarked before about exclusive points of view yet your whole approach is based on breaking the “.” and “___” relation of your example down into two (“From . point of view” and “From ___ point of view”) mutually exclusive points of view or sub components. We still might not agree with what you say but your assertion might carry more veracity if it were not so abundantly apparent that even you do not agree with what you say.
