My suspicion of points goes deep yet I could not comprehend the
relationship of regions to points, because regions often _still_
utilized points, which was maddening. But now I can see it: the
REGION becomes the primitives and any points are derived _as
necessary_ for their pragmatic purpose but no longer need to
serve as "fixed points in the heavens". === Oh
[1]
https://en.wikipedia.org/wiki/Sheaf_%28mathematics%29 is
something I like. The concepts 'clicked' well with me. Well, you
know _far far_ more about any of this than I do. I just
understand what I can. I'm not anywhere near your field of study
- or any really. But I like to comprehend the 'gists' of things.
I like to be able to understand just enough of a concept or set
of concepts to be able to be able to explain _some form_ of it
using a short story, a thought experiment, imagination, a
demonstration. I don't strive for precision, but I do strive for
accuracy. "Low precision, high accuracy". This way, if someone
is interested in learning more, they'll be in the correct
'range' but if not, I'll know that I at least hadn't misled them
to a precise but inaccurate direction. = ==thanks to andrei
who wrote== "From points to regions In standard general
relativity*and, indeed, in all classical physics*space (and
similarly time) is modelled by a set, and the elements of that
set are viewed as corresponding to points in space. However, if
one is *suspicious of points**whether of spacetime, of space or
of time (i.e. instants)*it is natural to try and construct a
theory based on *regions* as the primary concept; with
*points**if they exist at all*being relegated to a secondary
role in which they are determined by the *regions* in some way
(rather than regions being sets of points, as in the standard
theories). So far as we know, the first rigorous development of
this idea was made in the context of foundational studies in the
1920s and 1930s, by authors such as Tarski. The idea was to
write down axioms for regions from which one could construct
points, with the properties they enjoyed in some familiar theory
such as three-dimensional Euclidean geometry. For example, the
points were constructed in terms of sequences of regions, each
contained in its predecessor, and whose *widths* tended to zero;
(more precisely, the point might be identified with an
equivalence class of such sequences). The success of such a
construction was embodied in a representation theorem, that any
model of the given axiom system for regions was isomorphic to,
for example, R^3 equipped with a structured family of subsets,
which corresponded to the axiom system*s regions. In this sense,
this line of work was *conservative*: one recovered the familiar
theory with its points, from a new axiom system with regions as
primitives."
References
Visible links
1.
https://l.facebook.com/l.php?u=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSheaf_%2528mathematics%2529&h=UAQGEIDLk
