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 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.”