I Preliminaries
a/ The historical and scientific context
I propose a completely different interpretation, where what you call a 'point' can also be seen as a very small sphere, with content. This changes the perspective, the structure, and the logic we draw from it."
And yes, the founding fathers of mathematics and geometry could have realized this by making the simple drawing I did, but for now I want to add that for me, it's my observation that acts like a geometric and mathematical microscope. Do you understand? So, if we look for the smallest point, we see an infinite series of central points (because when we enlarge with a microscope, we realize that the infinitely small is actually large, and each point is no longer really a point, but, for example, an atom or a cell).
The founding fathers of ancient geometry (Euclid, Archimedes, etc.) defined the point as a dimensionless entity, a simple abstract "without content." This definition was philosophical and ideal, based on the perception and tools of the time. Today, with microscopes and fine observation instruments, what we call a point can no longer be perceived as a strictly null "point."
In physical reality, this "point" is a small sphere, a very small but non-zero volume, visible due to the resolution limit effect.
What you call a point is a misnomer because to the human eye, what is a sphere actually appears (in my example, with a microscope, for example) as a point, given its smallness, a bit like an atom.
What we call a "point" is often a misnomer or a misnomer, because at a certain scale of perception (human or geometric), a very small sphere looks like a point, but it is not a dimensionless point.
What must be understood is that the fathers of geometry in antiquity did not have access to a microscope, for example, or a magnifying glass, so what they describe as a point can, and I'm being kind, appear today only as a sphere and nothing else.
b/ Geometric and Conceptual Implication
The "mathematical" point is an abstraction.
In my view, this point is best represented by a sphere of very small but non-zero dimension.
This changes the view of nested geometric structures, where a point becomes a sphere, which contains an inner cube, which in turn contains other shapes, and so on. It's a way of reconciling the mathematical abstraction with observable physical reality.
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