Dear Gordon,
Thanks for the feedback and the opportunity to shed more light on possible grey areas on this topic.
1. Good to see a No answer. So in your opinion what's the difference?
2. FIRSTLY, there is no strangeness or originality in the extended point. As I discussed in my first essay, it dates back to the Pythagoreans and was a bone of contention between The Pythagoreans, Proclus, and partially Aristotle on the one hand holding the point to be the smallest possible finite limit to extension resulting from division, while Plato proposed that the point be of zero dimension. The inheritors of both schools of thought are for the former those holding a 'substantival' view of space, i.e. space exists independent of objects (Newton being a famous son of this family, and probably also Maxwell) and for the latter those of the view that space is a mere 'relational' concept that does not really exist (Leibniz, Mach and Einstein being famous sons of this school). Sometimes, there is an overlap and some straying. Witness for example that Einstein though of the latter school that says space is a fiction and a relational concept that does not exist comes round to propose that this unreal and non-existing thing when amalgamated with time becomes something real that exists, it can curve the path of moving bodies, and it can vibrate giving rise to gravitational waves which propagate at velocity, c.
So from the foregoing going further down the scale, for a space that does not exist, its fundamental unit would be of zero dimension and does not exist. That makes sense. For a space that exists, following the idea dating back to Democritus that whatever exists must have some fundamental unit that also exists, since a multitude of what does not exist cannot make up what exists.
SECONDLY, what does "having no parts" mean? This can have two meanings for a fundamental object. It could mean not existing, being of zero dimension and therefore not divisible and split into parts.
OR it could mean existing, have a non-zero extension (since a multitude of zero existence and extension is zero existence and zero extension, while a multitude of non-zero extension is an even greater magnitude of extension), but not further divisible physically into smaller parts beyond some limit. Thus it CANNOT have parts. It cannot be split. A smaller part of it has no meaning and does not exist. It is suggested from empirical evidence that this limit to the divisibility of extension is ~ 10-35m (Planck length).
So it really boils down to whether Space exists independent of objects OR Space does not exist but is merely a relational concept between objects, i.e. remove the objects in a location and nothing exists there anymore. As you can see it is an argument dating back centuries. There is an interesting entry in The Stanford Encyclopedia by Hugget, N. and Hoefer, C., Absolute and Relational Theories of Space and Motion.
You seem to belong to the Relational school. Good for dialectic as I have someone whose position I can attack and who can attack mine as well.
All the best,
Akinbo
*I think you may have misinterpreted Leibniz a bit given your point of view. Cant be extended can mean cannot be stretched. Only something that has parts can have a shape, i.e. central part and a border part or this side and the other side. An extended point has no sides, no central part and an outer part.
If you read the first 7 paragraphs, which I consider sufficient, you will see that the pure point of Leibniz is a 'simple substance', the true atoms of Nature. Can this simple substance and true atom of Nature be of zero dimension?