Greetings Akinbo,
I want to take a moment in this e-mail to address the key question you left me in a general way, here rather than on my forum. I've still not made it through your essay, but your intriguing and delightful questions bear some attention, and have been unavoidably a part of my contemplations of late. I'll talk here about learning how to count and measure. This is a key part of the cognitive science research I've been engaged in the last 8 years or so. As I point out in my essay, an early cognitive landmark is grasping object constancy, but children up to a certain age have an endless appetite for games of peek-a-boo, where you hide and then appear to the child's delight. So there is a transitional period for learning to distinguish clearly between none and one, and to understand the persistence of objects (and people).
Just so we are clear (addressing the topic of your essay); I've never quite bought into the point-particle concept, and have always thought things had to have an extent and/or underlying structure - to exist in spacetime. In order to exist, particles and composite objects must possess duration or extent in time, as well as being extended in space - in my view. So I don't adhere to that part of Plato's reasoning. So to continue...
Distinguishing none from one, while it is a prelude to counting, also evokes a different but related skill - the ability to distinguish none from one, or a few, or from many, and from a very large number - magnitude range estimation. During this same developmental stage, however, children are also learning distance range estimation - through triangulation. There is a natural connection in this to principles of constructive and projective geometry, getting a sense of whether various things have size or thickness or depth. Children must learn the rules of dimensionality. In a lecture I attended by Alfie Kohn; I heard a wonderful story about how a group of children learned by being guided to playfully discover for themselves about standard units of measurement. At first; the teacher didn't provide rulers or tell the kids how to measure, but instead they posed a challenge - the boat had to be big enough to fit everybody in the class - and let them figure out how to do it.
There is a connection between the developmental or learning processes above and the hierarchy of smooth, topological, and measurable objects and spaces. Smooth relations admit fields and waves, topology is for objects that have a face, surface, or hypersurface, and the property of measurability arises only when you have two or more such objects to compare sizes. So there is a natural progression from continuity to distinct and separate identity, and for the verifiability or knowability of same. In my view, the way we learn about these things parallels how nature unfolds form. And it is certain that learning the difference between none and one helps the child to then learn how to count as well as to estimate magnitude, but that is grasped in stages.
All the Best,
Jonathan
