Hello Akinbo
Speaking as an author of two books, I found it to be a beautiful piece of writing.
Your argument is fabulous, and I will be rating it as such (8) because it is foundational, and uses philosophical principles very well. In many ways your attributes of monads from i to vii align well to the fundamental interactions between boundary omnets in my model. I went very carefully through your essay, and am going to show my thoughts as I went through. They are not at any time intended to pull your ideas down, but to suggest how the argument needs to be generalized beyond geometry, and that there are several background assumptions that can't really be ignored in a foundational argument. I think that if one sees these in the spirit in which they are offered that your redevelopment will gradually or quickly converge on my own essay's foundations.
There are several aspects within the first seven clauses about attributes of monads that are presumptive. For example, the idea of compressibility is a human-centered concept, and presumes a background space without showing how it comes to be, or indeed what space is (except as an argument subject to infinite regress). I can guess various arguments in response to this, and there is no need to point them out, but each seems to lead to further arguments, so ought to be left. The items after that assume several aspects that presume pre-existing time, with no causal mechanism shown, and would become unnecessary in terms of the GPE causal model of my essay. For example, emergence and annihilation was introduced to make the monadic concept work, but the Harmony Set evolves and brings time with it, and change in higher dimensional interpretations of the Harmony Set are likely to imply change of position of peaks of the vector strengths as the interactions between null elements superpose, which is likely given that regularity of events (regularity of change in similar situations) is guaranteed by the GPE itself. So by Occam's razor, your work can be simplified, and the endpoint of simplification would be to simply accept the GPE (as a matter of skeptical commitment) and see where it leads.
Accepting Euclid's fundamentals is O.K. although all of mathematics is initially degenerate under the General Principle of Equivalence and the problem of bundling.
The definitions themselves are fine, but make assumptions that there are such things, and that they are in themselves as we define them to be based on experience. But this is why the FQXi website exists, because under Kant and others we can't really do so with confidence.
'Information' - is dark energy information? Not according to the definition.
'Have no part' - there is probably no need to refer to geometric objects. Why not just 'objects'? Then an object can be generalized to an 'omnet' and then the idea becomes global, which is necessary for a secure argument, I think.
'Point' - Here is the concept that forced me to build the Harmony Set, for if there is a point, how can a point, which can have no part, have any property that can be connected to any other point, for there is always difference between two points (assuming that points are actual objects, or actual omnets within the actual ontology (referring to my essay) and there is nothing that can connect them to make either a line or extension, unless there is some overarching principle that bring such points and their connections (see my essay). That is, the whole concept of point has a problem, which I initially could not identify. From the endpoint rationalist perspective, one finds that the idea of something being of zero dimension (not possible) is not the same as it being dimensionless. The null elements of the Harmony Set are dimensionless, in that they exist without dimension having any meaning. Rather, they bring dimensionality into existence through difference between each null element (even if this is just ontological dependence and priority - see my essay). Then these null elements, if monads, have no extension, whereas the difference between them might be monad-like in that they have extension brought about by the implied dimensionality that pops out in the structure. Of course, I have only generated the 1-space solution, and so it would not be correct to say that these 'monadic' omnets properly correspond to the equivalent forms in higher spaces, at least in the sense Leibniz meant. This is cognitively challenging, I know, but then, once one gets it, it frees up one's thinking on foundational issues (and trades it for harder problems, unfortunately).
The bundling problem applies, and so implies a unique origin for a world of points, or lines, or fluffy animals (meaning anything else - the problem is global). Moreover, how does a world of points that experiences evolution, achieve such evolution, for the points themselves must undergo an infinity of change, and each change would experience an infinity of changes, unless it is instantaneous (the method of such change occurs through the GPE as constructor in my essay).
'Monad' - Yes! But drop the idea of it being geometrical, for one can't trust a pre-existing spacetime or Euclidean (or anything else) background. There is no need for the geometrical for it should develop from the foundational aspects.
'Motion' - as change of place, this could be generalized to include Aristotle's idea of place, so that geometry is irrelevant. Consider living in a two dimensional world where the x-axis is measure in degree of redness (no red, to fully red, say) and the y-axis is measured in degree of temperature. Motion is then simply a change in redness and temperature. Same result, and the human mind would likely come to interpret it similarly to that of change of position (but it would be less interesting) in the same way that the brain can images upside down if wearing glasses that invert the image.
'Variable lifetime of monads': assumes that a monad is aware of, or affected by time in some way. But if a monad is windowless, then this would be a challenge to make consistent, for, unless the system is driven by a universal principle that acts on all at once, how does the world pick out a certain monad for existence or non-existence.
Note: your boundary is not my boundary. Yours is a geometric object. Mine only gains a geometric property by it relations to other boundaries. This is not a point of conflict, just a point of difference.
Whence time in your world model? What is the foundational cause of change?
The Weyl tile argument: Yes! But this argument is founded around the expectation that the real number line is valid. In the Harmony Set interpretation, geometry contains values that are slightly fuzzy - what I call 'block numbers'. In another post I show that the Pythagorean theorem implies that mathematics is cataclysmically inconsistent (view attachment). Block numbers fix the problem, and in doing so imply Heisenberg uncertainty, in that there is a minimum fineness of scale in a Harmony Set universe. Does this imply that your monads have a minimum extension? Depends what you mean by a monad.
In (c) 'Use for writing programs' the bother is that the shifting of an 0 from one place to another, requires that something act, already 'knowing' where the 0 has to be. But what caused the change within the mechanism that made it shift from one place to the next? This leads to an infinite regress, I believe. This is similar to Parmenides argument, and that of Zeno.
Best wishes
Stephen AnastasiAttachment #1: 1_A_problem_for_geometry.pdf
