[Bruno Marchal] But how do you relate that with you experience? You have not yet told me what is the duality in the nested structured duality. You need to try to explain this without using the terms “nested”, nor “duality”. You seem not trying to explain.

[rf] Try this. Think of 'duality' as a pointer to 'quanta' or 'multiple states', and consider that such quantum features are and must be inserted first in the foundationation of mathematics (even in arithmetic), rather than as a somewhat magical add-on after a classical, non-quantum, introductory prepi-cycle. In my first pass through this terrain back in 1975-1982 -- in the Bad Old Days, back in the Reagan eras, I was imagining building a tetrahedron using four rod magnets. With my wife's metal-working help I soldered a center connector together and then played around with the five states of the inner magnetic tetrahedron. Tetra- implies poly- and inner implies outer, so in a couple of moves on the gameboard I was considering the states of all inner and outer "bipolar polyhedral structures" (bps). So, if you follow, I upgraded the term from 'bipolar' to 'duality'. Originally, though, I came around to noticing that even forgetting about magnets and just making a structure out of anything, to my way of thinking there would always be a tiny, maybe what mathematicians might call "infinitesimal difference" between one end (half) of a radii or edge. One half could have a few more electrons or photons or quanta on one half than the other, So the ends are different, similar to what is overtly present with magnets, but now more subtle, tending to the point of practically indistinguishable.

That's the original meaning/origin of "duality" as in "nested structured~duality", to me. That's why I sometimes qualify it a "difference", and it could be a difference or duality in many different traits of features.

In the last few weeks, I have remembered that I used to think that ALL structures, even the highly idealized ones, are ~actually 'bipolar polyhedral structures' existing or having multiple states. That is, that there are not the two categories: {regular, bipolar}, but just the one category: {bipolar}.

I suspect mathematicians would opt for pure structures being pure and not having the so-called "bipolar' multiple states. But this, quibble, to me, raises what perhaps may be a fundamental issue in the foundations of mathematics. This being where and how "quantum features" are acknowledged.

Seems to me if such quantum features are fundamental or primitive, or however it's termed, then those features would be inserted/acknowledged as being in the initial step or first phase of all of the mathematical theory/modeling. And, all of this brings up the issue or question of "nested structure" in terms of the "proper" ordering of traits and features in the structures of mathematical structures.

I do not observe this feature is overtly included or considered in ~your "assume mechanism" elementary arithmetic storyline but you do mention arriving at "quantum logic" or having issues with "wave-packet reduction" later in your presentation. Also, what do I know about mathematics? So, perhaps you can explain the current partyline.

Over ~here in the weeds in approximate analog math, in my storyline, I suppose I might be able to invoke my N+1 rule where the NSD multiple-states of each number N, equals the count of N+1, which is also equal to the value of the next closest successor. Thus, visualizing each number as a count of center-to-vertex 'radii', each of which having one-half spin would have N+1 multiple states within itself which sort of means that each number has a representation inside itself ~equal to the value of its successor.

~This relocates "quantum features" into numbers at, I think, a more fundamental level of organization, giving them (numbers) more of a nested structured~duality flavor or flare.

Thoughts?

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