Hello Indra
Dare I say, 'Wow!' I think our work is strongly connected. The following detailed comments are provided to show the connection. My apologies if these are rather long. My question is, are the nodes in my Harmony Set the centres of your pearls?
Thoughts as I read through:
You say, 'Isomorphisms of mathematical structures will play a central role.' I agree. I agree because it follows directly from the general principle of equivalence (my essay, in which the GPE is 'the' fundamental) and argument by Kant, so it is philosophically acceptable to me. The issue then comes down to whether the mathematical structure with which one begins is fundamental and well-founded, or at least a trustworthy representation of reality.
You say, 'Could the world be made out of a small number of fundamental building blocks, by putting them together like we do with bricks when building a house?' I think, 'Absolutely.' It is these bricks that are the bones of my Harmony Set, derived from my principle of equivalence. Reading further it is surprising how closely your bricks seem to align to my own, though yours are derived as empiricist, while mine as rationalist. This is good.
You consider Euclid's and Hilbert's geometries. I have a significant problem with this (not your fault) because these geometries lead to a provably transfinite number of contradictions (let me know if you would like to see this). Riemann and Hilbert were also concerned about this. The problem must be embedded in the axioms, and no one has yet identified which and why, however the conclusion is that lines simply cannot be sets of points. For example, how many points cover the line? If you try this, the mathematics blows up. My work dissolves these problems and removes the need for axioms, but to do so is not part of this comment. I like your identification of automorphisms and find that my own work coincides with this.
My other concern as I reach the part on 'holographic fundamentalness' is that the connection between mathematics and physics has never been properly established, as shown by Wigner in his paper "The unreasonable effectiveness of mathematics in the natural sciences," available online, and also considered by other greats of mathematics and physics such as Einstein and von Neumann. My work provides a link but the Harmony Set is still quite basic, and points and lines are of a different nature, though they do produce a vector space. However, that's my work. I don't think this is fundamental to your next steps (having now read ahead).
You then identify that you are considering an ontological relativity, and I think, 'Yay! Someone who gets it! We are looking for mathematics that is fused to the underlying reality.' Then 'lo and behold' you identify equivalence as being somewhat at the centre of it all. Does it not then immediately occur to the writer that equivalence itself (symmetry, equivalence and all equivalent ideas) is the fundamental?
You then move into some high level mathematical physics which is essentially a literature review. Correct me if there is something new in that section. I note that my Harmony Set, raised into a 3-space topology (for which I have a basis, but not the detail yet) seems to produce at least some, possibly all, of the prerequisites for a Clifford algebra, but this would require a deal of work for me to connect the two. However, I will investigate this aspect, so you essay is valuable to me in this respect.
Indra's pearls do seem to be very similar to what I get if I rotate up my 1-space Harmony Set into a 3-space topology, at least as a skeleton (this is not in my essay). Rotating up the 1-space Harmony Set as mentioned does produce complex vectors as well, so I am getting a bit excited (though dreading the volume of extra maths that I will have to derive to show the holomorphism).
Now to the crunch. My model also had the 'problem' relating to it all happening at once, but if you follow the nature of evolution of the Harmony Set, the encoding within each pearl is evolving, and each step in the evolution is ontologically dependent on the previous state, so, while your description is quite reasonable, I would add to this in. You will note that this also gets around Parmenides' denial of change. Otherwise your proposal, as I read it, would be flawed for you need an ontological propagator, not an epistemological propagator (a la Feynman). Please, this is not really a criticism, because I like the treatment. I am just trying to add the missing part, which is in the essay that is precursor (https://fqxi.org/community/forum/topic/1904) to my present submission (https://fqxi.org/community/forum/topic/3041).
Finally, the entire universe being enfolded in a spaceless and timeless point is one solution to my Harmony Set. Some might call it the trivial solution, but that seems deflationary.
Fabulous paper. I rate it highly. If there was a book on this, your chapter would either be the chapter prior to or immediately after mine. You might consider my paper in the light of yours, and also my essay previous to this, which was in the It from Bit contest, which I think strongly relates. My question is whether the nodes in my Harmony Set are the centres of your pearls.
