Armin,
Thanks for the remarks and questions. Since most of my formal education and my "official" academic work is mathematical, I wrote this essay in an effort to help me begin a dialogue with competent physicists on topics I have thought about a great deal. I knew I would not get the style and focus precisely right at first, but I was hoping that some people could point out obvious flaws and things that required more or different explanations. Let me itemize my reply to correspond to your questions.
1. Binary relations on sets obviously play a central role in my approach, but there are a lot of "relational" theories, and I am not sure if you are referring to a particular one of these (or group of these) when you reference "relationism." For instance, prominent physicists like Rovelli, Thiemann, Baez, Smolin, Markopoulou, Loll, Ambjorn, Sorkin, Rideout, Bombelli, etc. all emphasize binary relations, but they all include assumptions in their work that I disagree with. These physicists work primarily on loop quantum gravity, causal set theory, causal dynamical triangulations, and a number of lesser known variants. Of these ideas, mine are most similar to causal set theory (Sorkin, Rideout, Bombelli, etc.) but there are multiple crucial distinctions that make the overall picture quite different.
2. There are "metric recovery theorems" (for instance, by Malament) that allow recovery of the entire metric structure of Lorentzian spacetime (including spacelike separation, etc.) from the causal structure and appropriate volume information. These play a prominent role in causal set theory; they imply that an appropriate causal set "looks like" a Lorentzian spacetime on sufficiently large scales. At the fundamental scale, you would define spacelike distance by counting relations; for instance, two unrelated elements with a common direct descendant are one unit of distance apart. Only at larger scales does this begin to resemble an ordinary distance function. My framework is more general because I don't assume a constant discrete measure, but the simplest versions still involve counting.
3. Frame-dependent order (relativity of simultaneity) is one of the most important points to understand because it highlights the new meaning of covariance (order rather than symmetry). In my approach (and also in some versions of the above theories), a frame of reference is a refinement of the causal order; i.e., an assignment of order to certain events which are not related in the causal order, just like a frame of reference in relativity assigns order to certain spacelike-separated events. The whole point is that the causal order carries the canonical information; the refined orders carry additional contextual information.
4. I think you point out a good way of comparing the Hilbert space version of quantum theory, in which classical states arise as an appropriate limit (correspondence principle), with Feynman's sum-over-histories version, in which the quantum picture is built up from classical alternatives via superposition. It is an interesting objection to the sum-over-histories version that the "building blocks" are classical; my view is to be grateful to Feynman for making the presence of a Hilbert space physically comprehensible; they're beautiful mathematically, but I prefer to see them arise from something primitive like superposition, just as I prefer to see manifolds arise from something primitive like binary relations.
Great questions; I hope that explanation at least somewhat answers them. Take care,
Ben
