Dear Israel,
After noticing some interesting comments of yours on another thread, I was drawn to read your essay. I must congratulate you on an excellent submission! I have a few comments and questions.
Let me first remark that while I don't dispute your arguments for treating the existence of the PSR as "not an issue of parsimony but of usefulness," I believe that there are good reasons, particularly in regard to the possibility of spacetime microstructure, to reinterpret the principle of covariance in such a way that the assumption that "all (suitable) frames of reference are created equal" is not only "not useful," as you characterize it, but actually "false."
1. Regarding your distinction among true, false, and useful assumptions, would you agree that an assumption that is "useful" for the purposes of one theory might be "false" for a more refined theory? For example, you mention the unsettled (and possibly unsettle-able) question of whether the universe is infinite, and in many contexts, the answer to this question is irrelevant to the derived predictions. However, there are discrete theories that use the assumption of a finite universe to try to predict the small nonzero observed value of the cosmological constant, and for these theories, the answer is not irrelevant. Similarly, for your other example of time-reversibility, some theories of spacetime microstructure require a fundamental local "arrow of time" and others require its absence (examples of both appear in this essay contest!)
2. I think we can agree that one of the most "useful" assumptions since the beginning of the scientific revolution has been the smooth manifold structure of space/spacetime. However, I believe that this assumption may be nearing the end of its usefulness. Riemann himself did not take this assumption for granted, and my own opinion is that it has persisted largely because the associated mathematical tools were quickly developed and reasonably tractable. Information-theoretic, order-theoretic, category-theoretic and graph-theoretic constructs, for instance, were largely inaccessible or nonexistent at the time.
3. It is possible to reinterpret the principle of covariance in the following way. In relativity there is a local causal order (partial order) on Minkowski spacetime defined by light cones. A given event succeeds the events in its past light cone, precedes the events in its future light cone, and is unrelated to the events outside its light cone (which is why the order is only "partial"). A frame of reference imposes a time order, which is a refinement of the causal order; in this refined order a given event either succeeds or precedes every other event except those in the same "spacelike section." Traditionally, a frame of reference is given by a coordinate system, and two (inertial) frames of reference are related by a Lorentz transformation. However, this requires the manifold structure to persist down to arbitrarily small scales. In many theories of quantum gravity, there is a nonmanifold spacetime microstructure, and frames of reference can no longer be viewed in this way. However, order-theoretic ideas still apply as long as there is a causal order, and a "generalized frame of reference" may be defined simply as a refinement of the causal order. The point of all this is that even if one restricts to refinements "resembling" inertial frames in relativity, it will usually no longer be true that all such frames are "created equal." For instance, arbitrary boosts may be problematic. Theories that exhibit some of these features include causal sets, causal dynamical triangulations, and deformed special relativity (DSR).
4. When you mention "dark matter" (section III part A) you suggest that this phenomenon is an aspect or property of spacetime. Now, I lean toward the same view, but my impression is that the mainstream view still regards dark matter as simply particles that don't interact electromagnetically. Is your statement here deliberately at odds with the mainstream view, or am I misinterpreting your statement?
5. I agree with your analysis that the concept of the "aether" would likely have been merely "reinterpreted" or "refined" rather than discarded if various high-energy and cosmological phenomena had been recognized at an earlier date.
6. The ideas you present in section IV B may be radical, but are certainly worthy of serious consideration. Of course, this would require more details than one can be expected to present in a short essay!
Thanks for the great read! Take care,
Ben Dribus
