Dear Jonathan,
I think your essay is right on target, and it rates very high in my opinion. Let me make a few remarks. First, let me say that I don't believe the manifold structure of spacetime persists to arbitrarily small scales, but this in of itself is hardly a radical position anymore.
1. I agree that one ought not to begin with mathematical models of time (or anything else!), but ought to begin with physical concepts, and then use whatever mathematics is necessary to get the job done. This may lead to mathematics that is "less convenient," but so be it! Choosing mathematically convenient but physically dubious models has caused too many problems in physics to even begin to list.
2. An excellent point you make: "Often more than one conceptual picture is described by similar mathematics." Likewise, there is often more than one choice of mathematical formalism to use in attempting to make a physical idea precise. Often the differences among these conceptual pictures or formalisms involve physical issues at the periphery of what is being considered when the theory is first developed. Only later are the distinctions recognized as important, and by this time it has often become "common knowledge" that a particular marriage of concept and formalism is the "only way to go."
3. You say "And then other physical laws, which also depend on there being a timeline (or rather, many), and behind them fundamental principles like cause and effect, which also depend on a timeline." Now, this is something I have thought about a great deal. Do cause and effect depend on a timeline, or does time depend on cause and effect? Or are they two ways of talking about the same thing?
4. You say, "Within the light cone, where events are in range of each other, there's a clearer sequence - one can say an event happens before another if it can influence it by getting a light signal there in time. This short range way of relating events has meaning, based on causality. But it doesn't mean there are long range time links across space, as in Minkowski spacetime." This, in my opinion, is the crucial point. The physical order is the causal order, and the "time-orders" given by choices of reference frame are not physical. They represent extra, noncanonical information added for mathematical convenience and should be given no weight when discussing issues of existence.
5. Continuing from 4, I believe another aspect of this false assumption is the "symmetry interpretation of covariance." Covariance in special relativity is, conceptually speaking, the statement that different inertial frames are "equally valid," and the usual way of making this precise is to invoke the symmetry group of Minkowski space (the Poincare group). I do not think this is the best interpretation, especially when one generalizes the discussion from special relativity to general relativity and then to the fundamental structure of "spacetime." I think a better interpretation is in terms of order theory. The causal order on Minkowski space is defined in terms of the light cones, and an event E is simply unrelated to events outside its light cone in terms of the causal order. Imposing a time order on Minkowski space refines the causal order by artificially relating E to most of the events outside its light cone (all those outside the same "spatial section.") Different frames of reference, then, are different refinements of the causal order. However, it is obvious that such a refinement carries no canonical physical meaning. The physical information is contained in the causal order, which does not imply a block universe.
6. These topics are a major focus of my essay, On the Foundational Assumptions of Modern Physics. Since you have evidently thought about these issues deeply, I would be grateful for any further thoughts you might have on the subject. I think that "spacetime" is essentially a way of talking about cause and effect, and that geometry is a very good approximation to this at currently observable scales.
Congratulations on an excellent contribution! Take care,
Ben Dribus