Dear Mark,
I enjoyed your essay! You have sound ideas, an excellent historical perspective, and a pleasing command of the English language. A few thoughts come to mind.
1. The statement by Parmenides that you quote on page 2 might possibly be the earliest formulation of the principle of "background independence," which you cite on page 3. Thanks for pointing this out; I had no idea this concept went so far back! Background independence is crucial to understanding unification and quantum gravity (at least, in my opinion!)
2. Coming from the math side, I can't resist pointing out that atomism and the continuum are only two of an uncountable number of different types of possible structures for modeling the physical world. Personally, I am bothered by some aspects of the continuum. What is a "continuum," after all? Well, it is a "linearly-ordered interpolative-complete set satisfying the least upper bound property." Sounds horrible, right? Personally, I think that two of these properties (interpolative completeness and the least upper bound property) have nothing to do with physics. Whenever I hear a physicist say "continuum or discrete?" I want to respond, "how about neither?"
3. If you want to know what I would put in place of atomism or the continuum, I say, "cause and effect!" If you are interested, my essay attempts to describe how the universe might be built out of cause-and-effect relations. In light of your introduction, I am a bit trepidatious that you might consider some my ideas "absurd," but I draw comfort from the fact that neither of us included a single equation in the body of our texts, as you note for your essay on page 9. So much the better! The ideas come first, and the mathematics comes afterwards. My view is that the physical principles should be simple and clear, and that the mathematics should be whatever it has to be to get the job done.
4. You point out that Making do without proper particles is not at all easy (page 8). Very true... I am still trying to figure out how to describe "particles" with causal structures.
5. You say that "A non-atomist causal perspective would instead be that there are two boundary conditions imposed on the field by emission and absorption events, which create a quantization of the field between them." This, and the ensuing discussion, is interesting and requires more thought. The music analogy is a good one.
Thanks for the great read! Take care,
Ben Dribus