Dear Tim Maudlin,
You emphasize that associating a mathematical structure with physical items is not the same as postulating that they are mathematical entities. I fully agree with this.
As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.
You then ask, what features must a physical entity have in order to display a geometrical structure? Perhaps this question has been less frequently asked because there is no obvious answer. You discuss topology as the standard basis of continuity based on open sets. I find open sets highly abstract and artificial, and difficult to relate to physical reality in the natural way that integers (counting) relate to discrete physical entities. It is therefore of some interest that you switch tools to your Theory of Linear Structures and ordered, i.e., directed lines. As this is, in my opinion, a preferred approach to geometry, it then suggests the question, "What physical feature of the universe might be responsible for creating lines?" Your suggestion that it is time that underlies geometry is both novel and fascinating.
Thank you for your essay presenting these ideas to the FQXi community. My own ideas seem compatible with your ideas. In my Automatic Theory of Physics (1979) I essentially identified directed lines as time. Once lines are associated with physical time as the basis of linear ordered events underlying geometry, one can then go to second-order and employ geometry to construct sequential switching (see my 'End Notes') to construct automata, which can then be mapped into any axiom-based physical theory, answering Wigner.
I invite you to read and comment on my current essay. I believe you'll find it compatible with your essay, while conflicting with your work on Bell. I'm interested in your comments.
Best,
Edwin Eugene Klingman