I wrote my essay with a certain perspective in mind. I am not locked into any particular perspective on this matter. That spacetime is continuous and smooth seems very much in line with the NASA Fermi and ESA Integral measurements, which involves a large ruler, or equivalently nearly zero energy, measurement of spacetime. Conversely a high transverse momentum measurement would result in chaotic fluctuations. A high energy of a particle in a tiny region of spacetime would by virtue of the Heisenberg microscope argument demonstrate a very chaotic or maybe "foamy" fluctuating structure. The quantum fluctuation is then a manifestation of the stochastic nature of quantum measurement.
The measurement of the smoothness of spacetime by this method is rather indirect, which is by virtue of no observed dispersion predicted by a foamy non-smooth structure. The measurement of wild fluctuations in the metric by high energy or transverse momenta would be more direct.
This reflects something odd about quantum mechanics. The founders touted the idea of a wave-particle duality, but in time it became clear that the particle is what is measured and the wave sort of "flaps in the breeze." Feynman even proclaimed quantum physics to be entirely about particles and not waves. It has only been recently that by weak measurement techniques with electrons entangled with an EM wave function have aspects of the quantum wave started to appear empirically. There is this odd asymmetry to nature. The field, wavy or continuum aspects of nature are difficult or more indirect to measure that the particle nature.
The "body-soul" duality I tend to advocate is something one can "wear" as needed. I might by virtue of some argument want to invoke a mathematical objectivity of sets, continuous spaces and even to the point of Platonism. At other times I may put this entirely aside. In my essay I largely put this aside.
Garrison Keillor has a feature on his show "Prairie Home Companion" on Guy Noir with the opening line, "On a dark night in a city that knows how to keep its secrets, one man seeks answers to life's persistent questions; Guy Noir, Private Eye." That is about my sense of the question about the relationship between physics and mathematics. We may never know for sure. Further, the universe may have a kernel of structure, symmetry and order to it that appears in a fractal-like form at different scales, but where nature also has this inherently chaotic or disordered nature to it as well, which I think is distilled down to the stochastic nature of quantum measurement.
Cheers LC