Dear Ben
Thanks so much for your comments on my submission discussing Riemann's concept of density. I believe everything in your submission, up to its last sentence on "energy density," reflects rigorously much of what I discuss at a different level. And it does so a hundred times better.
I've also often wished that Einstein had met Riemann. Einstein and Planck borrowed a lot from Riemann, from the zeta function to "their" term quantum. But I agree, they did not borrow enough. And, as you show, the problem lies in foundational assumptions. I discuss some of their assumptions in my European Journal of Physics article http://iopscience.iop.org/0143-0807/30/4/014. But I think your essay already identifies them.
You're absolutely right: Riemann himself did not take continuum manifolds for granted as a basis for physics. His unpublished notes reveal an approach closer to your paper's causal structures. I will spend more time on your immensely insightful proposal. It deserves better thoughts than the following. I'll send you an improved response but for now just a few "brainstorms:"
I believe Riemann's concept of manifold is not the one you reject as "the manifold structure of spacetime." I'm working on a paper like my one on the concept of density, except on Riemann's concept of manifold (Mannigfaltigkeit). Few people remember Cantor's Riemannian manifold -theory (Mannigfaltigkeitlehre), later known as set theory. I learned from your paper about "causal set theory." The term reminded me of Cantor's late research on set-theoretical "physics."
Like you, Riemann would also reject the "evolution of systems with respect to an independent time parameter." As he tells us, the reigning paradigm was mostly Kantian. If you view time as a way of talking about causality then you come close to his "neo-Kantian" approach, i.e. space and time as somehow mathematically observer-dependent. He never finished this late work.
As for the "commutativity of spacetime" I believe Riemann held space and time to be dual, i.e. just like the particle-wave duality, but somehow commutative-noncommutative. He did not have non-commutative geometry but perhaps came close.
As for your promising causal metric hypothesis, I believe you may find philosophical/foundational support in what Hermann Weyl wrote about Riemann, specifically in Weyl's recently reprinted books.
"In the universe of scientific thought, ideas from mathematics, philosophy, and the empirical
realm combine in the form of general physical principles, which crystallize into the formal
postulates of physical theories, while remaining colored and sometimes distorted by the interpretations
and prejudices of their intellectual environment." Riemann could not have put it better than that!
A few of Riemann's contemporaries did not formalize causality as an irreflexive, acyclic, transitive binary relation on the set of spacetime events. I think you implicitly mention them. As I read them, Gauss' reciprocity laws and Riemann's reciprocal numbers "arythmos," were interpreted as rythmic, oscillatory, cyclic, reflexive, causal (force-effect) relations inextricable from space, time or gravity. A more technical version of my paper would say that the inverse square and quadratic reciprocity law were not separate into "physical" and "mathematical" laws. One can see that just from the laws' names. ...(cont.)
