Christopher -
I appreciate that your essay is well written and clearly thought-out, but it seems to me you gloss over a basic difference between physics and mathematics. You acknowledge that the axiom of measurement is unusual, in that there's unavoidable complexity in specifying each particular measurement arrangement. But I think the issue goes deeper than that. What gets measured in physics is always a specific parameter such as mass or spin or distance or the gravitational constant. Each of these diverse parameters has a very different significance in physics. And even though the meaning of each parameter can be expressed through its mathematical relations with other parameters, this is no way reduces the special role each one plays in the physical world. Likewise each one requires its own particular kinds of measurement arrangements, so that measurement processes have the same kind of irreducible diversity.
Do you think there's anything in mathematics that plays a similar role? In geometry we have points and lines, lengths and areas, circles and triangles, etc. I can see the argument that since mass and electric charge, spacetime intervals and momentum, etc. all show up in the equations as mathematical quantities, they're essentially the same as geometric entities and properties, only more complex. But how lines and triangles work together in geometry doesn't really seem parallel to how atoms and molecules interact in physics.
I proposed in my essay that while the language of physics is surely mathematical, it takes a very special combination of many complex mathematical structures to support something like a physical world, where each parameter is definable and measurable in the context of other such parameters. So I don't disagree with your thesis, that physics can be considered a form of mathematics. But I think it's more important to understand how this unique kind of self-defining mathematics differs from the various systems we construct on the basis of undefined elements and formal axioms.
Thanks for the chance to respond to your quite interesting and intelligent paper.
Conrad