Yafet,
Really nice essay. I think it is important and most difficult to understand the relation of physics and mathematics by simple examples. I basically agree with all your are saying in your excellent essay.
One thing you left out in your analysis is the Kantian question: "What did we already accept in order to think that mathematics or physics is possible?"
You say that: "The truth of a mathematical propositions is independent of any physical phenomena." I would say, that in order to be able to do mathematics some physical conditions have to be met: constancy of certain phenomena. Maybe also the structure of time.
In physics the situation similar, when we say, that physics is an empirical science. The structure of time seems to be a precondition of experience.
Maybe also the conservation of energy (Time translation invariance): Once Heisenberg heard about an experiment that showed, that the energy was not conserved. He reacted instantaneously: "Impossible!" How could he know that? How comes that some physical (empirical) statements although empirical seem always to hold true in our experience? The Kantian response to that is: because they are preconditions of objective experience to be possible.
Again I think you wrote a great essay and I would love you could read mine and comment on it.
Best regards,
Luca