This is what I've been trying to say. Mathematics is a game having nothing to say about reality unless you interpret the results you obtain in mathematics and apply those results to the world. The interpretation process is how you 'hook' mathematical equations to the world. Science does this through the act of making hypotheses (often based on mathematical concepts) and finding empirical support. But, in no way can the real world concepts (the hook) be considered as derived from strictly formal processes. The derivation of real world concepts is not part of mathematics (mostly deductive reasoning), but is based on largely inductive approaches.
As you both know all too well, the history of science is full of a variety of inductive approaches to 'hook' these concepts into a mathematical underpinning. For example, in the 1961 both Yuval Ne'eman and Murray Gell-Mann used SU(3) symmetry to explain the behavior of the strong interaction as it was being observed in experiments. There was no formal concept that somehow mitigated itself in a classroom setting to explain the strong force. Rather, it was a trial and error effort to apply a successful pattern that could represent the observations (as well as predict new observations). This is what the eightfold-way accomplished.
Hence, including concepts of the world as if they are derivatives of closed formal systems is not the case. The close cooperation of mathematics and science only makes it appear that way. Mathematics is a game (quoting Wittgenstein). It takes science to give real meaning to that game.
Warm regards, Harv