Another way to say it would be: when do the inherent limits of expression in
formal models (in your first sense) impinge on the capacity for them to be
formal models (in your second sense)?
HP: There are at least two schools of thought (or two ontologies) that bear
directly on the form of your question. At the extremes there are the
Platonists who see physical laws as derivatives of abstract forms. "In the
beginning was the word" (John 1;1). And there are the Constructivists who
see the semantics of formal symbols limited by physical laws. "Words grew
out of the womb of matter" (Laotzu, Tao Teh Ching 1).
I am a Constructivist, so I would phrase your question conversely: when do
the limits of physical laws impinge on the capacity of formal symbols to
have meaning? That is, when do the purely syntactic expressions of formal
mathematics go beyond what can be measured or encoded by nature?
Also Constructivists, like most computer modelers, see no reason why the
formal concept of Turing computability should limit how we actually write
programs. Many programs for physical and biological models are not even
algorithms because they do not halt by themselves. In fact, strictly
speaking, our computers are not Turing-equivalent. They are finite memory
and finite state machines, and all theorems on computability depend on the
syntax of infinite sets.
This is a classical metaphysical controversy. Henri Poincare (the
constructivist) and Bertrand Russell (the formalist) argued this for years
and finally agreed to disagree. Rosen and I were closer to agreement than
Poincare and Russell, but there was no doubt Rosen was not as
constructivist as I am. We will not resolve this issue here, but I think
understanding why there is an issue might help the discussion.
Poincare's basic argument is that there is no empirical, ontological, or
semantic evidence for infinite sets and certainly none for transfinite
sets. He saw infinite sets as useful but purely syntactical games with
symbols. He saw Richard's and Russells's paradoxes (and would have seen
Goedel's and Turing's theorems) as just limits on how you can play
syntactic games, but semantically irrelevant for models of physical reality.
Of course, the constructivist agrees that there are formal symbolic models
(in my first, right-hand sense) that he cannot compute by Goedelian
conditions; but to convince a constructivist that this is scientifically
relevant requires an example of physically observable behavior that he
cannot adequately model (in my holistic second sense) by satisfying only
the Hertzian conditions.