Eric, Apologies right off, the following analysis is for myself and probably should be kept to myself. On the one hand, I am struggling to learn a particular formalism. On the other hand, I am struggling to get better at understanding what it is that people I admire appear to be doing. The formalism I am continuing to explore is the adjunction (Con ⊣ Lang) relating formal type theories to their categories of models[⊣].
You begin your exposition with the embodied description of a paddlist building up a set of experiences and (via signal-boosting?) arriving at a reasonably stable system of relations (restoring forces and spatial relations, say). That is, you extract the internal logic of some phenomenon from a principle *model* to a *theory* of types, relations and deductive rules. The derived type theory comes equipped with group theoretic relations capable of distinguishing what we define as a purely formal SOLID type from a purely formal LIQUID type. Isolation of a formal theory provides leverage reflected in the category of models: 1. The theory provides a means for producing a generic model, free of surplus meaning and yet preserving desired logical consequences such as P and S waves when reinterpreted in the principle model. 2. One can study the *shape* of the interpretations of the theory via morphisms from the generic model into the principle model. 3. As a corollary, group theoretic deductions of the theory are consistently embodied in the model, correctly assigning properties like solid and liquid to the intended scoped-patterns. As I understand you, "framed within a very partial description of nature." You then proceed to perform certain calculations within the context of the theory, creating types A and B (presented as propositions in the logic of the theory) and then pointing to the sorts of deductive exercises one might hope to perform with these types. Relative to the principle model, so long as our interpretations are valid we don't particularly care whether we are speaking of "a hockey puck, or an Evangelical Who Knows the Glory of God, or a heathen, or a psychologist". What matters from this perspective are the theory-preserving morphisms from our generic and arguably privileged POV model to our principle model. I feel like I am still not grokking what I can from your discussion of causality, so I will leave that (very interesting) exposition aside for now. One thing that strikes me as being meaningful is something about the nature of the generic model. For instance, there is Griffith's *famous grey box* in his text on *classical* electrodynamics wherein he states that *magnetic forces do no work*. What sometimes frustrates students of this theory[B] is that models constructed from classical electrodynamics give rise to insanely complex epicycle-like thought experiments involving electric forces actually *doing the work*. This isn't a criticism, of course, because what seems to satisfy physics students is then to enrich the classical theory to a quantum theory. All said of course, if one were to argue that you did none of the formal things I describe above, I fully accept that too. Jon [⊣] https://ncatlab.org/nlab/show/syntactic+category [B] https://www.youtube.com/watch?v=fHG7qVNvR7w
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