The gap between purely symbolic manipulation and the physical model it represents is a well-recognized challenge in mathematical physics. While narrative text is the traditional bridge, there are formal, semi-formal, and mathematical frameworks designed to make the underlying model, its assumptions, and its relation to the derivation more explicit.
Here are several approaches from physics, computer science, and applied mathematics that address this problem.
1. Variational and State-Space Frameworks (Mathematical Physics)
In classical and quantum mechanics, the "model" can often be completely specified mathematically before any derivation begins, reducing the need for narrative assumptions mid-derivation. This is typically done using variational principles or state-space formulations.
To specify a model precisely without narrative text, one can define:
The State Space (Manifold $M$): Defining the degrees of freedom (e.g., position coordinates, fields) and their boundary conditions.
The Constraints ($C$): Explicitly stating algebraic or differential constraints (e.g., holonomic constraints in classical mechanics, gauge conditions in field theory) as mathematical subsets of the state space.
The Action ($S$) or Hamiltonian ($H$): Encoding the entire physical dynamics and interactions into a single functional.
How this helps: Any subsequent derivation (such as finding equations of motion) is simply an application of the principle of least action ($\delta S = 0$) under the specified constraints. The physical model is entirely contained in the definition of $S$ and $M$. If a derivation is partially complete, knowing $S$ and the constraints is mathematically sufficient to complete it.
2. Physical "Types" and Dimensional Ontology (Computer Science)
In standard mathematics, a variable might be defined simply as $x \in \mathbb{R}$. In physics, this lacks the context of the model. Computer science addresses this using type theory and ontologies, which can be applied to mathematical physics.
Instead of relying on narrative to explain what $x$ is, one can assign it a highly specific "physical type":
Dimensional Typing: A variable is not just a real number, but an element of a vector space of physical dimensions (e.g., $L^1 M^0 T^{-2}$ for acceleration).
Semantic Typing: A variable is defined relative to a specific coordinate frame, reference point, or approximation regime. For example, instead of writing "assume $v$ is small," the variable $v$ is typed as $v \in [0, \epsilon \cdot c]$, where $\epsilon \ll 1$ is an explicit scaling parameter.
By formally defining these types at the start, any mathematical step that violates the dimensional or semantic rules (e.g., adding a velocity to a force, or applying a low-speed approximation to a relativistic variable) is flagged as mathematically or logically invalid.
3. Graph-Based Derivation Trees and Dependency Graphs
A derivation is rarely a linear sequence; it is a directed acyclic graph (DAG) of logical inferences. The relation between the model and the math can be formalized by structuring the derivation as a graph where:
Attributes on Edges explicitly state the physical assumption or approximation justification (e.g., "Adiabatic expansion," "Dipole approximation," "Neglecting $O(x^2)$ terms").
By formalizing the derivation as a graph, you can trace exactly which physical assumptions are required for which mathematical steps. If a derivation is partially complete, the graph shows which assumptions (the "model") must be carried forward to validate the remaining steps.
4. Categorical and Applied Category Theory Frameworks
Applied Category Theory (ACT) is increasingly used to formalize physical systems, particularly through string diagrams and monoidal categories.
Port-Hamiltonian Systems and Bond Graphs: These frameworks use category theory to model physical systems (mechanical, electrical, thermodynamic) as interconnected blocks. The connections represent energy transfer.
String Diagrams: These are visual, mathematically rigorous diagrams that represent physical processes. The geometry of the diagram is the mathematical representation of the physical model. The equations are derived directly by manipulating the diagram according to strict algebraic rules.
This approach replaces narrative explanations of "how components interact" with a rigorous topological and algebraic map of the system.
In digital humanities and mathematical knowledge management, frameworks like OMDoc (Open Mathematical Documents) and the MMT (Meta-Meta-Theory) language are used to represent mathematical physics.
These systems allow authors to:
Define "theories" (which represent physical models or frameworks, like "Newtonian Gravity" or "Ideal Gas").
Define "views" or "morphisms" that map abstract mathematical structures to specific physical interpretations.
Explicitly declare the domain of validity of an equation as metadata associated with the equation, rather than as narrative text.
Summary of How to Specify a Model More Precisely
If you are writing or programming a derivation and want to minimize narrative text, you can structure your document or code to include:
An Explicit Context Block: Before the math begins, declare:
Ontology: Mapping of symbols to physical quantities (e.g., $T \rightarrow \text{Absolute Temperature}$).
Approximation Parameters: Explicitly defined small parameters (e.g., $\epsilon \equiv v/c \ll 1$).
Domain of Definition: The geometric boundary conditions and state-space constraints.
Justification Tags on Equations: Instead of narrative paragraphs, label mathematical steps with formal justification tags (e.g., $\xrightarrow{\text{Taylor } (x \to 0), \text{ Order } 1}$).
An Action or Conservation Principle: State the fundamental variational principle or conservation law from which the entire system is derived, rather than introducing ad-hoc equations mid-way through.