Foundational.Reference:2.1 - Five subjects and equally legitimate entries

The subject of this composite field is the construction, justification and development of mathematical and physical accounts, computational ways of obtaining results, modeling correspondences and operative expressions. Here an operative expression is a formula, diagram, table, program, score, gesture or other structured expression with rules for reading or changing it. Such an expression may be read by a person or interpreted by another system; it need not be executable software. A construction can be developed for its own explanatory or theoretical consequence. Physical application, engineering realization and commercial benefit are not universal entry conditions.

The five branches retain their complete declared subjects. Their DPFs describe selected repertoires within those subjects, not complete textbooks for the academic fields their names may suggest.

• Mathematical Thinking constructs mathematical objects, operations and representations; develops arguments, witnesses, countermodels and bounds; and changes axioms or conjectures when the question requires it. Its result can be a theorem, a new construction or an impossibility argument. A proof can finish the work without a physical example. A computational search may help obtain a witness, but the mathematical claim retains its own warrant.
• Mathematical Modeling constructs and revises a mathematical question that retains the subject distinctions needed by the intended answer. Its subject may be a physical situation, a way of working, a computation or another mathematical construction. Formulation, observation models, inference, reduction, coupling and return to the subject are work in their own right. A model is not complete for a use merely because an equation has a solution, and the DPF does not cover every kind of modeling.
• Physical Thinking constructs and challenges an account of interactions, preparation, constraints, scales, evolution and observation. A physical explanation, discriminating observation or limit on possible transformation can be its primary result. Particular material laws and specialist physical knowledge can be supplied from outside the repertoire. Obtaining a numerical prediction is one continuation, not the definition of the branch.
• Computational Thinking develops algorithmics within computer science: effective operations, their representations and semantics, correctness and progress, resource demands and interaction. An algorithm, interpreter, lower bound or changed computational representation can be the result. A mathematical definition of an answer does not by itself supply a way to obtain it. This subject is wider than numerical calculation and narrower than all computer science or software engineering.
• Notational Engineering constructs and changes conventions for structured expressions: formation and binding, interpretation, transformation, translation, coordinated representations and the operations their readers need to perform. A usable notation or an improved reading and editing procedure can be the primary result. Its subject includes temporal and embodied notation, not only formulas and programs. Designing a score’s timing and reference rules does not perform the music; specifying a gesture’s frame does not supply the capability to execute the motion. General semiotics, perception and learning remain outside this declared subject.

These are equal entry points, not stages of one mandatory workflow. Their conditions determine which further contribution is needed. A mathematical question may lead to a more suitable representation; a notational ambiguity may lead to an algorithmic or physical question; a failed inference may require another observation rather than more computation. A familiar adequate contribution can be used directly, without opening the other four languages.