MATH.Preface:3.1 - Form objects, operations and representations

MATH.16 starts one step earlier, when several constructions seem plausible. Describe the maps the new object must support, then compare arbitrary allowed ways of supplying or processing its data. The resulting universal property distinguishes, for example, carrying both components from accepting either input, and arbitrary pairs from compatible pairs. It can also specify an object that represents a prepared function. A known construction can then supply the object and its maps.

MATH.1 starts with permitted elementary connections and constructs finite paths, identities and composition. Retaining a path can preserve the order and history that its final effect forgets. MATH.5 starts with assigned values for generators and obtains values for their composites while preserving the operations.

When several descriptions should count as the same input, MATH.2 tests whether the required operation is independent of the representative. For a partially available operation, its availability can matter as much as its output. A failed test supplies a distinction to restore.

MATH.7 addresses a reversible change of representation. It constructs the operations in the new representation and carries results back. Renaming the elements while keeping an unsuitable operation can change the problem; the transported operation supplies the repair.

MATH.17 makes the rules themselves available for construction and change. Select allowable operations, determine whether their composition stays allowable, then construct an operation that transforms them. Its required law follows the intended use: repeating a composite and repeating its stages separately can yield different answers.

MATH.18 compares descriptions with different primitives, allowed maps or equality. Construct the needed interpretations, establish what transfers, and inspect the return. A partial interpretation can be enough for one consequence; equivalence requires the corresponding comparisons in both directions.

These methods can be used separately. They also connect: form expressions from generators, interpret their operations, identify descriptions that preserve the desired answer, then choose a convenient representation for calculation. When the rule or the whole account changes, use MATH.17 or MATH.18 to construct and examine that change.