MATH.22:10 - Architectural Rationale

A proof states what follows from assumptions. A model makes their simultaneous satisfaction and separating consequences inspectable. Using both permits directed theory change: follow a needed construction to its assumptions, change them, and return the consequences to that construction’s use.

MATH.6 constructs countermodels, MATH.18 compares accounts through interpretations, and MATH.19 constructs missing arguments. Here these methods support a revised theory whose consequences can be used deliberately.

Logical consistency, mathematical fruitfulness and correspondence to an observed subject answer different working questions. The first controls what the axioms jointly permit; the second concerns the further constructions and questions they open; the third requires subject knowledge and observation. Their connection supports mathematical modeling without making one stand in for the others.