The Choice of Formal Realities: A Meta-Mathematical Argument for Explicit Foundational Context

Foundational discussions across mathematics, physics, and philosophy frequently proceed underan implicit assumption of neutrality with respect to formal frameworks. This paper arguesthat such neutrality is structurally impossible. Any sufficiently rich scientific or mathematicaltask presupposes a choice of a coherent formal reality—a package of axioms, logic, ontologicalcommitments, and admissible inferential practices. We present four meta-mathematical results,stated theorem-style, establishing the inevitability, non-uniqueness, structural cost, and stabilityconstraints of such choices. Together they yield a metatheorem: foundational completenessrequires explicit disclosure of the chosen formal reality. The analysis is prescriptive ratherthan ontological: it formalizes constraints on foundational practice using standard results (independence,non-categoricity, reverse mathematics, and constructive/classical contrasts), withoutproposing a new foundational program.