Mathematics Has Already Chosen Federalism: The Logic of Domain Separation

Mathematics, as actually practiced, operates as a federated system: practitioners work within autonomous domain-specific axiomatizations (geometry, algebra, analysis) and construct explicit bridges only when cross-domain reasoning is required. This organization is not accidental; it is a structural adaptation that safeguards local decidability and algorithmic efficiency.Yet the dominant foundational narrative still presents ZFC set theory as the universal foundation into which every domain ultimately reduces. We argue that this monolithic reductionism is pedagogically misleading and structurally inefficient. Embedding a decidable (tame) domain into an undecidable (wild) one imposes a substantial epistemic overhead: efficient, domain-specific decision procedures are buried under general proof search, and local structural immunities are lost.We introduce the Decidability Threshold — a litmus test based on Negation, Representability, and Discrete Unboundedness — to explain why mathematicians instinctively isolate tame domains from wild ones. Finally, we distinguish the Mathematician (builder of formal systems) from the Scientist (consumer modeling reality), and argue that federalism, through explicit bridges and domain autonomy, serves as the primary safeguard against inadvertently importing mathematical paradoxes into physical theories.