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Our interest in mathematical inquiry arises from its role as a model of precise reasoning rather than as a toolkit to be uncritically imported into other disciplines. Mathematics exemplifies how complex structures can be built from simple assumptions through explicit argumentation. We have drawn on this mode of thinking to clarify what it means for a linguistic theory to be well-formed, internally consistent, and explicit about its commitments. At the same time, we have cautioned against confusing formalization with explanation, insisting that mathematical structure must always serve conceptual understanding.