Difference That Preserves: From Transcendental Genesis to a Genealogical Foundation of Mathematics

Abstract This paper reconstructs the philosophical genesis of a foundational motif— difference that preserves —emerging at the intersection of ontology, logic, and mathematics. Through a genealogical arc spanning Fichte’s theory of self-positing, Hegelian mediation, Bergsonian duration, and the anti-psychologism of Bolzano and Frege, we identify a deep tension in modern foundations: how can a logic of differentiation account for identity across transformation? We propose that this unresolved tension structures both historical and contemporary foundational programs. To address it, we articulate a new meta-theoretical framework— Mnēmaic logic —which grounds preservation not in static identity, but in recursive genesis. This paper provides the philosophical foundation and conceptual justification for that framework; its formal development appears in a companion article (Ballús Santacana, 2025). We conclude by suggesting that difference-that-preserves offers a powerful alternative to existing models of identity, continuity, and foundation in mathematical logic.