<p>The p-adic conception of number, inaugurated by Kurt Hensel in 1897, redefined the meaning of completeness and proximity by replacing magnitude with divisibility. Born within number theory, it has since transformed the conceptual foundations of mathematics and physics alike. This paper traces the historical genesis and philosophical significance of this transformation-from the arithmetization of the continuum to the emergence of p-adic and adelic structures. Methodologically, we combine a concise mathematical exposition of p-adic and adelic constructions with a structural analysis of their role in modern mathematical physics, particularly in renormalization and scale hierarchies. We show how ultrametric topology and valuation naturally encode hierarchical regularity and yield convergent structures where Archimedean frameworks encounter divergence. Conceptually, we argue that the p-adic and adelic frameworks do not merely extend analysis but reconfigure the categories of infinity, unity, and knowledge. Infinity appears not as boundless extension but as hierarchical depth; unity arises from coherence among distinct local worlds; and knowledge unfolds as refinement rather than approximation. Taken together, these developments lead to an arithmetic metaphysics in which reality is defined by coherence across scales—an ontology of structure and harmony rather than of substance and extension.</p>