Abstract: When second-order boundary value problems are solved in finite-dimensional spaces, it is inevitable that some of the properties inherent to the corresponding continuous problem must be compromised. Although the literature on numerical techniques is extensive, relatively little attention has been paid to how the essential structural properties are preserved in finite-dimensional models. A reason for this is historical; The development of numerical techniques has been driven by the needs of various fields of physics, and hence, there has been less emphasis on the structures boundary value problems share and on their preservation on finite dimensional settings. In this work, we adopt a complementary perspective by first asking: what do second-order boundary value problems have in common? To address this, we begin by defining a general class of such problems and subsequently demonstrate how well-known instances can be derived from this framework. We then transition to the finite-dimensional setting, focusing on the approximations involved. As a result, we observe that widely used numerical techniques—such as the finite element and finite difference methods—are, in essence, variations of a common underlying approach rather than fundamentally distinct methods