Shane Keagan

Division by zero is excluded in mathematics to preserve algebraic consistency by equating to undefined. This exclusion leaves a metaphysically intuitive void in algebraic axioms that are typically handled by workaround structures such as limits in standard analysis, Nonstandard Analysis, or the extended reals. This paper explores the basis of undefined by reframing division by zero through a philosophical lens. We treat as an abstracted unit of place value such that division by zero is reframed as a scaling into infinity while preserving numerator ratios, yielding a satisfactory expression. By introducing a symbolic unit called positive boundary zero, distinct from 0, we can address axiomatic paradoxes from allowing zero-infinity reciprocity. This philosophical reframing accomplishes resolving certain paradoxes and exposes conceptual assumptions that algebra makes about zero, infinity, and undefinedness to shape itself. It does not aim to change arithmetic, but to propose a notational and ontological extension that integrates zero and infinity as complementary boundaries.