Mathematics of Existential Reality

This paper proposes a zero-free foundational framework called Existential Mathematics. I partition numerical entities into (i) Existence, representing physically existing quantities (positives); (ii) Anti-Existence, representing reversals of existence (negatives); and (iii) Nothingness, which corresponds to the symbol commonly denoted as zero. Nothingness is not admitted as a number and is excluded from arithmetic. I formalize an Existential Algebra in which addition is partial (annihilating pairs are not summed), multiplication is total, division requires a denominator in Existence or Anti-Existence, and factorial is defined only for positive integers (0! is undefined). I further outline a calculus without zero by treating zero as an annihilation boundary approached via infinitesimals from either side but never attained. Conceptually, this program aligns mathematics with physically interpretable states of being (existence and reversal of existence) and rejects calculations that require “nothingness” as an operand.