Describes the interconnection between integro-differential equation nonlinearity, self-reflexivity, e.g., rocket self-propulsion or mechanical self-force, and finite-time division-by-zero singularity. Discusses the 300+ years general failure of modern mathematico-science to solve general nonlinear integro-differential equations in analytical closed form, due to its failure to discover the kinds of transcendental functions that can satisfy such nonlinear equations. Discusses the failure and non-closure of all of our standard arithmetics with respect to division by zero, and the spurious nature of infinity as a valuation for zero divisions in physically meaningful mathematical models, and the 'full zero axiom' for finite resolution of divisions by 'empty zero' in the non-standard, 'Mu' arithmetic. Discusses the 'psychohistorical' origins of "The Nonlinearity Barrier".