This document states the relevant basic assumptions of Mathematics that lead to some of the salient features of Abstract Mathematics. These features are then deduced from these assumptions in a fundamental way. Infinitesimals and Infinitesimal Numbers are introduced and then used in an example of a Riemann sum to create a contradiction that motivates the introduction of Leibnizian Mathematics as a model for Mathematics that is supplemental to Abstract Mathematics. Leibnizian Mathematics is then introduced by stating its basic assumptions. Lastly a list of the meanings of some words that are common to both models, but describe properties that differ between the models, is given. The reader is then referred to the document LEIBNIZIAN MATHEMATICS, posted on the link stated in the Comments below, where this model is developed.