This book is a description of the examples and ideas that trace the progress of the author’s view of Mathematics from being subjected to the Scientific Method in division three, to being a technique of Ontology in section one. Section three was posted on the link http://vixra.org/abs/1501.0153 on VIXRA in 2015. The scientific Method requires that any model should be as near as possible to perceived reality. PART ONE of DIVISION THREE is concerned with the concept of “infinite decimal fraction” as used by Cantor in his well-known diagonal proof. This concept is contrasted to the same concept in perceived reality by exploring the differences between Cantor’s arguments and perceived reality, resulting in a rejection of the validity of Cantor’s proof. Next an analyses of the discrepancy between the cases where a line is assumed to be a string of points, and thus more than countable many points are required so that their lengths can add up to one, and the case where the points are limits of partitions of the interval and the lengths of only countable many points are required to add up to one. The standard number system is then extended to the Cauchy Numbers which comprises infinitesimal numbers, rated numbers and infinite numbers. Infinitesimals are also defined. The concept of cascades of differentials is introduced and the fundamental theorem of Calculus is derived for this context. It is then pointed out that the approach of Parmenides of Elea is in line with the system in which Cantor’s argument is invalid. The implications for Physics and the possibility of uniting the two approaches to the divisibility of Space is investigated. In DIVISION TWO the dichotomy, or discrepancy, studied in DIVISION THREE is further developed resulting in the introduction of the Euclidean Cosmology and the Leibnizian Cosmology. Leibnizian Cosmology required a closer look at the process of using numerals for depicting irrational numbers. This then resulted in a closer look at the concept of “infinity” in these two cosmologies. In DIVISION ONE The results of DIVISION TWO are refined to define an alternative paradigm to Abstract Mathematics. It is a transitional step towards the document “LEIBNIZIAN MATHEMATICS” that includes further insights of the author, and which extends the current paradigm of Mathematics to include Leibnizian Mathematics as complementary, and not an alternative, to Euclidean Mathematics.