DEVELOPMENT OF A NEW THEORY WITH GENERALITY TO UNIFY DIVERSE DISCIPLINES OF KNOWLEDGE AND CAPABILITY OF APPLICATIONS

Abstract

It is shown that ideas and methods of the axiomatic set theory, based upon ZFC axiom system can be used to establish a more powerful theory, named systems analysis. This new theory unifies different research interests, including topology and algebra, into one. It is not only a theory with its own meaning of existence, but also one with extremely interesting applications. As examples, the following recent applications are listed: (I) how to reduce the dimension of a system of ordinary differential equations to 1-dimension; (2) how Axiom of Regularity implies the existence of basic particles; (3) relation between Russell's paradox and the theory of science of science; (4) Δ-lemma and a new understanding of a social phenomenon; (5) why mathematics is so “unreasonably effective” in applications; (6) an impassable chasm is found between pure and applied mathematics; (7) an example, called vase puzzle, is given to show how different mathematical methods can lead to contradictory conclusions; (8) how Bellman's principle of optimality is incorrect and how a sufficient condition can be derived, under which the principle holds; (9) what impact the new theory has over some longstanding philosophical problems. In the last section, open questions are posed for future research. Seeking the answers to these questions will lead to both a more mature theory of systems analysis and more advanced applications in different areas, such as sociology, philosophy, epistemology, optimization theory, applied mathematics, material structure of learning, etc.