Historical records show that approximately no more than three thousand years the study of mathematics evolved into two distinct groups of mathematical structures namely: the geometric and the algebraic structures. The first is predominantly deductive while the second is inductive. Deduction implies starting from the general mathematical truths then arriving at the specific truths while induction starting from the specific truths then reaching the general mathematical truths. Many mathematicians believe that a mathematical structure is geometric if it is characterized by a special class of subsets whose basic properties are described in the axioms of the system. On the other hand, a system is an algebraic structure if it is characterized by one or more operations whose basic properties are described in the axioms of the system.

Simultaneously, parallel developments in the physical sciences assert that every physical observation is a relationship between the observer and the physical phenomenon being observed. The observers never directly perceived but only see by changeable spacetime positions and measure by macroscopic measuring tools that do not provide a direct contact with the observed phenomenon. Each personal observation is the product of two factors. One factor is contributed by the phenomenon. The other is contributed by the special location of the observer which poses a fundamental problem for finding a way to eliminate this personal special position from the equations. However, it is a common practice of modern science to build structures upon mixed structures. The outcome is an interweaving of geometric and algebraic structures and a shuffling of methods that hope to reveal the essence of a subject being studied. It can be said that geometry and algebra are two complementary aspects of mathematical systems analogous to the particle-wave duality of the universe. Particle corresponds to geometry and wave corresponds to algebra.