Mathematics on Non - Mathematics — A Combinatorial Contribution

A classical system of mathematics is homogenous without contradictions. But it is a little ambiguous for modern mathematics, for instance, the Smarandache geometry. Let F be a family of things such as those of particles or organizations. Then, how to hold its global behaviors or true face? Generally, F is not a mathematical system in usual unless a set, i.e., a system with contradictions. There are no mathematical subfields applicable. Indeed, the trend of mathematical developing in 20th century shows that a mathematical system is more concise, its conclusion is more extended, but farther to the true face for its abandoned more characters of things. This effect implies an important step should be taken for mathematical development, i.e., turn the way to extending non-mathematics in classical to mathematics, which also be provided with the philosophy. All of us know there always exists a universal connection between things in F . Thus there is an underlying structure, i.e., a vertex-edge labeled graph G for things in F . Such a labeled graph G is invariant accompanied with F . The main purpose of this paper is to survey how to extend classical mathematical non-systems, such as those of algebraic systems with contradictions, algebraic or differential equations with contradictions, geometries with contradictions, and generally, classical mathematics systems with contradictions to mathematics by the underlying structure G. All of these discussions show that a non-mathematics in classical is in fact a mathematics underlying a topological structure G, i.e., mathematical combinatorics, and contribute more to physics and other sciences.