There are 2 contradictory views on our world, i.e., continuous or discrete, which results in that only partially reality of a thing T can be understood by one of continuous or discrete mathematics because of the universality of contradiction and the connection of things in the nature, just as the philosophical meaning in the fable of the blind men with an elephant. Holding on the reality of natural things motivates the combination of continuous mathematics with that of discrete, i.e., an envelope theory called mathematical combinatorics which extends classical mathematics over topological graphs because a thing is nothing else but a multiverse over a spacial structure of graphs with conservation laws hold on its vertices. Such a mathematical object is said to be an action flow. The main purpose of this report is to introduce the powerful role of action flows to mathematics with applications to physics, biology and other sciences, such as those of extended Banach or Hilbert G -flow spaces, geometry on action flows or non-solvable systems of solvable differential equations with global stability, · · ·, and with applications to, for examples, the understanding of particles, the spacetime and population biology. All of these makes it clear that holding on the reality of things by classical mathematics is only on the coherent behaviors of things for its homogenous without contradictions, but the mathematics over graphs G is applicable for contradictory systems over G because contradiction is universal in the eyes of human beings but not the nature of a thing itself.
Key Words: Graph, Banach space, Smarandache multispace, G -flow, observation, natural reality, non-solvable equation, mathematical combinatorics.
AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35.