There are 3 crises in the development of mathematics from its internal, and particularly, the 3th crisis extensively made it to be consistency in logic, which finally led to its more and more abstract, but getting away the reality of things. It should be noted that the original intention of mathematics is servicing other sciences to hold on the reality of things but today’s mathematics is no longer adequate for the needs of other sciences such as those of theoretical physics, complex system and network, cytology, biology and economy developments change rapidly as the time enters the 21st century. Whence, a new crisis appears in front of mathematicians, i.e., how to keep up mathematics with the developments of other sciences? I call it the 4th crisis of mathematics from the external, i.e., the original intention of mathematics because it is the main topic of human beings.

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 story 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 survey is to introduce the powerful role of action flows, or mathematics over graphs with applications to physics, biology and other sciences, such as those of G-solution of non- solvable algebraic or differential equations, Banach or Hilbert−!G -flow spaces with multiverse,multiverse on equations, · · · and with applications to complex systems, for examples, the understanding of particles, spacetime and biology. All of these make 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, i.e., complex systems because contradiction is universal only in eyes of human beings but not the nature of a thing itself.

A complex system S consists of m components, maybe inconsistence with m ≥ 2 such as those of self-adaptive systems, particularly the biological systems and usually, a system with contradictions, i.e., there are no a classical mathematical subfield applicable. Then, how can we hold its global and local behaviors or true face? All of us know that there always exists universal connections between things in the world, i.e., a topological graph −→G underlying parts in S. We can thereby establish mathematics over a graph family −→G1,−→G2, · · · for characterizing the dynamic behaviors of system S on the time t, i.e., complex flows. Formally, a complex flow −→G L is a topological graph −→G associated with a mapping L : (v, u) → L(v, u), 2 end-operators A+ vu : L(v, u) → LA + vu(v, u) and A+ uv : L(u, v) → LA + uv (u, v) on a Banach space B over a field F with L(v, u) = −L(u, v) and A+ vu(−L(v, u)) = −LA + vu(v, u) for ∀(v, u) ∈ E −→G holding with continuity equations dxv dt = u∈NG(v) LA + vu (v, u) , ∀v ∈ V −→G, where xv is a variable on vertex v for ∀v ∈ E −→G. Particularly, if dxv/dt = 0 for ∀v ∈ V −→G, such a complex flow −→G L is nothing else but an action flow or conservation flow. The main purpose of this lecture is to clarify the complex system with that of contradictory system and its importance to the reality of a thing T by extending Banach or Hilbert spaces to Banach or Hilbert continuity flow spaces over topological graphs −→G1,−→G2, · · · and establishing the global differential theory on complex flows, characterize the global dynamic behaviors of complex systems, particularly, complex networks independent on graphs, for instance the synchronization of complex systems by applying global differential on the complex flows −→G L .