Hegel, "Phenomenology of Mind" (1807)
"EVERY SUFFICIENTLY STRONG AXIOMATIC THEORY IS EITHER INCOMPLETE OR INCONSISTENT"
Goedel, Incompleteness Theorem (1930)
NO FINITE SYSTEM OF STATEMENTS IS THE ULTIMATE TRUTH
It is clear that many of his ideas had roots in the past. Hegel considered that "The evolution of ideas occurs through a dialectical process - that is, a concept gives rise to its opposite, and as a result of this conflict, a third view, the synthesis, arises. The synthesis is at a higher level of truth than the first two views. The logic that governs this developmental process is dialectic."
Furthermore, the "dialectical method states that any movement, or process, or progress, is the result of the conflict of opposites. The thesis is an idea. Such an idea contains within itself incompleteness that gives rise to opposition, or an antithesis, a conflicting idea. As a result of the conflict a third point of view arises, a synthesis, which overcomes the conflict by reconciling at a higher level the truth contained in both the thesis and antithesis".
In 1979, I came to conclusion that classical coloring theory, based on the notion of edge, was incomplete and asymmetric. Therefore, using the notion of the conflict of thesis and anti-thesis, I first defined Mixed Hypergraph Coloring. In this language, the D-edge corresponds to a thesis, the C-edge corresponds to an antithesis, and the mixed hypergraph itself corresponds to a synthesis. One can say that philosophy of Hegel, especially the learning about contradictions, served as a source for the idea of mixed hypergraph.
Contradictons are everywhere and they are universal. For example, warmth and cold, light and darkness, positive and negative charges represent contradictory sides in the nature.
The process of natural selection in biology is based on contradictions. Two sexes - males and females interact as opposites. In this sense, the life of populations with sexual reproducton is pure dialectical process. If for a moment we accept the Hegel's point of view that there is no difference between reality and ideas, then anyone of us is an idea, a thesis; at some stage of our life we meet a conflicting idea, an antithesis, which is a person of the opposite sex whom we love. Our children represent a synthesis and this dialectical process is continuously repeated. The best match means that the respective ideas complement each other in the best way from the point of view of a system - human population. The population is an integral system having its superior goal and it is developing in space and time. The superior goal is to acheave the maximal lifetime = lifespace.
We can find the thesis and antithesis even in DOUBLED spirals of DNA molecules.
Politic parties in parlaments, competing companies on the market, represent contradictory sides in our society. Contradictions serve as base for any development. Soviet system failed for artificial lack of contradictions in the economic and politic life. In this sense, the FAILURE of such system was PREDICTED BY HEGEL more than 100 years before it appeared!
The two-party system in the USA works well and also proves that contradicitons between two parties provide the base for further development. We witness this every day. Generally, our history is the history of contradictions; wars are the worst "best" examples . Love and hatred, talent and stupidity, positive and negative numbers, even the hell and paradise represent contradictory sides in our thinking.
Contradicitons determine the development of mathematics. For example, one of the most famous theorem of 20th century, Godel's Incompleteness Theorem (1931)explicitely states that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. This eternal contradiction dooms us to dialectic in our search for truth. It completely fits into the scheme "thesis-antithesis-synthesis". It also can be regarded as mathematical interpretation of the famous Hegel's statement "THE TRUTH IS THE WHOLE" (1807) .
In a similar way, as many other opposite to each other things, the C-edges and D-edges interact, and this interaction brings a lot of new properties and qualities.
From the very beginning of coloring theory in 1852, it took about 150 years to get to this model.
Dialectical method states that new quality appears as the result of conflict between opposites. The contradictions between the two principles "to be the same" and "to be different" represent the basic contradictions in Combinatorics. The monograph provides a proof and shows the way how this method explicitly works.
For example, as a result of contradictions, the following new properties=qualities of colorings have been discovered since the introduction of mixed hypergraphs (in fact, at the very beginning there was a firm general prediction that we will find something NEW OF PRINCIPLE):
The mixed hypergraph coloring theory directly contributes to the mathematical comprehension of relationship between the two categories of contemporary philosophy: IDENTITY and DIFFERENCE.
Many philosophers, mathematician and not only, studied the relation between these categories.
For example, in [1, page 470, (italian translation)] Hegel describes a story about Leibnitz: in the parc of a castle, in order to prove that there are no two things that are identical, Leibnitz made ladies to look for two identical leaves among all the leaves on trees.
Another example, Alfred Nobel wrote that all science is built on observations of similarities and differences. He continued: "A chemical analysis is of course nothing other than this, and even mathematics has no other foundation. History is a picture of past similarities and differences; geography shows the differences in the earth's surface; geology, similarities and differences in the earth's formation, from which we deduce the course of its transformations. Astronomy is the study of similarities and differences between celestial bodies; physics, a study of similarities and differences that arise from the attraction and motive functions of matter. The only exception to this rule is religious doctrine, but even this rests on the similar gullibility of most people. Even metaphysics - if it is not too insane - must find support for its hypotheses in some kind of analogy. One can state, without exaggeration, that the observation of and the search for similarities and differences are the basis of all human knowledge."
All mathematicians work entirely in this area since every theorem, each formula (=equality) or any inequality contribute to this relation in universal cognition process. There is no more important and used sign in Mathematics other than "=". In this sense, ANY MATHEMATICIAN IS A PHILOSOPHER.
It is interesting to find an interpretation of mixed hypergraph coloring as a philosophical model. For example, as it is described in section 12.10 , this theory explicitly points out on the contradictory, dialectical nature of the concept of natural number which is the base of all Mathematics. Moreover, it shows for the first time that this concept is the MOST SELF-CONTRADICTORY concept that can be thought of using two philosophical categories: IDENTITY and DIFFERENCE.
Another observation is that Uncolorability may be treated as "thing-in-itself" and in this way we find the explicit indication on Immanuel Kant's philosophy. One more question concerns the interaction between formal and dialectical logic. By introducing C-edges, we explicitly introduce contradictions in set partitions, but we study them using traditional formal logic. Isn't it the first case of such kind?
However, main expectation is that the theory will allow to find new fundamental mathematical, namely, combinatorial relations in genetics on the level of molecular biology. Life is the property of materia to maintain the rest, i.e. to strive to be the same while everything is changing, i.e. is different.
Our HEREDITY is based on comparison and, being DISCRETE and FINITE, roughly speaking, begins with "having at least two the SAME features"="having at least two vertices of the SAME color". At this stage it generally looks like the meaning of existance (= supreme goal) of ANY finite discrete living system is to achieve the maximum lifetime, what, in the language of mixed hypergraph coloring, is the upper chromatic number. Thus the meaning of all our activity is a function of what is combinatorially recorded in our chromosomes. Many new questions arise and many new general conclusions are expected in the near future in this direction.
In this way, the idea of mixed hypergraph coloring, generated by the philosophy of Hegel, will be back to philosophy.