12.10 Coloring Mixed Hypergraphs: natural number interpretation

The language of mixed hypergraphs allows us to interpret a concept which is hidden in the notion of a natural number. This could be described as follows.

Any concept in our mind that can be modelled by a mixed hypergraph H=(X,C,D) is a result of interaction between two families of opposite constraints expressed by C and D. If H is colorable, then we have a solution of this interaction. However, if H is uncolorable, then we have no solution, and the respective concept can be treated as ``self-contradictory". The ``more uncolorable" H is, the more self-contradictory the respective concept is.

In real life any object is identical only to itself. Even if we have object A and object B which look identical (say, two points), they, in fact, are different because they have distinct positions in space. Counting may be viewed as a sequential naming of objects which are in fact pairwise different (= of different colors), but by definition are considered pairwise identical (= of a common color). In other words, when counting, we roughly speaking don't add cars and birds together.

Let us count the cars A, B, C, D, E, ... . When we look at the car A, we implicitly denote it by the simbol 1. This is the initial condition for counting rather than counting itself. Next we want to add another car, B: it belongs to our set of cars, so it is of the same color as the car A; on the other hand, it is a different car, therefore it is of a different color compared to A. (Here the word "color" means "something the same" rather than real color of a car). We have a contradiction here which can be modelled by the smallest uncolorable mixed hypergraph ({A,B}, {{A,B}}, {{A,B}}). This contradiction is appearing and is "solved" by denoting the situation by the simbol 2. Next we want to add the car C. It is a car, and therefore it is of the same color as A and B. At the same time, it is different from A and B. So, it is of a different color compared to A and it is of a different color compared to B. We get new contradictions here and the entire situation can be modelled by the uncolorable mixed hypergraph ({A,B,C},{{A,B},{C,A},{C,B}},{{A,B},{C,A},{C,B}}). The new set of contradictions appeared. We "solve" all of them simply by denoting the situation by the symbol 3. In a similar way, adding cars D, E, ... we get new sets of contradictional reasonings, "solving" them by introducing (=denoting by) the symbols 4, 5, ... . Since mixed hypergraphs obtained are uncolorable, no colorings exist, and our "solutions" are simply reducing to notations. (This is not the unique case when we use notation in contradictory situations: e.g. infinity, irrational numbers, basically have similar nature).

Generally, if X is an arbitrary set of objects, then the counting of X corresponds to a step-by-step construction of a mixed hypergraph U_n=(X,{{X}\choose {2}},{{X}\choose {2}}). The concept of the natural number n=|X| arises as a result of n\choose 2 elementary contradictions (= ``minimal uncolorabilities") generated by pairwise comparisons. As we have seen in Chapter 3, among all uncolorable mixed hypergraphs on n vertices, U_n is the ``most uncolorable" since each of its non-trivial induced subhypergraphs is uncolorable as well. Moreover, U_n is the unique such mixed hypergraph for every n. This means that whatever concept based on comparison we would have on X, the concept of the natural number n is the MOST SELF-CONTRADICTORY. This fact explicitly points out on DIALECTICAL nature of the concept of natural number which is, most likely, the very ingenious product of human mind.

In fact, real counting begins when we start comparisons, i.e. with the number 2. Thus for each n, we have n-1 such steps which coincides with the vertex uncolorability number \Omega_X (U_n).