1.Introduction

We are used to hear the word ”problem” in many various connections and it usually stands for an object (a part the objective reality) that appears to be in an undesirable state. This study introduces a general definition of problem, some of attributes of problems, both man-independent and man-dependent, and explains the exact validity of so called principle of problem subjectivity.

That a state of an object is regarded as undesirable does not mean that any other state is desirable - defining the object and its state (the undesirable one) is not a complete problem specification. We need to define the desirable state as well. Therefore we employ ordered couple of states of object as the general definition of problem. Some illustrative examples of problems can be found in Examples of Problems page.

The study consists of two logical parts: chapters two - problem algebra - deals with problems as abstract mathematical objects, there are lot of vectors, multidimensional phase spaces, trajectories in phase spaces, operations with problems and other staff. The chapter three - biology of problems - considers that the abstract problems from the chapter two can get incarnations in human brain, the model of pendulum dissatisfaction/contentment mechanism is introduced and finally the issue wouldn’t be complete without discussion of principle of problem subjectivity.

The primary application of the principle of problem subjectivity lies in the field of psychology (chapter four). In the future the principle could help with exact and fast solving of some types of the functional psychosis. Other attractive applications could lie in the computer science, neuron networks or artificial intelligence.

The version history was placed at the end of the issue to remind about stable history of principle of problem subjectivity. The basic definitions and assertions presented in this issue have not changed since 1994.

The principle of problem subjectivity is based on modern natural sciences widely accepted today - it uses terms of the set theory (see e.g. Ref.[1]), the theory of multidimensional functions, the theory of graphs (see e.g. Ref.[2])  (mathematics), refers to the atomic theory or uses examples from mechanics (physics, see e.g. Ref.[3]) and finally relies on knowledge about neurons and the central nerve system gained in biology and biochemistry (Ref.[4]). Since the study deals with a very complicated and disputable field it uses lots of simplifications that only help to be more exact and are believed not to be substantial for the deduced results.

To build up the system of problems, the basic term we will start from is ”objective reality”. The term ”objective reality” stands for the entire surrounding world and, as the basic term, cannot be defined but must be understood intuitively. Lets look at the general definition of object:

Definition 1 (object): Object is a defined part of the objective reality.

Two basic means commonly used for defining part of the objective reality are:

Examples of objects are: an electron, a proton, a neutron, a hydrogen atom, several electrons, a molecule of H₂O, a protein macromolecule, a red blood cell, a part of our body, our body, the air in this room, the photons (electromagnetic light radiation) in this room, this building, the 6-th planet in our solar system, all stars in the Galaxy, etc. (From the physical point of view, every object is composed of elementary particles - mostly protons, electrons and neutrons - and is carrying its energy).

Logical Operations with Objects:

We will use capital letters X, Y ,Z, ... as a substitute for well-defined objects. When thinking, we are logically (virtually) splitting objects into smaller ones, and consider these smaller objects as independent systems (with the rest of world reduced to boundary conditions). To describe this we will introduce logical composing/decomposing of objects.

Definition 2 (decomposition of object): Decomposition of object Z will be called such objects X and Y that every part of objects X and Y is also part of the object Z. The sign È (unification) will be used for composing of object Z = X È Y.

Note1: We use the term "logical operations" for composition / decomposition of objects and problems since these operations are not physical - nothing of the objective reality changes when we decompose an object - it is only a virtual (logical) decomposition, serving for later operations with problems.

A special case of object according to the definition 1 is the whole objective reality itself, lets mark it X₀. We will introduce the definition of "complete disjunctive set of objects" that will later serve for construction of the set of all problems.

Definition 4 (Complete Disjunctive Set of Objects): As "complete disjunctive set of objects" O will be called such set of disjunctive objects, that their composition is the whole objective reality: O ={X_i ; X_i Ç X_j=ø  for all i,j   and ÈX_i = X₀}

The center of gravity of every object is defined by its position vector (x,y,z) and its momentum by vector ===( p_x, p_y, p_z) where m is the mass of the object and stands for time derivation. Not taking into account other states of freedom the state of the object is exactly described by 6 numbers x, y, z, p_x, p_y, p_z . These numbers can be understand as components of a vector in a 6 dimensional space, usually denoted to as phase space where just the values of x, y, z, p_x, p_y, p_z are marked on individual axes. Any status of the selected object within given approximation corresponds to just one point inside the 6 dimensional phase space.

Figure 2: Description of object state in the phase space

Definition 6 (problem): As problem will be called every ordered couple of states of an object, in symbols

The square brackets on the right side of the definition stand for an ordered couple while on the left side they represent the symbolic notation used for problems. Let’s look at the phase space what is the representation of problem: instead of one point representing single state of object, problem is represented by two points in addition with defined order (Figure 3).

Figure 3: Representation of problem in phase space

This means that whenever we hear or think the word "problem" we can always find a well-defined part of the objective reality to which the problem is related. That part of the objective reality is called "problem object". There are two states of the problem object that determine the problem itself - the initial state - which is the unfavorable and often the ccurrent state  - and the final state being the favorable one, and  usually we would like to achieve that state in the future. Please have a look at Examples of Problems to understand more easily what the definition of "problem" as well as the following definitions mean.

Definition 7 (zero problem): As zero problem will be called every such problem X[,] that = (it is a_i = b_i for i = 1,2,...,n ), marked 0

Definition 8 (inverse problem): As problem inverse to problem X[,] will be called problem X[,]

Figure 4: Serial and parallel compositions of problems

Let’s look what the definitions 7,8,9 and 10 mean for the phase space of problem object. For the zero problem the points representing the initial and the final state are identical. The inverse problem is represented by the same couple of points as the original problem but understand in the opposite order. For inverse problem the state change vector = - (since= -= -(- )= -). The parallel composition of problems lead to transformation of phase spaces used for description - the original phase spaces X: n-dimensional with axes x₁, x₂, ... x_n and Y: m-dimensional with axes y₁, y₂, ... y_m transform into one n+m dimensional space with axes x₁, x₂, ... x_n, y₁, y₂, ... y_m.

According to the current knowledge, the total matter and energy of the universe is final. Therefore the complete disjunctive system of objects (in the sense of definition 4) contains a final number of objects. Let's mark this final set of objects O, each elementary particle in the universe belongs to just one element of O.

Figure 5: The set of all objects

Figure 6: Phase spaces, description of states of objects

If we have selected a different splitting of universe into objects we would get a formally different "object representation" i.e. different coordination system. Different values of object state coordinates in the phase space would represent the same objective reality. These transformations among object representations will not be discussed in details in this version of the issue.

Let's advance to the construction of the set of all problems P₀ (P-zero) . According to the general definition of problem (definition 6) every ordered couple of points in the phase spaces on figure 6 represents one problem. When we consider all the possible ordered couples of points in all the phase spaces on figure 6 we get the set of all problems P₀. Every problem we have ever solved or are ever going to solve has its mathematical counterpart in P₀. The logical steps that lead to the construction of the set of all problems are shown on figure 7.

Figure 7: Logical steps leading to the construction of the set of all problems P₀

We can try to understand approximately the size of the set of all problems P₀. We are going to use the sign || (norm) to specify size of a set i.e. number of elements. Each of ||O|| objects can be considered in states (average number of points in the phase spaces, S stands for phase space considered as set). Therefore the size of P₀ is:

Finally in this paragraph let's introduce the concept of actual and virtual problems. On the figure 6, there is just one special point in each of the phase spaces that represents the current (actual) state of the object X the phase space is attached to. Let's mark the current (actual) state vector as _0X (a-zero-X) for each object X.

Definition 11(actual problems): As "actual problem" will be called such problem X[,] that =_0X , i.e. the problem initial state is the current object state.

 Both actual and virtual problems are important from the point of view of thinking - we can be troubled and searching solutions also of the virtual problems. In addition virtual problems may later become actual as states of objects are changing. However only actual problems can be solved at the present moment.

Mathematical representation of solutions of problems is not yet included in this issue. Basically solution of problem X[,] is an external force affection that induces the object X to change its state from X() to X(). It is clear that such change occurs always along a continuous curve in the phase space. The curve's boundary points are and .

All the definitions and system of problems defined in previous chapters are pure theory - objects exist only in just one state at any time. If no thinking being existed the system of problems could still be defined, all the relations are mathematically valid but individual problems would have no real counterparts. Let’s have a look at what real-measurable objects would correspond to abstract ”problems” when we admit the existence of thinking beings.

We are surely able to notice and/or imagine an object in a state (e.g. a book lying open on our table). Certainly there is a specific activity of our brain behind such imagination. For simplicity let’s consider a group of neurons connected in a ”circuit” that is ”active” - nerve pulse is running periodically through the active neuron circuit (see e.g. Ref.[4] for a related issue). Please observe that such activity of the group of neurons is real-measurable phenomenon, the nerve pulse is carrying its energy. When a state of an object is considered not suitable we usually search for a more suitable state of the object. The imagination of the more suitable state may exist almost simultaneously with the original one and the couple of groups of active neurons will for our purposes represent the real-measurable counterpart of the abstract ”problem”. This specific activity of neurons will be further on called ”reflection” or ”incarnation” of problem in human brain.

Let’s have a certain time interval T. For simple explanation let’s use the value T = 3 minutes. We will investigate our ability to solve problems within the time interval T - changing states of objects from one state (the initial one) to another state (the final one). For sure there are some problems we are able to solve within 3 minutes - as opening the book lying on our table, going into the next room and many others. On the other hand there are problems surely not soluble within 3 minutes - e.g. moving 1000 km away. Every attempt of solving of problem X[,] within time T will always have an unambiguous result - either the problem will be solved within 3 minutes or not. There is nothing between. The ability to solve a problem is a man dependent property of problem (the value depends both on problem and on the person and is objectively measurable).

Let’s consider a function S_T, man dependent and defined on the set of all problems by the formula:

S_T(X[,]) =

0 if X[,] is not soluble for within the period T

1 if X[,] is soluble within the period T

The set of all problems soluble within interval T can be defined as

If we prolong the interval T, more and more problems will belong to the set S_T , finally the set of all for man soluble problems is defined as

and the corresponding function S defined on the set of all problems as will be called ”solubility for man”.

Of course we have got the ability of learning (and forgetting). Therefore the set of all for man soluble problems is time dependent:

Let’s mark the set of all at moment t for man insoluble problems as N(t):

Of course, superset of S(t) and N(t) is P at any moment t and their intersection is empty (the following assertions are valid):

Many people would agree that the most important value in human life is contentment. Original reasons for our activity can be diverse - from earning or having money, doing something for ones health, satisfying basic needs as hunger, being appreciated by another person, being in love with a person, sexual goals, goals in sport, ... etc. When an obstacle appears to the goal, dissatisfaction appears and it is the fuel for all our further activity. Without the dissatisfaction we wouldn’t be able to reach anything. After overcoming the obstacle the dissatisfaction disappears and we are again content. In a first approximation our activity is propelled by dissatisfaction and the imprinted tendency to reach contentment. Let’s have a more scientific look at the pendulum mechanism of dissatisfaction and contentment.

Nowadays science has got great achievements in the study of individual neurons. Neuron is nerve cell, having all the general properties of cells - it has its nucleus, external membranes and cell organelles. The specialty of neurons is their form and the ability to lead and integrate nerve pulses. Nerve pulses consist of flow of sodium and calcium ions (Na⁺, K⁺) through neuron membranes. There is quite sophisticated mechanism of regulation of the flow. The short projections called dendrites carry nerve pulses into neuron, after processed (integrated) in neuron body the nerve pulses are carried by the only long projection called axon out of the neuron toward other neuron(s). Both dendrites and axon are plentifully split at their ends. The connection between axon of one neuron and dendrite of another one is called synapse. Human brain contains about 15 milliards (15.10⁹) neurons and much more synapses.

In addition to the above-described neuron-level activity that in some parts of central nerve system probably correspond to individual imaginations, ideas and logical thinking, there is the higher-than-neuron-level activity. It has a more global character and consists of changing concentrations of diverse chemical compounds in various parts of the system [human brain], various electric and magnetic field changes. This higher-than-neuron-level (called further on as 2nd level) activity corresponds to emotions and feelings rather than individual imaginations and ideas, the most famous representative is the concentration of chemical compound adrenaline.

Imagine that we have devices able to measure (of course in a non-destructive way) all the characteristics of the level one and two of in all parts of the system. After averaging the measurements from lots of people it could be possible to define such emotions as fear, anxiety, nerves, fright as well as positive emotions, all based on physical, chemical and biological quantities (that on such occasions appear in our brain) and with exact mathematical definitions.

Most of time brain exists in its normal state corresponding to contentment, concentration of chemical compounds and electromagnetic activities have their standard values. For describing the level of discontentment we will introduce global state function (denoted to as GS function) by the following description:

Description 1 (Global state function): The global state function is biochemical quantity describing dissatisfaction objectively measurable in central nerve system. The values of GS are H(GS) = R⁺₀ (the positive real numbers plus zero), the value GS = 0 corresponds to normal values found in the status of contentment, higher values of GS correspond to higher dissatisfaction.

To exactly define the GS function we would need the measurement devices mentioned above to work out what is the role of individual compounds and measurable quantities in the dissatisfaction as we feel it. The reason why the description 1 was introduced is to remind of the matter and objective base of the following assertions.

The time dependency of the GS value is quite complex, there is number of things that effect human contentment -both status of external objects and internal neuron interconnections. An example of GS time development is in figure 9. 

Figure 9: GS (Global state) function time dependency example, at moments t_A and t_B serious obstacles (problems) appeared

When an object X changes its state to X() this fact may have an influence to the global state of our brain. This influence is objectively measurable. For example we have bought a new jacket and we have torn it up of a fence. This causes our dissatisfaction, global state of brain has changed, e.g. the adrenaline concentration can grow. On the other hand for example a fall of a leave from a tree lets our brain status unchanged in the state of contentment (GS = 0). The global state function GS described and discussed in the previous chapter can be understand as a function of states of object GS = GS(X()) defined on the set of all states of all objects and expressing the immediate reaction of human brain to object X appearing in state . This dependence will be below referred to as ”disadvantage of state X()” being objectively measurable and man-dependent biochemical quantity.

To progress further in the following chapters we will need to go little deeper - to the neuron level - and create a basic model of how our discontentment can be connected to the status of an external object. The model is very simple and with a good agreement with modern scientific knowledge about neurons and human brain.

We are going to assume that there is a specialized part of human brain that takes care of generating negative emotions and discontentment. This region of human brain will be further on called ”center of negative emotions”, CNE. Individual neuron groups that perform imaginations of objects in various states are connected more or less strongly, directly or indirectly to the center (CNE) using neuron based ”conductors”.

On the figure 11, if the object Y appears in the state it will bring a higher discontentment than if the object X appears in the state , since the neurons that realize the imagination of Y() are connected to CNE more strongly (the synapses involved have got lower resistance) then those realizing X(). GS(Y()) > GS(X()), disadvantage of state Y() is greater than disadvantage of state X().

It is fundamental to realize that the connections between every region realizing the imagination of an object being in a state and the CNE was generated during the process of thinking, may vary from person to person and the natural limits are as follows:

(*) GS_min (X()) = 0 -any synaptic connection between arbitrary couple of neurons has got the

ability to relax (completely disconnect)

evolve to maximum strength

Any disadvantage of state GS(X()) greater than GS_min (X()) is not necessary - it was created in our past by our own ”subjective” thinking. Therefore we will call the minimal necessary importance of problem X[,] the quantity

as the objective (objectively necessary) importance.

After using the equation (*) within (***) we get the general principle of problem subjectivity (GPPS) assertion:

”The importance of arbitrary problem is only subjective (the objectively necessary importance is zero)”.

The general principle of problem subjectivity de facto says that there exists an objective possibility (guaranteed by properties of synaptic connections of neurons) that our brain can stay in the basic state of contentment even if arbitrary object appears in arbitrary state.

The general principle of problem subjectivity even if it brings really objective measurable information has no practical significance but serves only as the base for the contracted principle of problem subjectivity.

In the last chapter we have discussed the general principle of problem subjectivity in the form

Figure 12: A - an arbitrary subset of P

The general principle of problem subjectivity is just one special case A = P among all the assertions about importance. All assertions about importance have been proved by the fact that the general principle of problem subjectivity is valid:

As it was mentioned in previous chapter the GPPS -even if it is exactly valid - is of no practical significance. If we regarded every problem that appears as important only subjectively (according to the GPPS) our entire dissatisfaction would increase - mostly it is of better advantage simply to solve the problem.

Note: Due to complexity it is recommended on the first reading to skip the following paragraphs and continue from "Assertion (contracted principle of problem subjectivity)".

1) Appearing of a problem [in our brain] referred here as ”activation of problem X[,]” has got the same probability for all problems from P

2) Only 1 (or 0) problem can be activated at any time

We will assume the following:

To explain this rather complex assumption 3 that directly leads to the contracted principle of problem subjectivity we will analyze GS = GS(t) function presented already on figure 9 now shown with more details on figure 13.

Figure 18: Subsets of P considered for the minimization of

where

P ...is the set of all problems

m ...the number of elements of P (here assumed final) = || P ||

s ...the number of elements of S (here assumed final) = || S ||

n ...the number of elements of N (here assumed final) = || N ||

To test the constructed expression let’s investigate the dependence =() . The fraction is portion of soluble problems among all problems. We will consider all variables but fixed and since s + n = m the substitution n = m - s will be involved:

The partial derivation by provides:

Since the derivation is negative the interpretation of the dependence =() is as follows: the higher fraction of problems we are able to solve the lower is the average contribution of randomly selected problem to our discontentment. This is the result that was expected.

Again we will start from the initial expression for derived from the table above:

We can have several word forms for the contracted principle of problem subjectivity (PPS):

-Any problem is either soluble [for man] at moment t or important only subjectively at t.

-If we are not able to solve a problem X[,] at moment t it only seems to be important at t.

The contracted principle of problem subjectivity (PPS) brings a relation between two objectively measurable properties of problems - the solubility for man (see 3.2 Solubility for Man, Sets of Problems) and the importance of problem (see 3.5 Importance of Problems). Since the solubility for man is time dependent quantity the PPS has got differential character in the time scale, it is related to one moment t.

The discussed assertion relies on the general principle of problem subjectivity (see chapter 3.6 General Principle of Problem Subjectivity) that in turn relies on some properties of neurons known from biology, mainly the basic ability of every neuron connections to relax (”disconnect”). The PPS is a special assertion about importance (special ”subset” of the GPPS) that leads to minimal integral or average discontentment (mathematically represented by integral or average value of the GS function). A prove of the special position of PPS has been brought out in the chapter 3.7 Contracted Principle of Problem Subjectivity. However it currently relies on rather complex assumptions, these should be simplified in future versions of this issue.

Since we believe that fundamental properties of neurons are stable and fixed the future validity of PPS is guaranteed. There is no contradiction in the validity of PPS as may seem during first reading (remember that it is bound to one moment t). Let’s have a look at some seeming contradictions:

What may somebody imagine after first reading of GPPS or PPS is the question: ”What if I would be endangered - this is of course an important problem and I may not be able to solve it. There is a contradiction in these assertions ...”. Lets come to the exact handling of such problems slowly.

After some practice with using the PPS we can virtually get closer to our ”abstract unavoidable death” finally we are able to imagine ourselves very close (for example 3 seconds) before our death. By the way that moment will not differ too much from presence in the sense that we will not be able to have more than normal contentment. Unless we will be able to avoid that death it needn’t be important to us - we’ll be able to be content.

The correct processing of our ”abstract unavoidable death” is just one example of usage of the contracted principle of problem subjectivity. Of course there are problems that may bother us more but this one was selected to remind of the fact that there is no contradiction in PPS when we investigate its validity even in the time limits near our death. The rather surprising result that even our death may become unimportant for us just few seconds before it occurs (in the case that we are not able to avoid it) is caused by assumptions we did at the beginning of the chapter 3.3 Global State Function that the global state of our brain at the present moment is the basic value all the other values are bound to using ”neuron-based conductors” with the relaxing ability.

Another interesting feature implied by the validity of PPS is dynamics of importance: since the solubility for man changes along time so does the importance of problems. Let's illustrate this on an example: it is very important to avoid car accident - disadvantage of the state "having ccar accident" is very high. We are trying to prevent it by all means, e.g. careful driving, complying all rules on roads, etc. However at the moment when we have one, "not having car accident" is no longer possible - the problem ["having car accident&qquot;, "not having car accident"] is not soluble. At the moment PPS asserts that "having car accident" is objectively unimportant (only seems to be important). The more quickly the disadvantage of the state "having car accident" becomes zero the better for us and for all the people around. When the consequences of the accident straighten it is logical that the disadvantage of the state "having car accident" becomes high again. The importance has been changing its value from high to zero to high again.

Since the principle of problem subjectivity is a fundamental piece of knowledge we will devote yet another brief chapter (this one) to exact understanding of its consequences. We will utilize two time-views for the discussion: the differential time-view of PPS and the sequential time-view of PPS.

The following diagram shows a standard cycle of the problem processing that in fact more or less consciously takes place in our brain - we are identifying problems, solving them or forgetting if they are not soluble. The part in the dashed line is not our job - problems are instantiated automatically. Nobody exactly knows what problem is going to arise tomorrow, next week and so on. The more precisely and more swiftly we are used to do the individual steps in the diagram including recognition of insoluble problems and solving soluble ones, the better for us (in terms of our discontentment).

Figure 21: Model of problem processing

This chapter is included in order to give a hint of what the principle of problem subjectivity can be good for. If we are talking about people suffering psychical illness in this chapter we will mean people who were quite normal having no physical brain injuries, who have however become psychically ill e.g. with schizophrenia, depressions, .... For an overview in this area the following references are recommended: Ref[5], Ref[6], Ref[7] and Ref[8]. Assisted learning of the principle of problem principle of problem subjectivity could be considered as kind of modern psychotherapy, aimed at complete recovery and   long-term immunity against emotional instability problems, while treatment with antipsychotics and other medications may still be neccesary from short term point of view, as an immediate relieve for the patient.

Though most of the time we do not notice it our psychical stability and contentment is bared by two basic pillars: the ability to solve problems and the ability to forget problems. It is hard to remember how many problems we have dismissed during the past 12 hours (since it cannot be remembered what was forgotten). A scheme of such ”two pillar” model is shown in the figure 22.

Figure 22: ”Two pillar” model of psychical stability

If either of the pillars is lowered our contentment will decrease. It is also not true that the more problems we are able to solve the more content we are.

To explain what is happening when the psychical illness is developing several steps will be presented. In the reality these steps can be interwoven:

- a problem or problems have instantiated that the person is not able to solve and are regarded as important

- the problem has been instantiated for long time and the connection to CNE is becoming stronger by positive potenciation of the synapses involved

- after overcoming certain threshold of time during which the problem is neither solved nor forgotten, the connection to CNE becomes yet stronger, other insoluble problems reminds the original problem and also cause discontentment

 - during the next period a number of other problems evolve a strong connection to CNE. Activation of any of these problems leads to high discontentment and further fixing of the connection to CNE by positive potenciation of the neurons involved

- the result is that the ability to forget problems is significantly weakened and the person is exposed to long term and very high discontentment

 - finally dealing all the time with insoluble problems leads to decreasing the ability to solve problems, even those everyday life depends on

 The long term exposition (weeks, months) to discontentment is big conflict since human brain has got the imprinted tendency to get rid of discontentment. There is number of external expression of this conflict: serious problems to sleep, to work, depressions, schizophrenia, aggression to other people or to the person himself/herself. Total destruction of personality can be the result if the psychical illness is not treated by medical care. Even forgetting of the original problem so that it finally becomes unimportant doesn’t prevent other ”secondary” problems and future problems to become cause of high and long term discontentment and continuation / regression of the illness.

Due to complexity of the whole system (human brain), it is possible to describe time changes of the global state function representing discontentment using analytical functions only under very simplified preconditions. The example of an analytical model below depicts existence of the threshold in the average discontentment (): under the threshold in the average discontentment tends to drop down to zero, above the threshold it tends to grow above all limits - corresponding to the state of psychical illness.

Let's assume that one problem is instantiated (appears in somebody's brain) at time t_i and has got assigned the initial discontentment GS_i.

Let's assume that the discontentment caused by the problem goes down exponentially with time:

                                                                                                                                                                  (A1)

The parameter K_DD will be called "coefficient of discontentment dropping" and describes how quickly the discontentment GS goes down, the higher K_DD the faster GS drops.  If K_DD=0, the discontentment will not drop at all. If K_DD goes to infinity,, the discontentment will go to zero extremely fast. After a time period during which the problem is active, the problem is either solved or forgotten. In both cases at moment the discontentment drops to zero.

The contribution of this problem to the total discontentment (=GS integral value corresponding to the area under the graph) is:

                  (A2)

The average discontentment experienced during the time interval when the discussed problem was active is

In reality, not only one problem, but many problems are activated in our brain, one after another. In order to simplify the analytical description we will assume that all problems will have assigned the same initial discontentment GS_i and the same duration :

The only parameter that may vary in this model is the coefficient of discontentment dropping K_DD. In real life, our ability to forget problems (modeled here by K_DD) depends (besides others) on our history - e.g. how many and what problems or stress situations we were exposed to. We could model this for example by having K_DD indirectly proportional to average of the historical discontentment :

, where K₀ is a constant.

Notice that this way the higher has been the average discontentment in the past, the lower is the K_DD value (and ability to forget problems). As it would be difficult to calculate average over a large number of problems, we will yet simplify this by making K_DD depend only on average discontentment from the previous problem:

                                                                                                                                                                                (A4)

Now we will investigate difference . If it is positive the discontentment tends to grow, if negative tends to drop, if D=0 it is the threshold mentioned above.

             Substitution

The function looks like the following in dependence on :

                                                                                                                                         (A5)

So in this simple model, if the average discontentment is lower than the threshold (A5), it tends to go to zero (and K_DD tends to go to infinite). If is greater than the threshold (A5), it tends to increase above all limits (and K_DD tends to go to zero) - this corresponds to the state of psychical illness.

Fortunately we are not quite helpless when facing psychical illness. In addition to number of various therapies that more or less convincingly work nowadays medicine offers number of drugs called antipsychotics that efficiently help people who are facing such problems. The principle of its functionality is a partial blocking of the neurotransmitters e.g. dopamine at the synaptic connections of neurons:

Of course there are drawbacks in this ”hardware” approach, e.g.:

- the drug influence is not ”selective” -even good knowledge of thus cured people becomes dimmed or lost

- there is high probability that the illness will regress after the drug supply is stopped

As it was described in chapter 4.2 the psychical illness is developed by pure thinking. Since the properties of neurons that take part in the process are reversible it seems to be highly logical and expected that there should be a pure thinking way to return from the state of psychical illness (when big number of insoluble problems is strongly connected to human discontentment) back to normal.

The way of course exists and it is the exact problem discrimination according with the principle of problem subjectivity  (PPS):

-the solubility of problem should be investigated / guessed for each problem (the question "Am I able to solve this ?")

-it should be recalled as quickly as possible for every problem that is not soluble for sure that it is unimportant (important only seemingly) at the present moment

-soluble problems on the other hand should be consistently and with all responsibility solved

A more exact scheme is presented in the chapter 3.9 Model of Problem Processing.

There could be a few potential difficulties in adopting the logic of PPS:

1) PPS vs. emotions conflict - it may happen that there is disagreement between what PPS asserts and what emotions are suggesting about importance of a problem. In such case it should be clarified that PPS assertion is right while emotions are subjective.

2) Modification of thinking patterns - adopting PPS may mean little modification to number of thinking patterns that has been adopted in the past. Example: A quite usual life concept is "It is important to be neatly dressed". Unfortunately it is not exact enough for patient suffering psychical illness. PPS modifies this concept to "It is important to be neatly dressed as long as one is able to be neatly dressed". The modified concept contains correct handling of the situation when one is not able to be neatly dressed (because then it is unimportant). It may be very difficult to change what we are once used to.

3) High level of abstraction - it requires high level of abstract thinking to identify the vast variety of things that can trouble us (ranging from feelings possibly connected to health, through interpersonal relations to tasks in occupation, etc.) as problems.

As it will be shown the seeming simplicity of principle of problem subjectivity disappears when hundreds or thousands of problems are being incarnated. To depict the usage of PPS in the time scale let’s have a certain period of time, e.g.T=5 minutes and let’s presume that during this period 6 problems were incarnated. In a basic model any of these problems can be considered as important with an attempt to solve the problem (+) or as unimportant without an attempt to solve it (-).

In this model there is 2⁶ = 64 ways how to handle these six problems. It is an extreme maze that is posed by nature on us - the actual rate of problems coming to our mind can be tens during several minutes. A hundred problems give 2¹⁰⁰ = 1,267 . 10³⁰ (a number with thirty zeros) ways how to handle them. The principle of problem subjectivity defines just one among them - the thick line example in the picture. With a small exaggeration we can say that going out of the proper way may lead back to the relapse of the psychical illness for somebody who is facing it.

The initial simplicity finally fades out when we discuss the abstraction that is necessary for correct understanding of the PPS logic. Let’s assume that we are troubled by a problem X[,] and we are not sure whether we are able to solve it. Then it should be regarded as ”it only seems to be important”. However it may happen that we are not able to get rid of thinking about X[,]. Then ”thinking about X[,]” rather than the problem X[,] itself should be understand as the problem we have. We can try some tricks how to stop the thinking about X[,] - thinking about something else, counting sheep and so on. If it doesn’t help then we are not able to ”get rid of thinking about X[,]” - according to the principle of problem suubjectivity there is an objective possibility that we stay completely content even if we are going to stay ”thinking about X[,]”, the problem is unimportant. Then we discover that due to ”thinking about X[,]” we are not able to fall asleep. We may try to breathe deeply, think about something nice and we probably succeed with sleeping for many hours. If we do not it should be recalled that it only subjectively seems necessary to fall asleep (since we are not able to do it),..... and so on.

It was not mentioned yet that the next problem can be the same as just the ”processed” one. The principle of problem subjectivity should be used as many times as the problems is coming - every time again the solubility should be investigated as quickly as possible.

Perhaps it is clear now that the abstraction necessary is extreme and the simplicity of the principle of problem subjectivity is only principial. A background for the exact validity of the PPS was discussed in chapter 3 - PPS is a relation between two objectively measurable properties of a problem incarnation with differential character in the time scale. The principle of problem subjectivity can be learned as any other normal knowledge.

Let’s return back to the neuron level model of the psychical illness and have a look at how the principle of problem subjectivity is able to ”cure” it. We will start from the picture describing high number of insoluble problems strongly connected to the center of negative emotions (CNE):

The time portion during that insoluble problems are active is close to 100%, soluble problems have got a weak or no connection to CNE.

Consistent and careful discrimination between soluble and insoluble problems and applying the PPS leads to decrease of time portion during that insoluble problems are active, while connections between insoluble problems and CNE are still strong:

Originally strong connections between insoluble problems and the CNE are not active. The lack of positive potenciation of synapses involved leads to weakening of these connections.

Yet further consistent and careful usage of the PPS leads back to normal state when the portion of time during that insoluble problems are active goes to zero. The average value of the global state function is close to zero. The importance of soluble problems is growing.

The normal state of human neural activity can be characterized by relatively quick forgetting of insoluble problems while most of the time we are seeking solutions and dealing with the soluble ones.

T.B.D.

The most widely spread view of interpersonal relations today is the liberal one: as long as it doesn't breach other people liberties, it is everybody's free will what he does, what he believes in and most of all what he thinks. We shall cherish the variety of opinions and be very happy, that it exists. Principle of problem subjectivity very well fits into this picture - it mostly provides means for mere description of the thinking process (based on problems) and in the case of the principle of problem subjectivity (PPS) defines the way of thinking, that leads to minimal discontentment. Principle of problem subjectivity does not suggest in any way to think according to the PPS -for example sometimes we don't want to be content at all (however this does not collide with the exact mathematical validity of PPS assertion!).