X2 X3...
XN=X0). Then set O =
{X1,X2,X3...XN}can be called set of all
objects.Principle of Problem Subjectivity in a Nutshell
Problem Algebra
During the process of thinking, we are splitting the
complex objective reality - the surrounding world - into separate objects.
Object is a defined part of the objective reality and will be further on marked by capital
letters X, Y, Z. Objects can be very small, like some
elementary particles, quantum of light, atoms, molecules, cells or very large like man,
house, city, country, planet, etc. An object Z can be considered as
objects X and Y instead when referring to the same part
of the objective reality in both cases, or the other way around Z instead
of X and Y - these operations are called decomposition
/ composition of objects. We can imagine whole universe X0 split into
objects X1,X2,X3...XN so that composition of
these objects is the whole universe (X1 X2 X3...
XN=X0). Then set O =
{X1,X2,X3...XN}can be called set of all
objects.
State of object is the arrangement
of the internal parts of the object as well as its position, velocity and other
quantities at one moment. If
the state of an object X can be described by quantities (physical,
chemical, biological or other) Q1,Q2,Q3...Qn
the values x1, x2, ... xn of these quantities can be
considered as coordinates in n-dimensional space called phase space.
Each point in the phase space represents one state of the object X.
Values x1, x2, ... xn are components of the state vector 
. Object X in state is marked by X(). With the approximation defined by the
quantities Q1,Q2,Q3...Qn the phase space
represents all possible states of the object X.
Whenever we are doing any work, having a trouble with something or would
like to achieve a goal, the purpose or substance of that can be exactly described by
defining part of the objective reality (so called problem object)
and its two states (initial - unfavorable and final - desirable) of that part of the
objective reality. Therefore problem is generally defined as ordered
couple of states of an object, in symbols X[,
] 
[X(), X()]. Vector is called the initial state vector, the final state vector, 
-(delta) the change state vector. In the
phase space of object X the problem is represented by two points with
defined order (sequence).
Problems can be composed and decomposed
serially ( X[,] + X[,
]
= X[,] ) or in parallel ( X[,] + Y[,
] = Z[
,
] ; XY=Z ). As inverse problem to problem X[,]
is called the problem X[,]. Zero problem is such
problem X[,] that = . Composition of problem and its inverse problem
is the zero problem (X[,] + X[,] = X[,]). If the initial state vector represents the current (present) state of the
object X, the problem X[,] is called actual,
otherwise it is called virtual. Solution of
problem X[,] is generally an external force affection that
induces the object X to change its state from to . Composition of solutions of individual
subproblems is one possible solution of the composed problem.
Biology of Problems
Abstract problems can get reflected
(incarnated) in human brain: imagination of an object X in a
state which we don't like or in a state which we would like to achieve,
or both, are implemented by a nerve pulse in our central nerve system. More specifically a
couple neuron loops with a nerve pulse (which is running through periodically) can be
considered as the problem X[,] reflection.
Because any attempt to solve a well-defined problem X[,]
within a given time period T does have an unambiguous result (yes/no) the solubility
of problem is measurable, man and time dependent quantity. Symbols S(t) and
N(t) are used for the sets of all soluble resp. insoluble problems.
Problems influence dissatisfaction or contentment of people. Dissatisfaction can be represented by a biochemical quantity GS (global state) definable based on various chemical, electrical and other quantities measurable in the central nerve system. GS can have values 0 (contentment) to GSmax (extreme discontentment), and can be seen as function of time GS = GS(t). On the example drawing problems appeared at moments tA and tB and caused some increase of discontentment:
Instead of time-dependent, the quantity GS (describing discontentment)
can be seen as function of state of object GS = GS(X()) defined for all states of all objects, because
different states of different objects bring us different discontentment. In human brain,
the connection object state X() <=> dissatisfaction GS(X()) is implemented by neuron connections between the brain
area implementing the object X in the stat and
the emotion related structures..
Importance 
GS of problem X[,]
is defined as GS(X[,])
= GS(X()) - GS(X()) and is man dependent and measurable quantity. It says something
like how much our contentment will increase if we solve the problem X[,].
The neuron connections implementing importance of problems were either inherited or mostly created during our lives by our own subjective thinking. In any case the minimum possible importance is zero, because any of these neuron synaptic connections has the biological capability of disconnection (relaxation). Therefore so called general principle of problem subjectivity is exactly valid:
GSmin(X[,],t)
= 0; 
X[,],
t
"Importance of any problem is only subjective" - in other
words the objective importance (defined as minimal subjective importance) of any problem
at any time is zero. This means that we can stay completely content at moment t even if we
do not solve the problem X[,].
General principle of problem subjectivity is of no practical benefit because considering all problems as objectively unimportant would increase our discontentment (it is mostly of better advantage simply to solve the problem instead). By minimizing contribution of random problem to the discontentment we come to so called contracted principle of problem subjectivity (briefly called "principle of problem subjectivity"):
X[,]
N(t)
GSmin(X[,],t) = 0; t
i.e. problem that is not soluble at moment t is important only subjectively (seemingly) at that moment (or alternatively every problem is either soluble at moment t or objectively unimportant at that moment). Reason for exact validity of this assertion is the objectively existing (biological) possibility of complete disconnection of the neural path which implements the "problem importance". The principle is called "contracted" because we contract (reduce) the assertion of the general principle of problem subjectivity only to the set of all insoluble problems. There are no contradictions in the assertion of the principle of problem subjectivity, not even close to our death because the principle has differential character in the time scale, and global state of our brain (contentment) at the present moment is considered the basic value all the other things are bound to using neuron-based connections.
We can remember the principle of problem subjectivity as follows:
or as follows:
And finally logical construction of the set of all problems:
