#  On the Analyzing of a system and its consciousness (I)

Kazem Haghnejad Azar  Department of Mathematics and Applications, Faculty of
Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.
<EMAIL_ADDRESS>
###### Abstract.

We investigate on the analysing of systems including machines and living
organisms. We investigate to understand that how living organisms or machines
react to their internal and external events and how do we explain the
performance of a function of a system. We need to explain how a system central
processes can bring information together and use in the facilitation of
various behaviours. In this paper, the processing of information within the
system and the output quality of any system information will be investigated.
We will identify and investigate on the components of a system which have
important role for the system consciousness experiences. As results, we will
analysis some concepts such as wisdom and intelligence with mathematical
models. Here we will discuss the consciousness of a system and we will study
the comparisons between the consciousness of different systems. The ability of
a system to analyze itself and other systems will be studied and evaluated. By
mathematical methods, we also will show that there is no comprehensive
definition of living things as systems that distinguishes them from non-living
things. The relationships between the conscious experience and the physical
domain of a system, and also the relationships between a system and its
environment, and correctness of a proposition within the system have been
studied and investigated. By mathematics methods, we prove that we are not
able to determine when a living being reaches the stage of conscious
experience ability from birth to death. We explain how a system could perform
the function of producing reports on internal states, how a system could be
appropriately affected by its internal states and use information about them
in directing later processes, how a system’s central processes can bring
information together and use them in the facilitation of various behaviors and
why physical processes give rise to consciousness.

###### Key words and phrases:

Analysing of systems, Consciousness experience, Phenomenal consciousness,
Phenomenal concept of mind, Philosophy of mind, Intentionality and intention.

section.1 section.2 section.3 section.4 section.5 section.6 section.7
section.8 section.9 section.10

##  1\. Introduction

In this manuscript, we call a machine or a living organism as a system.
Basically, in general, we mean a system as device that receives input
information and after processing, it has output information. Of course, the
output information may be the result of processing information that already
exists within the system. Given that living organisms, machines, or machines
with artificial intelligence follow this rule, which are examples of the
systems. Here, we put organisms that have some characteristics of animals or
plants in a similar category and call them systems. In this article, we call
an animal or a group of animals a complete system.

Unfortunately, there is no precise definition of living organisms, so a
complete separation of living and non-living organisms is not possible. Thus,
our mean for a system is not always mean a living being or an intelligent
being. Many philosophers like Cleland, C. E. (2012, 2019), Vera M. K. (2015),
Knuuttila, T. and Loettgers, A. (2017) have been studied on this subject, but
in general it is not possible to separate living and non-living beings. The
logical reason for this is that we put some creatures in a category as living
creatures, and we call the rest non-living, are a misconception, and such a
distinction is meaningless. Because we separate from the body of all beings a
group with certain characteristics and we want to say that the rest do not
have these characteristics. Our mistake here is that we do not have a precise
definition of this trait that distinguishes beings, and most importantly, we
do not have complete aristocracy over other creatures that lack this trait. In
the second chapter, we will discuss this issue in more detail with
mathematical methods. Given the above, we base our initial knowledge of living
things on them as an example of a system. So in the discussions that will have
in this article, we mean the system of living beings or so-called intelligent
machines.

In this article, we will try to answer questions about system self-awareness,
how to develop system experiences within ourselves and the ability of a system
to access its own internal states. In order to answer some of philosophical
questions such as the origin of the world and its functioning, we need to have
investigate and analysis the systems. We will try to analysis more precisely
and logically the concepts such as intellect, intelligence, illusion, etc.
that we are involved with on a daily basis.

In the first stage, every system with its sensors creates a connection between
itself and nature. The system receives some information through its sensor and
processes this information according to its potential. The processing that
takes place inside the machine or living organism system consists of different
components and leads to changes in the living organism potential. Here system
potential refers to the unique state of a machine or living thing that is its
inside, and this internal state leads to a specific reaction that can usually
be categorized but unique to each system. For example, humans in the face of a
natural phenomenon are usually in one category and plants in another. However,
each human or plant also deals uniquely with an external phenomenon or event.
In any case, we call the internal conditions, physiological conditions or
capability of a system its potential.

Information processing of a system may be external or internal information or
both. To better understand the information processing performance of a system,
we consider a simple process where a set of information enters the system and
after processing, the system will have an output. Of course, a system without
receiving information from the outside can have an output from its internal
information processing at any time. Thus, in the general, the output of any
systems depend on the both external and internal information and their
effective to each others. The stages of these processes lead to our experience
of consciousness, which I will explore symbolically in Chapter six.

After receiving information from the environment and processing it, the system
shows a reaction that we can consider as the output of this information. This
output depends exactly on the input information, how the sensors receive it
and how this information is processed. Here these two factors play an
important role in the output of information that can easily be represented by
a mathematical equation. In the third chapter, we introduce and examine the
internal components of a system which process the internal and external
information of the systems.

At each stage of receiving information by a system, the potential of the
system may change. In other words, at different times when the quantity of
system information changes, the quality of information processing may also
change. These changes within the system change the potential of the system and
cause us to have another output for the same message.

An important issue when receiving information is how a system receives this
information through its sensors and how it translates, given its potential. By
translation we mean how a system sees this information according to its
potential. For example, humans react to sound and light or smell in almost one
but unique category, which is different from the reaction of plants or
animals. Here we interpret this type of reaction that occurs within the system
as the translation of information. This initial translation of information
that occurs inside the system (after it enters the system by sensors) is
called by some philosophers as animals consciousness. Nagel (1974) has been
studied the consciousness of organism. As his description, conscious is
subjective aspect experience. When we see, for example, we experience visual
sensations: the felt quality of redness, the experience of dark and light, the
quality of depth in a visual field. Other experiences go along with perception
in different modalities: the sound of a clarinet, the smell of mothballs. Then
there are bodily sensations from pains to orgasms; mental images that are
conjured up internally; the felt quality of emotion; and the experience of a
stream of conscious thought. The processing of information according to
initial translation is done by the system and the basis of processing depends
on the type and manner of our initial translation. In the fourth chapter, we
will try to examine primary consciousness from different way and we will show
that primary consciousness has different functions in different systems. Here,
if there is no ambiguity, according to the philosophers, we call this initial
translation ”primary consciousness”. Primary consciousness has been studied by
many authors, but there are some problems for its standard definitions.

The mathematical concepts used in this article are very simple and at the
level of basic mathematical knowledge.

##  2\. Mathematical method for examining the distinction between living and
non-living organisms

One of the long-standing questions in origins-of-life research centers on how
the proteinaceous side chains and the protein backbone were selected during
the earliest phases of evolution. Moran Frenkel-Pinter and others authors
(2019) studied oligomerization reactions of a group of positively charged
amino acids, both proteinaceous and nonproteinaceous. Amino acids
spontaneously oligomerized without the use of enzymes or activating agents,
under mild, hydroxy acid-catalyzed, dry-down conditions. We observed that the
proteinaceous amino acids oligomerized more extensively and with greater
preference for reactivity through their $\alpha$-amino compared with
nonproteinaceous amino acids, forming predominantly linear, protein-like
backbone topologies. These findings provide a purely chemical basis for
selection of the positively charged amino acids found in today’s proteins. The
earliest life forms we know of were microscopic organisms (microbes) that left
signals of their presence in rocks about 3.7 billion years old. The signals
consisted of a type of carbon molecule that is produced by living things.
Microbes are just single cells with no organelles and no nuclei to package
theirDeoxyribonucleic Acid (DNA). Something revolutionary happened as microbes
began living inside other microbes, functioning as organelles for them.
Mitochondria, the organelles that process food into energy, evolved from these
mutually beneficial relationships. Also, for the first time, DNA became
packaged in nuclei. The new complex cells (“eukaryotic cells”) boasted
specialized parts playing specialized roles that supported the whole cell.
Cells also began living together, probably because certain benefits could be
obtained. Groups of cells might be able to feed more efficiently or gain
protection from simply being bigger. Living collectively, cells began to
support the needs of the group by each cell doing a specific job. Some cells
were tasked with making junctions to hold the group together, while other
cells made digestive enzymes that could break down food. These clusters of
specialized, cooperating cells eventually became the first animals, which DNA
evidence suggests evolved around 800 million years ago. Sponges were among the
earliest animals. While chemical compounds from sponges are preserved in rocks
as old as 700 million years, molecular evidence points to sponges developing
even earlier, see David H. and Hall K. (2022). In general, when we talk about
living things, we consider things that are similar to animals or plants. In
this regard, we should identify organisms that have characteristics close to
plants or animals. So we have to identify the things that have common
characteristics with plants and animals, which are acceptable from our point
of view as living organisms.

In this chapter, we will investigate whether we can identify a complete sample
of all living organisms that have common characteristics that satisfy our
demand.

Jianhui. L (2019) in his paper refer to the different definitions of living
things and their separation from non-living things that have been studied by a
number of philosophers and scientists such as Cleland, C. E. (2012, 2019).
Finally, after various studies, it has been concluded that there is no clear-
cut line between life and non-life. One of his reasons is that some organisms,
such as viruses, are between living and nonliving in many ways. As notice to
Jianhui paper, the definition of life thus may conflict with, or even
contradict, common-sense. Our common-sense concept of life is generally
related to the general characteristics of animals and plants, including
growth, reproduction, the ability to self-sustain, and responding to external
stimuli. However, if we want a definition for organisms, we need to consider
the characteristics of all types of life, including microorganisms, such as
bacteria and even viruses, viroids, and prions. The characteristics of these
creatures are very different from those of more commonly conceived organisms.
Physiology, for example, often defines organisms systems that perform
functions, such as digestion, metabolism, excretion, respiration, movement,
growth, development, and response to external stimuli as living systems.
Biochemistry and molecular biology often regard living organisms as systems
that can transmit genetic information encoded in DNA and Ribonucleic Acid
(RNA)., which can control the synthesis of proteins, which determine the main
properties of organisms.

Now suppose that we divide the creatures of the universe into two distinct
groups living and non-living creatures which are not similar to each other at
a point in time compared to the characteristics we defined for living
organisms and call them $A$ and $B$, respectively. The basis for this
distinction into two distinct groups is to define properties for members of
$A$. Now if our assumption is that we have correctly identified and defined
these characters, for this separation we need to make sure that all members of
$B$ do not have these characters, and for this it is necessary that we
recognize all members of $B$ that is not possible.

On the other hand, elements $A$ can easily leave group $A$ and enter group $B$
over time, and vice versa. All of this leads to the fact that we can never
give a precise and comprehensive definition of living things. But despite
these problems, we do not stop here, and we can take living things as an
example of systems that includes humans, animals, plants, bacteria, and
viruses.

Now, if we go so far back that there was no so-called living thing in world,
or more precisely, these special properties of objects did not belong to group
A, or in other words $A=\emptyset$, then the production of members of set $A$
at any given moment will lead to our revision to define the properties of
elements $A$. A paradox arises here as to whether it is the members of $A$
that lead us to define specific properties for the set of elements $A$, or
whether it is the properties of a number of objects in the universe that make
up $A$.

Now if we want to study the problem in mathematical language, suppose we have
a finite number of properties for living organisms that we denote by

$q_{1}(x),q_{2}(x),q_{3}(x),...,q_{n}(x),$

where some of the properties $q_{1},q_{2},q_{3},...,q_{n}$ holds for a living
thing $x$. On the other hands, if some of the properties
$q_{1},q_{2},q_{3},...,q_{n}$ holds for $x$, then $x$ is living creature. Put

$A=\\{x\in U:~{}~{}q_{i}(x)~{}\text{holds}~{}\exists i(1\leq i\leq n)\\}.$

As we said that $A$ is including of living creature. The main problem here is
that, it is the properties of $q_{i}$ where $(1\leq i\leq n)$ that determine
members $A$, or, conversely, the presence of members $A$ that determine
property $q_{i}$ for some $(1\leq i\leq n)$. There is an another problem in
the above definition that is a number of organisms which identified as non-
living may be members of set $A$. For example, artificial intelligence
machines may belong to $A$ or perhaps other non-living things. This makes us
more cautious in defining set $A$. Now in the following, we want to define the
set $A$ to be more precisely.

In order to define living organisms, we need their complete samples of living
organisms. In this regard, we only have a number of samples with similar
characteristics which including humans, animals, plants or a number of other
organisms. Let’s suppose we put the samples of living organisms into
categories and call them $X_{1}$, $X_{2}$, …, $X_{k}$ where we can assume that
$X_{1}$ be a group of animals, $X_{2}$ be a group of insects, $X_{3}$ be a
group of plants and others. The categories $X_{1}$, $X_{2}$, …, $X_{k}$ have
some similar characterization that we explain as follows.

Now, if we take time too far back or too far forward, $A$ will change in each
case and our samples of living things will be different at various times. We
have to using from notion of $A(t)$ which is refer to definition of living
thing at the time $t$ or in a neighbourhood of $t$. We assume that the set $U$
is the creatures on Earth and $B(t)=U\setminus A(t)$. Now there is an
important question that $A(t)\cap B(t)=\emptyset$? And the answer to this last
question leads us to the concept of how we defined characteristics for living
things as $q_{1},q_{2},q_{3},...,q_{n}$. Is there a standard logic for
determining these characteristics for living things? Consider, for example,
the properties of a category of materials such as gases. Some gases are liquid
or solid at certain temperature conditions that other gases may not be. On the
other hand, we know that all materials were gaseous in Universal in many
before. Can we really define a group of gaseous materials that have nothing in
common with liquid or solid categories?

In the definition of living beings, we are referring to the characteristics of
creatures that we called them $q_{1},q_{2},q_{3},...,q_{m}$ for
$X=\cup_{i=1}^{k}X_{i}$, we try to find these or similar characteristics in
other organisms as well, that is, we added $q_{m+1},q_{m+2},q_{m+3},...,q_{n}$
to the above characters. Thus we have a finite sequence
$\\{q_{1},q_{2},q_{3},...,q_{n}\\}$ of characterization of organisms and we
assume that this sequence is complete. Let
$B\subseteq\\{q_{1},q_{2},q_{3},...,q_{n}\\}$ be non-empty such that all
living things satisfies to all properties of the elements of the set $B$.
Assume that $B=\\{q_{k_{1}},q_{k_{2}},q_{k_{3}},...,q_{k_{t}}\\}$ where $1\leq
t\leq n$. Our main that $x$ satisfies in the properties $B$ is equivalent to,
$x$ satisfies in the property $q_{k_{1}}\wedge q_{k_{2}}\wedge
q_{k_{3}},...,\wedge q_{k_{t}}$ where $1\leq t\leq n$, equivalently
$q_{k_{1}}(x)$, $q_{k_{2}}(x)$, $q_{k_{3}}(x)$,…, and $q_{k_{t}}(x)$ hold for
all $1\leq t\leq n$. Here, our goal in defining set $B$ is to try find a
standard way that does not cause the above problem for set $A$. In other
words, non-living organisms do not belong to set $A$. Suppose $U$ is the
collection of elements of the whole universe, we define the set $A$ as
follows:

$A=\\{x\in U:~{}x~{}\text{satisfies ~{}in ~{}the~{}properties}~{}B\\}.$

Now let $b_{(k_{1},k_{2},...k_{t})}=q_{k_{1}}\wedge q_{k_{2}}\wedge
q_{k_{3}},...,\wedge q_{k_{t}}$. So $b_{(k_{1},k_{2},...k_{t})}(x)$ holds
means that $x$ is a living thing or equivalently $x\in A$. It is clear that we
can display the set $A$ as below

$A=\\{x\in U:~{}~{}b_{(k_{1},k_{2},...k_{t})}(x)~{}\text{holds}~{}\\}.$

The basis of our assumption is the existence of a definition for living
organisms, having a common point is necessary for living organisms, because it
is not possible to separate them from non-living things. Thus for solving this
problem, it must $B\neq\emptyset$.

In the above relations, $B$ is dependent on $A$ and vice versa. On the other
hands, the set $X$ is including of all living organism if and only if
$B\neq\emptyset$. So $X$ and extensive of $B$ dependent on each others. In
other words, the characteristics that we mentioned above for living organisms
are exactly dependent on the samples of living organisms, but these samples,
especially the complete samples of living organisms are selected based on
their characteristics. As a result, the characteristics of living organisms
and the selection of a complete sample cannot be independent of each other.

Unfortunately, $B=\emptyset$. The wooden chair we are sitting on, a strand of
hair, a bone or an egg have certain genetic characteristics, but we call them
non-living things. An egg (zygote) is not a living thing, but when a bird sits
on it for the period of time, it becomes a living thing. Let’s assume that
before the bird sits on the egg, the egg has a chemical and biological
characteristic, which we call these characteristics with $a$, and after the
egg turns into a chick, we call these characteristics $b$. Now we define a
function $f$ from $[a,b]$ onto $[0,1]$ where $f(a)=0$, $f(b)=1$ and $f(x)$ is
a number which shows the distance between simple eggs and a stage of embryo
formation. To be more precise, the moment which the bird sits on the egg for
the first time we show by $t_{a}$ and we also represent the moment when the
chick hatches with $t_{b}$, then the position $a$ occurs at time $t_{a}$, the
position $x$ occurs at time $t_{x}$ and the position $b$ occurs at time
$t_{b}$. We define $f(x)=d(a,x)=\frac{|t_{x}-t_{a}|}{|t_{b}-t_{a}|}$ where $d$
is a distance from $a$ to $x$ for all $x\in[a,b]$. The function $f$ is
continuous, and so we cannot determine when the number $f(x)$ must be in order
for $x$ to be living in our definition. If we define the set

$\omega=\\{f(x):~{}x~{}\text{is}~{}\text{a}~{}\text{living}~{}\text{organism}\\}.$

Then $1\in\omega$. So $\omega$ is non-empty and bounded below. Let
$\inf\omega=\alpha$. Then there is $x_{0}\in[a,b]$ such that
$f(x_{0})=\alpha$. It follows that for all $x,y\in[a,b]$ with
$f(x)<\alpha<f(y)$, the position of $x$ indicates that the egg is not alive
(or $b_{(k_{1},k_{2},...k_{t})}(x)$ not holds) and the position of $y$
indicates that it is alive (or $b_{(k_{1},k_{2},...k_{t})}(y)$ holds). We can
take $f(x)$ and $f(y)$ (or $t_{x}$ and $t_{y}$) so close together that the
counteractions of $x$ and $y$ do not differ (regarding as the common
characterizations mentioned in $B$), so that $B=\emptyset$. This conclusion is
true for almost all living things in our category and this argument also
applies to proteins that become unicellular as a result of evolution.

On the other hands, there is an another fundamental contradiction in the
definition of live organisms, that is, if we want to know all samples of
living organisms, it is necessary to know their characteristics that
contribute to being alive, but for knowing the characteristics of all living
organisms, it is necessary to know all the samples. So paying attention to the
arguments presented, we generally conclude that the definition of living
organism is not possible.

The fact is that by looking at several specific states in a number of beings,
we can not reach a general consensus. And most importantly, the samples whose
properties are evaluated are not complete and are different at various times
and depend on the chemical composition of the environment. We can easily say
that there is not and will not be a precise and complete definition of the
creatures which called living beings.

##  3\.  The processing information in the systems

As we stated in the introduction of this article, we call a case or a group of
animals or insects a complete system. In order to study complete system
accurately, we need to have more detailed information on simpler systems, such
as artificial intelligences or plants.

In a simple way, how information is received by a system and processed will be
investigated. We will introduces a system function which will works for all
systems and we will divide some systems into different categories such as
plants, animals or intelligent machines.

Falsifying a theory of consciousness is based on funding such mismatches
between reported experiences and predictions. Kleiner J. and Erik Hoel. (2021)
showed that falsification is especially problematic for theories of
consciousness. They have been studied the relationships between inference and
prediction for a human system and showed that if inference and prediction are
independent, it follows that any minimally informative theory of consciousness
is automatically falsified, or if inference and prediction are strictly
dependent, it follows that a theory is unfalsifiable.

In the following, we have introduced different parts of a system that are
effective functions for the output of a system, which we call system
components. In the Chapter 6 we will examine the system components in more
detail and try to answer a number of Chalmers, D. J. (2009) questions.

A topological structure of a system:

  1. (1)

$X_{1}=$ Sensation set (Reactivity of molecular reaction).

  2. (2)

$X_{2}=$ Primary consciousness set (An initial translation of the input data).

  3. (3)

$X_{3}=$ Awareness set (Knowledge and understanding that something is
happening or exists).

  4. (4)

$X_{4}=$ Analysing set (Review and data analysis).

  5. (5)

$X_{5}=$ Memory set.

  6. (6)

$X_{6}=$ Character mentality and mood set (Emotion, mood, and traits of
character).

  7. (7)

$X_{7}=$ System components coordinator set (The central part of the system for
communicating components).

  8. (8)

$X_{8}=$ Quality of will to accomplish an output set (The origins of the
concepts of intentionality and intention).

  9. (9)

$X_{9}=$ Curiosity along with perception set (A kind of deep sense and
exploration of a phenomenon or concept without analysing).

  10. (10)

$F=$ Other molecular and physiology or chemical and physical conditions set.

In the above, we have tried to define a pattern of system components that
describes the functioning of a human-like system. Of course, simpler systems
such as artificial intelligence, plants, or insects may lack some of these
components. Here, it is not necessary that the components of a system are
separate from each other. For each $1\leq i\leq 9$, the component $X_{i}$
works as function that we will describe in the details in the Chapter 6.

In this section, we will try to describe the components of a system, examine
their relationship with each other and their role in the output quality of a
system. As a result, for humans, the way of understanding the input
information and its processing is examined. We will examine how this
information is processed by the components for humans.

It may seem that the above components work the same in some cases, but after
describing we will find that their function is different. They are all
interdependent, and if we ignore one of them, we will not have a favourable
output in a system like humans. Of course, in some systems, all components are
not active, which we will talk about later. It can be said simply, all of
components are physical items of a system. All system questions and system
curiosities depend on these components.

Sensors play an important role in the operation and processing of information
in a system. The component $X_{1}$ is the collection of primary system sensors
that receive information from outside or inside of a system. Here the system’s
ability to receive information plays an essential role in the operation of
other components. Not only the other components dependent on $X_{1}$, but the
system output also depends entirely on it. For example, a deaf or blind person
would be fundamentally different from an ordinary person in outputting
information. Therefore, our understanding will depend on the performance of
the sensors.

Translating of the input information in a system plays a key role in the
system output and the operation of other components. The type of translation
causes the other components that depend on it to have different functions.
Some information such as color, smell, or sound enters the system environment
by sensors and is initially translated by primary consciousness. Our memory or
perception information is also translated by primary consciousness. These
translations vary by primary consciousness for different systems.

Suppose the sensors of a system receive a message $\alpha$ from the
environment. We can assume that $X_{1}$ receives this message as a function
and transmits it to $X_{1}(\alpha)$, in this case $X_{2}$ will also affect on
$X_{1}(\alpha)$ as a function. Finally, the initial translation of the message
will be in the form of $X_{2}(X_{1}(\alpha))$, and the process of transferring
the message to the initial translation will be as follows:

$\displaystyle\alpha\Longrightarrow X_{1}(\alpha)\Longrightarrow
X_{2}(X_{1}(\alpha)).$ (3.1)

In the (3,1), the sign ”$\Rightarrow$” means a result of transferring the
message to the next stage and $X_{1}(\alpha)$ means that the set of the
sensors have received the message $\alpha$. $X_{2}$ determining which category
this information belongs to it. In human system, for a person with
schizophrenia problem, $X_{2}$ does not work properly compared to normal
people. In other words, in some cases, the initial translation of input
information by the sensors is different from ordinary people and this improper
translation will cause the other components to malfunction. This component is
interesting by many philosophers which has been studied in many books and
papers. In the next chapter, we will discuss this component in detail, but our
discussion will be more on the primary consciousness of an arbitrary system.
We emphasize again that we consider primary consciousness to be an initial
translation of information entered into the system by the sensors of a system.
This initial translation helps us to classify the input information by the
sensors without any logical analyzing.

The operation of $X_{3}$ is similar to a function that receives information
from outside or inside of a system and send them to other system components
such as memory, system components coordinator or curiosity along with
perception. Sometimes this component of the system is confused with $X_{2}$,
but in order to activities of the system awareness, the initial translation of
the input information is necessary. A simple and symbolic diagram of this
component function is in the following:

$\displaystyle\alpha\Longrightarrow X_{1}(\alpha)\Longrightarrow
X_{2}(X_{1}(\alpha))\Longrightarrow X_{3}(X_{2}(X_{1}(\alpha))).$ (3.2)

In the first stage, when a person faces an issue or a problem, she/he may not
have a standard rule and logic to solve it. She/He may also use her/his
illusions to analyze the problem. Now, if the system uses an acceptable rule
and logic to analyze issues or solve problems, we say that the system follows
the rule of mathematical logic (shortly, we call $ML$). So for the component
$X_{4}$, it is not necessary that the system analysis follows $ML$. Now, if
our analysis is based on $ML$, it means that we have restricted the function
$X_{4}$ which its domain is dependent on $ML$. In a human system, the
contradiction that occurs in the output of information is such that some of
the information that enters the system becomes basic principles and the system
tries to prove and justify these baseless and inconsistent principles by using
all the tools such as $ML$. Here, symbolically, we can describe the function
of component $X_{4}$ inside a system. According to the (3.2) in a simple
diagram as below, the analysis of the messages in the system will be as
follows:

$\displaystyle\alpha\Longrightarrow X_{1}(\alpha)\Longrightarrow
X_{2}(X_{1}(\alpha))\Longrightarrow X_{3}(X_{2}(X_{1}(\alpha)))$
$\displaystyle\Longrightarrow X_{4}(X_{3}(X_{2}(X_{1}(\alpha)))).$

This diagram is a part of the message decomposition diagram in a system. Of
course, above diagram is not complete for analyzing a message, and we wanted
to explain the message transmission process inside the system. The role of the
rest of the system components cannot be ignored for analyzing a message and we
will try to make complete diagram later.

The memory of a system, component $X_{5}$, not only records information, but
also emotional states and processing methods are recorded. The software that
is created from learning a skill in a system is stored in the system memory.
Most of the system activities carried out by its components pass through the
system memory and are stored in it while passing.

Component $X_{6}$ can be seen more in animals and humans, which is more of an
individual temperament or a personality trait. The application of this
component is for emotional issues, love, hate or attractiveness of a
phenomenon or subject and individual mood in the relation to oneself and
others people. Component $X_{6}$ reveals its characteristics better in
different conditions where the system is placed. To describe the component
$X_{6}$, we give a clear example of human as a system. Suppose two people in
the $X_{1},\cdots,X_{5},X_{7},X_{8},X_{9}$ components have the same value but
have abnormally different outputs. In other words, one is interested in others
suffering (sometimes for the spacial people) for her/his benefits, but the
other is the opposite. Here, if we take the component $F$ too small, $X_{6}$
is a very important role for two different outputs, and this component can
easily affect on the other components $X_{1},\cdots,X_{9}$ and $F$. In many
cases, it can be seen that this component has a great impact on a person’s
reasoning. This phenomenon can be easily observed in humans and this is
sometimes part of the main problem of humans in the relation to each other. In
other words, sometimes the $X_{6}$ creates a state in the system that the
$X_{4}$ be a function of it. In other words, a person who goes through her/his
life and facilities for improving the condition of other people, or vice
versa, a person who goes through the life of other people or their facilities
to improve her/his own condition can depend on this component. Here, social
conditions or one’s ability to analyze data on the base $ML$ may not allow
component $X_{4}$ to be a function of component $X_{6}$. Of course, the
potential of system components and how they relate to each others, play a
fundamental role for a system output. This component is not exists in the
artificial intelligence, so in examining this component, we easily notice the
difference between animals and artificial intelligence.

The components $X_{7},X_{8},X_{9}$ can be further observed in animals, insects
or intelligent machines. If we want to concentrate our thoughts on a topic or
a point for a moment, the effective components for this will be
$X_{7},X_{8},X_{9}$, whenever we ignore the component $F$. We do not need to
use $X_{2},X_{3},X_{4},X_{5},X_{6}$ information for focusing our thoughts on
one topic or point. Of course, these components are effective, but they can
play an indirect role.

Here the component $X_{8}$ plays an essential role in coordinating the
relationship between the components and the processing of information by them.
$X_{8}$ will play the role of central control for coordinating system
components. On the others words, $X_{8}$ with the help of component $X_{3}$
has an important role in identifying the components and regulating the
relationship between them. The ability of a system to execute an output is
determined by $X_{8}$. For example, when animals make a decision after
processing information that it is a kind of system output, they will need this
component in order to demonstrate it in action. In fact, this component shows
the willpower of the system to perform an activity which has already self-
awareness or unconsciousness decided to do.

Here we may confuse in some cases, we can consider the performance diagram of
$X_{9}$ as follows with $X_{2}$, but they are different. $X_{2}$ is an initial
translation of input information by sensors, but $X_{9}$ is meaningful
curiosity. In the face of this component, we may make a momentary guess or be
motivated in the face of information that may be of internal or external
origin. We may be curious for a moment and pursue an issue and a concept. In
some cases, we can consider the performance diagram of $X_{9}$ in the symbolic
form, as follows:

$X_{1}\Longrightarrow X_{2}\Longrightarrow(X_{6}\vee X_{9})\Longrightarrow
X_{4}.$

$X_{6}\vee X_{9}$ means that at least one of the components $X_{6}$, $X_{9}$
in the system is active. The above diagram shows that when $X_{6}$ or $X_{9}$
are activated, it leads us to quickly analyze information and have an output.
Here’s the important point that the roots of delusional thoughts in ordinary
people depend on $X_{2}$, $X_{6}$, $X_{9}$ and $F$. Most hallucinatory
thoughts occur in humans due to the function of the components $X_{6}$,
$X_{9}$, $F$ or the inability of the component $X_{2}$. Here our sense may
come from an abnormal event that invisible forces are angry or trying to hurt
us, and we may exhibit delusional behaviour. Now, if we can analyze this
seemingly unusual event and find its root, we will no longer resort to
illusory thoughts to solve this issue. Here the role of the $X_{2}$ component
in the analysis of faith perceptions is very important. Of course, delusions
have many roots, but the scope of our discussion is for a normal person or a
healthy system. Of course, most of these discussions are related to the
psychology of man or society, but we also refer to its philosophical
discussion. A system typically uses the information it stores to process input
information. Now, if this stored information has no logical basis and is
rooted in the illusions of previous systems, it can change the psychological
structure of a human-like system in such a way as to create a kind of
dependency. This psychological dependence on illusory concepts causes us to
follow principles in the analysis of input information that are in the
structure of the booklet of basic principles but have no logical basis. In
other words, incorrect and illusory information stored in our mental structure
and memory sometimes act as basic principles in the system. These basic
principles play a key role in analyzing the input information of a number of
systems. And what’s more interesting here is that we sometimes try to show
them correctly with the help of $X_{2}$, $X_{6}$ and $X_{9}$.

Suppose that a data enters a system by sensors. In this case, the performance
of the system according to the following diagram will depend on the
performance of the components. In the following, for convenience, we represent
$F$ with $X_{10}$.

$\displaystyle X_{1}\Longrightarrow X_{2}\Longrightarrow g(\sum_{1\leq i,j\leq
10}(X_{i}\Longleftrightarrow X_{j})),$ (3.3)

where $g$ is a function related to the system. The above digram shows that
when sensors of a system have been received an information, $X_{2}$ has been
translated it in system inner. In the next step, the other components receive
this translated information from $X_{2}$. In the formula (3.3), notion
$\sum_{1\leq i,j\leq 10}(X_{i}\Longleftrightarrow X_{j})$ means that a kind of
combination of the effect of the components on each other. In other words, the
message is passed by the system components and in every transfer from one
component to another, changes occur on it. Finally, after some steps, the
message transmission may be stopped and recorded as the final output in the
memory. Here, we do not have detailed information processing steps by the
information system, but we know that a message within the components of a
system such as (3,3) is translated, analyzed or creates chemical and physical
molecular reactions. In Chapter 6, we will try to examine the details of this
issue. Now, with the help of the following example, we will check the
relationship between the system components.

When we see an accident, first $X_{1}$ receive informations by eyes, ears and
other sensitizers. Secondly $X_{2}$ has been received these informations and
translates this information into an initial concepts where we show this
process by $X_{1}\Longrightarrow X_{2}$. Other components, such as awareness,
character mentality and mood, system components coordinator (the central part
of the system for communicating components), quality of will to accomplish an
output (the origins of the concepts of intentionality and intention),
curiosity along with perception (a kind of deep sense and exploration of a
phenomenon or concept without analysing) and other molecular and physiology or
chemical and physical conditions receive these basic concepts immediately and
after processing these basic concepts, they send them to memory. Finally, the
analyzing component gives the system an output that is transmitted to memory.
Of course, this is a raw process of input information into the system, which
may not be accurate, but it is important to note that all components work
simultaneously after receiving the initial concepts. The symbol
$X_{i}\Longleftrightarrow X_{j}$ means that in information processing, the
components regularly transfer them to each other and the symbol $\sum_{1\leq
i,j\leq 10}(X_{i}\Longleftrightarrow X_{j})$ indicates a combination of
transferring information to each other and processing them. Of course, as we
said, the information process in (3.1) is not accurate and is given only as an
example to understand the function of the component.

The important point in the information process by the systems is that in many
systems the role of the $X_{4}$ component in their output is very important.
However, due to the fact that the components affect each other, the function
of the component $X_{4}$ in many systems may depend on other components. This
is the fundamental difference between human and artificially intelligent.
Clearly a simple machine with artificial intelligence has the following
diagram:

$\displaystyle X_{1}\Longrightarrow X_{2}\Longrightarrow g(\sum_{1\leq i,j\leq
4}(X_{i}\Longleftrightarrow X_{j})).$ (3.4)

Here $X_{4}$ depends only on the $X_{1}$ and $X_{2}$ components and a logical
process is performed to analyze the information. In some humans or animals,
$X_{4}$ may be dependent on other components, and the output of the system may
be a different process than that of artificial intelligence. In other words,
we may accept a misconception and force $X_{4}$ to subconsciously have a
function to justify it. If we consider the human being as a complete system
that includes all the components of the system, it is easy to say that the
performance of these components is the result of physical and chemical
reactions around the system that create special characteristics for these
components over time. Of course, by studying very simple systems, it is
possible to better understand the performance of system components.

When a system receives a message $\alpha$ at the time $t=t_{0}$, then the
decision or output of the system is a function of the following formula:

$\displaystyle w(\alpha)=f(X_{1},\cdots,X_{10})(\alpha).$ (3.5)

Here, the function $f$ determines the relationship between the system
components. Now if this system showed by $\Lambda$, then we define the
function $f$ from $\Pi_{i=1}^{10}X_{i}$ into $\Lambda$ at the time $t=t_{0}$.
The above formula is a system output that occurs at time $t_{0}$, and the
components on which the system output depends occurred on time $t\leq t_{0}$.
$w$ is working for different system in the different way that we can observe
some of them as in the following:

  1. (i)

For plant, we have $w=f(X_{1},X_{2},X_{10})$.

  2. (ii)

For artificial intelligence, we have

$w=f(X_{1},X_{2},,X_{4},X_{5},X_{7},X_{10}).$

  3. (iii)

For animal, we have

$w=f(X_{1},\cdots,X_{10}).$

The components $X_{1},X_{2},\cdots,X_{10}$ sometimes work in the one time,
that is, all of them work together in an instant. In essence, this is $w(t)$,
which represents a potential for a system, and the knowledge of the
environment by the system will depend on the potential of the system. We call
the function $w$ as system function or shortly ”system”, if there is not any
ambiguity.

By observing the above functions, we see that there are many systems with few
components, but thinking to a system with more components then animals is very
hard. As the previous chapter, it is very difficult to understand that we can
identify which chain of components is unique to living organisms.

Our behaviour, way of thinking, knowledge of the world, feelings are all
dependent on the function $w$. On the other hand, for humans, concepts such as
intellect, intelligence, the truth and falsity of a statement have been
proposed since ancient times and have many applications, but these concepts
have never been properly defined, and we have never had the means to justify
the correctness of a phenomenon by using logical principles and assumptions.
The principles and rules of logic that guide us properly to a phenomenon are
established by our system, and there is no absolute measure for their own
evaluation and these logic principles and processes vary from system to
system.

Some system as humans often believe that they are fully aware of their inner
and outer environment and they think that this awareness is increasing moment
by moment and they can have more accurate and complete information about their
inner and outer environments. Unfortunately, this is not the case, and humans,
as a system, are in the first place dependent precisely on our sensors, which
enter information into the system, and in the next stage we are dependent on
the $w$ function. Beyond that, whatever it is will never be achievable and
does not exist for us. Consider, for example, a cat or ant that we want to use
to the best of our ability to give information about the universe and the
stars, which is by no means possible, because their system have no sensors or
components to receive this information and make initial translation. Humans
have different outputs from the same phenomenon at different times. The reason
for this issue is that our potential under the $w$ function has been different
at the various times. For a phenomenon, humans in ancient times had a
melancholy output, and now our output may be more accurate and have a
scientific basis. The reason for this change in our output is the change in
system potential that we represent with $w$.

According to the above, humans destiny and understanding of the world and
himself are exactly dependent on his potential, especially his sensors and
primary consciousness. This shows the importance of the $w$ function, and as
we have said, our output, which includes our understanding, curiosity,
aptitude, and abilities on the one hand, and our normal and abnormal
behaviours all depend on the $w$ function and our environment. The environment
around us enters information into our system, and it is $w$ that determines an
output for it. And it is interesting that our mental illusions and
abnormalities also follow this rule.

As an example, humans have many questions about the universe and why they came
into being and the existence of creatures, especially humans. In essence, the
questions that a system has from within or around itself, or to questions that
are curious, depend exactly on the components $X_{1},\cdots,X_{10}$. As
looking to above diagram, we see that everything of systems depends on these
components and almost all components depends on $X_{1}$. After $X_{1}$, the
component $X_{2}$ has important rule for the potential type of a system.

The questions raises about the performance of a system have been introduced in
the book by David J Chalmers (2010) is in the following that how a system
could perform the function of producing reports on internal states. To explain
internal access, we need to explain how a system could be appropriately
affected by its internal states and use information about them in directing
later processes. To explain integration and control, we need to explain how a
system’s central processes can bring information together and use them in the
facilitation of various behaviours. We have some answers to these questions
that you can see in chapter 6.

It is also that our general perception depend on above components, that is
including of $X_{1},\cdots,X_{10}$. The combination and relationship of these
components will give us an overview of a concept which we can evaluate as an
output. Of course, it is not important to understand this concept correctly or
incorrectly. The issue of right or wrong output of a system as an
understanding of a concept is not currently discussed here, but its importance
is that the system has an image of it as a true proposition in its primary
consciousness component, $X_{2}$.

David J Chalmers (2010) has studied the corresponding systems. On page 23 of
this book, he points out some points for examining the correspondence of
systems in this regard. Here he points out that: if the causal patterns of
neural organization were duplicated in silicon, for example, with a silicon
chip for every neuron and the same patterns of interaction, then the same
experiences would arise. According to this principle, what matters for the
emergence of experience is not the specific physical make up of a system but
the abstract pattern of causal interaction between its components. This
principle is controversial, of course. I have presented another way of looking
at this issue in Chapter 6.

In this manuscript, we call that two systems are corresponding whenever they
have corresponding components. On the other hand, two components are
corresponding, if they have almost the same function for processing
information. For example, most humans translate red as one form of
translation, and other creatures as another. In other words, humans correspond
in the experience of colors. Of course, some people may correspond to most
components, but this is not the reason that their systems are corresponding.
If we display the correspondence between two systems or components with
$\approx$, then for two systems $w$ and $w^{\prime}$, $w\approx w^{\prime}$ if
and only if $X_{i}\approx{X^{\prime}}_{i}$ whenever $X_{i}$ and
${X^{\prime}}_{i}$ is component for $w$ and $w^{\prime}$, respectively, and
$1\leq i\leq 9$. Here, if there is no ambiguity, we reduce the role of
component $X_{10}$. For the two corresponding systems, the output information
will be approximately the same.

Now consider an arbitrary system with function $w(t)$. If we give a numeric
value for $X_{i}(t)$ and $w(t)$ at time $t$, $\Delta w=w(t)-w(t_{0})$ an
output difference or difference in our decision distance for this system at
the time $\Delta t=t-t_{0}$. We assume that $w$ is continuous function and if
it is derivable in $t_{0}$, then we define

$\frac{dw}{dt}=\frac{df}{dt}=\sum_{i=1}^{10}(\frac{\partial w}{\partial
x_{i}})(\frac{dx_{i}}{dt}),~{}~{}\tau(t_{0})=|\frac{dw}{dt}|_{t=t_{0}}$

where $x_{i}\in X_{i}$. $\tau(t_{0})$ shows our behavioural or decision
mutation at time $t_{0}$. If $\tau(t)$ is too small at moment $t=t_{0}$, no
behavioural changes or system decisions are noticeable. But when it is large,
then the behaviour or decision of the system has changed fundamentally.

Let $w$ and $w^{\prime}$ be system functions (if there is not ambiguity we
call them systems). We define $w\leq w^{\prime}$ if and only if $X_{i}\leq
X_{i}^{\prime}$ where $1\leq i\leq 10$. The notion $X_{i}\leq X_{i}^{\prime}$
means that $X_{i}^{\prime}$ has a potential quality relative to $X_{i}$. Here,
in mathematical terms, we can say that systems as set are partial ordered. For
example, when a child grows normally, the quality of its components also
increase. Now if we have the relationship $w\leq w^{\prime}$ for two systems,
then $w^{\prime}$ has more or equal potential quality with respect to the
system $w$. We express this relation about humans in such a way that if two
people have functional systems as $w$ and $w^{\prime}$. So the relation $w\leq
w^{\prime}$ means that the person with the function system $w^{\prime}$ has a
higher or better mental and psychological quality than a person with systemic
function $w$. For simplicity, we display two people with $L_{w}$ and
$L_{w^{\prime}}$ corresponding $w$ and $w^{\prime}$, respectively. Here we may
consider a system in such a way that over time its potential, or in other
words its capabilities, increases. If we represent this system at time $t$
with $w$ and at time $t^{\prime}$ with $w^{\prime}$ where $t\leq t^{\prime}$.
It follows that $w\leq w^{\prime}$. Now we can check the correctness of a
statement here in relative terms. In other words, a statement as output that
is true for the person $L_{w^{\prime}}$ will always be true for the person
$L_{w}$, but vice versa is not always true. This discussion can easily be true
for two any different systems. However, the correctness or incorrectness of
the system outputs is generally relative.

##  4\.  The domain and range of a system

A system may not be able to translate all the messages which receives from the
outside environment, in other words, the relation (4.1) may not be established
for any message. Let’s assume that the messages that enter the system from the
outside system in a period of time $a\leq t\leq b$. Then the relation (4.1)
define the following set:

$A=\\{\alpha:\alpha~{}\text{is}~{}\text{a}~{}\text{message}~{}\text{from}~{}\text{outside}~{}\text{of}$
$\text{the}~{}\text{system}~{}\text{and}~{}\text{the}~{}\text{relation}~{}(4.1)~{}\text{holds}\\}.$

On the other hand, let’s assume that $B$ is all the output information of the
system in the mentioned time period. Here it is clear that the sets of $A$ and
$B$ are finite, but $B$ is much wider than $A$. For example, we observe a
finite number of objects in our experience in nature, which are elements of
$A$ as counting numbers. In other words, in our experience of counting numbers
in nature, we deal with a finite set of numbers such as $\\{1,2,3,...n\\}$
where $n$ is natural number. But the situation of numbers in $B$ is different.
In $B$ we deal with special properties of numbers that are the result of our
system processing. Infinite or uncountable sets, prime numbers, complex
numbers and other properties of numbers, which are the result of mind
processing and there are not in $A$. In very simple systems, sets $A$ and $B$
are very close to each others, but in complex systems such as humans, set $B$
is much larger than $A$. The extent of $B$ is exactly dependent on the
potential of the system and the set $A$. For example, this issue can be easily
observed in systems such as plants, different animals and humans. Of course,
if we consider a group of humans as a system, the development of $B$ will be
much wider. The important point is that if there were a few thousand people on
the planet, then $B$ would be smaller than what we see now and vice versa, if
there are more people with high potential quality, then to what extent the $B$
collection will be developed. Since $B$ is the result of the set $A$ and the
potential of the system, we consider $B$ to be a range of a function like
$\varphi$ from $A\times w$ into $B$ where $w$ is a function which we defined
in (3.5). In other words, we define $\varphi(A,w)=B$. Throughout history,
humans have had findings as output which all of them have not a logical basis.
Now, the basis of our assumption is that we can separate logical and illusory
or imaginative findings. So we are looking for a subset of our findings that
are based on $ML$. Here we are looking for a subset of $B$ such as $C$ whose
members obey the following rules.

  1. (1)

The function $w$ has the potential to follow the rules of $MA$ in processing
information inside or outside it.

  2. (2)

At the same time or under the same conditions, a proposition and its
contradiction should not be inside $C$.

  3. (3)

The elements of $C$ are the result of a $MA$ process.

Of course, it may seem that condition (2) is not required, but according to
relation (4.1), we are not sure that one message may be translated in several
ways in one system. Components within a system are similar to software defined
within the systems. For example, a number of components in the human system
are genetically present, but as a result of social life, some potentials enter
the system in the form of education and they become parts of the system where
may have a significant impact on the performance of the system and system
components. Now, we represent the mathematical logic device with $ML$ and if
the base on the principle that the system follows the rules of $ML$, in the
analysising of information at time $t_{0}$, then the function $w$ will also
depend on $ML$. If we assume that $ML$ is one of the components of the system,
we define the function $w_{0}$ at the time $t_{0}$ as follows.

$\displaystyle w_{0}=f(ML,X_{1},X_{2},\cdots,X_{10}),$ (4.1)

or we can write as follows

$\displaystyle
w_{0}=f(X_{1},X_{2},X_{3},X_{4}(ML),X_{5},X_{6},X_{7},X_{8},X_{9},X_{10})$
(4.2)

Basically, $w_{0}$ is a restriction of the function (3,5) and if there is no
ambiguity, we will use again $w$ instead of it, so we write $\varphi(A,w)=C$.
Here we assume that the input information $A$ is the same for a set of
systems, so the sets $B$ and $C$ are dependent on the function $w$, and if
there is no ambiguity, we write $B_{w}$ and $C_{w}$ instead of $B$ and $C$. Of
course, our assumption here is that the function $\varphi$, which is used to
connect the components of the system, works the same in this group of systems.
The interesting thing here is that if we have two people with systems $w$ and
$w^{\prime}$ such that $w\ncong w^{\prime}$, then maybe $C_{w}$ and
$C_{w^{\prime}}$ are different. It is also clear that the role of the function
$\varphi$ is effective for the output of the system. Two people may have the
same analytical power and in better words $X_{4}\cong X^{\prime}_{4}$, but
$C_{w}\ncong C_{w^{\prime}}$. Of course, there is no such difference in output
for machines with artificial intelligence. The exact cause of this problem
depends on the function $\varphi$ and the relationships between the components
$X_{8},X_{9}$ and $X_{10}$ under $\varphi$, which are not predictable. For
example, in a war, maybe a soldier at a critical stage between sacrificing
herself/himself and saving her/his comrade or saving herself cannot see a
difference in her/his decision. In this decision, we have two different
outputs, but we cannot easily say which decision is more logical or correct.
The interesting thing here is that both of these decisions are made by one
person at the almost same time. Now if we set $\Sigma=\cup_{w}C_{w}$, then
$\Sigma$ will be a complete output of observations of all systems of the
world. The extent of our knowledge of the world depends on the maximum size of
$\Sigma$. The important point here is that our view of the world depends on
the type and size of $\Sigma$. Production of $\Sigma$ is exactly depend on the
stricture and potential of systems. Of course, the structure of a system
depends on the environment around the system and how it is formed. Because of
this, any number of systems in a group can produce different $\Sigma$.
Absolute accurate of informations or complete knowledge of the world is
completely meaningless. Because the factor that creates the systems is a small
part of the world that created the components of the systems and they are the
components of the systems that determine the size and type of $\Sigma$. In
this process, the systems act like a mirror, and so the type of mirror and its
range of images determines the size and type of $\Sigma$. In other words, if
we represent the set of all phenomena and events in the world for all times
$t\leqslant t_{0}$ with $U(t_{0})$. There is a subset $\Lambda(t_{0})\subseteq
U(t_{0})$ that create the collections of systems such as
$w(t_{0})=(w_{\alpha}(t_{0}))$ such that $\Phi(\Lambda(t_{0}))=w(t_{0})$ for a
function $\Phi$. In this case, it is clear with this symbol that
$\Sigma(t_{0})=B$. It follows that
$\varphi(A(t_{0}),\Phi(\Lambda(t_{0})))=B(t_{0})=\Sigma(t_{0})$. With notice
to the above relations, $\Sigma(t_{0})$ is exactly dependent on $A(t_{0})$,
$\Lambda(t_{0})$, $\varphi$ and $\Phi$. In other words, $A(t_{0})$ and
$\Lambda(t_{0})$ is the domain of our observation and $\Sigma(t_{0})$ is our
observation of the world under the functions $\varphi$ and $\Phi$. The rule of
the functions $\varphi$ and $\Phi$ are transformation of domain into our
observation of the world $\Sigma(t_{0})$. Due to the physical and chemical
event that happened in the world, it has led to the formation of these
functions and finally $\Sigma(t_{0})$. The process of each parts of
$\Sigma(t_{0})$ is the result of physical factors. The concept of prior
decision making or planning is meaningless in this process. In other words, in
the plant system, when the leaves react to light or cold, it does not mean
that the plant has made a pure decision or planning, but it is the molecular
and chemical reactions of the plant that lead to these reactions or, in
simpler words, when water freezes in the cold, in this case the water does not
decide to freeze, but a chemical and molecular process takes place. For
complex and complete systems, the output result of a system is the same
molecular process (chemical and physical). Our answer to this curiosity that,
why and how these processes started and continues, it will be within the
domains of $A(t_{0})$, $\Lambda(t_{0})$ under the functions $\varphi$ and
$\Phi$. Simple systems behave almost according to fixed and specific rules in
a period of time, but complex and complete systems have a type of
characteristic that they analyze their phenomena and their internals every
moment. In this regard, complete systems seek to make meaningful; are their
own findings. They need construction for their existence, the reason for their
existence and their behaviors. These constructions can be built based on basic
principles and $ML$ that to minimize paradoxes and contradictions in it.
Finally, the set of these structures will be a part of $\Sigma\subset U$ by
complete systems.

##  5\. The primary consciousness of a system

In the last few years, a number of works have addressed the problems of
primary consciousness within the framework of cognitive science and
neuroscience. The contents of awareness are to be understood as those
information content that are accessible to central system and brought to bear
in a widespread way in the control of behaviour.

Awareness is a purely functional notions, but it is nevertheless to primary
consciousness experience. In the defining the output $w$, we must be careful
that the components $X_{1},\cdots,X_{9}$ and $F$ interact exactly with each
other, and this effect is done continuously, and we may not be able to
understand the distance of the effect.

Given that there is currently no precise definition of primary consciousness,
we consider primary consciousness to be the initial term that is either
entered into the system by the sensors or created by the components we have
mentioned. Because of this, we can also primary express consciousness for
simple systems such as plants, where the plant system, after receiving a
message from outside through its sensors, shows an function that this primary
consciousness may be a reaction within. Calvo P (2017) and Anthony Trewavas,
František Baluška, Stefano Mancuso and Paco Calvo (2020) have been studied
about consciousness of the plants. In these manuscripts, maybe for a plant
this consciousness works as an intracellular reaction and at the molecular
level. Sensing of their ambient temperature allows plants to regulate
germination and flowering time and to adjust their architecture. Extreme
weather events such as suboptimal temperatures, altered water availability,
and high soilion content, result in abiotic stress which can lead to decreased
growth. In fact, this reaction of plants in front of environmental changes
around them shows a kind of consciousness of plants. In the following
chapters, we will explain the structure of systems, which shows that the
consciousness of plants or viruses is logically similar to the consciousness
of complete systems, and they differ only in quantity and quality.

To better explain primary consciousness for animal, David J Chalmers (1996)
has been described in his book that when human perceive, think, and act, there
is a whir of causation and information processing, but this processing dose
not usually go on in the dark. There is also an internal aspect, there is
something it feels like to be a cognitive agent. This internal aspect is
primary conscious experience. In this section, we mean the consciousness or
experience of a system, the same as their initial part, i.e. $X_{2}$.

In some of the information that humans receive as a system through their
sensors and are translated into primary concepts as consciousness, the process
is more evolved than a similar process in animals, and similarly it is more
evolved than insects and lastly in plants. In humans themselves, the initial
translation of input information is different. Therefore, it will make it very
difficult to find a precise and comprehensive definition of consciousness.

At a certain point in time, we may experience a sense of happiness, sadness,
or worry for no reason and without understanding the information in our
awareness, and think that there must be something in this regard that creates
this state in us. It may be that we feel right and it is something we have
forgotten, and the connection of our awareness system with memory can not be
effectively established, but an issue here can make us think that the
consciousness component plays an important role in this. In some discussions,
this kind of real feeling, which is related to an issue and subject that our
mind is not aware of at that moment, is attributed to the subconscious mind,
while here for this feeling, components $X_{1},X_{2},X_{6},X_{9}$ have
important rule for creating this real feeling.

In ancient times and in some cultures, this issue was attributed to
transcendental forces, so consciousness component plays a very important role
in the justifying this issue. We must consider that our consciousness and
other system components are often interrelated. Even we have this connection
when we have no awareness of ourselves. For example, when we are asleep, the
awareness may not work, but these connections exist.

As we said, consciousness is an initial translation that the sensor of a
system receives information from outside or inside, and this may be with or
without awareness for different systems. To describe and understand
consciousness, we need to examine it for simpler systems, such as insects or
animals specific, and then compare it to human consciousness. If we consider
animals or insects, the $X_{2}$ component (consciousness) works differently.
This initial translation, which occurs in $X_{2}$, has different functions in
different systems. For example, when human, other animal or insect sensors
observe or inhale a color, different translations for them occur in $X_{2}$.
The important point here is that when we understand a concept, it occurs when
we perceive an image or the original translation in $X_{2}$ that does not
occur in other animals or insects.

Now if we consider human as a system, the image that all components create in
$X_{2}$ is different. Consider that when we derive a so-called correct result
from a subject or problem, a different image emerges in $X_{2}$ as opposed to
a contradiction created by a subject in our minds. The quality of other
components has a major impact on the type of image creation or initial
translation in $X_{2}$. Here not only is the component $X_{2}$ dependent on
other components, but all components are interdependent and interact. Even in
a human-like system, different images of a phenomenon in $X_{2}$ are created
for different people. This leads us to not have a precise definition of
consciousness.

Now, if we want to define consciousness for all systems, the situation becomes
much more complicated. If we consider simpler systems such as plants, we can
more easily understand the input and output of a system. When plants react to
sunlight and some of them change the direction of their leaves to receive
sunlight, we can easily observe the function of the $w$ function in this
simple system. In this system, the plant’s sensors transmit information into
the system, received by $X_{2}$ and translate it initially, and its output
changes the direction of the leaves in the direction of sunlight. Of course,
it should be noted that the consciousness of plants in the face of other
phenomena such as temperature change, in the plant system causes another
output. Here plant consciousness is the simple molecular reaction that is
simpler than animal consciousness. But they have fundamental similarities, and
the difference between them is that we can explain that animal consciousness
is a generalization of plant consciousness.

Likewise, for many phenomena, the function $X_{2}$ for human is an extension
of animals in the some parts. If we look at this generalization purely, we can
say that this generalization continues and this is a natural process that can
happen easily. For higher generalizations we can never get information but as
we said, they are possible and that makes it even harder to find the exact
definition.

Here we can consider the following models for the consciousness component
$X_{2}$, which are different for each systems.

  1. a.

$s_{1}=$ Reacting of molecular reactions

  2. b.

$s_{2}=$ Phenomenal or psychological concept of mind (Primary translation by
mind).

  3. d.

$s_{3}=$ First translation by artificial intelligence

In this way, for some of different system consciousness component $X_{2}$ is
working in the following way:

  1. (i)

For plant, we have $X_{1}=X_{1}(s_{1})$.

  2. (ii)

For artificial intelligence, we have $X_{1}=X_{1}(s_{3})$.

  3. (iii)

For animal, we have $X_{1}=X_{1}(s_{1},s_{2})$.

In describing the consciousness that we described as the primary translation
of input information, the important point is that we also translate the
initial or imaginary for all mental events by the component $X_{2}$, such as
the correctness or in-correctness of a proposition or an deducing or
contradiction. As we said, not all concepts can be originally translated in
all systems, even if it receives the sensors of an information system, the
$X_{2}$ component may not be able to translate it into the original. In other
words, in any system, $X_{2}$ is not capable of initial translation of all the
information input by the sensors. Some input information may be initialized by
$X_{2}$, but others may not. Of course, it is also important here that our
sensors, indicated by the $X_{1}$ component, can not receive much information
and transfer it into the system. Consider the artificial intelligence as a
system, which is unable to understand emotions. The reason is obvious. First,
the $X_{2}$-component in artificial intelligence is not able to translate
emotional issues as well as animals. Second, it does not have the necessary
components to understand the emotions. It is clear that the role of $X_{2}$ in
the processing of input data is very fundamental. Without primary translating
the input information, the other components will not be able to any functions.
When two systems receive a message from the outside, they may have different
types of initial reception and translation. Now, if they are in the same
category, for example, they belong to human systems, the type of receiving the
message and its initial translation will be almost the same. We can observe
the reason for this message similarity in the output of the systems. For
example, almost all people react to happy, sad or emotional songs in the same
way, and the difference are mostly in the intensity or quantity of the
reactions. Therefore, the output of receiving the message, i.e. $X_{1}$, and
its initial translation, i.e. $X_{2}$, in the systems that are in the same
category, are almost the same.

##  6\.  sensors theory and structures of a system experience

Our consciousness experiences depend on stimuli which arrive at our various
sensory organs from the external world and these stimuli cause changes in the
states of our brain, ultimately causing us to feel a sensation which may be
pleasant or unpleasant. The desire of animals to achieve a goal makes them
move their bodies in a certain way in a certain direction to achieve what they
want, but how is it possible that conscious experiences can arise out of an
inert lump of gray matter endowed with electrochemical properties? How does
someone’s desire cause that individual’s neurons to fire and his muscles to
contract in exactly the right manner? In this section, we want to
fundamentally examine how the experiences of a system were created and
according to what criteria its activity and behavior are set. In this regard,
the theory of sensors may be a reasonable answer to these issues from a
philosophical or perhaps a biological point of view.

Chalmers, D. J. ( 2009) raised the problem of human or animal consciousness as
follows.

The easy problems of consciousness include those of explaining the following
phenomena:

  1. (1)

the ability to discriminate, categorize, and react to environmental stimuli

  2. (2)

the integration of information by a cognitive system

  3. (3)

the reportability of mental states

  4. (4)

the ability of a system to access its own internal states

  5. (5)

the focus of attention

  6. (6)

the deliberate control of behavior

  7. (7)

the diﬀerence between wakefulness and sleep

As Boly, M (2013), consciousness depends on the integrity of certain brain
regions and the particular content of an experience depends on the activity of
neurons in parts of the cerebral cortex. It is not known why the cortex
supports consciousness when the cerebellum does not, despite having four times
as many neurons, or why consciousness fades during deep sleep while the
cerebral cortex remains active.

Giulio Tononi (2004) has been studied main problems of consciousness of
animals. In this paper, the first problem is understanding the conditions that
determine to what extent a system has conscious experience. For instance, why
is our consciousness generated by certain parts of our brain, such as the
thalamocortical system, and not by other parts, such as the cerebellum? And
why are we conscious during wakefulness and much less so during dreamless
sleep? The second problem is understanding the conditions that determine what
kind of consciousness a system has. In this paper, he presented a theory about
what consciousness is and how it can be measured.

In this section, we will try to give a kind of answer to some of the above
questions. To explain the preceding problems for consciousness, we need to
symbolically explain the process of forming a system and its operation. We
also discuss on the other mathematical aspects of consciousness for a complete
system. We mean from ‘conscious experience’ (or shortly, experience) is
denoted all of impressions, feelings, thoughts, perceptions, etc., which an
experiencing of a complete system through at a particular instant of time. If
we want to provide a precise definition for human experience, it is necessary
to have a precise definition of the experiences of all living beings, which
will not be possible due to the indefinability of living beings. On the other
hand, human experiences can depend on her environmental and social conditions.
Suppose humans lived almost in isolation like some animals, or there were no
plants called grains on the planet that lead to the formation of communities,
then the experiences of humans would be different. Because of this,
environmental conditions, materials around us and different social conditions
lead to the creation of different experiences. Therefore, the collection of
all human experiences cannot have a precise and definite structure that we can
recognize all of them or determine one or more characteristics for all of
them. However, we can understand our experiences and study and research them.
To answer a number of questions, we need to know how a complete system works.
Therefore in the below, we will symbolically examine how a system comes into
being and evolves which ultimately leads to the emergence of the experience of
consciousness in beings.

We know that all materials, including chemical substances, react to changes in
the environment around them. These reactions can be in the form of basic and
molecular changes in chemicals materials. In other words, chemicals can be
sensitive to changes in their surroundings. Now, we consider a chemical
material called $\gamma$ that is sensitive to changes in its surrounding
environment. Here, these sensitivities are mostly in the form of molecular,
chemical or physical reactions. Various messages such as temperature changes,
sunlight and substances in the surrounding environment affect on the chemical
reactions inside of $\gamma$ in the over time. This process maybe to increases
its sensitivity to the environment (its cause is the change in the structure
and molecular reactions of $\gamma$ that occur in a period of time). So the
increasing $\gamma$-sensitivity creates changes within $\gamma$ chemically
that sensitivities become detectable. Now if $\gamma$ develops chemically to
recognize the messages, it can compare them by its sensitive. We denote this
set of sensitivities of $\gamma$ with respect to our environment by $X_{1}$.
Over time, as a result of $X_{1}$’s activities, secondary sensitivities are
created in the system, which shows sensitivity to $X_{1}$ group. We denote
this new set of sensitivities by $X_{2}$. The elements of the set $X_{2}$ have
sensitivity to the elements of the set $X_{1}$, in other words, each element
of $X_{2}$ shows a certain sensitivity to an element or a number of elements
of $X_{1}$.

The important point here is that $X_{1}$, in addition to being sensitive to
the surrounding environment, is also sensitive to internal reactions, even
$X_{2}$, and this process or cycle continues regularly. We can show the
process (4.1) as below

$\displaystyle\alpha\Longrightarrow X_{1}(\alpha)\Longleftrightarrow
X_{2}(X_{1}(\alpha)),$ (6.1)

where $\alpha$ is a message received from environment. In the continuation of
this process, another new generation of sensitizers are created that are
sensitive to the changes and performance of $X_{2}$, we show an example of
these sensitizers in the form of a category with $X_{3}$. Again, we are
reminding that all these sensitivities are molecular, chemical, and physical
that happen inside $\gamma$. $X_{3}$ elements react functionally depending on
the changes of $X_{2}$ elements. Of course, in this process of the so-called
evolution of $\gamma$, $X_{3},X_{4},X_{5}$ may appear in parallel as next
sensors inside $\gamma$. At this stage, the chemical composition $\gamma$ will
look like a simple system and we call it a system. Now here, the components of
the system including $X_{1},\cdots,X_{5}$ sets have a special sensitivity to
the their elements that have been created over time. As we said, these
sensitivities started step by step, which are caused by environmental factors
outside the system and chemical and physical reactions inside the system. The
process of the forming a system depends on environmental factors, the chemical
composition within the system and the effect of these two factors on each
other in a long time process. The process of evolution of a system according
to its chemical and physical state and its relationship with its internal and
external environment is started by the sensitizers of the elements of the set
$X_{1}$. The process of the changes and behavior of $X_{1}$ elements over time
leads to the creation of second-generation sensitizers to $X_{1}$, which we
called $X_{2}$. Assume that the sensor elements of the sets
$X_{1}=\\{A_{1},A_{2},\cdots,A_{n}\\}$ and
$X_{2}=\\{B_{1},B_{2},\cdots,B_{n}\\}$. We assume that for each element
$A_{i}$, its correspondence element in $X_{2}$ is $B_{i}$ for all $1\leq i\leq
n$. In other words, $b\in B_{i}$ is a sensitivity that is sensitive to the
interactions of $a\in A_{i}$. Now we can define a function $f_{12}$ from
$X_{1}$ into $X_{2}$ such that $f_{12}(A_{i})=B_{i}$ for all $1\leq i\leq n$.
For animals, let $A_{1}$ and $A_{2}$ be the set of auditory and visual
sensors, then $B_{1}$ and $B_{2}$ have been translated to the experiences of
mind, respectively. Here $b_{1}\in B_{1}$ and $b_{2}\in B_{2}$ are sensitive
to changes and molecular reactions of $a_{1}\in A_{1}$ and $a_{2}\in A_{2}$,
respectively. Therefore, according to the above explanation, there will be
similar functions from component $X_{2}$ to other components. Now let
$X_{3}=\\{C_{1},C_{2},\cdots,C_{n}\\}$, there is a function such as $f_{23}$
from $X_{2}$ into $X_{3}$ where $f_{23}(B_{i})=C_{i}$ for all $1\leq i\leq n$.
The function $f_{23}$ determines our awareness of the elements of the set
$X_{2}$. In fact, the elements of the primary functions that connect the
system components (these transfers can be caused by nervous systems) are as
follows:

$\displaystyle\beta=\\{f_{ij}:X_{i}\rightarrow X_{j}\mid~{}~{}1\leq i,j\leq
n\\}.$

The combination of elements $\beta$ represents the activity of the system and
converts input messages or internal processes into an output message. Here,
the components of the system are interdependent. For example, the performance
of the elements $X_{5}$ depends on the changes that occur in the elements of
$X_{2}$ and $X_{3}$. The performance of the elements of some components of a
system can depend on the elements of several components. For example, consider
the sensitivity of an element of $X_{4}$ that simultaneously reacts to changes
in elements $X_{3}$ and $X_{5}$. More simply, assume we experienced two sounds
as $c_{1},c^{\prime}_{1}\in f_{23}(f_{12}(A_{1}))$. These experiences can
become the process of another experience in $X_{4}$, as sensitivity to two
components $c_{1}$ and $c_{1}^{\prime}$, where this sensitive so-called
comparison between two experiences which is created in $X_{4}$, we call it the
logical sensitivity of the system. The collection of the logical sensitivities
of the system, which are recorded in different ways in the memory component,
are the basis of analysis component, i.e. $X_{4}$. The activity of logical
sensitizers can be the result of events that occur in nature. In fact,
sometimes system sensitizers create a one-to-one correspondence between
themselves and events in nature. Now, the collection of all sensors, namely
$X_{i}$ where $1\leq i\leq 10$ and the functions of $\beta$, are a basis for
the system’s activity. Diagram (4.3) is a complete symbolic diagram of the
activity process of the components of a system by combining these functions.
We should pay attention to is that the activity of first level sensors, i.e.
$X_{1}$ causes the emergence of second level sensors, i.e. $X_{2}$. The first
level sensors also respond to the activities of the second sensors and this
process favors their development and subsequently this process improves the
potential of the second sensors. This process continues over time for the
development of next-generation sensors and the process of developing the
system components. These process of the system development is actually the
evolution of the system. Sensors or components of a system are connected by
nerve cells, and this connection has led to the evolution of next generation
sensors by stimulating next generation sensors (as Steven Laureys and Giulio
Tononi (2009), 100 billion neurons, connected by trillions of synapses,
emerges our conscious experience of the world and of ourselves). Within a
system the important point is that if sensor $\alpha$ is created by the
activity of sensor $\beta$, and so the activity of sensor $\beta$ can lead to
the evolution of sensor $\alpha$, and this process can continue.

When we are facing a blank screen that is alternately on and off, and we have
been instructed to say ”light” when the screen turns on and ”dark” when it
turns off. A photodiode a very simple light-sensitive device has also been
placed in front of the screen, and is set up to beep when the screen emits
light and to stay silent when the screen does not. The first problem of
consciousness boils down to this. When you differentiate between the screen
being on or off, you have the conscious experience of ”seeing” light or dark.
The photodiode can also differentiate between the screen being on or off, but
presumably it does not consciously ”see” light and dark. What is the key
difference between you and the photodiode that makes you ”see” light
consciously? (Tononi G. (2004)).

The most accurate answer that can be given to this question is that in
addition to the fact that the quality of our sensor inside $X_{1}\cup X_{2}$
is different from the photodiode sensor, we have other sensors that have a
fundamental effect on the quality of our output.

Sensors have the role of translating (such as primary translation, analyzing,
psychology reaction and others) or storing input information in a system.
Input information may be transferred from one sensor to another sensor after
changes as (3.4) diagram. Compatibility between the sensors and the input
message inside the system leads to a kind of satisfaction of the system.
Compatibility between the system and its input information means that the
system can identify the message or provide a way to identify it based on the
potential of its sensors (the initial potential and the experiences it has
acquired and led to the change or improvement of the system’s sensors).
Identifying a message means that the system’s sensors perform their normal
actions within a defined range after receiving the message. Inconsistency
between the system’s sensors and incoming messages can lead to system
confusion and abnormality. As example, sensors of component $X_{2}$ and
$X_{9}$ may immediately send message $\alpha$ to sensors $X_{4}$ and $X_{6}$,
here if there is no compatibility between $\alpha$ and sensors of $X_{4}$ and
$X_{6}$, then disturbances will occur in the system. In this regard, other
sensors will be affected by other components until the sensors of the system
may create some kind of adaptation by increasing their potential, or they may
lose their natural balance and go out of the so-called normal state. Easier
examples for this are the experiences of a infant seeing a new animal or the
experiences of an employee of his promotion or demotion.

Our conscious experience can be classified as follows:

  1. (1)

Experiences that are directly caused by the effects of materials or events in
the nature (we show by $R_{1}$).

  2. (2)

Experiences that arise within the system according to the information within
the system and they are not directly the result of environmental factors
outside the system (we show by $R_{2}$).

$R_{1}$ is a result of the sensitivity of the sensors to the environment
around the system, and $R_{2}$ is the sensitivity of the sensors to the
internal information of the system and the behavior of the sensors in the
system. It is obvious that $R_{2}$ at a particular time is a function of
$R_{1}$. When we experience an apple falling ($R_{1}$), we understand the
force of gravity and design mathematical formulas for similar events
($R_{2}$). On the other hand, when two people analyze each other’s behavior to
find common ground, the quality and extent of $R_{2}$ plays an important role
to achieve this goal. The spread and large size of $R_{2}$ in humans has led
to the difference between humans and other creatures. There are the activities
of the second (or next) generation sensors that lead to such a distinction.
The development and expansion of $R_{2}$ can lead to changes in the
experiences of $R_{1}$ and even expand its quality and scope. The result of
the development of $R_{2}$ has led to the compatibility and consensus of
humans and the formation of large communities which is the result of the
performance of the next generations of the sensors.

The experience of self-awareness lies within $R_{2}$. The question of the
awareness of a complete system for itself is the result of the relationship of
the system sensitizers with respect to each other. As two systems in one place
can show sensitivity to each other and the environment, that is, they can be
aware of each others, so the integration of these two systems and the
formation of a new system consisting of them can explain the self-awareness of
the system. In other words, consider two systems $A$ and $B$ that can be aware
to each other. Now, if these two systems are merged and their sensors react
with respect to each other, then the awareness of the new system to itself can
be clearly seen. Suppose that the set $\Sigma$ is all of functions that make
connection between two systems sensors $A$ and $B$. Let $\varphi$ be a
function from $\Sigma$ into $[0,1]$ where $\varphi(f)$ shows the quality of
sensors connections between $A$ and $B$ for all $f\in\Sigma$. For example, if
$\varphi(f)$ near to zero, the connection between to sensors $A$ and $B$ is
weak. The quality and activity of the function $\varphi$ and elements of
$\Sigma$ can change at different times, so it affect to the connection between
two systems. A system that has self-awareness about itself follows this rule.
In other words, such systems work similarly to integrated systems. Controlling
our behavior and will can also follow this rule. Due to the fact that
sensitizers show sensitivity and reaction to each other, there is a
sensitivity which, in addition to reacting and showing sensitivity to other
sensors, is also sensitive to the output of the system. If we call this
sensitizer the ultimate sensitizer, in fact it is that makes us aware of our
behavior and thus this process leads to self-awareness. When a system becomes
aware to itself, this awareness process is an experience that we call by
$\mu$. Now again, the system can be aware of both $\mu$ and itself, and this
experience will be different from the experience of $\mu$. This process can be
continue. Therefore, the problem of merging two systems can be the best
solution for system self-awareness. In fact, systems that have the ability to
self-awareness are similar to systems that are composed of the integration of
two or more systems. This correspondence explains the ability of a system to
access its own internal states, how the will of a system to process itself and
produce an output, how the will of the system reacts in the face of a problem
and how the system makes a decision in a moment. Now, in order to explain the
consciousness of humans towards themselves or their awareness towards their
own consciousness, it follows this rule of integration of systems. Of course,
the communication between systems, $A$ and $B$ may not be established in some
conditions or at any moment, so in this case, the system created by the
combination of two systems $A$ and $B$ may not be aware of itself. According
to the above explanation regarding the evolution of system sensors, it is
clear that each system has undergone a special evolution. Therefore, a system
consciousness depends on the potential of the system and how its sensitizers
evolve. Sensitizers evolve according to the molecular conditions of their
systems and the influence of the external environment on them. These changes
do not follow any specific rule and as we said, they are dependent on
environmental factors and molecular conditions of the system. In fact, these
events do not have a special rule in nature, for example, if we put fifteen
beads in a bag and all five of them are of the same color, the probability is
that we will take out five beads of the same color in a row is very weak, but
it is possible with more repetition. In other words, when we randomly draw one
die from the bag at a time, the probability that five consecutive dice will be
of the same color will depend on the number of repetitions. Here, as a result
of repetition, all the expected situations can happen, and in some situations,
it seems as if there is an order in the selection of beads. The conditions in
nature that have led to the creation of sensors in systems have caused the
sensors to have some kind of compatibility with these conditions, which has
led to the complete system defining the order of this situation. In other
words, the order we observe is the compatibility of the behavior of the
environment with the behavior of the sensors, which was created by repeating a
natural process. The consciousness of systems has also been created as a
result of this process, and it can be said that different natural processes
can create different consciousness according to the activity of sensors.
Therefore, we cannot have a standard and complete definition for
consciousness. To prove this claim and to prove the contents presented above,
we consider the following axiom.

Consciousness axiom 1: If a being has not any experience of consciousness
ability at time $t$, there is $\epsilon>0$ such that this being has not any
experience of consciousness ability in the perioud of time
$[t-\epsilon,t+\epsilon]$.

Consciousness axiom 2: In the normal conditions, if a being has the experience
of consciousness ability at time $t$, there is $\epsilon>0$ such that this
being has the same experience of consciousness ability at time $t-\epsilon$
and $t+\epsilon$.

In the Consciousness axiom 2 our main of normal conditions means that
momentary events such as instant death, instant senseless, and the like do not
occur for this creature in a neighborhood of time $t$.

A spermatozoon joins an ovum to form a zygote, in this case, we represent this
state (which includes all biological, chemical, molecular or physical
characteristic of this substance) by $a$ and the condition that a human or an
animal (for convenience, we denote by $\gamma$) has the consciousness
experience by $b$. In this case, $\gamma$ moves the distance between $a$ and
$b$ so that its consciousness ability is complete in the stage $b$. Therefore,
$\gamma$ does not have any consciousness experience in the stage $a$ and has
optimal consciousness experience in the stage $b$. Now we let state $a$ and
$b$ occur at times $0$ (or $t_{a}=0$) and $t_{b}$, respectively. Now assume
the time $t_{x}\in[0,t]$ represents the state and characteristic of
$x\in[a,b]$, then we define a function $f$ from $[a,b]$ onto $[0,t_{b}]$ where
$f(x)=t_{x}$. We define a set as

$\Delta=\\{t_{x}:~{}\text{the~{}consciousness~{}experience~{}
exists~{}for}~{}x\in[a,b]\\}.$

Since $t_{b}\in\Delta$ and $\Delta$ is bounded from below, we put
$\inf\Delta=t_{c}$. By Consciousness axiom 1, it is clear $0<t_{c}$. Now for
all $t_{x},t_{y}\in[0,t_{b}]$ with $t_{x}<t_{c}<t_{y}$, $x$ has not
consciousness experience ability, but $y$ has. According to the above axioms,
we encounter a contradiction when $t_{x}$ and $t_{y}$ are considered close
enough. This conclusion shows that we are not able to determine when a living
being reaches the stage of conscious experience from birth to death. This
argument is also valid for the transformation of proteins into single-celled
organisms and finally into intelligent multicellular organisms. Therefore, in
the evolution of the chemical composition $\gamma$ that we discussed above, we
are unable to determine when $\gamma$ is capable of experiencing
consciousness. This issue is a strong proof of the theory of sensors that we
presented in this section.

We have a lot of information stored in our memory, but we don’t use all of
them at the same time (or in a period of time) or we are not awareness to all
of them in the period of time. The duty of extracting this information from
the memory is available to the sensors. In one of these cases, environmental
factors stimulate the sensors, and the sensors extract related information
from the memory in this reaction. In another case, chemical changes in the
body can stimulate the sensors, and as a result, essential information can be
extracted from the memory. Of course, another way of extracting memory
information can be directly due to the communication of our system components
by sensors. Due to the fact that the sensors are active and changing
consciously or unconsciously, and this activity exists even in sleep, as a
result of these activities, part of the memory information can be extracted.
For example, if we are supposed to wake up in the middle of the night and do
something important, the sensors will take over this role. We may feel that
there is a something that worries us or makes us happy, without touching the
reason, and after we think carefully about the experience we may find the
cause, which can be a real issue or event. For example, for a moment in a
ceremony, our mind gets confused and then we realize that something happened
or that we have to do something, of course, this situation can also happen in
a dream. In any case, all events are caused by the activity of sensors. If we
think about a topic and then leave it and focus on other topics, our sensors
may follow the first topic in our subconscious mind. In general, when we
encounter a new experience, the sensors can extract the memory information
that is related to new experience and send them to other component of our
system such as awareness.

As we said, system sensors can be developed, and when messages enter the
system memory, they are regularly processed by sensors. These processes and
the performance of sensors can produce other messages that are different from
the original message. The images that human can create in her/his mind can be
related to this issue. In human societies, when humans can live in large
groups, their sensitizers show more activity, and this issue causes a very
rapid development of sensitizers, including the creation of special skills and
talents in humans. An important point to pay attention to in the discussion of
consciousness is the degree of consciousness of a system. The degree of
consciousness of a system is exactly dependent on the physical conditions of
the system’s sensors and the relationship between system components. For
example, a person’s level of consciousness is exactly dependent on the
physical conditions of the sensors in her/him brain.

The clear point here is that during sleep, the activity of the sensors are
very limited and the relationships between our components are minimized. For
example, when the connection of the $X_{3}$ or $X_{8}$ component with other
components of us becomes very weak in the during sleep, we cannot control and
be aware of the output of our thoughts, and we may enter into involuntary
dreams, which are called dreaming. Basically, sleep or some states with less
alertness are the result of decreasing the activity of our components and
weakening their connection. Of course, when we are awake, our level of
consciousness is adjusted according to the relationship of our components with
each other and the performance of sensors. Yuval Nir and Giulio Tononi (2010)
have been studied dream phenomenology. In that paper, authors showed that
dream consciousness is remarkably similar to waking consciousness, although
there are several intriguing differences in volition, self-awareness and
reflection, affect and memory, and there is considerable variability between
individual dreams. The neurophysiology of REM sleep, and in particular recent
insights into its regional activity patterns, offers a useful starting point
for relating dream phenomenology to underlying brain activity. However, the
initial equation of REM sleep with dreaming has been shown to be inaccurate.
Thus, it is time that researchers moved beyond sleep stages when trying to
link dream consciousness to neuronal events, and focused on more subtle
features of brain activity in space and time. One’s disconnection fromt he
external environment when dreaming poses a central unsolved paradox, the
answer to which might be instrumental for understanding dreams. Converging
evidence from multiple fields of study, including phenomenology, development,
neuropsychology, functional imaging and neurophysiology, support the notion
that dreaming might be closely related to imagination, where brain activity
presumably flows in a ‘top-down’ manner. Viewing dreams as a powerful form of
imagination can help explain many of their unique features, such as sudden
transitions, uncertainty about people and places, poor subsequent recall and
disconnection from the environment, and offers testable predictions for future
studies.

Consider the components of a system. The quality and potential of the
components of the systems are different and depend on various times. For
example, our ancestors had different components than us hundreds of thousands
of years ago and maybe our components be different in the distant future than
today. Neanderthal human had different sensors compared to modern humans and
animals have different sensors. Therefore, we have not a general framework to
formulate or express a specific structure for all sensors of a complete
system. In fact, the reason for this is related to the chemical and physical
changes of a system at different times and its effect on the sensors of the
system.

We may wonder why a person who knows logic shows contradictory and illogical
behavior in the face of problems. In other words, even though a person knows
logic, she/he may not have a desire for logical requests that are contrary to
her wishes and interests. In fact, the cause of this issue is related to the
performance of our sensors. No matter how much we teach logic to a person, it
is possible that in the end she/he will show illogical behavior where her
interests or special psychological conditions demand. It is the structure of
the sensors and the relationship between the system components that ultimately
makes the final decision, and this is exactly the answer to the mentioned
problem.

The reason living things look for food or reproduce is rooted in the molecular
evolution of sensors. In the evolution of chemical substance $\gamma$, the
molecules inside $\gamma$ are constantly affected by their external substances
in a stage of evolution, at this stage a generation of sensors is born that
are sensitive to this process. The activity process of these sensors leads to
the tendency of $\gamma$ to absorb more specific substances from around
$\gamma$, and the continuation of this process causes the creation of other
generations of sensors, which leads to the search for food sources and the
reproduction of $\gamma$. Of course, when introducing the components of a
system, we put this sensor inside $F$. For convenience, we call this sensor
$Y$. Several factors may lead to the emergence of the sensor $Y$, for example,
the energy requirement of previous generations of sensors can lead to the
activity of this sensor. All these steps have occurred during the hundreds of
thousands of years of sensors’ activity, and as we explained the possibility
of choosing five beads at the same time, this evolutionary process for
$\gamma$ has worked out for millions of $\gamma$ due to the repetition of
these events. The activation of one type of sensor leads to the activation of
other types of sensors. For example, as a result of the activity and evolution
of the sensor $Y$, the next generation sensors that have the task of
replicating the system are produced. This explanation is a symbolic
explanation of the evolution of $\gamma$-sensors, but its detailed explanation
is related to biology. The evolution of a system’s consciousness is directly
dependent on the evolution of a system’s sensors. The activity rule of the
sensors of a system is a method to explain the different operational codes of
the system’s consciousness. Consider the component $X_{7}$, it consists of a
classification of sensors that are sensitive to almost all the system
components activity which is a justification for the integration of
information by a cognitive system. Of course, the component sensors $X_{7}$,
may appear after the evolution of most sensors or evolved in parallel with
them.

In the meantime, the point that is important and we should know it is that the
concepts that we are interested to them and important to us are the results of
the activities of the sensors that have led us to these desires over the years
of evolution. The activity of the sensors does not guarantee us to always
choose a logical and so-called correct way. Illusory or melancholic thoughts
can also be the results of the activity of the sensors and this is not
abnormal. Of course, the quality of a system depends on the activity of the
sensors, and if the activity of some sensors is disturbed due to improper
feeding or accidents, we can see the effect of this change on the quality of
the system. Voluntary and involuntary behaviors differ only in the activity of
some sensors and their relationship. Our willpower and our decision-making
depend on the behavior and quality of the sensors. The activity and quality of
our sensors can increase or decrease under the influence of the environment
and conditions of training or others experiences. The effectiveness and
performance of sensors defines the difference between humans and artificial
intelligence. For example, when someone asks us a question, the purpose of
this person may not be the answer, and we may understand it. In fact, our
understanding of the purpose of this person depends on the complexity and type
of activity and function of our sensors, while artificial intelligence does
not have this capability.

The concept of time and space as an experience was created by the reaction or
response of the sensors to the environment. According to the previous
explanations, the sensors react or show sensitivity both to the environment
and to each other. The result of this operator has led to the creation of the
concepts of time and space as the experience of a system. The experience of
space and time is different in various systems. For example, compared to
humans, other animals such as insects and fish have a different experience of
space and time. Human sensors have different functions from birth to
adulthood, and therefore our experience of time and space during childhood is
different from our experience during adulthood. In fact, the performance of
our sensors leads to the experience of time and space, and this experience
depends on the behavior of our sensors. As mentioned, the experiences of a
system are the result of the activity of the environment and chemicals inside
and around it. Therefore, different chemical compositions and environmental
conditions can create a system with different experiences. Knowing the world
as an experience also follows this role. In other words, environmental changes
and the chemical and physical state around a chemical substance $\gamma$ lead
to the activity of its sensors, and in a time process, different generations
of sensors are created, which lead to the creation of the system and thus lead
to the creation of the system experiences. We know that the concepts of point,
line, plane and space cannot be defined in mathematics and the concepts of
matter and space in physics. Our experience of these concepts is a primary
experience and not the result of other experiences. These experiences are the
result of the activity and reaction of our system’s sensors and we can put
them in the category of our dreams. The real world that we like to see and
touch is within our experiences, which are the result of the activity of our
sensors. Absolute talk of a real or unreal world is meaningless and it is the
sensors inside a system that create a representation of a world for the system
and this display is seen differently in various systems. And if we ask
ourselves this question, what is the real world like outside the category of
our sensors and experiences? This question would be pointless, because any
answer would depend on our experiences and therefore on the activity of the
sensors. The physical laws we have made for the universe are no exception.
These rules are built based on the activity and reactions of our sensors and
work in this domain. In other words, they are defined according to the
correspondence between our sensors and our surrounding environment, and
perhaps these rules can be defined differently in the systems of different
categories. All organisms, living or non-living, have sensors that can respond
to environmental changes to some extent. A question that scientists or
philosophers usually ask is whether an artificial intelligence has experience.
According to the explanations we have given regarding the behavior of sensors,
systems that are in the same category can have many common experiences, but
when the distance is greater, the experiences or, more precisely, the activity
of the sensors will be different. According to our definition of conscious
experience, an artificial intelligence can also have conscious experience. In
fact, the sensors of a system can respond to themselves and the environment in
the form of molecular or physical reactions, this process leads to the
creation of a consciousness experience in the systems that works differently
in each system. A point that may be interesting is that when we experience a
phenomenon in nature, this phenomenon is really the same as we have
experienced or it is in a different form. In the first stage, we may claim
that it must be a phenomenon in nature that we experienced, but when we have
schizophrenia or take certain drugs and drinks, this claim may not be true.
Now, if more people experience the same phenomenon and are confident about
their health, then our claim can be almost true. Now, regarding the phenomenon
that we experience, our knowledge is in the domain of our experiences, so any
result and explanation regarding this phenomenon will be within the framework
of the class of our experiences. Sensory theory can explain many questions,
including how we desire and enjoy life, why we sometimes lose this desire and
pleasure, why we want to live forever and why we afraid of death and
destruction? According to the theory of relativity, when an object accelerates
relative to us, the space and time we have around us (we call this state $A$)
is different from the space and time we observe inside the object (we call
this state $B$). In other words, space and time are contracted inside the
objects that accelerate towards us. Our experience of evaluating the state of
space and time in states $A$ and $B$ is related to changes that occur in our
sensors or physical changes in state $B$ lead to it. In other words, does
acceleration cause changes in the activity of sensors that lead us to see
space and time as relative, or can acceleration directly cause physical
changes in space and time? If we want to be more clear, let’s raise this
issue:

Do physical changes within $B$ leads to our experience of relativity or the
changes that occur in the our sensors?

##  7\. On the Mathematical methods for studying of the consciousness
experience

Kleiner, J. (2020) have been studied some parts of the structure of the system
as models of consciousness by mathematical formalism. The basis of his work
was to go beyond the mathematical representation of experience, and in fact
consider formal hypotheses about how conscious experience relates to the
physical domain, i.e. formal models of consciousness. He give an account of
what warrants mathematical representation of phenomenal experience, derive a
general mathematical framework which takes into account consciousness and
which mathematical structures some of the key characteristics of conscious
experience imply, showing precisely where mathematical approaches allow to go
beyond what the standard methodology can do.

Integrated Information Theory (IIT) has gained a lot of attention for
potentially explaining, fundamentally, what is the physical substrate of
consciousness. Tononi, G. (2008) has been studied Integrated information and
he defined the function $\phi$ as the amount of information generated by a
complex of elements. The foundational concepts behind IIT were extremely
innovative, and it has been very exciting to see certain predictions being
upheld in experiments. It aims to describe both the quality and quantity of
the conscious experience of a physical system, such as the brain, in a
particular state. An IIT aims to specify for each system in a particular state
its conscious experience. As such, it will require a mathematical model of
such experiences. Kleiner, J. and Tull, S. (2020) have been presented the
mathematical structure of this theory. In this manuscript, the basis
relationships of the physical system class and experience spaces have been
investigated. They provided a definition of a generalized IIT, that is, they
studied which exact structures the IIT algorithm uses in the mathematical
description of physical systems and in the mathematical description of
conscious experience, on the other hand.

Many books and articles such as Chalmers, D. J. (1996, 2003, 2009) and others
have been written to examine consciousness or its exact definition. There is
an important question that which physical system can generated the
consciousness, the quantity or level of consciousness of a complete system.

Kleiner, J. (2020) has been design mathematical models for describing
conscious experience and its relation to the physical domain of brain. He
defined a mathematical space that represents conscious experience. He has
shown that this mathematical representation of experience allows to quantify
the ambiguity involved in any reference to experience precisely. Kleiner and
Erik (2021) described both the quality and quantity of the conscious
experience of a physical system, such as the brain, in a special status. In
this contribution, they raised the mathematical structure of the conscious
theory. They have been introduced a class $P$, including the elements $p\in P$
as particular physical system which has some specific physical configuration,
state, or dynamics. $P$ might comprise the set of long-range cortical
connections that make up the global workspace of the brain. They provide a
formal framework for experimentally testing a particular class of theories of
consciousness. The class that they consider makes predictions about the
conscious experience of physical systems based on observations or
measurements. This information is used to create datasets such as functional
networks, wiring diagrams, models, or transition probability matrices. To
formalize this process, they denoted by $\mathcal{O}$ all possible datasets
that can result from observations of $P$, that is, each $o\in\mathcal{O}$ is
one particular dataset, the result of carrying out some set of measurements on
$p$. They denoted the datasets that can result from measurements on $p$ with
$obs(p)$ where $obs$ is a function from $P$ into $\mathcal{O}$. They defined
another function such as $pred$ from $\mathcal{O}$ into $E$, denotes the space
of possible experiences specified by the theory under consideration where
$pred(o)$ is a a subset of all experiences of a theory models for all
$o\in\mathcal{O}$. In that paper, two functions $pred$ and $obs$ are
correspondence functions.

In this part, we try to find some relationships between the conscious
experience and the physical domain of a system. In the following, we show that
all of physical signals occurs in the brain is not bounded. We know that for a
consciousness experience in mind there exists at least one physical signal
occurs in the brain. Let $\mathcal{P}$ and $\mathcal{E}$ be the all of
physical signals occurs in the brain and consciousness experience in mind,
respectively. So there exists a function $\Gamma$ from $\mathcal{P}$ onto
$\mathcal{E}$ where for any consciousness experience set in mind
$B\in\mathcal{E}$ there exists $A\in\mathcal{P}$ such that $\Gamma(A)=B$. It
is clear that consciousness experience set in mind is not bounded. For
example, for each natural number $n$ we have consciousness experience such as
$\alpha_{n}$ and there is a relationships $n\longleftrightarrow\alpha_{n}$
between them . Since the natural numbers is infinite, our the set of
consciousness experience from numbers is infinite. Thus all of physical
signals occurs in the brain is not bounded. This result shows that the
information stored in our memory is not bounded. And as a result, the
processing and analysis of information in our minds will not be limited.
According to the consciousness experience information set $\mathcal{E}$, we
get some information about how the physical domain of brain, $\mathcal{P}$
works. Now if we put the elements of $\mathcal{P}$ that lead to an experience
in the form of a class, then $\Gamma^{-1}$ can be a function. The set
$\Gamma^{-1}(\mathcal{B})=\\{\Gamma^{-1}(B):~{}B\in\mathcal{E}\\}$ is complete
physical domain of brain in the relation with the consciousness experience set
in mind. Considering that all the physical signals of the brain have not
result in the form of conscious experience in this case
$\Gamma^{-1}(\mathcal{E})\neq\mathcal{P}$. It follows that
Domain$\Gamma\subsetneq\mathcal{P}$. So we restrict the function $\Gamma$ on
the its domain and we put Domain$\Gamma=\mathcal{A}$ without any ambiguity. So
we can assume $\Gamma^{-1}(\mathcal{E})=\mathcal{P}$. In this case, there
exists a one-to-one correspondence between $\mathcal{P}$ and $\mathcal{E}$.
Now as notice to (3,5), obviously that $\Gamma^{-1}(w(\alpha))\in\mathcal{A}$
where $\alpha$ is a massage that the system has received. More simply, if we
experience a massage $\alpha$ as frequency of light that is translated to a
color, $X_{2}(X_{1}(\alpha))$, then $\Gamma^{-1}(X_{2}(X_{1}(\alpha))$ will be
a physical domain this color in the brain. The definition of a function with a
specific rule between $\mathcal{P}$ and $\mathcal{E}$, is not possible. Due to
the fact that the set of consciousness experiences is limitless and does not
have a specific structure, so we can not define a rule for this function
between two sets. In order to obtain a rule for the function $\Gamma$, we must
to restrict our domain and range.

Unfortunately, as following, from a mathematical points of view (Set Theory),
such sets and functions can lead to contradictions, of course the mathematical
representation of all our experiences as a set is contrary to the principles
of set theory. Maybe the limiting the domain of our discussion can solve this
problem.

As the above discussion, there is one-to-one correspondence function from
Domain$\Gamma=\mathcal{P}$ into $\mathcal{E}$. $\mathcal{P}$ itself is an
experience, so there is a $p\in\mathcal{P}$ such that
$\mathcal{P}(p)=\mathcal{P}$ that as mathematics looking is meaningless. To
solve this problem, we have to limit the Domain and Range of the function
$\Gamma$ and consider a finite time interval for the set of experiences.

On the other hand, the collection of all the experiences of a complete system
cannot be a set. By way of contradiction, we assume that $\Delta$ is the set
of all the complete system experiences. Then $\Delta$ itself is a experience
of the system, so $\Delta\in\Delta$.

There is another way for making the collection of system experiences. If we
consider all humans as a system $w$, then each human have a finite number of
experiences in a very small time span such as time interval $[t_{1},t_{2}]$.
If $\Delta(t)$ is the every experiences of $w$ for all $s\leq t$, then
$\Delta(t)$ is finite. Now, it is clear that
$\Delta(t_{1})\subseteq\Delta(t_{2})$ whenever $t_{1}\leq t_{2}$. Now we
define $\Delta=\cup_{t}\Delta({t})$, which implies that $\Delta$ is countable
set. It follows that $\Delta$ is a possible experiences for the system. Since
$\Delta$ itself is experience, $\Delta\in\Delta$. Considering that the last
result is contrary with the principles of set theory, so $\Delta(t)$ can not
be as a set. It follows that $\Delta(t)$ is infinite collection of our
experiences. Unfortunately, this last result is contrary to our observations
from the collection of our experiences in very short period of time.

In the following, we will try to define the set of experiences of a complete
system more precisely. Let $A_{0}$ be a set of experiences that do not result
from other experiences such as seeing different colors, feeling different
pains and hearing different sounds, $A_{1}$ be a set of experiences that
resulted from the experiences of $A_{0}$. We can define a function $f_{1}$
from $\cup_{n=1}^{+\infty}\prod_{n}{A_{0}}$ into $A_{1}$, that is,
$f_{1}(x_{1},x_{2},\cdots,x_{n})=y$ means that the experience $y$ follows from
the experiences $x_{1},x_{2},\cdots,x_{n}$ where $x_{i}\in A_{0}$ for all
$1\leq i\leq n$ and $y\in A_{1}$. Here, the comparison of two colors, the act
of counting or the reaction against a pain will be members of $A_{1}$. We
continue this process and assume that the set of experiences $A_{n}$ is
generated by the set of experiences $\cup_{i=1}^{n-1}A_{i}$ and the function
$f_{n}$. Now, it is clear that $\Delta=\cup_{i=1}^{+\infty}A_{i}$. If $A_{0}$
is a set, so $\Delta$ will be a set. The problem here is that we had a
difficulty for defining $\Delta$ as a set, so how does the preceding relation
can hold? Now, if we have a function of the process of our brain’s behavior to
the set of experiences $A_{0}$, we can finally expand this function to all our
experiences. Now we can limit the function $\Gamma^{-1}$ defined above to
$A_{0}$. Thus $\Gamma^{-1}(A_{0})$ is a set of brain processes that are caused
by external stimuli. Therefore, all activities and behaviors of the brain that
lead to a experience are obtained from the extension of the function
$\Gamma^{-1}$ to $\Delta$. On the other hand, according to the definition of
the component $X_{2}$, it is clear that $X_{2}=A_{0}$. Since $\Delta$ is not a
set, $X_{2}$ is not a set. The reason for this problem can be that we have
different experiences from the same phenomenon. For example, when we see red
color with different concentrations and imagining it in a dream make many
different experiences. Therefore, to solve this problem, we put the phenomenon
of seeing, imagining or any experiences the color red in a class as one
element of $A_{0}$. The set of all possible experiences for a complete system
in a time interval can be of any arbitrary size. For example, if we represent
the white color and the black color with $w$ and $b$, respectively, there is a
function $f$ from $[w,b]$ into $[0,1]$ where the elements of set $[w,b]$ are
combinations of colors between $w$ and $b$, that is, $f(x)$ is a the degree of
color indicates how close to white or black when $x\in[w,b]$. $f$ is a one-to-
one and surjective and we define $d(x,y)=|f(x)-f(y)|$ for all $x,y\in[w,b]$
where $d$ is a distance between $x$ and $y$. It follows that $f$ is an
isomorphism. Since each $x\in[w,b]$ is an experience, from a mathematical
point of view, all our possible experiences of seeing or imagining colors
between white and black have the cardinal $c$, but these experiences are not
possible for any system. So, if our experiences starting from an almost
absolute brightness and ending in absolute darkness, and this experience
occurs continuously (Like the sunset moment by moment), we cannot claim that
we have experienced this entire period. Because the role of the time process
from complete light to complete darkness is important. The process of our
experience from light to dark may occur in a minute, an hour or a day, and our
experiences will be different. In other words, our experiences from seeing a
light to seeing complete darkness depend on the time when this process occurs.
We have the same problem with our emotions like love. Here, for an experience
of smell or pain, there is also a continuous interval from zero to the last
upper bounded that such as above have cardinal is $c$. In general, sensors do
not always follow specific rules for their activities, and due to
environmental and internal chemical changes of systems, unpredictable changes
occur for the experiences of systems. Sensors constantly react to changes in
the environment, and their activity does not have a specific rule to create
experience, and their fluctuations are not bounded. Considering that we do not
have a specific rule for all the environmental changes around us, no specific
rule can be predicted for the reaction of sensitizers in general. These
examples show that it is difficult to talk about the mathematical
representation of all our experiences.

In order not to create more confusion, we take $\Delta$ as a collection or
class of experiences. Now, to avoid such ambiguities, here we mean set,
subset, and function, respectively, class, subclass, and relation.

Most philosophers and scientists are looking for rules and methods that can
fully analyze the human experiences. Unfortunately, a system cannot completely
analyze itself and make a rule for it. Of course, in chapter Three, we stated
that the systems with higher potentials can analyze the systems with lower
potentials.

Here, our mean for raw experiences is that these experiences are not obtained
by logical reasoning, for example, we consider seeing a color, liking or
smelling a flower, or listening to a piece of music as raw experience. In this
section, we mean the language of the system, including all of system raw
experiences and the methods of deducing a system that connects experiences and
we mean the language of mathematical logic, including mathematics tool and
mathematical reasoning methods to express a mathematical conclusion. Of
course, it may not be necessary to separate the methods of reasoning and
conclusions from our experiences because they are also a kind of conscious
experiences. A mathematical model for an intuitive or mental concept is the
most standard model that all related systems have the same output. The process
of our understanding of an event, phenomenon or concept may have different
outputs in different expressions. For example, when two people see the same
color, we cannot find a reason that they see that color in a similar or
different way, but in the examination of mathematical concepts, the same
observations are acceptable to some extent. Therefore, if our observations
have a mathematical basis, our perceptions can approach each other to an
acceptable extent. In mathematical logic, as in traditional logic, deductions
and proofs are central objects of investigation. However, it is the methods of
deduction and the types of argument as used in mathematical proofs which are
considered in mathematical logic. Since we have experience to both languages,
the language we use in mathematical logic is a subset of a complete system
language. In other words, if we represent the language of mathematical logic