[ieee 2012 international conference on open source systems and technologies (icosst) - lahore,...

2012 International Conference on Open Source Systems and Technologies (ICOSST)

978-1-4673-3097-8/12/$31.00 ©2012 IEEE 7

Can Mathematical Cognition Formulate

Consciousness?

Fahad Ahmad

School of Computer Sciences,

National College of Business Administration & Economics,

Lahore, Pakistan.

[email protected]

Quratulain Khan

School of Computer Sciences,

National College of Business Administration & Economics,

Lahore, Pakistan.

[email protected]

Abstract---Mind is a compound information handling and

processing mechanism and consciousness – (the processed

information) is the core component of mind. The spirit of

consciousness is definite correspondence between substance with

some rules or laws. Information processed in the mind

approaching from both the outer world (physical world) and the

inner world (mind). The information can be represented in many

ways. Everything which embodies information is a model and the

process is known as modeling. This paper will present that how

can we formulate consciousness and discussion about two

cognitive mathematical models. Firstly, a hierarchal model of

human consciousness will be discussed, Secondly, the application

of ‘homomorphism of a group’ to the model of transformation

from physical structures to the psychological structure. The

proposed model is a robust combination of the above mentioned

models and provides the mathematical representation of each

hierarchal module in terms of homomorphism. This paper will

try to provide a sound justification for subjectivity of human

experience, in term of proposed model. Furthermore, the

capacity of this model is evaluated by putting some examples

before this model to formulate them. And finally the paper

concludes that consciousness exists when: (a) the units of system

certainly map the entities outside the system; (b) the system

operates by some rules; and (c) the rules definitely map some

laws or relations in the entities.

Keywords- mind; consciousness; mathematical models;

information;-models;- models;- models; homomorphism;

information; entity

I. PROBLEM STATEMENT

In this paper we want to review the previous models of consciousness, propose a model which will be able to formulate the continuous process of consciousness and implement it as discrete process in different phases of an artificial agent‟s life.

II. OBJECTIVES

The objectives of this research include formulating the consciousness, formulating consciousness as natural phenomena of phases like in humans and implementing this continuous process with the help of group theory because machines can work only with discrete data

III. INTRODUCTION

The explanation and understanding of consciousness is one of the supreme challenges ever faced in the history of the advancement of human consciousness. It is still a mystery for

which any human consciousness had asked, but failed to find a successful answer, even by world famous mathematical modelers like Sir Isac Newton and greatest philosophers like Aristotle.

In Taylor‟s own words, “Confusion abounds in present consciousness research, in spite of the high level of present activity. These confusions may arise due to the many deep problems facing us when we try scientifically to understand consciousness [1]. These include: Merging subjective experience, the core component of consciousness, with objective knowledge, the root component of scientific methodology.

Attaining interaction between the self, apparently having a non-relational character, with the contents of experience, as contained in so called ‟qualia‟ which apparently have a relational structure as the phenomenological content of our experience.

The explanation of many diverse states in which it is suspicious either consciousness can be said to be present or not like in the normal awake state, in dreams and possibly in deep sleep, under drugs, in trance state, in meditation, in mental ill-health, such as schizophrenia, Alzheimer‟s disease, and so on.

The selection of models of consciousness from the wide range that have been proposed, with no present consensus amongst them.

To determine which theoretical framework should be taken inside which a theoretical model of consciousness could be made.[2]

To overcome all these difficulties a model is established, which defines the reality, would be a better solution, by considering all these difficulties. Models are in no way equal to the reality so; it is possible to find a new model which can define the reality more perfectly, than any standing model [3]. Here it is assumed that; human consciousness is based upon models and these models are also involved in the advancement of consciousness from the very basic level. Now an interesting relevant question arises. How it could be possible that extremely complex phenomena grow from a simple null state? To avoid the aforementioned difficulties and answer the raised questions, it is better to define some boundaries first. The following rules of definition are proposed to remain within some certain boundaries.


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Definition of consciousness must be accorded with the everyday meaning of the word “consciousness”, that is, defining consciousness is not voluntary

Definition of consciousness must be acceptable both for philosophy and the science. The definer should be clear that the definition of the word “consciousness” is just one as both the philosophers and the scientists.

The concept of consciousness must be common for any type of the matter to be tested by the definition-standard, in other word, no matter human, animal or machine it is, if it satisfies the definition-standard of consciousness, it should be regarded as being conscious.[4]

Norman Malcolm described the common meanings of consciousness that “being conscious” is used in the two different ways. In the first use consciousness takes an object so it is called transitive, because it is consciousness of something. In the second use consciousness has no object so it is called intransitive [5].

From this viewpoint, it can be concluded that the main sense of the word “consciousness” is transitive. The consciousness can be defined as the spirit of consciousness is transitivity of an information system, and transitivity can be defined as a certain correspondence between the system of information and the outside matters. For example, the word “sun” correspond the entity sun. Correspondence can be shown in mathematical expression which represents the correspondence.

It is very difficult to define new ideas of an object without investigating all its characters phenomena. It demands pre-existing ideas in our consciousness. The characteristics of mind are information accepting, information processing and information generating. New information enters in mind from the sensory organs. Consciousness is the final output of all information processing activities which takes place in the mind. Despite of similarities in the processing of information, the final output may vary from person to person because of the variations in the accepted information and the handling of the information internally. Some associated evidences are available that consciousness of a child would be null if information is entirely blocked from entering in the mind instantly after the birth for a sufficiently large period of time [6].

IV. HIERARCHAL MODELS OF CONSCIOUSNESS

In hierarchal models conscious develops and mature in different stages with the passage of time. These stages of consciousness maturity named as hierarchal models like α-

model, β-model and-model.

Here in this model axiomatic approach is used to study consciousness. To develop any mathematical theory it is good to define it through axioms. In a theory, axioms are taken as provable statements, and consider as true. Axioms provide a simple but strong start to a theory. Consider the following axioms.

The model is the basic unit of the information which is processed in the mind.

Mind compares, classifies, characterizes and differentiates several models.

The existing models are used to form new models.

Mind associates some value or preference with each model.

The first axiom used to define the complex problem about consciousness into its basic units. To study the internal processes of the mind which contains consciousness second and third axioms are used. The fourth axiom starts a study of our mind in terms of the subjectivity and objectivity.

Among the three kinds of models, α - models are relatively simple and γ - models are most complex. When a child born α – models begin, after that when child starts to speak β- models come into being, then there are γ- models which are highly complex and developed to explain the experiences of mind. Preference is associated with all the three models, which causes subjectivity.

V. THE HOMEOMORPHIC MODEL OF CONSCIOUSNESS

Mathematical Modeling of human consciousness with the help of homomorphism of a group was made by Jean Piaget, who verified the thought about transitivity of consciousness of Norman Malcolm. Jean Piaget believed that Transformation of the physical and psychological structures can be expressed by a model of transformation of a group and this model of consciousness is used in transitive essence, in accordance with the thought of Norman Malcolm.

The mathematical model of homeomorphism of group which expresses the physical structures without human interference, the structure is according to the operational structure of our mind and the model is considered as conscious in transitive spirit, as defined by Norman Malcolm.

The definition of homeomorphism of group widely accepted in mathematics is:

Let (X, ) and (Y,*) be the two groups, f: X Y is a mapping,

if for any element a, b X, there must be:

bfafbaf (1)

Then f: X Y is defined as homomorphism from (X, ) to

(Y,*).

Therefore, in reference to the above definition of homomorphism of group, consciousness can be defined as:

Let F(X, ) be an information system, X is information and be an operator of information, Similarly F‟(Y,*) is the system of entities, Y is the set of the entities outside of the information such as the substance, * is the type of movement or relation of the entities, there is a set F, there is an element f in F such that

YXf : Is a mapping Where

Xxxx n ,, 21 and n>0

i.e, nxfxfxfxfxxxf 321321

(2)

Here the system F is taken as the consciousness system, i.e., F is conscious or F has consciousness. The spirit of consciousness is the transitivity that is correspondence between information and entity.


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VI.HOMEOMORPHIC HIERARCHAL COGNITIVE MATHEMATICAL

MODEL OF CONSCIOUSNESS

In H-H Model, the consciousness has divided into its three fundamental hierarchal phases/models which are called as α-Model, β-Model and γ-Model. The Processes appear in above mentioned three hierarchal models of consciousness will be defined in terms of homeomorphic functions. „Homeomorphism of a group‟ is used here to transform the physical structures into the psychological.

A. Models

It is observed that a born baby or even in early stage of development a baby can only listen a sound, respond to light, taste and touch. A baby receives these basic forms of information with the help of its sense organs. These basic pieces are processed as models of information in the mind. These models are referred as α - model. Mind can handle only α - models in the early phase of development, but use of α - models does not stop in the later phases. A born baby cannot observe and differentiate these phenomena so his consciousness is null at that time. He accepts each and every piece of information unconsciously exists in any form. In this phase subjective experience not exists, it is spiritless phase therefore, α – model is also named as Meek model. Because a baby at its initial stage does everything unconsciously so it can be called as objective model.

The spirit of consciousness is the transitivity that is correspondence between information and entities, whereas a born baby is unable to observe the correspondence between information and entities. This is because of the absence of the operator, who operates over information and mapped it on entities. On its initial stage, the operator is not mature enough to operate and establish correspondence between information and entities.

Mathematically in terms of „Homeomorphism of a group‟ the α-model can be represented as

nn ifififiiif 2121Null Conscious =

Unconscious (3)

Here niii ,, 21

are components of information system and f is a link between information and entities. The above representation is meaningless because operator is unavailable and without operator function cannot establish correspondence between information and entities. This is just like a phenomenon that we try to open an engine without any tool. This is a fact that α–models have formed and preserved in our mind but have its source in the external world.α-models can independently exist without any human conscious interference. These models can be termed as pre-linguistic models.

B. β-Models

In second phase of mental development, mind starts processing of basic phenomena like identification of already familiar models, comparison of two models, differentiation and classification of different models etc. All these mental processes have diverse difficulty levels.

These potentially elastic models form firsthand experience in the mind. The language begins when conscious seek for how to express experiences and states of mind to the environment that it observe. Some models signify ideas, objects and phenomena with the help of one to one correspondence in

external world. An object represented by a word is an example of one to one correspondence. These models are referred as β - models. β-model is formed when a person see something and then assign it a name. Like when we see a Lion and pronounce „LION‟. Therefore, every language is a group of β- models and is structured according to some rule. In this phase subjective experience is present but only at initial stage that it can only create One to One correspondence between information and entities and cannot process complex mental activities. So according to the spirit of consciousness transitivity starts in this phase. Because in this phase mind experiences everything subjectively and assign only a single component of information to a single entity so, it is also called as Unambiguous phase.

Mathematical representation of β-model in terms of “Homeomorphism of a group” is:

nifififf 21n21 )()iii( (4)

= Conscious matures enough to establish One to One correspondence between information and entities.

For the explanation of this phenomenon an example can be quoted here. Like the mapping of the sentence “the glass is on the table” by the homeomorphic model is:

i1: “the glass” (as the information about entity) i2: “the table” (as the information about entity) i3: “on” (as a word) i4: “is” (as a word) f(i1): the glass (as the entity) f(i2): the table” (as the entity) f(i3): the relation that an entity exists on the level or surface of another entity f(i4): current state Indicates combination of the words according to grammar

* Indicates relation of the entities of the cup and the table.

f (The glass, is, on, the table, ) is conscious for that

f(The glass is on the table)

= f

(the glass) * f

(is) * f

(on) * f

(the table) f (a b) =f (a) *f (b) =the real circumstance that the glass is on the table (5)

The brain of human, cognize f, satisfying f ( ) = * f (), so the brain is conscious. The aforementioned example indicates that the spirit of human language is ability of processing symbols, the laws of processing correspond the natural laws.

C. γ-Models

Consequently, model with diverse and infinite dimensions starts is called γ-model which is highly complex and developed to explain the experiences of mind in more refine manner although the backend processing is very complex. Stories and poems in literature or a theory in mathematics or physics is an example. Speaking in this sense, all real time phenomena which require attention for understanding are γ-models. The problems of this phase are very complex to understand and require extra subjective attention then pre-existing models, so these models can also be called as Ambiguous model.

Although in this phase the function maps one component of information to one entity but iteration of the function is


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performed here to solve the complex problems and to make the output refine. When iteration is performed then a single piece of information is associated with more than one entity, either the entity is physical or non-physical.

Mathematically these phenomena in terms of „Homeomorphism of a group‟ can be defined as:

)]()()([)]([ 2121 nn

nn

ifififiiif ff

=Conscious matures enough to map one piece of information to the more than one entity. (6)

It can be explain as: suppose we have (a word Chair) as piece of information in our mind, if we consider β-model here then our mind associate (a word Chair) as information with a single physical entity “Chair” but in γ-model because of the iteration of function our mind associate single piece of information (a word Chair) with more than one physical entities of different shapes, color and material.

Again a more comprehensive example can be quoted here to describe these phenomena completely. Utilizing the tools is the action with consciousness.

i1: information of the tool

i2: information of the object operated by the tool

1if: The tool

2if: The object operated by the tool

f )( 21 ii : The result of intention utilizing the tool upon the

object

11 if * 21 if

: The result of operation utilizing the tool upon the object : Combination of the information of the tool and the object *: Operation by the tool upon the object

In fact)(1)( 211211 ififiif

,so utilizing the

tools is the action with consciousness.

Making the tools is the action with consciousness.

Let

1i : Information of the material for making the tool

0i : Information of the tool (i.e., the primary tool hands) for making the tool

1f ( 1i 0i ): The intention making the tool

)()( 0111 ifif : The result of making the tool utilizing the

material by the hands.

)](2)(2)]([])([ 2101221 12 nn ififiiffiiiiff o (7)

: The action from making tool to utilizing the tool can be denoted as

Obviously, utilizing the tool made by the person is more complex phenomenon and requires more mapping then utilizing the tool only.

VII. IMPLEMENTATION

The proposed model and mathematical formulism can be implemented in agents or intelligent machines as a phenomenon of continuous conscious learning in phases so that when an agent cross a particular threshold level of conscious learning in a specific phase then it will be shifted to the more mature phase and its learning capability will increase as compared to the previous phase. In this way artificial agent can consciously learn and categorization or hierarchy in different modules pushes her to learn fast in next module as compared to theprevious to enhance its capabilities.

Figure 1. Homeomorphic Hierarchal Cognitive Mathematical Model of

Consciousness

VIII. DISCUSSION

The proposed model is a combination of the Hierarchal model and Homeomorphic model. The Hierarchal model describes that human consciousness grows in three phases which are α-models, β-models, γ-models, but the mathematical representation of these Hierarchal model is not given. On the other hand Homeomorphic model presents the mathematical representation of consciousness but does not describe the hierarchy of consciousness. The presented Homeomorphic Hierarchal Cognitive Mathematical Model of Consciousness is an attempt to eliminate these above mentioned flaws and combines the properties of these models to enhance the ability of formulating the consciousness. We can implement it by using the properties of group theory which is discrete in nature and till now all machines which have invented have discrete nature. Firstly Hierarchal model can help in designing the different phases of mental development in artificial agents or cognitive machines and homomorphism can help in mapping of all natural phenomena in artificial agents or interactive machines.

IX. CONCLUSION

“Homeomorphic Hierarchal Cognitive Mathematical Model of Consciousness” presents that all three models are connected with each other and will be used in subsequent phases. With the passage of time these models become mature and use in more comprehensive manner. Similarly Phases of proposed

IX. Mr. FahadAhmad, is an M.Phil. Scholar at

NCBA&ELahore, Pakistan and working as

Hardware Engineer in Services Institute of

Medical Sciences. His area of Interests is

Artificial Intelligence, Fuzzy Modeling and

Network Security (E-

mail:[email protected]).

X.

VII. Mr. FahadAhmad, is an M.Phil. Scholar at







VIII.

V. Mr. FahadAhmad, is an M.Phil. Scholar at







VI.

III. Mr. FahadAhmad, is an M.Phil. Scholar at







IV.

I. Mr. FahadAhmad, is an M.Phil. Scholar at







II.


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model demonstrate that mind becomes ambiguous and more conscious about complex phenomena with the passage of time. These elastic phases enables mind to map more complex problems with the help of compound function which will operate over information to get more and more refine correspondence between information and entities.

The subjective experience is dominant in human consciousness and there is hardly any experience which is completely objective. So without any subjective experience study of consciousness is incomplete. Subjectivity is a preference which we associate with a model. This association is unconscious most of the time. It can be observed when we consider two or more models concurrently.

Models which are ambiguous such models can be interpreted in numerous ways. Ambiguity can cause subjectivity. Ambiguous models represent different things for different consciousness. This gives freedom for a consciousness to interpret the model accordingly.

Conscious can be stated by H-H model as the correspondence between information in the inner parts of the system and entities outside the system. Consciousness occurs when (a) The information units of system definitely map the entities outside the system; (b) The system operate by some rules; and (c) the rules definitely establish some relations in the

entities. Finally, the spirit of consciousness is firm correspondence according to the some rules or laws and Mathematical Cognition have potential to formulate consciousness.

ACKNOWLEDGMENT

This research work was carried out in the Computer Science department of NCBA&E, Lahore, Pakistan. We must acknowledge the support of our respected teachers and research group fellows for their supporting and encouraging behavior.

REFERENCES

[1]. J.G Taylor, "Mind and consciousness: towards a final answer," Elsevier, no. Physics of Life Review, 2005, pp. 1-45.

[2]. R. Kumar, "Modeling of consciousness:classification of models," vol. 2, no. Advanced Studies in Biology, 2010, pp. 141-146.

[3]. A. Yu. Khrennikov, "Toward an adequate mathematical model of mental space: conscious/unconscious dynamics on m-adic," Consiousness, p. 26, Nov. 2006.

[4]. Z. Yinsheng, "The mathematic model of consciousness," 2ndAsia International Conference on Modelling & Simulation, 2011, p.5.

[5]. D. M. Armstrong and Norman Malcolm, "Consciousness and Causality: A Debate on the Nature of Mind," no. Nature of Mind, 1984.

[6]. Wikipedia. (2011) Wikipedia. [Online].

http://en.wikipedia.org/wiki/Feral_child

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