意识可以由数学模型表示,其本质是同构。
2. The Mathematic Model of Consciousness
Modeling consciousness has been started for a long term. The clear declaration that consciousness can be modeled as homomorphism of a group was made by Jean Piaget which proved the thought of Norman Malcolm about transitivity of consciousness
Jean Piaget believed that there are some physical structures independent on human, and the structures is in accordance with the operation structures of our mind. Transformation of the physical and psychological structures can be expressed by a model of transformation of group. How ever, Jean Piaget has not given a mathematic expression, and he has not taking the model for the common model consciousness.
We are hereby trying to give the mathematic model of homeomorphism of group for expressing the physical structures independent on human, the structures is in accordance with the operation structures of our mind, and we take the model as one of consciousness in transitive essence, the meaning of consciousness is defined by Norman Malcolm.
We first give the definition of homeomorphism of group. The define is widely accepted in mathematics.
Let X,g Y,* be the two groups, f: X Yis a mapping, if for any element a b X , there must be
f(agb) =f (a) *f ( b)
(1)
The essence of consciousness is the transitivity, that is correspondence between information and entity, i.e., for the system, information x1 x2 ……x ncorrespond the mapped entities f1 (x1) f1(x2) ……f1 (x n) . Besides the primary correspondence, there is a higher function, that is, information of the system can be processed (operated, the results as x1gx2……x n), however, the operation is not meaningless or lawless, the result of operation f (x1gx2……x n) is restrict by f (x1) * f (x2)……f (x n)which is movement or relations of the mapped entities. Here, movement of the mapped entities is generally considered as permutation in a time serial, relations of the mapped entities is generally considered as a new state of action of the mapped entities.
Mapping f is corresponding, it is a representation in essence rather than an objective existence. The symbol Fÿis a reflection or cognition of the set of f, just Fÿis consciousness. In this sense, consciousness Fÿis the characteristics of the third person.
As the homomorphic model of consciousness is the psychological model based on the experiment methods and the theoretical abstract based on philosophy of many viewpoints of Norman Malcolm and the others, so it meet the regulations of definition 2. For the model is based on any information system, so it meets the regulation of definition 3.
then, f : X Y is defined as homomorphism from X,g to Y,* .
Therefore, in reference to the above define of homomorphism of group, we define consciousness.
Let X,g be an information system F ’, X is information, g be an operator of information, Y,* is the system of entity, 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, among F, there is element f : X Y is a mapping, if there is the elements x1 x2……xn X | n>0, so that
f (x1gx2……xn) = f (x1) * f (x2)……f (x n)
(2)
3. The Examples of Analysis Using the Model
Example 1. The analysis of the sentence of “the cup is on the table” by the model.
Answer.
x1: “the cup” (as the words) x2: “the table” (as the words) x3: “on” (as a word) x4: “be” (as a word)
f (x1) :the cup (as the entity) f (x2) :the table” (as the entity)
f (x3) : the relation of the space structure that an entity exist on the level or surface of another entity f (x4) : null
g: permutation of the words according to grammar * : space construction of the entities of the cup and the table
We take the system F ’ cognizing the set F as the
f (the, cup, table, on, is,g) is couscious for that consciousness system, i.e., we say F’ is conscious or F ’ (3)f (the cup is on the table) has consciousness. = f (the cup) * f (the table) * f (on) * f (be) The interpretation is as following.
=the real circumstance that the cup is on the table