Semantic Space Construction

An object of any science is described through a definite set of attributes of its paradigm, (they can be physical, sociological, psychological etc.) with the indication of their intensity:. It is clear that formalization of the objects' mental representation can be most naturally performed in a vector space, where the object would correlate with a vector in a certain - dimensional space of characteristics or attributes.

Let us juxtapose vector to each object in the attribute space with the orthogonal  basis: , where identity vector is correlated with the independent  - property, and - with its intensity and .

Let us define extended semantic space in the basis of , where  is a unit vector orthogonal to . It characterizes variability (dynamism) of vector in the space of characteristics. Thus, and is a semantic vector that produces an adequate/complete object description. Vector components determine the objects' semantic coordinates while determine relative semantic coordinates. Here   , therefore,  

Since perception of features or characteristics is done sequentially, we will be defining them through semantic coordinates by projection sequence onto the corresponding planes (the plane of -th characteristic), that is by vectors with   coordinates or , where , and .

The -relation received determines the standardized -th characteristic or angle at which we perceive object  in the relative characteristic space. The matter is that by vector representation of an object, it is the angular coordinates and not linear ones that have to be correlated with characteristics. If characteristics were added in the manner of linear coordinates then two identical pieces of chalk would be twice as white as one piece, and the summary speed of two bodies moving parallel at the same speed would be twice greater. It is the vector lengths that are added in both of the mentioned cases and rigidity of these properties that is increased, and not the degree of their expression/explicitness.  So, when identical vectors are added, their coordinates evidently remain unchanged. Since characteristics that are measured for their intensity are actually correlated with a certain standard, we will accept value as the absolute intensity of a characteristic, where is the ideal, or a limiting value of the characteristic/property, and . Thus, the limitation of physical speed by the limiting value - the speed of light is not the law of nature, but simply the result of a specific subject-generated categorization of that nature.  Devoid of an option to grade our sensations by reality itself, we find ourselves in a situation similar to that of the ancient astronomers' who were making a space map on a celestial sphere of an arbitrarily fixed radius , by angular coordinates .

Since , то . While ,  therefore .

Or .

As noted above, the presence of finite ideals on the subjects' mental map makes it limited in every situation and practically applicable for describing any part of the world, and it actually enables the subject to perform this description in terms of finite concepts. Nevertheless, it costs scale non-linearity, which is especially increasing with approximation to the ideal. While

and the last identity formally defines the familiar correlation:, where и - is a hyperbolic cosine and sine, respectively.

Since , the sum of   and characteristics will be determined by the formula:

.    

This expression completely coincides with the rule of velocity composition in relativist mechanics. However, while most problems in physics are solved with the use of a classical setting , in sociology and psychology consideration of non-linearity is a must, due to a lower exponent rigidity. In particular, it leads to the fact that a simple averaging of experimental data (by characteristics) will always result in distortion especially significant in the proximity to the ideals.

We can draw some parallels to paradoxes in cartography when practical necessity makes us reflect the Earth in flat Euclidean projections topographical maps. So far our studies have been staying within the boundaries of small areas where this method of area description developed, and there have been no paradoxes. But it is evidently impossible to construct an overall flat picture of the whole world without distortions that increase with approximation to specific points (usually the poles of the Earth). It is clear that these distortions can be expressed as a precise objective law, although it is more closely related to the way reality is mentally represented than objective reality itself.

Thus, the scales' boundedness by limiting values results in their non-linearity, which is especially increasing with approximation to the ideal. It can easily be noticed about expert evaluation of athletic achievements etc. Our mentality subconsciously perceives this as a natural increase in difficulty, the closer one is to perfection: every next step is harder than the previous one, while the difference in achievement is hardly noticeable.

Let us define the addition rule for :

It should be noted that there are other biological perception peculiarities not yet studied within our research method. They are also the ones that determine the structure of our mental map. Firstly, we dont only record primary, elementary sensations, but we also localize their perception in a three-dimensional physical space. This way, every characteristic can be immersed into a certain subspace that has from one to dimensions. Secondly, what we record are not simply features and objects, but also their alterations, that is we perceive them it time as well as in space. Evidently enough, an object can not be assigned by a point in a semantic space, but it would be determined by a certain range of admissible values of characteristics, and should the object cross its barrier, we perceive it as object metamorphosis.  Thirdly, the buffer sensory memory retains a complex of perceived sensations for several milliseconds during which a sense analysis of the incoming info takes place. Perception actually works in definite temporal segments or quanta. This makes it possible for us to perceive a sequential row of static movie shots as a continuous movement. That means that we actually perceive a certain integral function of the world characteristics instead of those features themselves.

If we ascribe a certain set of semantic space properties to some spatial volume in physical space , we are actually defining a physical object. (A social object will be defined likewise). In a physical space restricted to only four dimensions (including the time), properties can only differ in terms of types of processes that they define. For example, if property defines the process or , it corresponds to a regular rectilinear body motion. If process   is not decomposable into components in the physical space-time, then coordinate represents a wave front, and  defines the wave with -period and  -wave-length, sphere with  -radius representing the coordinates of the wave. Physical space coordinates are actually observable parameters here, through which we define the properties. In reality we tend to correlate any properties with specific characteristics or interaction process peculiarities in the organism-environment system, while those properties are used for marking a mental map - our image of external reality. Dynamism index , by definition, characterizes the velocity of this process that we can perceive as a change in some object parameters that are time-dependent (for example, coordinates).

Lets assume that any process can be described through a specific ordered sequence of forms of one - type (for instance, ).

Thus, for the purpose of various process descriptions we can define a finite set of mutually exclusive sets that characterize all possible alterations in properties of -object, and also a denumerable linearly ordered set (time) that indicates a specific sequence of these forms defined by a certain mapping of    onto . This mapping is given by a relevant operator that determines the rules and sequence of form selection from as well as defines process matter. Cartesian product   creates a form space where any process could be described.

Let us take a situation where we are perceiving two identical processes defined by a sequence of the altering parameter and characterized by different degrees of intensity (and ) of a certain property . Let us introduce the notion of absolute time as an index that is assigned to all the processes that take place in the world.  Since process dynamism depends only on and is defined by the speed of indexing of parameter   (speed of time flow for the subject), indexing time of the first process perceived by the subject would be defined as , while indexing time of the second one will look like .

Taking advantage of the transformation received for we obtain:

.

Consequently, . Given that property   is speed, , whereas defines time transformation while transferring into another inertial system.

For a wave process with period   we have:

.  

Since , then . Here is the angle at which we perceive a wave..

Thus, the rules of velocity, time, coordinate transformation, Doppler effect are essentially the result of a definite method of reality representation operating within our mentality, which results in paradoxical perception of reality once beyond the range of natural biological adaptation and  proximate to the ideal (given that  ). While mapping different types of physical space properties onto semantic space characteristics, obviously enough, we do lose information on coordinates, because object definition works independent of those. All the above is also true for any other types of processes (psychological or social, which types are correlated with characteristics). Generally there should be a distinction between the inertness of object characteristics, which is to be correlated with , and inertness of the object, to be correlated with . As issues from our analysis, the concept of observer  (subject) can not be fundamentally excluded from any scientific paradigm for quite objective reasons, as the world description depends on his system of observation (or reference frame in physics).

Let us clarify the sense of semantic vectors. Since  , and , therefore:,  

where .   is a mass or property rigidity of an object with intensity , - that is what corresponds to rest-mass in physics (or object rigidity with a zero property intensity), while is an impulse or the absolute -th object semantic coordinate.

Therefore, introducing the notion of an impulse, physicists implicitly turn from object description through their properties to their adequate/complete semantic representation. Inertness of property   of object is directly proportional to the module of its absolute semantic vector. Value is actually a projection of some property subspace unknown to us or not taken in this process, onto dimension , while the object is understood as a certain factor that links -properties in a certain stable relationship. It should be noted that rigidity of characteristic/property of object is connected with  speed of the time flow , which determines the process dynamism. Thus, the process where the subject perceives the external reality is largely analogous to astronomic observations whereby we perceive only angular star coordinates and have to define a radius of a celestial sphere for their adequate representation on the map. To find out the real location of stars we need to multiply these coordinates to the ratio that represents a sort of inertness index, since the change of angular coordinates in identical celestial objects (which by analogy corresponds to their properties) for their equal linear shifts would be inversely proportional to their distance. Let's point to the fact that for increase in characteristic intensity their rigidity degrees also increase в times. The reasons for boundedness of characteristics or properties by limiting values become evident, assuming this method of mental map construction, because they are the angular measure of a corresponding quality.

The addition of two semantic vectors will also yield a semantic vector (a complex object):, where is an angle between and . The difference actually determines connecting energy of the two objects. It is clear that this kind of interaction will bring forth a new compound object, defined by vector , where the part of mutually compensated characteristic and  intensities would define  . Since the new object must possess parameter space coordinates common to and in describing a process (for example, in physical time-space), the balance of forces of interacting objects operates at a certain (in psychology psychological proximity/closeness should be defined, although in a number of cases it can be correlated with the physical proximity).

Let us take a classical case where it is possible for one of the properties to change, for example property  of the complex object defined by vector (with initial ), with no change in other properties . . Consequently, all the semantic vector projections onto plane must remain unaltered, except for the one defined   , assuming there is a sequential change in semantic coordinates of the object to  (Fig. 1)

Fig. 1 Semantic invariants (comments in the text)

Evidently enough, it is possible only by prolonging semantic vector , where . Let us express in terms of properties:

Thus we obtain:  or the well-known physical relativist formula for energy-impulse relation. Therefore, the laws of conservation for impulse and energy are adequate to the conservation of the sense definition of the object, and in their universal form they are common to all sciences. Characteristic intensity can basically be treated as a quantitative measure of tension within the context of a specific quality, while object semantic description can be treated as a summary tension for a number of qualities that reflect the internal energy of this object.

It should be noted that adequate calculation of interrelation of compound objects and defining their tolerance for external impacts requires absolute semantic coordinates of objects instead of relative values (family, major and minor groups, ethnicity and others can serve as objects).

In principle, it is possible to introduce negative values and define antiparticles as well as existence conditions of an elementary object and even to treat the whole Universe as a single object to conduct the appropriate semantic analysis.

EXAMPLE 1

EXAMPLE 2

Our task does not involve specific treatment of physical categories. Instead it offers model testing within the semantic space of the correlations obtained. Let us sum up all the above said at this intermediate stage.

We have demonstrated that semantic space makes it possible to give an adequate and relative simple representation, description and analysis of objects of various nature, as they naturally appear in our mentality. Our method does not contradict the world image perceived by us (the latter is exemplified with physics).

Our studies have also revealed that relativist paradoxes of perception spring from peculiarities of reality representation on our mental map, while the laws of conserving energy and impulse are the effect of sense conservation (semantic definition of the object) within a system. Hence they are not specifically physical, but they also hold for any science.

At present a regular factor space of the studied characteristics or properties is declared to be a semantic space [6]. As we have indicated above, it does not present a complete object description, containing a number of drawbacks.

First of all, it does not take rigidities of characteristics into account, and therefore, it does not allow to either study the kinetics of the relevant phenomenon, or introduce laws of conservation etc., that means to give an adequate/complete description of a phenomenon, and consequently, model and predict it. Secondly, it does not consider non-linearity of the semantic scales obtained, which can result in serious distortions and errors in quantitative calculations. Thirdly, the findings of studies in various mentalities  within the factor space become practically incomparable.

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