Semantic model and optimization of creative processes at mathematical knowledge formation

doi:10.4236/ns.2010.28113

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Vol.2, No.8, 915-922 (2010) Natural Science

http://dx.doi.org/10.4236/ns.2010.28113

Semantic model and optimization of creative processes

at mathematical knowledge formation

Victor Egorovitch Firstov

Department of Mechanics and Mathematics, Saratov State University, Saratov, Russia; firstov1951@gmail.com

Received 26 February 2010; revised 24 April 2010; accepted 29 April 2010.

ABSTRACT

The aim of this work is mathematical education

through the knowledge system and mathemat-

ical modeling. A net model of formation of ma-

thematical knowledge as a deductive theory is

suggested here. Within this model the formation

of deductive theory is represented as the de-

velopment of a certain informational space, the

elements of which are structured in the form of

the orientated semantic net. This net is properly

metrized and characterized by a certain system

of coverings. It allows injecting net optimiza-

tion parameters, regulating qualitative aspects

of knowledge system under consideration. To

regulate the creative processes of the formation

and realization of mathematical knowedge,

stochastic model of formation deductive theory

is suggested here in the form of branching

Markovian process, which is realized in the

corresponding informational space as a seman-

tic net. According to this stochastic model we

can get correct foundation of criterion of opti-

mization creative processes that leads to “great

main points” strategy (GMP-strategy) in the pro-

cess of realization of the effective control in the

research work in the sphere of mathematics and

its applications.

Keywords: The Cybernetic Conception; Optimization

of Control; Quantitative and Qualitative Information

Measures; Modelling; Intellectual Systems; Neural

Network; Mathematical Education; the Control of

Pedagogical Processes; Creative Pedagogics;

Cognitive and Creative Processes; Informal Axiomatic

Theory; Semantic Net; Net Optimization Parame-

ters; The Topology of Semantic Net; Metrization;

the System of Coverings; Stochastic Model of Cre-

ative Processes at the Formation of Mathematical

Knowledge; Branching Markovian Process; Great

Main Points Strategy (GMP-Strategy) of the Crea-

tive Processes Control; Interdisciplinary Learning:

Colorimetric Barycenter

1. INTRODUCTION

“The book of nature is written by the mathematical lan-

guage” [1]. This Galilei`s manifesto determined the me-

thodology of natural science development on the basis of

observations and experiments, the result of these obser-

vations and experiments are interpreted within the

frames of corresponding mathematical model. The latter

implicitly assumes the mechanism of science integration

thus, this methodology of development have been suc-

cessfully realized during the last century beginning with

famous Newton’s “Mathematical Priciples of Natural

Philosophy” (1687) [2]. The reason of this success is

easily explained by L.

Boltzmann s

′

assertion in his

book [3]:” The theory is more practical, it is quan-

tum-essence of experiment.” One can understand in the

sence that quantum-essence is the system of postulates,

which is in the foundation of theoretical model of the

object under consideration.

In 1948 N. Wiener [4] had stated main cybernetics’

positions. He had done it on C.

Shannon s

′

information

theory (1948) [5]. According to this theory, any process

of control system represents some transformation of

systemic information. Information is basis notion of cy-

bernetics and has an abstract quantitative measure. It

allows to measure both material and non-material as-

pects of the object under consideration. Thus, possibili-

ties of mathematical modelling have extended including

processes in social and humanitarian fields [6].

With the appearance of computers in the middle of the

century, a separate direction connected with the

realization of intellectual systems (IS) has been formed

in cybernetics [7,8]. Accents in this case focused on

learning of cognitive processes. In 1980s this led to the

building of learning ACT-theory [9] and neural network

associative model of J. Hopfild [10]. Modern level of IS

development represents adaptive learning systems [11].

A half century experience of IS development shows

that their possibilities are determined by mathematical

model parameters, which are in the ground of IS. Thus,

the questions of IS efficiency lead to the parameters op-

V. E. Firstov / Natural Science 2 (2010) 915-922

916

timization of corresponding mathematical models which

describe given cognitive processes. Formally, it assumes

the investigation of topology of semantic nets which

realize given cognitive processes. This investigation

allows determining the system of net parameters with the

help of which one can influence the knowledge system.

Thus, optimal control by cognitive processes is carried

out, including creative search and interdisciplinary

learning.

2. SEMANTIC MODEL OF INFORMAL

AXIOMATIC THEORY

Let S = (M;

) is a mathematical structure, where M =

{M1;…;Mk} is a system of ground sets, representing

main objects of structure S, and

= {

;...;

αα

} is a

system of axiom, describing ground relations between

them. The informal theory Th(S) of the structure S

represented denumerable set, the elements of which are

ordered by definite rules of conclusion. One can speak of

the theory Th(S) as of the structural space information

[12], which is formed within given concluded rules in

the form of constructively infinite recursive procedure,

that is to say, the space Th(S) is some kind of

Herbrand s

′

universe analogue.

Theory space Th(S) is given by the digraph

()

ГS



representing the mathematical structure S in the form of

semantic net, which realizes the passing of definite ob-

ject information. At this, the set Th(S) determines the

nodes of the object net domain

()ГS



, and its arcs

(orientated edges) are given with the help of the com-

mutator I – set of functions f of following type:

T ;...;TT

→

, (1)

where

()

T;T;...: TThS

∈

, and symbol

→

means in-

formal logical consequence of the statement T from

T ;...;T

. Digraph

()ГS



is appeared by the pair (V; E),

where the set of nodes V and the set of arcs E is defined

by the expressions:

V= Th(S)I; E(Th(S) I) (I

), (2)

where

is the complement of the system axioms

till Th(S), and, without community restrict, the axiom

system

may be considered independent. For a given

digraph

()ГS



, the system of nodes

Th(S)

represents the sources and, thus,

()ГS



represents of the

semantic net, which determines the information space

structure of axiomatic theory Th(S).

3. TOPOLOGICAL PROPERTIES OF THE

SEMANTIC NET

()

 

ГS

In order to investigate semantic net properties

()ГS



topological presentations, realized according to two di-

rections, are used [12].

The first direction determines routes, distances and

connection between subject nodes of the semantic net

()ГS



. Let on the digraph

()

ГS



(2) the nodes are dis-

tinguished

; ;...;

vvv V

∈

, which form the sequence of

arcs

001 121

( ;...;):( ; )(;)...(;),

n nn

Lvvvvv vvvE

−

∈



(3)

where

( ;)( ;)(;);

iii iii

vvv ffv

;

∈

0; 1in= −

Thus, the oriented route is given, connecting the node v0

with vn (it means, that the node vn is reached from the

node v0), and the nodes vi ,

0; 1in= −

, are called inter-

mediate ones on the route (3), and this fact is expressed

in the form of:

0in

v vv

. The route length (3) and

the distance from the node v0 to the node vn are, corres-

pondingly, determined by correlations:

( ;...;)

Lv v



| = n ; |

(;)

rv v



| = inf |

(;)

Lv v



|, (4)

where |

(;)

Lv v



| is the set of all route lengths, connect-

ing the node v0 with vn.

The second direction of semantic net

()

ГS



investi-

gations is carried out with the help of special system of

coverings, which is formed by the following way. For

arbitrary node T Th(S) the set is determined:

U(T)={Ti|TiT Ti = T, Ti;T Th(S), iN}

Th(S), (5)

the elements of which represent the nodes, for which

the node T is reached on the digraph

()

ГS



. The set U(T)

is called the sphere of dominating of the node T in the

space Th(S), and its power |U(T)| determines the capacity

of the dominating sphere U(T). Capacity |U(T)| in given

case represents topological of information quantity ac-

cording to N. Rashevsky [13]. The following topological

properties are set up for dominating spheres:

()( )(),T UTUTUT∈⇔ ⊆

(6 )

moreover, the equality

( )()UT UT

TT≠

equivalent to the fact, that the nodes

;

are linked

by the cycle.

2) If

() ()UT UT∩ ≠∅

and sets

( ),()UT UT

are

not linked by implication, then among nodes

T∈

() ()UT UT

∩

, at least one of them, is a point of net

branching

()ГS



Let

(; )LTΣ



is the set of routes, leading from axioms

to the node T. Then with the help of (4), the distance

from

to T and the diameter of the sphere U(T), are

correspondently, determined:

(; )rTΣ



| = inf |

(; )LTΣ



|, d(U(T))= sup |

(; )LTΣ



|. (7)

Within the conception capacity (5) and distance (7), a

recursive procedure is determined, for forming the sys-

∪

⊂

∪

⊂

∈



∨

∈

⊂

V. E. Firstov / Natural Science 2 (2010) 915-922

917

tem of inclusion coverings in the theory structure Th(S).

Theorem 1. In the space Th(S) one can always under-

line a denumerable assemblage of subsets, which form

the chain

( )( )( )...,Th SThSThS⊃⊃⊃

(8)

thus, the chain of inclusion coverings of the space Th(S)

is formed:

h(S)

⊂



()hS

⊂



()hS

⊂

… , (9)

where h(S) =

{ }

( ):( )U TTTh S

∈

, 

()

{ }

():() ,

UTTThS= ∈

{ }

( ):(),(;),1;2;...

ThSTTTh SrTii=∈Σ≥ =



Thus, algorithm of knowledge generalization is given

in the semantic net

()ГS



: if a certain section

()

chosen in chain (9), then an object sphere of knowledge

Th(S) is covered by the system of spheres

()

UT∈

()

and in each of such sphere the dominated node

()

T UT∈

presents the generalization of the rest ele-

ments of sphere

()

. As a result, the spheres

()

are classified by some system of characteristics in order

to form an appropriate conception. Accordingly, the

property (9) in the process of generalization realizes the

principle of “matryoshki”, and as a result the level of

abstraction s

′

conception is gradually increasing. It

means that the process of getting knowledge is con-

nected with the development of intellect.

4. OPTIMIZATION OF THE DEDUCTIVE

CONCLUSION ON SEMANTIC NETS

Let among the statements T

∈

Th(S) there is finite set of

positions

;...;,

for which in the net

()

ГS



there is

the only function

fI∈

with the range of definition

Dom

{

;...;

}. This function realizes the in-

formal logical conclusion:

;...;k

→

T. (10)

If Dom

⊄Σ

, then each of the nodes

;...;k

Dom∈

fΣ

similarly (10) has a corresponding I-node

and also occurs the result, one-valued following from

corresponding positions of predicate universe Th(S) and

so on, till we come to the proofs:

1111 12221r

:;:;...; :,

fTfT fTΣ→ Σ→Σ→

( 11)

where

;...;

ΣΣ ⊆Σ

5. Thus, in general case, procedure’s

proof of the statement T

∈

Th(S) is a partially ordered set

B(T), which is made up of predicate nodes, structured

through functions (10),(11). It is evident, that B(T)

⊆

U(T) and, hence, the sphere of dominating U(T) present

a union of the all possible proofs` of the statement T.

Let

(; )LTΣ



is a set of routes from the axioms

the node T in the proof B(T) (10),(11). Then, the length

()



| of proof B(T) presents critical way on B(T) and is

determined through recursion:

()bT



| = max (|

()bT



|;…;|

()



|)+1= d(B(T)), (12)

where d(B(T))―the diameter of the proof B(T), which is

determined similarly (7).

Apart from the length (12), the proof B(T) is characte-

rized by the capacity |B(T)|. Through regulation by these

parameters of the conclusion the tasks of the optimiza-

tion while forming the knowledge, in the form of the

theory Th(S) are considered in the net

()

ГS



. Let

B1(T);…;Bj(T) are different proofs of the statement T

∈

Th(S), having the lengths |

()bT



|;…;|

()



| and capaci-

ties |

()BT

|;…;|

()

|. Then, the following tasks of the

optimization are determined on the net

()

ГS



B0(T) = opt(B1(T); …; Bj(T))

⇔

()bT



| =

min(|

()bT



|; …; |

()



|), (13)

B0(T) = opt(B1(T); …; Bl(T))

⇔

()BT

| =

min(|

()BT

|; …; |

()

|) (14)

Each of the tasks (13), (14) present the optimization of

the statement proof T

∈

Th(S), accordingly, by the mini-

mization of its length or capacity. On the whole, these

tasks may be considered together. This statement of op-

timal tasks means the simplification of the proof through

reducing the volume of the analysed information. On the

whole, it is coordinated with the principles of the infor-

mation theory.

For example, the optimization of Pythagorean theo-

rem proofs was carried out according to criteria (13);

(14), through the analysis of existing variants in school

geometry: Euclidean classical proof, the proofs of in-

dian mathematician Bhascara and vectorial method of

proof with the help of scalar product [12]. The parame-

ters of the proofs of Pythagorean theorem Т in the Euc-

lid’s, Hillbert’s and Weyl’s axiomatics are given in the

Table 1.

Table 1. Metric characteristics of main versions of proofs Pythagorean theorem in different axiom system.

Proof Axiomatics i |

()



| |

()

Euclid

Euclid (IV B.C.)

Bhascara (proof-I, ~1150)

Bhascara (proof-II, ~1150)

D. Hilbert (1899)

vectorial method

H. Weyl (1918)

V. E. Firstov / Natural Science 2 (2010) 915-922

918

As it is seen from the Table 1, the optimization of

proofs according to the criteria (13), (14) the preference

is on the side of the vectorial proof in Weyl’s axiomatics.

However, the rise of abstraction level happens in the

teaching process, which is in the inverse dependence

with didactic principles of accessibility and visual

teaching aids. This fact has its reflection in Russian

teaching literature on elementary geometry. The analysis

of this literature for the period of 1768-2000 shows, that

in the second half of the XIX

century, Euclid’s classic-

al proof is practically nowhere met in school textbooks

and

Bhascara s

′

proofs are mostly used, because they are

more vivid and accessable [12].

5. FORMALIZATION OF CREATIVE

PROCESSES DURING THE

ASSIMILATION OF THE SPACE Th(S)

AND THE RANGE SIGNIFICANCE OF

THE ELEMENTS IN THE NET

()



ГS

One can regard creative processes in the training as the

generalization of knowledge at the metalevel with the

help of heuristics. The link between metalevel know-

ledge and the known object sphere of knowledge is for-

mally expressed in the fact, that the space Th(S) is

formed with the help of endless recursive procedure and,

hence, one can say of current statement

()ThS

, for

which

() ()

ThSThSΣ⊂ ⊂

, then knowledge metalevel is

introduced with the help of partition

Th(S)

() ()Th SCTh S

= ∪

, (15)

where

()CTh S

is the complement

()Th S

till Th(S),

which determines the knowledge metalevel in the form

of nodes. To these nodes relation of incidence in the net

()ГS



is heurictically established. The partition (15) in

the net

()ГS



induces set partition of functional nodes

I ICI= ∪

, where

is the set of functions, realiz-

ing the conclusion to metalevel:

11 1

: ()()CITh SCTh S→

. (1 6 )

Correlations (15) and (16) present a formalisational

description of the knowledge generalization procedure to

metalevel in the process of axsiomatical theory Th(S)

formation. The optimization procedure of this process is

forming in the range of presentation about node signi-

ficance in the semantic net.

Let U(T) is the sphere of dominating of the statement

T and B1(T); …;Bn(T) are possible proofs of this state-

ment, having lengths |

()bT



|;…;|

()



|. Let us call the

quantity

(; )DTΣ=

min (|

()bT



|;…;|

()



|) (17)

some logical distance from axiom`s

Σ⊂

Th(S) to the

statement Т. In fact, logical distance

(; )DTΣ

coincides

with algorithmical determination of information quantity

according to A.N. Kolmogorov [14].

Formally, any significance presents a partial order in

the space Th(S) and it is given in the form of domination

relation according to Pareto:

TT⇔

()UT

≥

()UT

∧

;T )

≤

;

), (18)

where one of the inequalities is strictly done. In the case

of defining (18), the statement Т is considered to be sig-

nificant than

and it means that more important ele-

ments of the space Th(S) are more influential (the first

inequality in (18)), and they are closer to the information

system sources

(the second inequality in (18)). One

can also interpret the significant elements as huge main

points of the net

()

ГS



, which are closer to its sources.

6. GMP-OPTIMIZATION STRATEGY OF

CREATIVE PROCESSES IN THE

FORMATION OF MATHEMATICAL

KNOWLEDGE

The reasons of relationships significance (18) is carried

out in the language of

theorys

′

random processes [15].

As in the process of creative searching the moments of

revealing time t of new theory form statements Th(S) are

not determined, then its creation presents a random

process Th(S;t) with continuous time t

≥

0 and with de-

numerable set of the states, the moments of transititions

between them are distributed in the interval t > 0 by

chance.

Theorem 2. The random process Th(S;t) is a hetero-

geneous branching Markovian process.

Heterogeneous in time, the branching Markovian

process Th(S;t) is determined as a process, the transition

probability

(;)

of which satisfy the Kolmogorov-

Chapman equation with branching condition:

Pik(

;t) =

(;)(;)... (;)

lr lrlr

+...+r =k+i(l-)

PtPtPt

ττ τ

∑

, (19)

where Pik (

;t) is the probability of the condition having i

elements at the moment

, to the moment t it will con-

tain k

≥

l = |

| elements and the evolution

Pik(

;t) is described by Kolmogorov system of differen-

tial equations for the heterogeneous branching Marko-

vian process [16].

The reason of Pareto-optimization procedure by signi-

ficance criterion (18) means that when the information

space theory Th(S) is mastered by chance and is realized

by branching Markovian process Th(S;t) then more sig-

nificant theory statements Th(S) have higher probabili-

ties of transitions between the process statements Th(S;t)

and the result of this is as follows: let

(;)

ik ik

PPt

V. E. Firstov / Natural Science 2 (2010) 915-922

919

the solution of Kolmogorov equations under condition

(19). That will be enough to examine an example k = i + 1

and then from the branching condition (19) you will get:

1 ,1

(1 )

i,il lllllll

PiPPiPP

−−

==−

. (20)

With the increasing the length of the interval

(;)t

the meaning Pll

→

0 and the function Pi,i+1 is increasing

in the wide range l

≤

ln−

. Since the meaning i in

this case is linked with the quantity Th(S), which cor-

responds to i-state of the process Th(S;t), then it goes

without saying, that, under other similar conditions, the

dominating sphere U(T) with more capacity|

()UT

| has

more chances to widen, because the probability of proof

of new statements at forming theory Th(S) is increasing.

Thus, the reason of the first inequality in the Pare-

to-optimization procedure (18) is given. However, the

growth of transition`s probability

(;)

in this model

occurs not only with the growth i, but also with the de-

crease

, because in this case the interval length

(;)t

is increasing. The

decrease is equivalent to the decrease

of logical distance (17) that leads to the reasons of

second inequality in Pareto-optimization (18). Thus, the

conception of the range significance of the statements`

theory Th(S) by means of Pareto-domination (18) has a

sufficient reason. According to this reason, the optimum

control by creative processes in training comes to effec-

tive control of definite random process according to the

criterion of significance (18). In this case, the effective

strategy of theory formation Th(S) in the creative process

leads to the conception of “great main points” in the net

()

ГS



or GMP-strategy. This conception intends the

investigation, coming out from significant statements

T01;…;T0k

∈

Th(S), have been chosen according to the

net criterion of the significance (18). The inductive hy-

pothesis Н, given on the base of these statements, has

more chances “to be materialized” as a logical generali-

zation of starting positions. The idea of existing “great

main points” is in tune with the modern psychological

conceptions in the field of intellectual theory [17]. Ac-

cording to these conceptions, “great main points”, which

have an increased sensitivity to certain semantic influ-

ence, can qualitatively change the character of under-

standing` problem situation.

7. GMP-STRATEGY AND HILBERT’S

MATHEMATICAL PROBLEMS

A vivid illustration of optimization creative search with-

in the framework of GMP-strategy presents the solution

of well-known problems in the theory of numbers, and

also

Hilbert s

′

problems (1900) [18]. The chronology of

their position and solution is exactly known (Table 2).

The analysis shows, if the period of decision

Goldbach s

′

Waring s

′

and

Fermat s

′

problems in the

theory of numbers makes up hundreds of years, then

Hilbert s

′

problems has a unique result, which occurs by

1-2 less. It is important to say, that the choice of 23

problems from a wide manifold of mathematical ones,

appearing in the XIX-XX

centuries according to

Hilbert s

′

report at the II International Congress of Ma-

thematicians (1900), supposed quite certain “rules of

selection”. The sense of them is the realization

GMP-strategy. Those problems are interesting, the deci-

sion of which is possible at a given level of mathemati-

cal development. These problems can give a further

progress to mathematics.

8. THE EXPERIMENT OF

GMP-STRATEGY IN

INTERDISCIPLINARY LEARNING

The possibilities of GMP-strategy are not limited only

by the optimization of the mathematical researches but

are spread over the interdisciplinary training level within

the framework of

morphisms

′

category. In this case,

the author ’s experience shows, that the realization of

GMP-strategy usually takes place on the basis of some

Table 2. The problematic optimization of the research work in mathemati cs.

Problems

Problem Decision

Period of problem decision,

years

Name

Statement Time

Author(s)

Date

Last Theorem of

Fermat s

′



1630 A.Wiles 1995 365

Goldbachs

′

problem: n=p1+p2+p3

1742

I.M. Vinogradov

1937

195

Waring s

′

problem:

n=p1s +…+ pks

1770

D. Hilbert

1909

139-172

G.Hardy-J.Littlewood

1928

Ju.V. Linnik

1942

Hilbert s

′

1900

20 problems are solved during the period of 1901-2007

1-107

V. E. Firstov / Natural Science 2 (2010) 915-922

920

problems

general scientific methodology for example, in the

course of a canon or the conception of centrism.

The conception of canon development is often ob-

served in social sciences for example, in the states theory.

As a canon here, the ideals of a democracy state justice

can be taken into consideration dating back to the repub-

lic of Ancient Rome (510/509 B.C.) and

Platons

′

di-

alogues (427-347 B.C.). In this case, GMP-strategy is

started through the axiomatization of the justice canon,

on the basis of which the deductive theory is formed.

Nowadays, this theory represents a separate sector in

mathematics, known as cooperative games, within the

framework of which, it became possible to explain the

peculiarities of modern democracy [19].

The centrism conception represents a general principle

of methodology, the grounds of which dates back to an-

cient times, having a reflection in antiquity doctrines,

traditions and religions. According to Archimedean in-

terpretation, this principle leads to the conception about

center of gravity (barycenter) of a material corpse. On

the ground of this principle, modern classical mechanics

was formed. GMP-strategy on the basis of barycenter

conception is formed by the following way. In 1827,

А.Möbius gave mathematical grounds to the barycenter

of the

system s

′

material points, that is, he came to the

grounds of

barycenter s

′

coordinates, which turned out

to be projection ones [20]. Thus, mechanic concep tion of

barycenter acquired an abstract interpretation within

projection geometry. Furthermore, GMP-strategy can

have a lot of variations, for example:

1). With the help of

barycenter s

′

coordinates is given

the interpretation of Hardy-

Weinberg s

′

law in the

genetic population [21,22] and on this ground, a power

interdisciplinary direction in the form of mathematical

genetics has appeared [23];

2). In the end of the ХХ

century, a famous american

specialist in the field of psychology of the arts R.Arnheim,

propounded a thesis [24], according to which, the psy-

chology of professional artists differs by rather intuitively

sensitive color perception and as a result, they, anyhow,

see the distribution of color shades on the pictorial field as

a balanced one. This thesis is proved by the experience

within the conception about the colorimetric barycenter of

paintings, developed in the works [25-28], where it is in-

troduced through mapping:

→

W, (21)

which to every point of the pictorial image Im, depend-

ing on its color F, is brought to conformity with a

non-negative number from W set, which is designated

the colorimetric mass of this point. The mapping (21)

determines structural colorimetric of a pictorial work.

The colorimetric barycenter of the picture, with the posi-

tion of which the compositional peculiarities of a given

picture are connected, can then be calculated on the basis

of the well-known mechanics formulas. As investiga-

tions have shown [25-28], the colorimetrical barycenter

is a balancing color point of a pictorial work and this

fact has been proved by the formation of barycenter en-

semble for the large collection of paintings. For this,

barycenter ensemble coordinates of pictures are mapped

on the unit square and dotting image of ensemble re-

ceived like this gives an idea about barycenter dispersion

relatively to central position. Figure 1 shows such colo-

rimetric barycenter ensemble from 1174 pictures of

painters of the ХХth (white dot indicates the average

position of the barycenter for the ensemble). It is evident,

that artists representing a variety of compositional ge-

nres in many cases try to avoid significant deflections

from equilibrium of colorimetric mass in the picture.

This GMP-strategy within the conception of colorimetric

barycenter represents an important component in teach-

ing mathematics in the field of humanity education [29].

In pedagogics GMP-strategy is carried out on the ba-

sis of cybernetic conception [30]. Objectively, it is

caused by the fact that in the field of didaktikos, peda-

gogics is based on the theory of cognitive processes,

which realize transformation and transfer of information

from generation to generation. Cybernetics promotes the

development of pedagogical science helping solve aris-

ing contradictions between its content and form not only

through experience, but also within the category of

morphism with the help of modeling and optimization of

pedagogical processes. The realization of cybernetic

conception while making up the fundamental theory of

mathematical models in order to regulate effectively

cognitive processes in teaching comes from information

nature of pedagogical processes. The control in this case

can go on through aim influence on quantitative or qua-

litative aspects of information, realized in the teaching

V. E. Firstov / Natural Science 2 (2010) 915-922

921

Figure 1. Colorimetric barycenter ensemble from 1174 pic-

tures of painters of the XXth century.

process.

Models, describing the control by cognitive processes

through aim influence on quantitative information aspect

of corresponding educational content, are formed on the

basis of metric functions, which, in this case, have an

explicit mathematical form. The procedure of optimiza-

tion in these models has a universe character, as abstract

quantitative information measures are in its ground and

the optimal control in this case leads to the improvement

of the systems organization of the teaching process

through the search of optimal configuration of informa-

tion nets and flows in this process according to mini-

mum criterion of information entropy. The class of basic

models, the control of which is carried out according to

the quantitative measures of information, includes:

Socrat s

′

dialogue, testing, class-lesson system of teach-

ing, organization of group cooperation during the teach-

ing process and the procedure of subject planning of the

teaching process [31-34]. Given models make up the

basis of a number of information technologies, which

have been approved in the teaching process. In particular,

at optimization of group cooperation in the teaching

process the observations have shown the rise of progress

standards of education contingent up to 20-25%.

Models, describing the control of cognitive processes

through aim influence on qualitative (semantic) informa-

tion aspect of educational content, are formed within ac-

cepted model of knowledge interpretation. In accepted

cognitological model (point 2) the system of knowledge

is introduced within informal axiomatic theory in the

form of semantic net. This net is properly metrized and is

characterized by a definite system of coverings in point 3.

It makes possible to introduce net parameters of optimi-

zation, controlling qualitative aspects of the knowledge

system under consideration (points 3-6). To the class of

basic models, the control of which is carried out by in-

fluence on information semantic aspect, models of for-

mation of education content, creative pedagogics and

realization of teaching on interdicipline level are included.

The criteria of such control optimization are formulated

in points 3;4 and at optimization of deductive conclusion

leads to reducing of volume of analyzed information

within criteria (13),(14), and at optimization of creative

processes they are controlled by criterion of significance

(18). These theoretical statements are confirmed not only

by experienced data of Tables 1, 2, but also are effec-

tively realized by the author in his teaching work while

training teachers of mathematics at Saratov State Univer-

sity after N.G. Chernyshevski (Russia) since 1997.

9. CONCLUSIONS

As a matter of fact, GMP-strategy based on Pare-

to-optimization (18) is treated as a control by definite

random process Th(S;t), modeling the creative search

while developing the space Th(S). This search turns to be

more effective, if it comes from more significant posi-

tions. This thesis is acknowledged not only by theoreti-

cal premises (points 3-6) and by the success in solving

Hilbert’s problems (Table 2), but by the whole process

of historical mathematical development. It is difficult to

overrate Pithagor’s theorem in Euclid’s geometry, De-

sargues’s and Pascal’s theorems in projective geometry,

Euclid’s algorithm in the theory numbers, Viete’s theo-

rem in the algebra of polynomial, Cayley’s theorem in

the theory of groups and so on.

Nowadays the significance of specified states is actual

as well. Recently, in the work [35] within the framework

of GMP-strategy for determining of Pythagorean triples,

a special matrix transformation semigroup’s transforma-

tion of primitive pairs has been made up. Thus, a close

link between Pithagor’s theorem and Euclid’s algorithm

is stated and original approaches to the solution of the

oldest mathematical problem about the power of a set of

twin primes numbers. It is important to say that, except

the original mathematical results, an effective training to

the methods of the mathematical creation takes place

within the framework of GMP-strategy.

The possibilities of GMP-strategy are not limited only

by the control of creative processes in the sphere of ma-

thematical education. In the morphism’s category, GMP-

strategy is spread on the level of interdisciplinary teach-

ing. Thus, the integration of mathematical knowledge in

the field of natural and humanitarian sciences takes place.

In this case, GMP-strategy realizes a certain variant of

common scientific methodology (for example, canon or

centrism). This variant is axiomatized and then theoreti-

cal model of the object or phenomenon is formed. It is

important to say, that in the process of interdisciplinary

GMP-strategy, effective training to the methods of ma-

thematical creation takes place. So, the original reflexive

conception in the fields of creative pedagogics is rea-

lized on the basis of GMP-strategy.

10. ACKNOWLEDGMENTS

The author thanks Alaitseva V. V., an English teacher of Gymnasia №5

(Saratov, Russia), for the help in translating this work paper.

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