What is an intellect? There isn’t this word into the World Book Encyclopedia
from 20 bands. Here is a big enough article about intelligence. It does not say what is
intelligence because most psychologists are concerned with a measuring of intelligence
by the testing it with help of Intelligence Quotient, I.Q., rather than by the trying to
define it. However, the World Book Dictionary from two bands contains the word
‘intellect’ because there isn’t place enough for an explanation I.Q. that hasn’t some
kind of merits but it is the first attempt to introduce a quantitative measure. We can see
here a very short definition:”the power of knowing;understanding”. Authors of the
dictionary give further an example:”Isaac Newton was a man of intellect”, obviously
understanding very well that this definition can help us to lose any way in broad
daylight (power, knowing, understanding). The bright dazzle of understanding blazes
up into our mind, but the following below remark:”Elephants are intelligent animals”
immerses it into darkness again.
The development in this area has gone very slowly, and it is impeded by two main
causes. First, this field lies at the junction of modern mathematics, information theory,
neurophysiology, and psychology. Second, a conception of intellect does not have a
conventional quantitative measure. The used in psychometrics literature IQ doesn’t fit for
this goal because the creative process depends very strongly on the freedom conditions. It
is impossible to estimate the intellect within the bounds of psychological experiment, and
the level of universality and complexity of this conception is very high. It is impossible to
estimate the intellect in a direct way. Finally, it is necessary to provide the independence
of estimation from the specific peculiarities of the intellect and its activity realm.
In this brochure I would like to present my own approach to a notion “an intellect
“. I offer to estimate not an intellect, but the intellectual product. The following
statements underlie this approach:
1.The mathematical dimensionality is the most general characteristic of any
2. The dimensionality of the product changes during the intellectual process and
characterizes the intellectual strength.
3.Napier number ( e )is the proper unit for a measuring intellectual efforts.
4. The intellectual effort is a function of three independent dimensionalities.
However, I must begin with a definition too. It is necessary that the definition
include main property of intellect and a quantitative measure for its estimation. I offer
the following one,”The intellect is a capacity to make distinctions manifesting in front
of it definite measure of intellectual power.” This definition has only a scientific value
because we don't’ have any measure for it up to now. I offer else one more
practical,”The intellect is a capacity to make distinctions and get products that contain
an information of positive intellectual value.” I am not very satisfied with them, but I
only hope for more understanding later.
A notion “intellect” is one of the most popular concepts in the scientific world.
Computer semantics operates with concepts such as “intellectual effort” and “intellectual
strength”. The informational systems and the systems of artificial intellect are being used
in computer science and control theory.
People, animals, and even computers could be the intellect carriers. An intellectual
system can consist of either one (a sculptor) or many intellect carriers (an aircraft
research center) .As any system, the intellectual system has the input and the output (as a
rule, multivariate) that are connected via an informational-intellectual channel. This
connection is based on two main concepts, the information and the intellect.
Unfortunately, these concepts are not proportionate from a point of view of their
scientific significance .The concept of the information has the quantitative characteristics
that have been introduced by C. Shannon, the information measuring unit, and the
information amount. Information theory wouldn’t be without them. The concept of the
intellect did not have a conventional quantitative measure up to now. Whereas it is
known, that the science principles and the science regularities could be established
mainly through the quantitative relations.
Of course, the information as a term is not very clear either. .A.N.Kolmogorov had
analyzed three approaches to the definition the amount of information in one of his last
works .He pointed out the necessity of introducing a new concept about the value of
information .The problem of defining the intellect as a term is even harder .
The intellectual system can interact with any physical or spiritual (tangible or
intangible) object in the surroundings. In other words, the object in the input of
intellectual system could be defined either in a physical or an informational manner. The
object undergoes some changes inside the system so that a new object appears in the
output .We will refer to this object as the intellectual product. The final product may
represent a problem solution, an art, a technical or architectural project, a dream or
clairvoyance, etc. This product in output may be represented physically or informatively,
and it is peculiar pabulum for other intellectual systems.
The intellectual process is the process of modification of object from its state in input
to the final state in output. The modification takes a place in the informational-intellectual
sphere and, from the point of view of real psychology, it represents a specific triad of the
behavioral pattern: estimation of object in input; modification of object in immersion
stage; estimation of final product in output (Figure 1).
Object INTELLECTUAL SYSTEM Product
Input Immersion Output
Xm Stage Yp
feedback Feedback feedback
The feedback between the input and the output provides, according to the theory of
the advanced reflection, an influence of future result as a stimulus. The final product must
correspond to the data about the object in the input. Thus, the product is the universal
factor that defines all intellectual process in the sense, that it is the goal (stimulus) of
intellectual process and it has an influence on the choice of the optimal dimensionality.
The object changes its dimensionality in the intellectual process .It belongs to one of
three spaces in the consecutive order: the input space is denoted as X; the immersion
space is denoted as U; the output space is denoted as Y. We will denote the
dimensionalities of these spaces, respectively, by the numbers, m , (m + k) ,and p .In
many cases, a mathematical spaces (a dimensionality and a topology ) are not sufficient a
priory. Intellectual system just must make this work, and a change of the object
dimensionality will characterize the intellectual strength of the working process. There
are two types of knowledge: direct, inner, which has a term ‘gnosis’, and external,
natural, which has a term ’science’. Each of them has itself methodology of knowledge:
mysticism and rationalism. If we demarcate them very hard then their possibilities will
decries. Therefore, it is better to combine them in unite frames. We can’t put them one
into another, but we can alternate them in certain consequence.
The first stage must be rational because we usually have an object-subject situation in
the beginning. Here we separate a part from the whole and lose some relations of object
inside of whole. That’s why a logical way of solution could come in the died end. In this
case we must switch over mystic way (the second stage). Here we examine our object in
the space of more dimensionality. A perception of the object in the space of more
dimensionality is possible due to association of different feelings and consciousness
levels. People can conceive a reality with high dimensionality because they have six
feelings and six soul states. The third, an ending state, can be rational because we must
put the result into form that will be understandable for other intellectual systems.
Thinking demands a great energy. People obtained the fundamental intellectual
achievements in a realm both of scientific and mystic experience in the age from 19 to 36
years old. A brain can come out on the highest energetic level namely in this period of
2.1 Topological Dimensionality.
We had seen above that the notion of dimensionality is the key concept for us. It had
been introduced in mathematics by L.Brauer and A.Lebeg in the 1920s. Any closed
bounded set, lain in the n-dimensional space, has a integer-valued dimensionality m, m <
n. The object with the dimensionality m has a finite covering of multiplicity (m+1). This
is a so called topological dimensionality. The idea of dimensionality based on covering is
very valuable not only as a quantitative measure, but also in a cognitive sense. Namely, a
set of the totally defined covering elements (selected by us) is an original tool of
knowledge. We get some representation about an unknown object as a result of a simple
operation of superposition. And when I write about an unknown and odd notion as a
measure of intellect, I am just trying to cover it by a set of the semantic neighborhoods of
the known already notions. In the applied disciplines the dimensionality of the object is a
number of independent variables or factors. In our case, the dimensionality is the number
of degrees of freedom, which are essential by the interaction between the object and the
intellect. Moreover, the optimal dimensionality is the dimensionality, which provides the
minimal intellectual effort that is necessary for the product producing. The last is the
most widely used notion here.
It is known from the dimensionality theory that empty set has the dimensionality
m = -1.The isolated point has the dimensionality m=0. A curve has the dimensionality
m=1 in a space of any dimensionality. A surface has the dimensionality two. A solid is
the 3-dimensional object. The mass with variable density has the dimensionality m =4. If
the temperature of this mass is a essential factor for us, then it would be the object with
the dimensionality m = 5,and so on…According to the dimensionality theory ,the
dimensionality for the sum of some objects is the same as the greatest dimensionality of
one of them.
We can make the different transformations with the m-dimensional object in the n-
dimensional space. How does the number m change in this case ? The dimensionality is
the most important topological invariant. It doesn’t change if a transformation of the
object is bijectivel and mutually-continuous. If the transformation doesn’t satisfy these
requirements then the object may change its dimensionality. For example, sea surface by
good weather has the dimensionality m = 2. However, the dimensionality of sea surface
increases, m > 2,when white horses appear on the crests of waves. The surface becomes
2.2 Hausdorff’s Dimensionality.
During the last century mathematicians came up with many different notions of the
dimension. Several of them are topological in nature; their values are always natural
numbers and don’t change for topologically equivalent objects. Other notions of
dimension capture properties, which are not at all topologically invariant. The most
prominent one is the Hausdorff dimension . Hausdorff dimensionality of the
scaleinvariant object may be fractional value, and it is always equal or more than the
topological dimensionality. Physics and mathematics use this concept widely and
fruitfully. It’s about the time to do that in other fields of knowledge. A dimension
where ‘s’ is the reduction factor, and ‘a’ is the number of pieces of divided structure, is
called the self-similarity dimension.
There are different mathematical methods to estimate the dimensionality:
---theorem for ordinary physical objects;
--factorial statistical analysis for stochastic objects;
--Takens procedure for objects with chaotic dynamics.
Usually, it’s very hard to estimate the topological or Hausdorff dimensionality using
their mathematical definitions directly. We need some specific measurements and
software for this. Often the dimensionality problem is solved approximately because any
intellectual system; first, attempts to reduce its intellectual expenditure; and, second, is
able to make mistakes. However, this doesn’t lead us necessity to the unacceptable
results. In fact, an intellectual system estimates a number of essential variables
(coordinates or factors) of object, which the system is able to observe. Leibniz said: “The
movement is only there where a change accessible for the observation takes a place;
where a change is not observable there is no any change”. The observability of the
changes is provided by both organs of sense (hearing, sight, smell, touch, taste,
gravitation) and inner states of soul (love, faith, freedom, will, conscience, fear ), and in
the same way by special measuring apparatuses . An intellectual system can record more
than ten variables, but usually our experience doesn’t spread beyond 5-dimensional
3.1 Estimation of object in input.
As a rule, an object in X-space is open-closed. That means it doesn’t have the bounds for
some its variables. Besides, either an object has enough indeterminate structure or doesn’t
have it in general. Finally, the object in X-space has only an eigenmodification. Thus, the
object in the input is not restricted enough, not structural enough, and not variable enough.
In other words, the object has the maximum indeterminacy and presents a badly- posed
problem for an intellectual system. Therefore, a situation of “chaos” usually arises in its
input. When we have a problem, which has been stated by somebody, then the number of
variables of object is finite and known. However, in practice, this number is often infinite,
and we don’t know what variables are essential and independent. Of course, there is one
limitation in this situation: the dimensionality of the object in the input can’t be more than
the number of variables accessible for observation.
The intellectual system must choose some variables x1 ,x2 ,…,xn for the description
of object and estimate its dimensionality m. Variables x1 ,x2 ,… xn are known as the
variables of order . The dimensionality m may be less than n, if the variables of order are
dependent according to the data about the object in the input .The intellectual system
determines the number m of essential and independent variables, if the object is
topological. If the object is fractal then it determines Hausdorff dimensionality .Any
intellectual system attempts to reduce the number m using different ways for that :first, a
principle of decomposability of a multi-dimensional object; second, a principle of assembly
of some variables of order for a generalized variable .
Let’s begin with the simplest case. There is an amusing problem .Two friends D and
B run into each other at the bus station .D knew that B had three sons, but didn’t know
their ages .B answered that product of their ages is equal to 36 and sum is equal to the
number N of the bus that was about to leave. D looked at the bus and said after a small
hesitation that he needed some additional information .Then , B told him in a humorous
manner that his oldest son is red –haired. D had solved the problem in a moment .Now, we
must solve this problem . However ,it is more difficult for us than for D ,since D knew
the bus number and we don’t know it.
Let’s estimate the object in the input .As is easily seen the variables of order are three
ages x ,y ,z .The object can be described as the following condition : x y z = 36 ; x + y +
z =N ; x > y and x > z .The first of these conditions connects our variables, therefore ,the
object’s dimensionality is m =3 –1=2. We will temporary move on to other problems and
later return to the redhair one.
Now consider a luminous body, which can change its shape and position in space
practically with no inertia (UFO). Any celestial body has three space coordinates x 1, x2,x3
and the time x4.The variable of luminosity is denoted by x5 .It needs yet one variable x6
for notation of the density .It is know that variables x5 and x6 are connected by Einstein’s
relation for a mass and an energy , x5 = f( x6 ) .We find the dimensionality of an UFO
m =6 –1 =5.
How do we estimate a cloud?
It depends on what our goals are. If you just admire a cloud at the sunset, you don’t
puzzle .If you are a painter and want to express your feeling in the picture, then you need to
estimate the shape and the color of cloud .For this, it is enough to have three variables.
However, the cloud isn’t an ordinary body .It is fractal which has Hausdorff
dimensionality3 < m < 4. If you are a scientist and want to find a regularity of formation of
rain clouds, then you need the variables of temperature and pressure .The process of
formation of rain clouds is dynamic chaos; therefore, it needs the Takens procedure for the
estimation of cloud.
Thus, the intellectual system, estimating object’s dimensionality, takes into account
only those variables that are essential from the point of view of getting a future result. The
final solution depends on the intellect .The same object may have different dimensionalities
in different intellectual systems. However, any product has its optimal dimensionality m in
the sense that deviation from it to either side increases the intellectual expenditures .It is
obvious in the case m’ > m .In the case m” < m, the intellectual expenditures increase in
the immersion stage . The last will be shown in chapter 6.
3.2 Estimation of object in the stage of immersion
The immersion is the most complex stage of the intellectual process. It may take place
both in the sphere of consciousness and in the sphere of subconsciousness. Speaking
figuratively, it is a suspended bridge in a fog between two rocky banks. The U –space is
very complex .It is a stratified space as a result of product X –space by k-dimensional
subspace .The immersion provides maximum freedom for universal modifications of the
object .Assume ,you must think over some idea or situation .The German philosopher,
I.G.Fikhte, said : “A necessary way that is capable of accomplishing this action is inherent
to the nature of the intellect and does not depend on any arbitrary rule .It is something
necessary ,however ,that could be accomplished only in the free action and by the free
action ;something found ,finding something ;however ,it depends on the freedom”.
Measure of this freedom, obviously, is defined by dimensionality of U –space . In this
stage, an object with dimensionality m immerses in U-space with dimensionality (m+k)
.All these variables u1 , u2 ,…,um+k are functionally independent ,we would define
them as the variables of immersion .The intellectual system adds k (k.0) new variables
without assistance .These additional variables are introduced at that time when all
attempts of algorithmic solution of the problem have turned out unsuccessful. The
intellectual system is at a logical dead end .The additional variables are introduced in
different ways: due to the release of some permanent parameters of the object; due to the
using of new variables from the spaces adjacent to X- space, for example, the variables of
soul state. Time variable and random variable appear among the additional variables more
The variables introduced in the immersion stage, at first,
restore the missing variables of order in the description of the object; second,
remove a logical inconsistency of information about the object in the input; third,
provide a long-range (global) connection between variables of the m-dimensional
object in U-space.
Let’s get back to the problem about the “red-haired’’. We can’t decide it until we
transfer a fixed parameter N (the number of bus ) in a free state ,that is a range : 38 ,16 ,14
,13 ,13 ,11 ,10 .Thus, the immersion stage is characterized by the number k = 1 in this
case. Now we can see what D needs additional data for .The fact is that the number 13
has two expression : 6+6+1 =13 and 9+2+2 =13. Last condition of the problem (B has an
oldest son) removes this indeterminacy.
3.3 Estimation of object in output .
The problem about the red-haired has a solution: x = 9, y = z = 2. It is a point in 3-
dimencional space. It is known that dimensionality of point is p = 0. The estimation of
dimensionality of the output is realized just as in the input. Moreover, the object in the
output should be estimated in addition as a product. The ready product must contain the
novelty, the originality, the perfection, and the social utility. First of all, the intellectual
system must do it itself. It is also necessary to estimate the product with the help other
system (the commission of experts ) which has higher level than first.
Object in output is some (conceptual, physical, mathematical, and so on ) model
which appears as a result of some ultratransformation of the object from U-space into Y-
space .The variables of model y1,y2 ,…yp , p m + k ,have very complex or just implicit
connection with the variables u1 ,u2 ,…,um+k. Ready product has a rich structure ,the sign
inequality in the formula p < m + k just is a direct evidence of this. From the topological
point of view, the object in Y-space is more closed than in X-space .The consistency of
ready product to input data is tested by means of feedback.
3.4 Basic principles for estimation of dimensionality.
The object has its dimensionality in each stage of the intellectual process, but the
estimation procedure of numbers m, (m + k) , p is whole and indivisible .This unity is
based on the two objective principles .
First principle: The intellectual system strives to minimize its intellectual effort .
Both a physical system strives to minimize its energy and a intellectual system strives
to minimize its expenditure .The concept “optimal dimensionality “ of the object, that was
introduced chapter 2, corresponds to this principle .
Second principle: The structure of the object remains invariant in the intellectual
process. The structure of the object is some totality its characteristic properties which are
invariant relatively the given class of the transformation .The structure of the object in the
intellectual process remains invariant exactly in the same way as the amount of energy in
the physical process. If the intellectual system doesn’t keep to these principles, then there
isn’t a product.
- 10 - 10
CALCULUS OF DISTINCTIONS.
4.1 Introduction into calculus.
The calculus of distinctions is a mathematical tool on the basis of G. Spencer Brown’s
laws of form. The calculus has only one operator of differentiation of a region from the
whole. This operator is denoted by the symbol and called ‘Distinction’. An action of the
operator on any object x indicates a transition from x to the complement of x with
respect to the whole space, i. e.
x Cx , (1)
where Cx is the complement set of x.
It is a very simple and ‘soft’ operator. We can use it everywhere because a field of its
application doesn’t have any restrictions. An object, which one the distinction operator is
applied to, doesn’t also undergo any modification or variation. The operator selects only
some object from the whole and estimates it along a separating boundary with point of
view of distinction fact from all remaining. A notion of distinction doesn’t have here any
qualitative sense or quantitative measure. These simplicity and softness are both merit and
a lack of the operator. It is a good enough tool for an identical simplification any logical
expression, but it is less useful for a creative thinking, for the last isn’t connected with
global homeomorphic transformation. Set theory and topology are more powerful tools
undoubtedly, but they contain an imperfection of our language and way of thinking in
larger degree and reduce us to paradoxes sometimes.
It is clearly that the operator itself can be by object. If x= then we have , and it is
the lack of any distinction whatsoever. This special state is denoted by symbol and
called the ‘void’, i. e.
The void is an infinite expanse of undifferentiated substance, a state where are not
distinctions. It is infinite and continuous.
It is usual to consider (for example, into arithmetic) that a sequence of identical operators is
equivalent to one of them, i. e.
The calculus bases upon two main theorems.
Theorem I (Invariance),
x x , and
Theorem V (Variance),
These theorems could be proved easily by application of the expression (1) and (2). See
also a website [http://www.transcendentalphysics/ 1 calculus.htm].
4.2 Extension of calculus.
It is readily seen that the calculus of distinctions is less reach than, for example, the set
theory. It is possible that a golden middle between them will be better. Therefore I offered
to Edward Close to complicate the operator by the way of a binding it with one of the
fundamental topological notion, the mathematical dimensionality. Our new notion,
’Dimensional Distinction’ has a symbol n . The using of the superscript ‘n’ to the right of
distinction symbol means
x n Cxn, (4)
- 11 - 11
where Cn is the set of elements with dimensionality ‘n’. We focus now our attention not on
a common distinction but on the distinction according to ‘n’ some properties. Distinction is
grown by dimensional one, and it extends the possibilities of calculus. We also remain a
primary operator in its usual sense.
Now we have a problem of an interaction of the dimensional operators with the different
dimensionalities. To develop some algebra for this case, preserving the basic conditions of
the existed calculus, we propose one axiom of inclusion,
x n x m, mn. (5)
and the formal rules for the calculating of final dimensionality into complex expressions:
n m m and abc c-(b-a). (6)
It is readily seen that the axiom (5) doesn’t contradict with the modern set theory. Let’s
k , if k0, (7)
i.e. the objective distinctions are perceived outside consciousness only into space with a
positive dimensionality k>0.
Theorem I is formulated now as theorem GI.
Theorem GI (General Invariance),
xmx n , if mn.
Proof. We have according to the definition (4) x m= Cxm, and x n= Cxn. It is following
from axiom (5)
Cxn Cxm, if mn.
Now we can write
x mx n= Cxm, x n Cxm, x m= Cmm= ,
where C is the whole in the m-dimensional space.
It is necessary to remark that theorem GI is obvious according to the formal rule (7), and
theorem I is a partial case of theorem GI.
Theorem V also is reformulated.
Theorem GV (General Variance),
x n m x, if m>n, and x n m x, if m<n.
We will prove only the first part of this theorem, for the second part is proved identically.
Proof. According to axiom (5) we have
x n x m , m n.
Using the operator m for the both parts of upper expression, we will have
x nm x mm.
I must remark that symbol ‘’ is changed to the symbol ‘’ because, according to the set
theory, if the sets A and B are connected by AB then complements of them are connected
by the opposite relative, i. e. CACB. How it is x mm=x, so we have finally
x nm x, mn.
- 12 - 12
4.3 Expression for intellectual process.
Now we can write the intellectual process, that was considered into chapter 1, in a form
of mathematical expression:
An estimation of object in output
An immersion of object k+p
An estimation of object in input m+k
x m y. (8)
Object in the input is denoted by xX, and product is denoted by yY. A left part of
expression (8) is a three-stage process of making distinctions. The first stage,
means a narrowing of indeterminate distinction to the optimal dimensionality ‘m’ by the
way of examination the properties of object in input. Let’s to denote Cxm=. The second
means distinction into a extended complement of by the outside way, i. e. by the way of
examination properties which are manifested into complement set with dimensionality
m+k. The third stage,
means a narrowing of distinction to product dimensionality ‘p’. Namely, the formal rule
xmm+kk+p x k+p-(m+k-m) x p.
CHARACTERISTICS OF INTELLECTUAL PRODUCT
5.1 Unit of intellectual effort.
The information is measured in the bits .The children already know about binary code in
elementary school. What units is the intellect measured in? The children have known about
it too (the marks, the scores, grades, per cent). However ,the adults have not agreed with
them for this once ,because these scales are very short and the units don’t have a causal
relationship with the intellectual effort .To choose a more natural unit, let’s examine a
computer .A man is too complex for us in this respect .
The main body of any computing system is the processor. Usually ,the processor has the
following composition of parameters :
d-digitment of the data bus [ bit ] ;
q-clock frequency [ MHz ] ;
v-maximum speed of data transfer [ Mbyte/s ] ;
w-capacity of a processor [ MIPS =10 oper/s ] ;
s-number of system operation .
Using theorem for analyses of physical dimensionality of this composition, we
derive two dimensionless ( pure ) variables
L=d q/v and P=w/s q .
The parameter of the organizational structure is denoted as L and the relative
capacity of the processor is denoted as P. A functional relation P=f ( L ) is showed in fig.2
- 13 - 13
1 2 3 4 L
Figure 2 .
The curve was obtained as the result of a statistical processing of a random sampling
.The sampling contained about 30 processors of different types .As shown in fig.2 ,the
maximum relative capacity takes place by L*=2.8 .Now ,let’s take L*=2.718 ( number of
Napier e ).
According to the principle of minimization of the intellectual expenditure, any
intellectual system must work in the neighborhood of point of the maximum capacity L*.
Most likely, this value would not be equal to e for a brain, though an analogy between brain
and computer is well known. Many scientists assume that consciousness is connected
directly with the structure of the brain .However ,the question of choice of a unit of
measurement is not the question of a principle ,but a consent in any area of knowledge.
That’s why we may take the number e as the unit of the intellectual effort .
Many people assume that a thought has an energetic equivalent. When we’ll have an
instrument for measurement of this energy then, probably, we’ll introduce yet one physical
unit .Maybe ,a relation between the structure and the energy will be taken in theory of
intellect the same place as the relation between the energy and the mass in theory of
5.2 Quantitative characteristics.
The following quantitative characteristics may be introduced for any intellectual
T-intellectual effort necessary for producing a product;
G-threshold of understanding or intellectual effort necessary for using a product.
The level of the intellectual effort is a function of many variables, for example,
T=F( m ,k ,p ,e )
where k 0 , p m + k , e=2.718 .In addition ,we demand that this function satisfies
the conditions :
F ( * ) > 0 ; and F ( 1,0,0,e ) =e .
Further, the formula can be written as
T= ek ( e m + p e-1 ) =F(m,k,p,e) (9)
Needless to say that a process of understanding of the final product must satisfy the
conditions : k=0 and m=p .Hence ,from (9) we find a threshold of understanding
G= p (e + e-1 ) =3.086 p (10)
For any intellectual product T > G .If T = G ,then this product is a copy ( or an analogy )
of another product .
We want to add some explanations for the formula (9).The value em characterizes
the effort of the intellectual system in the stage of estimation of the object in input .We
suppose that this process is fulfilled with the maximum capacity in all m variables,
- 14 - 14
because the maximum indeterminate object requires an analysis of a very big set of the
possible alternatives .
The value pe-1 characterizes the effort of the system in the state of estimation of the
ready product in output .The final product exists in one variant .Its structure is known .We
must estimate only this single variant .Therefore ,the checking condition does not need the
maximum capacity .It needs the maximum accuracy .The transformation ,which was used
for the obtaining of a product ,is known too .Usually ,we need the inverse transformation
for checking .We suppose that the structural parameter of the system equals e-1 in this
Further, we take just a sum (em + pe-1 ) as X-space and Y-space don’t have a
common boundary. They are combined by means of U-space .The value ek characterizes
the efforts of the intellectual system during a creative condition in the immersion stage
.Each of the additional variables increases both the set of variants and the complexity of
each variant . We use here the k-positional operation of multiplication for the structural
parameter e. Further ,we take just a product ek ( em +pe-1 ) as U-space borders on both X-
space and Y-space .
Of course, these explanations can’t satisfy us .I just describe an heuristic method of
the obtaining of formula (9) .The scientific basing must rely on :
1.Research of theoretically basis for this formula ( Section 5.3 and 5.4).
2.Statistical test of hypothesis about an agreement the calculate and experimental data
( Chapter 6 ).
3.Testing of a capacity for work in a singular, limiting case ( Chapter 7).
4.Testing of logical and own consistency of formula (9), as a calculate expression and
as an intellectual product ( Chapter 8 ).
5.3Differential equation for intellectual effort.
The function F(*) is continuous with respect to its arguments since we make use of the
notion of the dimensionality in the broad sense of the word (the dimensionality may be also
fractional ).Hence ,we can differentiate the function (9) and suppose that it is a solution of
some equation of mathematical physics .It is known ,the models of physical processes are
based on the principles of conservation of an energy ,a momentum ,or a number of particles
.The model of the intellectual process is based also on the principle of conservation of the
structure ( Section 3.4 ).Hence ,it may belong also to this area .
In fact, the function (9) answers Helmholtz’s equation
where - Laplacian, with boundary condition:
F/mk=0 =e and F/pk=0 =e- 1 . (12)
The equation (11) describes the process of spreading the electromagnetic waves in a
homogeneous medium. It is possible to assume that the intellectual process also has a
wave nature .The conditions (12) show that the velocity of variation of the intellectual
effort is a constant both for the input and the output. These velocities characterize a
receptivity of the intellectual system and don’t depend on the object .We may agree with
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5.4 Interaction matter and intellect.
We will use a material from chapter 4 of ‘The Undivided Universe’ by D.Bohm&
B.J.Hiley. Let’s see an interaction a partikle with mass m and an intellect. The particle is
described into four-dimensional space with help of the wave function p(r1,t), where
r1=(x,y,z). The intellect is described into three-dimensional space with the function of
where r2=(k,m,p). We will consider the case where a common wave function (r1,r2,t) can
be written as the product
The quantum potential in this case has two terms,
Q(r1,r2,t)=Qp(r1,t) + Qe(r2,), and Qe(r2)=- h2/2m F(r2)/F(r2).
According to the formula (3) of this chapter we have
Qe(r2)= - h2/2m . (13)
The quantum potential (13) will be a very small in classical physics because m is big
enough. However, if mass m of particle is small then value (13) is essential. This potential
doesn't influence on the particle moving because (Qe)=0. However, this potential will
provide a phase shift according to the quantum Hamilton-Jacobi equation . And I think
that this fact is correlated with Heisenberg’s uncertainty relation.
Nils Bor has sad, “If man does not understand a problem he writes many formulas and
when he has understood what is the matter only two formulas are left in the best case”. In
our case maybe we have these two formulas. Denote the second formula by a special sign,
T=ek( me +pe-1), (**)
and show interconnection between (8) and(**) whith help of Fig.3.
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