Institute of Information Transmission Problems, Moscow, Russia
Note: This manuscript was received from Russia as translated in Russia. The text has not been changed expect for minor words such as "a" to "the", etc. To communicate with the author, Valerii Polesskii, e-mail to: bst@ippi.ras.ru with Valerii Polesskii as the subject. For suggestions as to the improvement of the English text, please send them to: spircom@emf.net.
ABSTRACT
To answer these questions we refer to the author's conjecture [1,2] of the existence of a subtle structure of the human thinking. In turn this three-level structure follows from the esoteric Tibetan's teachings [3,4]: it corresponds to a structure of basis of the hypothetical universal mind. Here are the answers:
1) They are concepts corresponding to a manifestation model of the basic abilities of the thinking into a mathematics beginning. This model (the mathematics beginning) is an initial mathematical form clas that has the simplest individual quality of any forms -- an individual distinguishabilty.
2) They have a three--level structure whose elements are three groups of mathematical concepts. The concepts of the first group correspond to a manifestation model of the two initial abilities of the thinking into the mathematics origin. This model contains initial mathematical symbols. The concepts of the second level correspond to a manifestation model of the three principal basic abilities of the thinking into the mathematics beginning. This model is a form class (using the initial symbols) manifesting the common quality of any forms -- the quality to unite -- and closed under basic intellectual operations. The concepts of the third group correspond to a manifestation model of the seven basic abilities of the thinking into the mathematics beginning. This model is an individual distinguishability of the initial mathematical forms -- the simplest individual quality of any form.
3) They arise in the process of realization of these three models.
As a result, we discover that the initial concepts are the simplest models of all basic concepts of mathematics that were established in the course of a long historical development. Such a phenomenon demonstrates the subtle esoteric nature of mathematics beginning and leads us to esotericism. Since the basic esoteric ideas illuminate the hidden nature of the mathematics beginning, we must ponder on these basic esoteric ideas.
INTRODUCTION
The initial (simplest, basic, etc.) concepts of mathematics were established in the course of very long (over the span of at least 20 centuries) historical development. These concepts are usually given by the axioms of set theory and mathematical logic. They are contained in the axioms.
From the original we do not need to think about the complexities related with the naive concept of set as "any collection of identifying things". We can avoid the complexities later by a well known adaptation of set axioms. From the bottom of mathematics we must give more attention to possible mental origins of the initial mathematical concepts. Also we must try to understand how new initial mathematical concepts are generated by our mind on the basis of existing concepts.
The initial concepts of mathematics can arise from different origins. As a rule, these concepts are deduced from a representation of numbers, space, time, and motion. Or we can say that these concepts have their roots into human consciousness or philosophy. Although the intuitionistic mathematics and the constructive mathematics open (see [5,6]) the mental origins of these concepts and lie in discrete mathematics, i.e. mathematics of finite or innumerable sets; they suppose that the question about the nature of these basic mathematical concepts is an open question. Also nobody questioned the deep mental cause of their arising.
We know that mathematics is fundamental to how we understand ourselves, as human beings and hence start ab ovo in order to answer the three questions. In order to deal effictively with our three problems, we must firstly establish a fundamental background -- an esoteric concept of the subtle structure of the human thinking from [1,2].
A SET OF THE BASIC INTELLECTUAL ABILITIES OF THE THINKING AND ITS SUBTLE STRUCTURE
Presented below is a description of such a set along with the subtle structure of its subsets:
First level. Two initial abilities of "the thinking that had existed before forms" :
Ability to display activity.
Ability to initial discrimination.
Second level. Three principal basic abilities of "the thinking of forms" (basic abilities of the thought--idea):
1. Ability to display purposeful activity including creation and perception of meanings.
2. Ability to create and perceive form qualities and form inter--relations including creation and perception of specific (essential) qualities and inter--relations.
3. Ability to synthesize and perceive forms as syntheses of other forms including the synthesis for meanings and qualities manifestation.
Third level. Four additional basic abilities (manifesting the principal abilities) of "the thinking of complicated and perfected forms" :
4. Ability to create and perceive beauty of forms.
5. Ability to perform an analysis of forms.
6. Ability to create and perceive similarity of forms.
7. Ability to perform ordered manipulation of forms.
We refer the reader to [1,2] for some illustration of these abilities and details. Our esoteric approach to a mathematics origin will be founded on this subtle structure. It is a moment of principle.
In this paper we show that the initial concepts of mathematics are concepts corresponding to a manifestation model of the basic abilities of thought into a mathematics beginning. This model (the mathematics beginning) is an initial mathematical form class that has the simplest indivividual quality of any forms -- an individual distinguishability.
Therefore, the nature of the initial mathematical concepts is esoteric and they have a subtle structure. It is a three-level structure whose elements are three groups of some initial concepts of mathematics corresponding to the three levels of the subtle structure of the human thinking.
The concepts of the first level correspond to a manifestation model of the two initial abilities of the thinking into the mathematics origin. This model is some initial mathematical symbols. We describe them in part I.
The concepts of the second level correspond to a manifestation model of the three principal basic abilities of the thinking into the mathematics beginning. This model is a form class (using the initial mathematical symbols) manifesting the common quality of any forms -- the quality to unite -- and closed under basic intellectual operations (singleton-- operation, one--element expansion operation, and substituition operation). Such initial mathematical forms are finite mathematical objects. We describe them in part II.
The concepts of the third level correspond to a manifestation model of the all basic abilitiies of the thinking into the origin of mathematics. This model is an individual distinguishability of the initial mathematical forms -- the simplest individual quality of any form. We describe them in part III.
In the process of realization of understanding the three ideas we obtain the simplest models of the basic mathematical concepts: concept of set, of mathematical operations, of unordered pair, of ordered pair, of function, of order, of equivalence, of 1-1 correspondence, logic concepts and rules (we write the arising "mathematical assertions" by means of symbolic forms and so "mathematical language" arise), etc., and lastly, the concept of the positive numbers.
It is wonderful, but all basic concepts of mathematics arise automatically in such process. Let us emphasize -- once more -- that the subtle esoteric structure of the human thinking gives us a key for answering the three questions.
By the way, such a phenomenon provides valid scientific proof of the fundamental character of the hypothetical basic abilities of the thinking, and also a verification of the basic ideas of the Tibetan's teachings.
We notice that the initial mathematical form class can be considered as a constructive base for constructing more perfect and complex mathematical forms.
PART I.
INITIAL CONCEPTS CORRESPONDING TO A MANIFESTATION MODEL OF THE TWO INITIAL ABILITIES OF THE THINKING INTO THE MATHEMATICS BEGINNING
INITIAL MATHEMATICAL SYMBOLS
From the basic principles of mathematics the initial ability to display activity is manifested as an ability to create some initial material for constructing different mathematical forms. In a simple manner the initial ability to display activity is manifested by a creation of symbols -- some initial material or a basic stuff for the symbolic mathematical forms.
Symbols are always used to construct any mathematical form: "Am Anfang ist das Zeichen" (D. Hilbert, [6]). From the bottom of mathemtics it is better to use the most abstract symbols, i.e. letters of the alphabets. They give us a possibility to use the most refined thinking for understanding of our three questions. In mathematics, the preference for the use of such abstract symbols was realized long ago, when the concept of number was being developed.
Thus, we use the letters of the concrete alphabet and some other symbols as initial symbols. We admit any reproduction of these symbols.
The non--intersection of the basic symbols are their (initial) discrimination. By this we use the discriminative quality of mind. Of course, we use the discriminative quality of mind for creation and perception of any mathematical forms also.
For example, intuitionism declares [5] that their concept of the positive numbers is founded on:
1) an ability of thought to represent any object as an abstract being by ignoring all its properties. In accordance with [5] "concept of the being gives us the possibility to perceive the object separately from the rest of whole".
2) an ability of thought to copy this being with no limit.
It is clear that these abilities are exactly the two initial abilities of the thinking from the introduction, i.e. the ability to initial discrimination and the abilty to display activity.
PART II.
INITIAL CONCEPTS CORRESPONDING TO A MANIFESTATION MODEL OF THE THREE PRINCIPAL BASIC ABILITIES OF THE THINKING INTO THE MATHEMATICS BEGINNING
THE COMMON QUALITY OF ANY FORMS -- THE QUALITY TO UNITE
FORM CLASS (INITIAL MATHEMATICAL FORMS) CLOSED UNDER BASIC INTELLECTUAL OPERATIONS
Contents
1. Initial mathematical forms
2. Simplest initial mathematical forms -- multisets. Generation process, singleton--operation, and one--element expansion operation
3. Initial mathematical forms of general kind. Substitution operation
4. Summary
1. INITIAL MATHEMATICAL FORMS
The esoteric teaching (see [1,2]) supposes that the idea of the thinking is an idea of forms, i.e., the idea of "unions of some discriminated objects inside a whole". Such a definition is better than the above naive concept of set as any collection of identifying things since the esoteric definition empasizes the quality aspect, ab ovo the simplest (it does not permit any generalization) "quality to unite".
What are the mathematical forms ? Of course, we use our mental abilities for the creation of infinite mathematical forms. But the forms are founded on finite mathematical forms. The last forms are a basis for a deeper mental abstraction. Ab ovo it is helpful to understand:
Are there initial mathematical forms ?
What mathematical concepts do we need in order to define such form class ?
We know (see [1,2]) that in our mind any form (including mathematical forms) arise as a potentially unlimited variety of forms. The nature of our mental abilities is the cause of such phenomenon. Therefore, we must have a process for the creation of hypothetical initial mathematical forms by a synthesis of forms from forms and using some initial material.
The concept of form includes the concepts of whole, of union, and of elements. Let the empty symbol be the empty strip (bond, row) limited by a pair of braces. Now we understand that the simplest mathematical symbol of whole is the empty symbol. We will use the empty symbol, the initial symbols, their initial discrimination, and, of course, the discriminative quality of human mind for constructing the simplest initial mathematical forms.
The elements of any initial mathematical form and the whole will be symbols. The "mathematical" union of the elements--symbols is the placement of them inside the symbol of whole. Any empty strip allows such a placement potentially. It is important that the elements--symbols are placed without their intersection. Such non--intersection of the symbols is their discrimination. It is necessary to use the discriminative quality of our mind for the constructing of the initial mathematical form class. Thus, any initial mathematical form (a symbolic construction) will be a symbol and this symbol can be an element for other initial mathematical form.
From the basic principles of mathematics it is natural to create a form class closed under basic intellectual operations. The idea of a form class closed under singleton--operation, one--element expansion operation, and substitution operation will give us the roots of the first initial mathematical concepts.
It is very important to realizethat the roots lie before any concepts of mathematical language and logic! In fact, we can define our initial mathematical form class without the language's concepts and logic concepts.
Of course, our classics realized such a phenomenon: "mathematics does not depand on language: in mathematics words and word bundles are used only for an expression of truth. The roots of mathematical ideas lie more deeply than the roots of language ideas" (L. E. J. Brouwer, [7]).
2. SIMPLEST INITIAL MATHEMATICAL FORMS -- MULTISETS.
GENERATION PROCESS, SINGLETON--OPERATION, ONE--ELEMENT EXPANSION OPERATION
Let us construct the simplest initial mathematical forms -- multisets. We will construct them by induction.
The simplest multiset is the empty symbol. The empty symbol is a specific initial mathematical form that does not contain any element. It is the simplest prototype for the well-known concept of empty set.
Then we have the singletons consisting of single initial/empty symbol. Any singleton is discriminated with its element. Any singleton is a simplest initial form. However, generally any of its element is an elementary particle of primordal "mathematical matter". We have a first initial mathematical operation -- singleton--operation.
Further. Suppose we have a multiset. Our generating rule of the inductive step corresponds to the concept of a new multiset that is obtained by addition of some initial/empty symbol to the constructed multiset. It is also an initial mathematical operation of synthesis. The new operation allows us to construct new forms from old forms. We say that the form obtained by an one-element expansion from the given form is its one--element expansion. The adding element is replaced on the line between the braces and, of course, the new element should not cross other old elements. For this goal we use the traditional symbols, commas.
It does not matter which multiset we have. We can always get its one--element expansion. Any element of any non-empty multiset is an initial/empty symbol. However, generally any multiset itself is not already an initial/empty symbol. It is possible that elements of a multiset are not distinguished with respect to the usual individual distinguishability of the initial/empty symbols). However, we discriminate the elements of this multiset. We discriminate also the union, the multiset itself away from its elements, and its elements themselves.
The initial proper order of form is manifested in the following way: the elements-symbols are inside of the symbols of the whole as parts are inside of the whole. We have also the simplest inter-relation of the elements of form. Such inter--relation is the union of these element for the form.
It is known that discrete mathematics considers "multisets" as the natural--valued functions on the finite sets. In the sequel, we will see that our multisets are the natural--valued functions of our simplest (finite) sets.
3. INITIAL MATHEMATICAL FORMS OF GENERAL KIND. SUBSTITUITION OPERATION
Now we can define the initial mathematical forms of the general kind. The simplest initial mathematical forms of the general kind are the multisets.
The generating rule of the inductive step corresponds to a new initial mathematical operation of substitution of one initial mathematical form or an initial/empty symbol instead of another initial mathematical form or an initial/empty symbol.
Suppose we have an initial mathematical form. If we use some specific meta-symbol then we can label any of its element. The labeled element is destroyed by the operation of its deletion. We say that the initial mathematical form obtained by the deletion of an element from the given initial mathematical form is its one-element restriction. The one--element restriction is also an initial mathematical operation that allows us to construct forms from forms. We say that the form obtained by a sequential deletion of elements from the given form is its subform.
Let us return to the substitution operation. Suppose we have some initial mathematical form and let some of its element be labeled. We delete this element and add some other initial mathematical form or an initial/empty symbol as a new element (or equivalently, first we add the new initial mathematical form or the new initial/empty symbol and then we delete the old labeled element). The result will be symbolic form. It is also an initial mathematical form.
Now we have the class of initial mathematical forms. We can add any initial mathematical form or any initial/empty symbol to any initial mathematical form. We can delete any element from any initial mathematical form (if the last form is not the empty symbol). We will always obtain [[The result will always be]] an initial mathematical form.
It is easy to see that any initial mathematical form is a graph-rooted tree. Every terminal vertex of this tree corresponds to an initial symbol or the empty symbol. These symbols are attached to the terminal vertecies.
The initial mathematical forms are very simple symbolic forms. However, in what follows we will see that such class contains the simplest models of many fundamentally known mathematical structures.
4. SUMMARY
On the second level of our mathematics beginning, we have the following pattern for the manifestation of the model of the three principal basic abilities of the thinking by the initial mathematical forms:
1. The generation of the initial mathematical form class is the simplest manifestation of the basic ability 1.
2. It is known that all forms manifest properties. The simplest -- the common quaility to unite (such property does not permit any generalization) is manifested by the class of initial mathematical forms. It is a manifestation of the basic ability 2. Thus, the simplest property of any mathematical form is the simplest property of arbitrary forms.
3. The basic ability 3 is manifested by the inductive generating process for synthesis of the initial mathematical forms when the forms are the result of themselves and the primordal mathematical material. Such a process uses basic intellectual operations: the constructing of singletons, the one-element expansion, the one-element restriction, and the substitution. It is well known that induction is the initial archetype of mathematical thinking.
It is easy to describe "structures of intelligence" that correspond to these mathematical concepts. We must substitute some concrete objects for the initial symbols for this purpose. For example, a knapsack (that contains some parcels that contain some things, etc.) is analogous to some initial mathematical form.
PART III
INITIAL CONCEPTS CORRESPONDING TO A MANIFESTATION MODEL OF THE SEVEN BASIC ABILITIES OF THE THINKING INTO THE MATHEMATICS BEGINNING
THE SIMPLEST INDIVIDUAL QUALTY OF ANY FORMS -- THE INDIVIDUAL DISTINGUISHABILITY
INDIVIDUAL DISTINGUISHABILITY FOR THE INITIAL MATHEMATICAL FORMS
Contents
1. Individual distinguishability of the initial mathematical forms as the origin of mathematical language, logic, and other initial mathematical concepts
2. Initial mathematical relations for small multisets
2.1. Initial mathematical relations generated by unordered pair. One--element choice operation and ordered pair
2.2. Initial individual distinguishability
2.3. Initial membership and functions
2.4. Initial order
2.5. Initial sets
3. Individual distinguishability for multisets
3.1. Initial equal cardinality and cardinality preference
3.2. Initial infinite set: the positive integers
3.3. Models for the positive numbers
3.4. Multisets as natural--valued functions on initial sets
3.5. Power operation for initial mathematical form
3.6. Initial relations for initial sets
3.7. Initial relations for multisets
4. Individual distinguishability for initial mathematical forms of general kind
5. Summary
1. INDIVIDUAL DISTINGUISHABILITY OF THE INITIAL MATHEMATICAL FORMS AS THE ORIGIN OF MATHEMATICAL LANGUAGE, LOGIC, AND RELATED INITIAL MATHEMATICAL CONCEPTS
In the previous second level of our mathematics origin we realized the common quality to unite by constructing of the initial mathematical form class. Now we recognize individual qualities of the forms. We see that the idea of individual distinguishability of the initial mathematical forms is an origin of the initial concepts of mathematical language and logic and many other related mathematical concepts. Such a phenomenon confirms the well known opinion that the logic principles have a pragmatic foundation ("according to historical evidence, classical logic was abstracted from the mathematics of finite sets" (H.Weyl, [7])). Also it confirms that "the positive numbers are an immediate emanation of rules of pure mind" (R. Dedekind, [7]).
Thus, on the third level we begin to explore individual behaviour of the initial mathematical forms. Swedish philosophist Henry Laurency declared that "the individual has a natural right to exist, to be different from all others, to be an individual with individuality".
We have a class of forms. An individuality of a form from this class is a singularity that selects this form. This individual singularity of the form needs its name.
We will speak of individual initial mathematical forms and their elements in terms of their names. We need different names for the elements of this initial mathematical form and a name for the form itself. This new distinguishability of names is not the discrimination of the initial symbols.
The simplest individual inter--relation of the initial mathematical forms is manifested on this level. It is their individual distinguishability. We speak of names of the initial mathematical forms when we speak of their simplest individual property, i.e., the distinguishability. We found the distinguishability on the usual individual visual distinguishability of the initial/empty symbols.
We have three stages on this level. They correspond to the structure of the class of initial mathematical forms.
FIRST STAGE. We consider multisets on this stage. So far, we do not have the positive integers and can not use the natural numbers as names. At first letters of some alphabet, say the Latin alphabet, will be individual names of the elements of an initial mathematical form and a name of the form itself.
But we must consider only "small" initial mathematical forms at this stage (generally, we will consider only small multisets). Every small initial mathematical form has at most 26 elements. Each element of such form can have an individual name. Such a name will be a Latin letter. The names of the elements of such form are distinguished by the individual distinguishability of Latin letters. Thus, we have a coding of the elements of any small initial mathematcal form by Latin letters.
In the sequel, we will use the small Latin letters for names of the elements of small initial mathematical forms and the capital Latin letters for names of the forms themselves. In section 2.3, we need sometimes one additional (twenty seventh) letter for a variable.
The relation of individual distinguishability of elements in the small multisets is realized at this first stage. It requires also a membership relation of the initial/empty symbols to the small multisets and an order relation for the small multisets. We will use a one--element choice operation for a definition of some ordered initial mathematical forms (in particular, for a definition of ordered pairs) and we will introduce some initial logical mathematical forms and operations. Also the result will be initial functions, initial sets, and direct or Cartesian product operation, in particular, operation of square (cube, etc.) taking for any initial set.
SECOND STAGE. The relation of individual distinguishability for any two multisets is manifested at the second stage. We will use a matching operation for any two multisets and we will introduce the positive numbers. Also we obtain the natural--valued functions on the initial sets, a factor--set operation, well--known set operations, and multiset operations.
THIRD STAGE. The relation of individual distinguishability for any initial mathematical form is manifested at the third stage.
Also the result will be other inter-relations between the initial mathematical forms and the inter--relations between corresponding logical forms. Thus, the result will be automatically the simplest models (since we use a minimum amount of resources for the manifestation of the basic abilities of thought) for the initial mathematical concepts, including the simplest model of the positive numbers.
Also we will open the main role (for the initial mathematical forms and the initial mathematical relations) of some specific initial mathematical forms. They are the initial sets. The initial sets are specific small multisets. The elements of such set are initial/empty symbols and all these symbols are distinguished by the individual distinguishability.
We suppose that the more complicated sets of mathematics (for example, the real numbers) can be obtained by thought that starts from the initial sets. Our mind uses the positive numbers and the initial mathematical operations for these purposes.
2. INITIAL MATHEMATICAL RELATIONS FOR SMALL MULTISETS
2.1. INITIAL MATHEMATICAL RELATIONS GENERATED BY UNORDERED PAIR. ONE--ELEMENT CHOICE OPERATION AND ORDERED PAIR
Unordered pairs are the two--element initial mathematical forms. Let us consider a simplest unordered pair, i.e., a multiset containing two symbols. It has the individual distinguishability of their elements. For example, the elements of multiset {o,o} are not distinguished.
We have a very important phase of our genesis. We need a concept of ordered pair to manifest such individual distinguishability by a logical form. It is well known that our language uses only ordered propositions.
The thinking of forms has an ability to perceive and to create an order relation on classes of forms. Such ability is motivated by the following cause. Elements of any forms are inside the forms as parts are inside whole. We will see that there is a one--element choice operation and such operation corresponds to the generation of ordered forms.
Our prototype of the concept of ordered pair is founded on this one--element choice operation. In fact, any ordered pair corresponds to a choice of one from the two elements in the corresponding unordered pair. The ordered pair can be realized as a specific unordered pair. The elements of this form are the chosen element (the chosen element is the "first" element) and the initial unordered pair.
We use an unordered pair {a,b} for names of the elements for the given unordered pair A (A is its name). We emphasize that the names of elements of forms or the names of forms do not identify with these elements and these forms accordingly. However, any set in mathematics is a set of names for some objects usually. We argue about mathematical phenomena in terms of such names. For our case, however, we can make such an identity for names of elements if these elements are distinguished Latin letters themselves.
The choice of an element from two--element multiset A corresponds to the choice of name of this element from the corresponding two--element multiset of the element names of multiset A.
"Unordered pair {a,b} together with chosen element a" is the ordered pair {a,{a,b}}. It is an initial mathematical form and it has traditional name (a,b) (and this name is not a Latin letter). We will suppose that (a,b) is a name for given multiset A together with a chosen element (this element has name a). In the sequel, analogical agreements will be supposed also.
Initial logical forms are propositions:
1) about the individual distinguishability of two elements for unordered pairs--multisets,
2) about belonging of element from such unordered pair to this multiset,
3) about an order for ordered pairs that correspond to such unordered pairs.
We have the following proposition of the individual distinguishability and the following proposition of the individual undistinguishability of the elements from multiset-unordered pair {a,b}:
"element a is distinguished with element b",
"element a is not distinguished with element b".
We have also logical forms "a not= b", "a = b", where "not=" and "=" are symbols of the individual distinguishabilty and the individual undistinguishability respectively.
Also ordered pair (a,b) corresponds to a belonging relation between a and {a,b}. Indeed, a belonging of part to whole could be explained by the one--element choice operation. This part is deleted from the whole, but the whole is not changed by the choice operation, since a new copy of chosen part arises.
Belonging of a to {a,b} is represented by proposition "element a belongs to unordered pair A" and by logical form "a in A", where "in" is a symbol of the belonging.
Our ordered pairs correspond to an order relation. Such relation results from the following informative proposition "if element a is chosen then a is more prefered than b" and by logical form "<(a,b)" or by "a < b". The empty set, the one--element multisets (the singletons) and the unordered pairs-multisets with distinguished elements are initial sets.
Now let us consider the following unordered pair. Their elements are an initial or the empty symbol a and a two--element multiset A. Let A be {b,c}. We have a relation of belonging or a relation of nonbelonging for unordered pair {a,{b,c}}.
"Element a does not belong to unordered pair A" (the proposition is represented by logical form "a notin A") if the next compound proposition about the individual distinguishability
"element a is distinguished with element b and element a is distinguished with element c"
is true. This compound proposition is represented by logical form "(a not=b) & (a not=c)", where "&" is a symbol for word "and".
If the next compound proposition of the individual undistinguishability
"element a is not distinguished with element b or element a is not distinguished with element c"
is true then we can assert that element a results from the choosing from unordered pair A, i.e. a in A.
The above proposition is represented by logical form "(a=b) v (a=c)", where "v" is a symbol for word "or".
2.2. INITIAL INDIVIDUAL DISTINGUISHABILITY
Let A be a small multiset. Generally, we have many initial propositions of the individual distinguishability for all possible unordered pairs of its elements. Let us realize these logical forms by an economical way. The ability of the thinking to create and to perceive beauty of forms requires it.
First let us construct an initial mathematical form 2A of all ordered pairs of discriminated elements from multiset A and the corresponding logical form of all initial propositions of the individual distinguishability of the elements from multiset A.
Now we need a concept of variable element. It is simply some specific symbol, for example, x. If we delete element x and write an initial symbol a instead of x then we have an initial logical operation "x | a" of substitution of a instead of x. If we substitute an arbitrary element from A instead of x then we declare that x is a variable element for A. Here variable x is just a symbol; its role is that one can substitute other symbols for a free variable.
(x,y) is a variable element for 2A, where x and y are variable elements for A. For our purpose we can not substitute the same symbol, say a, for x, y simultaneously. Therefore, they are not independent variables. The individual distinguishability on A results from the following propositional forms "element x is distinguished with element y" and "element x is not distinguished with element y" and these forms are realized by logical form "x not= y" and logical form "x = y" respectively. We have initial propositions of the individual distinguishability "a not=b", "a = b" after substitution a,b from A instead of x,y accordingly.
Therefore, our individual distinguishability on any small multiset is founded on the individual distinguishability of the initial and empty symbols (also on the initial discrimination).
We can construct a logic of propositions of the individual distinguishability, where the synthesis of compound propositions of the individual distinguishability is performed by induction.
Recall that we deal with a fixed small multiset A. The simplest initial propositions of the individual distinguishability are the initial propositions of the individual distinguishability for all possible unordered pairs of its elements. The generating rules of the inductive step correspond to initial logical operations that admit to construct new compound propositions of the individual distinguishability from constructed old compound propositions of the individual distinguishability. These rules indicate:
1) how we can construct a new compound proposition of the individual distinguishability from an old compound proposition of the individual distinguishability. The new proposition is called the negation of the old proposition. If the last proposition results from row B then its negation results from ~B,
2) how we can construct a new compound proposition of the individual distinguishability from two old compound propositions of the individual distinguishability (the last propositions are realized by rows C and D). The new propsition is called the conjunction, the disjunction, and the implication of the old propositions. The new proposition is realized by row "C & D", "C v D", "C --> D" respectively.
The corresponding unformative compound propositions of the individual distinguishability are formed by an unioning of the unformative initial propositions of the individual distinguishability and by means of words--bunchs "not", "and", "or","follow" (we use logical symbols ~, &, v, --> for these bunchs). The true meaning of the new compound proposition of the individual distinguishability is determined by the true meanings of the old compound propositions of the individual distinguishability and by well known tables of truth.
We can also construct a logic of predicates for the individual distinguishability. Such logic is inter--related with the logic of propositions of the individual distinguishability, since we use the compound relations of the individual distinguishability for an economic description of some logical forms. The elements of these logical forms are compound propositions of the individual distinguishability. These compound relations of the individual distinguishability are constructed by induction and such induction corresponds to well--known logical generating rules.
2.3. INITIAL MEMBERSHIP AND FUNCTIONS
The one--element choice operation that chooses one element from the given initial mathematical form is an initial mathematical operation. Recall that we already have the singleton operation, the one--element expansion operation, the one--element restriction operation, and the substitution operation. They are also initial mathematical operations.
We have a small multiset A and we have chosen its element a. The one--element choice operation generates a two--element initial mathematical form [a,A]. The elements of this unordered pair are the chosen element (its name is a) and the initial multiset A. Let us recall that such choice of one element from any multiset--unordered pair has determined an initial ordered pair.
We emphasize that this initial operation of element choice corresponds to the basic ability of the thinking to manipulate forms by algorithms. The ability creates ordered initial mathematical forms. It is known that any algorithm works by ordered forms. There is a membership relation between a and A for the initial mathematical form [a,A]. We realize this relation by true proposition
"element a belongs to multiset A" and by logical form "a in A".
The complete choice of all elements from A is motivated by the basic ability of the thinking to harmony. As a result, we will obtain a membership relation between the elements of A and A itself. We will also obtain a corresponding logical form for the initial propositions of the belonging.
We apply a minimal economic description for this logical form. Therefore, we must use a variable element x for multiset A and a variable element (x,A) for an initial mathematical form "A x {A}" whose elements are all possible ordered pairs with the first element from A and the second element A.
We realize the membership relation by propositional form "element x belongs to multiset A" and logical form "x in A".
Our psychological--mathematical analysis implies that mental origin of concept of function is the membership relation. We realize the initial function by initial mathematical form "A x {A}" and propositional form "x in A". More concisely, the function is realized by one construction "in: A --> {A}".
We obtain a concept of direct or Cartesian product "A x B" of multisets A,B and a concept of function "f: A --> B" by using of generalization.
Now we have an unordered pair whose elements are an initial or the empty symbol a and a small multiset A. Then proposition "element a does not belong to multiset A" (this proposition is realized by logical form "a notin A") is true if the following compound proposition about the individual distinguishability "element x is distinguished with element a for every element x from A" is also true. This compound proposition of the individual distinguishability is realized by logical form "forall x in A (x not=a)", where "forall" is a symbol for quantor "for every". We note that if A has 26 elements then we need an additional twenty seventh letter for the variable.
Let negation " ~forall x in A (x not= a)" of compound proposition of the individual distinguishability "for all x in A (x not= a)" (equivalently, compound proposition of the individual distinguishabilty "there is an element x of multiset A such that x is not distinguished with a" (this proposition is realized by logical form "exists x in A (x = a)", where "exists" is a symbol for quantor "there is")) is true. Then we can assert that element a is obtained by the choice from multiset A, i.e. "a in A".
2.4. INITIAL ORDER
Let A be a small multiset and A have at least two elements. Let us describe:
1) a procedure for enumerating of elements of A,
2) a procedure for constucting of an ordered form for A.
We denote by B + z and B - z the z-expansion and the z-restriction of initial mathematical form B respectively.
Let us choose some element a from A and form multiset [a,A]. We will obtain an ordered pair if A is a two--element multiset. We write its two elements a,b on a row (the first element on this row is the chosen element a) and both our procedures are over in this case.
If A is not two--element multiset then we form multiset A - a by deleting of a from A. We have chosen elements from A and they are written left to right on row a,b,...,d. We also have a two-element initial mathematical form [d,[...,[b,[a,A]]...]]. Finally, we have multiset B that was obtained from A by deleting of chosen elements and let B have at least two elements.
The inductive step coincides with the basic step. Such a procedure is completed when we obtain a two--element multiset {e,t}. Let us choose some element t from multiset {e,t} and form the resulting sequence a,b,...,t,e elements from A.
We say that two--element initial mathematical form [t,[...,[b,[a,A]]...]] is an ordered multiset or a chain. It has also traditional name (a,b,...,t,e).
Such ordered multiset corresponds to a relation of order for multiset A. The basis for the order relation is the initial propositions of following order for all possible elements a,b "element a is more prefered than element b if element a is chosen before b for the given enumerating".
This proposition is realized by logical form "a < b", where < is a symbol of the order relation. For an economic description of the initial propositions of order relation we use an initial order relation "x < y", where (x,y) is a variable element for multiset 2A. For the order relation an aspect of form is manifested by the form of its truth. We select all ordered pairs of elements from A that correspond to the truth propositions of order and let this initial mathematical form have the name {(x,y) in 2A: x < y }. We say that this form arises by the scheme of separation from initial mathematical form 2A. We also will say that we have a small multiset A with the order relation. We have an analogy for the relation of initial distinguishability and membershiprelation. We realize by induction a synthesis of compound propositions of the order and compound relations of the order. This process is analogous to the process for the logic of individual distinguishability.
We select the universally true compound relations of order (and the universally true compound relations of individual distingishability). The initial mathematical forms of their truth coincide with the domains of their definition. Theorems about the order (and the individual distinguishability) correspond to such forms. In the sequel, we can use these theorems as a basis for a definition of relations of equivalence and of order by a generalization.
Also we have the relation of direct order:
"element x is preferred directly more than element y if element x is chosen directly before element y (for the given enumerating)" is inter--related with the relation of order.
2.5. INITIAL SETS
We will create the sets of mathematics on a base of initial mathematical forms. These form are the initial sets. We believe that "all usual mathematical sets" can be constructed by our thought from the initial sets.
Let us introduce the initial sets. We say that a small multiset A is an initial set if the proposition "forall (x,y) in 2A (x not = y)" is true. It means that initial mathematical form {(x,y) in 2A: x not= y} is 2A. The truth of the negation of the last proposition, i.e. proposition "exists (x,y) in 2A (x = y)" means that there are at least two elements from A that are copies of the same initial/empty symbol.
It is clear that the initial sets are certain small multisets. We apply a new term "initial set" since we would emphasize an important difference between the discrimination and the individual distinguishability.
For the initial sets, the initial mathematical relations are simpler than for the multisets. For example, we know that the initial order x < y and the initial individual distinguishability x = y have the domain of definition 2A for multiset A and we can substitute any elements from A to variable (x,y). However (see section 2.2), we can not assume that x and y are independent variables for A. We have an another situation for initial set S. Every element a from S generates ordered pair-doubleton {a,{a}} (with name (a,a)). We say that the initial mathematical form D(S) of all such ordered pairs generated by the elements of S is the diagonal of S. The identity of elements--copies for S x S transforms it to initial mathematical form S^2. The elements of last form are elements from 2S and the elements from D(S). We say that the individual distinguishability for the initial sets is an initial inequality.
Initial mathematical forms D(S) and S^2 have also a new inequality "notapprox" generated by inequality "not=" on S. Let x,y,z,w be independent variables for S. Then (x,x),(y,y) and (x,y),(z,w) are different pairs of variables for D(S) and S^2 respectively. Let "(x,x) notapprox (y,y)" if "x not= y" and let "(x,y) notapprox (z,w)" if "(x not= z) v (y not= w)$. Then the elements of D(S) and S^2 are distinguished. For example, the names of elements of 2A are distinguished in section 2.2. We say that the initial mathematical form S^2 is the square of initial set S. Analogically, the cube S^3 of initial set S has also some inequality, etc.
Thus, we can construct more complicated sets on foundation of the initial sets.
3. INDIVIDUAL DISTINGUISHABILITY FOR MULTISETS
3.1. INITIAL EQUAL CARDINALITY AND CARDINALITY PREFERENCE
The procedure of enumerating of elements of multisets A and B generates a relation of equal cardinality and a relation of cardinality preference between the two elements of unordered pair {A,B}. Such comparison corresponds to a matching operation.
Let us choose some elements a and b from A and B respectively and form the unordered pair {a,b}. If both multisets A - a and B - b are not empty then we go on to the next step. The procedure is over if at least one from these multisets is empty.
Now let some elements from multisets A and B be chosen. Let we have:
1) initial mathematical form H whose elements are unordered pairs of
elements (one element is chosen from A and other element is chosen from B),
2) nonempty multisets A^ and B^ that are obtained from A,B by deleting of some their elements.
Our inductive step coincides with the basic step. We choose some elements f and g from A^ and B^ and form initial mathematical form H + {f,g}. If both of multisets A^ - f and B^ - g are not empty then we will continue our procedure. Otherwise, the procedure is over.
Our procedure is complete, when at least one from the multisets is exhausted. As a result we construct:
1) initial mathematical form C{A,B} of unordered pairs,
2) initial mathematical forms C(A) and C(B) that are obtained from multisets A,B by deleting of the elements.
We say that initial mathematical form C{A,B} is the cardinality comparison (or the matching) of multisets A and B. We say that initial mathematical forms C(A) and C(B) are the rests of multisets A and B respectively.
If both multisets C(A) and C(B) are empty then multisets A and B are in the relation of equal cardinality. The corresponding proposition is realized by logical form "A sim(C) B".
We note that C{A,B} corresponds to two reverse functions f(C):A --> B and f^{-1}(C):B --> A. So, for our psychological--mathematical analysis the first one--to--one correspondence arises between elements of one multiset and elements of another muliset.
If only one multiset from multisets C(A),C(B) is empty then we say:
"multisets A and B have not the same cardinality" and we realize this proposition by logical form "A notsim(C) B".
If multiset C(A) is empty and multiset C(B) is not empty then we say that "multiset B is more cardinality prefered than multiset A" and we realize this proposition by logical form "B >(C) A".
We have injection f(C) of multiset A in multiset B for this case. The comparison result of A and B does not depend on C. Therefore, we can realize the above relations by logical fors "A sim B" and "B > A".
If multisets A and B are alloted by the initial order or by the individual initial distinguishability then we can obtain a more interest comparison {f,f^{-1}}. It is an isomorphic embedding (an isomorphism if function f is a bijection). It admits to consider these initial mathematical relations up to the isomorphism acuracy for the class of initial mathematical forms. However, we must remark that our assertion about the functions are only applied for small multisets A, B since we have not yet the positive numbers to label many elements.
3.2. INITIAL INFINITE SET: THE POSITIVE INTEGERS
The ability of the thinking to harmony generates a minimal class of initial mathematical forms closed with respect to an initial mathematical operation, for example, with respect to the one--element expansion operation (and with respect to its dual the one--element restriction operation since we would have the minimal class). For example, we obtain next sequence of multisets:
{ }, {o}, {o,o}, {o,o,o}, ... (*)
{ }, {n}, {n,n}, {n,n,n}, ... (**)
Let us fix such sequence, for example, (*). We accept a conjecture about existence of the positive numbers (*) (as having been constructed series) by a pure mental act.
It is a moment of principle for mathematics.
Such act is founded on basic abilities 1, 3, and 7 of thought. Russian philosophist A.F.Losev said: "the positive integers is an incontrovertible proof of creative nature of the human thinking".
We do not realize such "infinite" form by "writing" of all their elements. We can do it only for any initial mathematical form. But it is known that, if we have (*), then another infinite mathematical form arises from (*) by means of mathematical operations.
The elements of the positive numbers are the natural numbers. They are initial mathematical forms. Our model for the positive numbers uses some initial mathematical forms. Generally, the elements of these forms are multisets using a single initial symbol and we use the matching distinguishability as the distinguishability. Of course, there exist other models of the positive numbers whose elements are sets and these models use another distinguishability (see following section 3.3).
We mainly explore the positive numbers on the following levels of mathematical knowledge after our mathematics origin. Of course, some concepts for the positive numbers can be obtained by a generalization of constructed concepts for the initial mathematical forms.
We accept also the principle of complete induction. Such principle admits to say about properties of the positive numbers. For example, we can say that the elements of the positive numbers (*) are distinguished with respect to the relation of equal cardinality and we also have a relation of order on the positive numbers.
For our first level of mathematical knowledge (the initial concepts of mathematics) it is essential that the natural numbers can be names for the elements of initial mathematical forms and names for the initial mathematical forms themselves. Therefore, now we obtain a possibility to realize the individual distinguishability for any initial mathematical forms.
We would emphasize the following aspect about the positive numbers:
For the esoteric approach the positive numbers are necessary components of our mathematics origin -- a variety of symbolic mathematical forms manifesting the subtle structure of the basic abilities of the thinking. The approach to mathematics adds this new standpoint on the positive numbers to standard ideas about the natural numbers. We could not perform our genesis of all initial mathematical concepts without the natural numbers. But we could not create the positve numbers themselves without having been arised concepts of mathematics. So, the positive numbers are a (very essential) part of the initial concepts of mathematics.
3.3 MODELS FOR THE POSITIVE NUMBERS
Let us consider some well known models for the positive numbers. Possibly the first model (1) is not well known one, but we must keep in mind that all models are isomorphic. We will emphasize their psychological aspects corresponding subset from the set of basic intellectual operations, i.e. the singleton--operation, the one--element expansion operation, and the substitution operation. All models must use the singleton--operation.
POSITIVE NUMBERS ARE MULTISETS
USING A SINGLE ELEMENT SYMBOL
This model uses the singleton operation and the one--element expansion operation only.
{ }, {/}, {/, /}, {/, /, /}, .... (1)
Here the positive numbers are multisets. We have the matching distinguishability for the multisets.
If we introduce the notion of multiset from the basic principles of mathematics then it is the simplest psychological model for learning.
POSITIVE NUMBERS ARE SINGLETON SETS
This model uses the singleton operation and the substitution operation only.
{ }, {/}, {{/}}, {{{/}}}, ... (2)
Here the positive numbers are singletons. We have an inequality for the singleton sets for second model (2). An antitransitivity takes place for (2), i.e. "((x in y) & (y in z)) --> (x notin z)". Therefore, x is differed with z as a nonbelonging element and y is differed with z as a belonging element.
This reflexive model is a hard psychological model for the first step of learning.
All other models use all three basic initial intellectual operation, i.e. the singleton--operation, the one--element expansion operation, and the substitution operation.
POSITIVE NUMBERS ARE LINEAR
ORDERED SETS
Let us consider the following model:
{ }, {/}, {/,{/}}, {/,{/,{/}}}, ... (3)
Third model (3) uses the one--element expansion operation for consructing of first ordered pair {/,{/}} only. Other ordered pairs are obtained by the substitution operation in (3). We have an inequality for sets for (3).
Usually model (3) is represented as a sequence of certain words on single--element alphabet (for example, on alphabet / ) since we make an agreement to represent such linear ordered sets by means of the words:
0, /, //, ///, ... (4)
It is the classic model used for mathematics school education. It uses the matching operation for the distinguishability of the words from (4).
POSITIVE NUMBERS ARE SETS
Let us consider the following model:
{ }, {/}, {{ },{/}}, {{ },{/}, {{ }, {/}}}, ... (5)
We have the matching distinguishability and the inequality for sets simultaneuosly for fourth model (5). Transitivity takes place for (5), i.e., "((x in y) & (y in z)) --> (x in z)". Therefore, x and y are differed with z as belonging elements.
3.4. MULTISETS AS NATURAL--VALUED
FUNCTIONS ON INITIAL SETS
There is a concept of segment in the positive numbers (also in every ordered form). It is a subform generated by two natural numbers n and m in accordance with the well known separation scheme.
There is the relation of equal cardinality of any multiset A and some standard segment ]o,n-1[, where "o" is the name for an empty set. We realize such relation by logical form | A | = n. We say that the natural number n is the cardinality of multiset A. Thus, for our genesis of initial mathematical concepts the cardinality (quantity) is a secondary concept after concept of the natural numbers.
Let a be an element of multiset A. Submultiset {y in A - a: a = y} is the class of a--undistinguished elements from A; of course, we join element a itself to such class. Let us construct an initial set A^o whose elements are some representatives of these classes for the all distinguished elements of A. We say that initial set A^o is the factor--set of multiset A. The forming of factor--set is an initial mathematical operation. This operation connects the class of multiset and the class of the initial sets.
Multiset A can be considered as a function f_{A^o} on factor--set A^o of multiset A. Its value belongs to the positive numbers N and f_{A^o}(x) is the cardinality | A(x) | of class of x-undistinguished elements from A, where x is a variable for A.
3.5. POWER OPERATION
FOR INITIAL MATHEMATICAL FORM
The same basic ability of the thinking to harmony generates also a specific initial mathematical form generated by a given initial mathematical form A and closed with respect to a new initial mathematical operation of taking of subform. We have the power operation for any initial mathematical form. We obtain the initial mathematical form P(A). The elements of this form are all possible subforms of A (including empty symbol also). The form is called the power of A. Obviously that P(A) > A. In the sequel, after our mathematics origin, we will apply this operation to the positive numbers, etc.
3.6. INITIAL RELATION FOR INITIAL SETS
If we realize a belonging of a variable element x of initial set S to the initial sets from P(S) by the distinguishability (inequality) on S, then we obtain a logic of belonging for initial sets.
If proposition "forall x in S (x in A) --> (x notin B)" is true then we say that the initial sets A and B are disjoint.
If proposition "forall x in S (x in A) --> (x in B)" is true then we say that the initial set A is embedded in the initial set B and realize this proposition by logical form "A subseteq B".
If proposition "(forall x in S ((x in A) --> (x in B))) & (exists y in S ((y in B) --> (y notin A)))" is true then we say that the initial set A is strictly embedded in the initial set B and realize this proposition by logical form "A subset B".
Lastly, if proposition "(A subseteq B) & (B subseteq A)" is true then we say that the initial sets A and B are undistinguished (equal) and realize this proposition by logical form "A = B".
Thus, different multisets A and B are distinguished at least by one element, i.e. there is an element in multiset A nonbelonging to multiset B or there is an element in multiset B nonbelonging to multiset A.
The relation of non-crossing, the relation of embedding, and the relation of distinguishability have certain well known properties. For the initial sets A and B the logic of belonging generates union "A + B", intersection "A & B", "A - B", and symmetric difference "A simdif B". Also we have a concept of disjoint (direct) union.
3.7. INITIAL RELATIONS FOR MULTISETS
Let R be an initial mathematical form whose elements are multisets. Multiset A from Re can be considered as ordered pair (A^o,f_{A^o}). We extend function f_{A^o} on the common domain of definition for an establishment of initial relations between such ordered pairs.
Let R^o be the initial mathematical form whose elements are the factor--sets of multisets from R. The elements from R^o are initial sets. It is possible that there are two such undistinguished sets. Let F(R^o) be the factor--set of set R^o. Initial mathematical form F(R^o) is a set whose elements are initial sets.
Let us form the common support S, i.e., the join of sets from F(R^o), i.e an join of sets that are indexed also a set. We have a new initial mathematical operation of set collecting by a set. Let x be a variable for S. We extend functions f_{A^o} on S by setting f_{A^o}(x) = 0 for x in (S - A^o) and use former symbols f_{A^o} for the new functions. Now functions f_{A^o} are defined on common set S for every A from R.
Multisets A and B are disjoint if their factor--sets A^o and B^o are disjoint.
Multiset A is embedded in multiset B (we have a logical form "A subseteq B") if "A^o subseteq B^d" and "f_{A^o} < or = f_{B^o}", i.e. f_{A^d}(o) < or = f_{B^o}(x)" for every x in S.
Multiset A is strictly embedded in multiset B (we have a logical form "A subset B") if "A^o subseteq B^o" and "f_{A^o} < f_{B^o(x)}", i.e. "f_{A^o} < or = f_{B^o}" and there is x in S such that f_{A^o}(x) < f_{B^o}(x).
Multisets A and B are undistinguished if their factor--sets A^o and B^o are undistinguished as the initial sets, i.e. "A^o = B^o" and the corresponding classes of undistinguished elements have the same cardinality, i.e. f_{A^o}(x)=f_{B^o}(x) for every x in S. We realize this phenomen by logical form "A = B".
In particular, such undistinguishability implies that any multiset does not depend on a sequence of its constructing. It also implies that, if these multisets consist of the same initial/empty symbols and the cardinality of their copies coincide, then they are not distinguished. Such distinguishability is the usual well known extensional standpoint. Last, from such a point of view we can conclude that there is the unique empty set.
Concluding, we remark that we also have operations on multisets, i.e. on natural--valued functions on the initial sets. Thus, we have the concepts of union, (including the direct union) and intersection (recall that we have also had direct product) for multisets. As a result, we obtain the arithmetic operations of taking sum, product, maximum, and minimum.
4. INDIVIDUAL DISTINGUISHABILITY FOR INITIAL MATHEMATICAL FORMS OF GENERAL KIND
We can define the major initial mathematical relation of the individual distinguishability of the class of initial mathematical form. We use the distinguishability of multisets and the belonging of initial or empty symbols to multisets as a basis for this purpose.
Let E, F be any initial mathematical forms. Since we have the positive numbers, we can determine a concept of level (or rank) of any initial mathematical form (the level l(E) of initial mathematical form E is the number of levels of its graph--rooted tree). We apply the sum E,F--levels as a parameter for inductive definition of E,F--distinguishability. The consideration is omitted. We have a full analogy with the undistinguishability for multisets from section 3.6. We will also extend the relation of belonging "E in F <--> exists X in F^o (X = E)", "E notin F <--> forall X in F^o (X not= E)" on the class of initial mathematical forms. Now all constructed concepts will have a general character within the class of initial mathematical forms.
Thus, we conclude that, in fact, the individual distinguishability of the initial mathematical forms requires the all basic mathematical concepts, including logical ones.
Let us emphasize that the individual distinguishability is founded on the discrimination, the individual distinguishability of the initial and empty symbols, the distinguishability between element and initial mathematical form that contains it, and, lastly, on the matching distinguishability. It is easy to see that the individual distinguishability of initial mathematical forms is the isomorphism of their graphs--weighted rooted trees.
5. SUMMARY
On the third level of the mathematics beginning we have the following pattern for the manifestation model of the seven basic abilities of the thinking by the initial mathematical relations.
1. Ability 1 to display purposeful activity is manifested by creation of the simplest individual mathematical relation -- the individual distingushability of the initial mathematical forms and related individual initial mathematical relations.
2. We add the individual mathematical relations -- the distinguishability, the membership relation, the order, the relation of equal cardinality, the relation of cardinality preference, the nonintersection relation, the inclusion, etc. to the common quality to unite elements that we had on the second level of the mathematics beginning.
3. We add the one--element choice operation, the matching operation, the power set operation for initial mathematical form, the initial set operations, the initial logical operations, etc. to the basic mathematical operations that we had on the second level of the mathematics beginning.
4. The basic ability 4 to harmony is manifested by a minimum amount of resources for our symbolic constructions, by a completness of the use of the initial mathematical operations, and by a closeness with respect to such operations.
5. The basic ability 5 is manifested by the logic of propositions and predicates for the initial mathematical relations (including the logic of belonging).
6. The basic ability 6 is manifested by the initial mathematical forms that can be individual undistinguished forms, by some common properties for initial mathematical forms (in particular, by their isomorphic embedding with respect to the individual distinguishability and the order). By the way, this is a prototype for the cardinal and ordinal numbers that arise after our mathematics origin. Also the ability 6 is manifested by the principle of mathematical induction.
7. The basic ability 7 is manifested by the creation of the ordered initial mathematical forms and by constructiveness of mathematical forms on this level.
It is not difficult to describe "structures of intelligence" that correspond to the three stages of part III. For this purpose we need (also see the summary of part II) replace the initial symbols by some concrete objects. Then new intellectual operations will correspond to our initial mathematical operations. For example, we have an assignment of names to different objects, a mental correspondence of concept of ordered pair, an individual distinguishability for concrete objects. We can list, label, count, or compare.
We have also an initial logic. The well known logic of propositions and classes is founded on our simplest prototype, i.e. on the logic of the initial mathematical relation for the class of the initial mathematical forms. We can investigate the corresponding psychological pattern.
CONCLUSION
In conclusion, let us compare our esoteric approach to the initial mathematical concepts with the intuitionistic approach and the constructive approach to mathematics beginning (see [5,6]).
The initial intuitionistic concepts are a notion of the positive numbers and a notion of equality for them. The initial constructive concepts are a notion of constructive objects and also a notion of equality for them.
With the constructive point of view on mathematics, the intuitinistic positive numbers are words on a single--letter alphabet. The positive number n is fixed by a material performance. They connect, for example, a stroke or point on a paper sheet with every element of the performance.
Such positive number is a specific form (a union of some discriminated objects inside a whole). In fact, the strokes or the dots are placed together (for example, they are not placed on different sheets): the symbols are placed without any intersection.
Also the simplest constuctive objects of constuctive mathematics are words for single--letter alphabet.
However, it already has an order and, therefore, it is not the simplest finite object. For the basic principles of mathematics it is better to start from our initial mathematical forms, i.e., the multisets; they are simpler objects and they are constructive in the broad sense of the word.
Intuitionism and constructivism did not uncover the key role of individual distinguishability for mathematical form class since they did not know the subtle nature (structure) of human thought and the basic abilities of thought. We have shown that the individual distinguishability for the initial mathematical form class generates all initial mathematical concepts including mathematical language concepts and logic concepts.
The intuitinistic notion of equality for their positive numbers are founded (see [4]) on "a before mathematical" ability of thinking to compare the positive numbers by means of "a simple observation". In fact, the intuitinistic equality is founded on the matching operation.
REFERENCES
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