INTRODUCTION
Modern mathematics contains such domains as set theory, mathematical logic, theory of models, theory of algorithms, and category theory. These domains were became reality. Axiomatic method secures its power (also its role was realized). Arguments of beauty (symmetry, closeness, completeness, minimality, etc.) were realized also. Development of intuitionistic mathematics and constructive mathematics opened mental origins of basic mathematical concepts and structures. It is possible to speak of the basic features of mathematics only after a very long process of accumulating of a lot of actual data and after arising of the modern mathematics.
The basic features of mathematics are subtle features -- they are hidden deeply inside under the cover of technical components. A key for their recognition is the basis of the thinking, i,e., the basic esoteric ideas.
It is natural to understand "the basic features of mathematics" those ones of mathematics which could be interpreted as a manifestation of the basic abilities of the thinking into mathematics. The work demonstrates such basic features of mathematics . They give a beautiful holistic systemic compact pattern of mathematics and, as a consequence, can be a base for the classification of different schools for mathematics philosophy. Approximately twenty years ago J. Dieudonne (see [6]) said about the difficult birth of concept of "mathematical structure" that we can assert it is only a part for the more general concept "basic features of mathematics".
MANIFESTATION OF TWO INITIAL ABILITIES OF THINKING
MATHEMATICAL SYMBOLS
In a simple manner, the initial ability to display activity is manifested in mathematics by the creation of symbols -- some initial material or "basic" stuff for the symbolic -- "mathematical" forms of mathematics. The letters of some concrete alphabets and some other symbols are usually such basic symbols. In mathematics any reproduction of these symbols are admitted. The non--intersection of the basic symbols are their (initial) discrimination. We use the discriminative quality of mind for creation and perception of any mathematical forms. For example, Intuitionism declares [6] that their concept of the positive numbers is founded on:
1) an ability of thinking to represent any object as an abstract being by ignoring all its properties. In acordance with [7] "concept of the being gives the possibility to perceive the object separetely from the rest".
2) an ability of thinking to copy this being with no limit.
It is clear that these abilities are exactly the initial abilities of the thinking of section 2, i.e., the ability to initial discrimination and the abilty to display activity.
Such a coincidence is not surprising; however, it is very pleasant to know that Intuitinism itself opened the initial abilities of the thinking.
MANIFESTATION OF BASIC ABILITIES 3-7 OF INTELLECT
REALIZED MATHEMATICS
MANIFESTATION OF BASIC ABILITY 3
MATHEMATICAL FORM CLASS
The ability 3 to synthesize and perceive forms as a synthesis of other forms including the synthesis for meanings and qualities manifestation corresponds to the following systemic question:
"What other objects, and in what way, make the object under consideration ? What kind of structure does this category of objects have "?
Firstly, we recall the philosophical school of set theory that emphasizes the aspect of form and supposes that mathematics deals with some properties of sets (mathematical forms) only. Of course, the set theory makes the concept of properties and sets a more precise one.
Also the philosophical school uses the fact that our intellectual abilities produce sets whose elements are sets, etc. We have indicated that the Ego deals with infinity since our thinking creates or perceives any form only into a class of forms reflecting the intellectual operations.
In mathematics mainly the ability 3 corresponds to the axiomatic method and the representation of mathematics as a family of different mathematical structures defined by means of axioms.
For such approach any mathematical structure is one set or sets connected by means some type of relations. All properties of these objects and relations (needed for development of the theory of given structure) must be fixed by means of the axioms. The axioms do not concern any concrete nature of the objects and the relations.
In accordance with the axiomatic method, other (uninitial) concepts are defined by means of a description of their properties using initial concepts. Any property formalized in terms of this language can be used for the definition of such objects (if it does not imply a contradiction).
In 1936, N. Bourbaki proved and proved that if we accept the Zermelo-Fraenkel axioms for set theory (and some principles of logic) then we can construct the existing mathematics.
Let us emphasize that Zermelo and Fraenkel do not point out any logical principles since they considered them out of mathematics. Also N. Bourbaki said that "our logic is a language/grammar only mathematicians use". In our terms of such phenomena, ability 5 is another (but also basic) ability of our thinking than ability 3.
It is well known that such understanding of mathematical structure (i.e. in sense of the theory that is covered by means of some theoretical set axiom system) is essentially wider than the concept of formal deductive theory. K. Godel has proven that any deductive theory can not exhaust even the problem variety for number theory.
Now we understand that the aximatization is only a deep organization of large sections of mathematics. Of course, we can select any axiom system (from many possible equivalent axiom systems). Also the use of axioms is not restrictive.
H. Weyl said that the axiomatization gives an accuracy for mathematics and organizes it. The axiomatization performs a function of classification and gives a simlipicity for mathematicians.
MANIFESTATION OF BASIC ABILITY 4
BEAUTY OF MATHEMATICAL FORMS
The ability 4 to create and perceive beauty of forms corresponds to the following systemic question:
"What kind of beauty is realized by the object, the beauty being a measure of simplicity of manifestation of the object" ?
Manifestation of this ability in mathematics corresponds to (also see the description of ability 4 in section 5) the beauty of mathematical forms, geometric figures, proofs, which have unexpected simple decisions (such proofs usually correspond to minimum amounts of mental resources).
A completeness, a closureness, etc. can be viewed as a manifestation of beauty in mathematics. It is well known the key role of transformation group concept for symmetry and harmony. We refer all readers to specific papers and books describing the beauty in mathematics.
MANIFESTATION OF BASIC ABILITY 5
ANALYSIS OF MATHEMATICAL FORMS
Ability 5 to perform an analysis of forms corresponds to the following systemic question:
"What kind of theory provides for a logical analysis of the object" ?
Let us recall (see section 5) that ability 5 includes an ability to dissect, to separate, and to choose subforms from the forms. At once we realize that the choice axiom for infinite sets is a manifestation of the ability.
When we speak about some sequential correct analysis, logical rules used in proofs, logical assertions, a mastering to see lacuna into given proofs, we speak about a manifestation of this ability in mathematics. Ancient greeks realized a deductive nature of proof for every new fact. Of course, we can apply the axiomatic method to deductive systems also. Frege suggested such an approach to logic. Aspects of the axiomatic approach are detailed by the attraction of ability 5: we obtain the noncontradictory, the independence, and the aspect of concept of existing as the concept which does not imply a contradiction.
The formalization of logic corresponds to the axiomatic method but it is very helpful to use symbolic notation and logical formulas when we write symbolically usual human logic assertions.
We understand perfectly clear that the formal deductive theories (i.e. theories developed from some finite set of axioms by means of chains of arguments (combined from links belonging to a finite set of fixed -- for the given theory -- elementary ways of logical inference) give to us only a formal logical tools for development of mathematical theories. Let us emphasize once more that mathematics is not only the family of formal deductive theories and inuitive realized ideas fall outside the limit of mathematical proofs. K. Godel's theorem says that any considerable axiom system does not prove (or refute) all theorems of mathematics domain for description of which the axiom system was created.
MANIFESTATION OF BASIC ABILITY 6
SIMILARITIES OF MATHEMATICAL FORMS
The abilty 6 to create and perceive similarity of forms corresponds to the following question:
"What properties does the the object share with other objects ? And what are those other objects" ?
We apply ability 6 when we note in mathematics a generality into an individuality and an individuality into a generality. Namely this ability gives the origin of a new concept when we meet its model (example), while we investigate different mathematical objects. Then we simply use its properties for the definition in accordance with the axiomatic method.
One of the most important concepts of mathematics, namely the isomorphism concept,corresponds to ability 6. It means that the theory of mathematical structure can be applied to any object system and their relations, if they both satisfy the axioms of this structure.
One mathematical structure can have a few of its models (even nonisomorphic) in other mathematical structures. The Lowenheim-Skolem theorem says that axiom systems admit much more different models than its creation supposes. The homomorphisms for algebraic objects and the imbeddings of one mathematical forms in other mathematical forms correspond to ability 6.
The well known abstraction, generalization, and mathematical induction are founded on this ability 6 to similarity. We know that the mathematical induction axiom is applied to any domain of mathematics which deals with the positive numbers. Poincare said that mathematical induction is a general principle for obtaining new results and since any logical rule does not include an ifinite number of arguments then the method of mathematical induction is not followed from any logical principles.
Generally, we can say that the well-known historical discussion about logical principles and mathematical induction corresponds to realizing of such phenomen: basic ability 6 is another basic ability of the thinking than basic ability 5.
MANIFESTATION OF BASIC ABILITY 7
ALGORITHMICAL ASPECTS OF MATHEMATICAL FORMS
The ability 7 to perform ordered manipulation of forms corresponds to the following systemic question:
"What are the algorithms needed to manipulate the object " ?All that is connnected with the concept of algorithm corresponds to the manifestation of ability 7. A. Church gave a precise sense for an effective procedure or algorithm when he introduced the concept of recursive (computable) function. At present the algorithm theory is an interesting mathematical theory.
The intuitionistic, constructive and formalistic approachs to mathematics emphasize the algorithmical ability 7.
When we speak about ability 7, we speak about the algorithmical concepts, an algorithmical abiltity to simplify sybolic mathematical exressions, to solve equations, etc.
MANIFESTATION OF BASIC ABILITIES 1-3 OF INTUITION
UNREALIZED MATHEMATICS
Manifestation of ability 3 has already been considered above; however, we will consider the manifestation of this ability at the next section again.
MANIFESTATION OF BASIC ABILITY 2
ESSENTIAL PROPERTIES AND RELATIONS OF MATHEMATICAL FORMS
The ability 2 to create and perceive form qualities and inter-relations including the creation and perception of essential qualities and inter-relations corresponds to the following systemic question:
"What are the specific (essential) properties of the object? What are its specific (essential) relationships with other objects "?When we speak of the manifestation of this ability in mathematics, we speak of a creative ability of the Ego to distinguish and select ideas, to choose the esssential properties and the inter-relations from the vast variety of potential properties and inter-relations for a given object. Namely, they are the individual major characteristics of the object: "thing is that, that is".
It is natural that such properties and inter-relations possess a great harmony, a coherence with the surrouding world. It is known that intuition "keeps in mind" the arguments of beauty. It is natural for mathematicians to search for a decision of any hard natural mathematical problem between suppositions characterizing of beauty.
Let us emphasize once more that there are many inter-relations: we could not ennumerate them in a lifetime for the enumeration of them. But let us recall: we usually react on very strong perceptions, for example, on loud sound and bright light, etc. It is natural to suppose by analogy that our Ego reacts also on the most important properties and relationships, i.e., that influence our intellect most strongly.
The realizing of the essential inter-relations corresponds to a general synthesis (see ability 2) that unites different elements into a harmonic whole. Hadamard (see [5]): "If I do not envelope mathematical proof as some general whole, I do not feel its understanding". The understanding can appear after a time.
Plato, Leibniz, Gauss, Hermit, G. Cantor, D. Hilbert, G. Hardy, J. Hadamard, A. Church, K. Godel, some members of Bourbaki group, etc. asserted (they were all "platonists") that mathematical concepts and properties exist in some objective sense (realm) and they can be realized by the human mind.
If we admit the hypothesis (we admit it for our spiritual (esoteric) philosophy of mathematics), then mathematics is only opened by people and we speak about a human knowledge of mathematics. In accordance with this philosophy, if we will think more deeply we can extract mathematical facts. Our mathematical assertions will be a record of intellectual and intuitive perceptions of these transcendental objects. Wrong assertions and contradictions arise from inadequate attempts.
Let us illustrate such position by two quotations (see [5]). G. Cantor: "I share the Plato's philosophy and believe that in a surrounding world there are ideas independent of man. In order to realize a reality of these ideas, we simply need to think about them".
J. Hadamard: "Although we do not know the truth, it pre-exists and prompts for us a way on which we must go."
MANIFESTATION OF BASIC ABILITY 1
MEANINGS OF MATHEMATICAL FORMS INTO MATHEMATICAL FORM CLASS
The ability 1 to display purposeful activity including the creation and perception of meanings corresponds to the following systemic question:
"What is the meaning of the object" ?We have indicated that the answer depends on what variety of objects we will estimate the meaning. Ability 1 has the most subtle manifestation among all the basic abilities of thought. If we omit the applied meanings of mathematics (usually such meanings dominate) then we will speak about meanings of mathematical ideas for the mathematical ideas. Then every mathematical concept must have its specific meaning in a network of mathematical ideas. Mathematical ideas are represented by a tightly connected hierarchical network of interlocking nodes. The nodes of the network are systemic representations (giving the basic systemic aspects) of some representing mathematical ideas.
At present we do not understand such networks yet at all. Uncovering of such representing ideas, relationships between them, and constructing of their systemic representations depend on special research. It is a difficult problem. It demands a large qualification and intuition. But famous mathematicians could solve the problem, if they would.
At present such problems have not even realized by scholars.
8. SPIRITUAL DEVELOPMENT AND MATHEMATICS
THE BASIC ABILITIES OF THE THINKING AND CLASSIFICATION OF SCHOOLS OF MATHEMATICS PHILOSOPHY
It is natural to suppose that every philosophical school of mathematics emphasizes some basic abilities of thought from 1-7. Let us indicate a sketch of such classification of well known schools in terms of the basic abilities of thought (sometime we indicate a second variable or a key additional word). This classification (possibly, it should be improved) is easily followed from the previous sections.
Empiricism -- 1 (the external application meanings),Platonism -- 2 (1), Intuitionism -- 2 (7), Topos theory -- 2 (3),
Set theory -- 3 (5),
"Beauticism" -- 4 (mathematics is a search for austere forms of beauty),
Logicism --- 5,
Theory of models -- 6,
Formalism -- 7 (5), Constructivism -- 7 (3).
We think that the formal functionalism of S. MacLane is a school that is the nearest school to our esoteric pattern of mathematics. In accordance with [8]: "mathematics is an array of forms, codifying ideas extracted from human activities and scientific problems and deployed in a network of formal rules, formal definitions, formal axiom systems, explicit theorems with their careful proof and the manifold interconnections of these forms. More briefly, mathematics aims to understand, to manipulate, to develop, and to apply those aspects of the universe which are formal".
We have indicated above that we have not the hierarchical network of representing mathematical ideas generated by human intuition (such ideas must be thoroughly selected, cleared, and organized). S. MacLane emphasized that we must give more attention to the possible origins and develoment of mathematical concepts, how these concepts are generated by mathematics itself and other domains of human activity. Of course, external applications of mathematics are very important, but they hinder the understanding of the key inter-relations of mathematical ideas. For the constructing of the hierarchy of mathematical ideas it is necessary to understand how the mathematical concepts can be generated by mathematics itself; how new ideas are generated on the basis of existing ideas!
BIRD'S EYE VIEW
The fact that the hierarchy of mathematical ideas exist was realized by N. Bourbaki -- after the refinement and classification of Mathematics that was made by mathematicians of 19th and 20th centuries.
N. Bourbaki recognized the following pattern of large domains of mathematics; we can see such domains from a bird's eye view. Bourbaki discovered the direction from general (abstract, simple) to specific (concrete, complex) corresponding to the hierarchy. Let us fly together with Bourbaki (see [9]).
Firstly, we see the origin of mathematics: a set of ideas contained in the axioms of set theory and mathematical logic. After this origin we see mother or "generating" mathematical structures, namely -- algebraic, order, and topologicaal structures. After this core of mathematics we see mathematical structures that are combined from generating structures. For example, we see the functional analysis using general algebraic and topological ideas. Further we see specific mathematical structures, where the elements of sets have a more complex individuality. Bourbaki said that classical mathematical domains are only "places", where the general mathematical structures are met.
Also S. MacLane is a big bird and he saw some inter-relations of mathematical structures from above, i.e. his category theory indicating an importance of the consideration of object morphisms.
However, this very beautiful pattern of mathematics has mach lacuna. It is only a bird's eye view. We need to land in order to construct a more detailed version of the hierarchy.
It is well known that the origins of the generating mathematical structures lie in discrete mathematics. This mathematics is algebra, topolgy, geometry, etc. J. Dieudonne said [10]: "I have the feeling that we don't understand at all the extraordinary interplay of combinatorics and what I would call "conceptual" mathematics".
Thus, in order to construct a more detailed hierarchy of mathematical ideas, we must call our attention to discrete mathematics.
For example, the author has constructed (see [3,11]) a hierarchy of the initial mathematical concepts (ideas) for the origin of mathematics for Bourbaki's pattern.
SUBTLE ESOTERIC NATURE OF INITIAL CONCEPTS OF MATHEMATICS
The initial (simplest, basic, etc.) concepts of mathematics were established in the course of a very long (over the span of at least 20 centuries) historical development. Now these concepts are usually given by the axioms of set theory and mathematical logic. They are contained in the axioms. But from the bottom of mathematics 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 our attention for 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 about numbers, space, time, and motion. Or they say that these concepts have the roots in human consciousness or philosophy. Although the intuitionistic mathematics and the constructive mathematics open the mental origins of these concepts lie in discrete mathematics, i.e. mathematics of finite or denumerable sets, they suppose that the question about the nature of these basic mathematical concepts is an open question. Also nobody opened the deep mental cause of their arising.
We use the concept of the basis of the thinking for answering the three following questions:
1) "What are the initial concepts of mathematics" ?,
2) "What structure have they" ?,
3) "How do they arise" ?.
Here are the answers (see [3,11]):
1) They are concepts corresponding to a manifestation model of the basic abilities of the thinking into a mathematics beginning -- an initial mathematical form class that has the simplest individual quality of any forms -- an initial individual distinguishability of the initial mathematical forms.
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 beginning. This model contains some "initial mathematical symbols". The concepts of the second group 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 (singleton-- operation, one--element expansion operation, and substituition operation). 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 initial individual distinguishability of the initial mathematical forms -- the simplest individual quality of any forms.
3) They arise automatically in the process of realization of these three models. As a result, we obtain the simplest models of all basic mathematical concepts: concept of set, concept of mathematical operations, concept of unordered pair, concept of ordered pair, concept of function, concept of order, concept of equivalence, concept 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 the all these concepts 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. We notice that the initial mathematical form class can be considered as a constructive base for constructing more perfect and complex mathematical forms. They arise from the initial form class by the use of the intellectual abilities.
ROLE OF MATHEMATICS FOR SPIRITUAL DEVELOPMENT
Mathematics is fundamental to how we understand ourselves, as human beings. We can consider mathematics as a tool for thought development. It gives a contact with the inner subjective world and extends immensly the limits of informative channels related to our sense agencies. Our mental instrument requires training in order to give an ability to realize the subjective world and mathematics helps us to develop such ability.
Let us confirm these philosophical conclusions by stating that some famous people (see [5]).
Goethe: "The true subject of investigation for mankind is man itself".Laplas: "Mankind pays less efforts for its progress than for the understanding of itself".
Poincare: "We discover mind in its inner essence in mathematics".
Poincare: "In studying geometric thought we may hope to read what is most essential in man's mind".
Hilbert: "When we penetrate more deeply into axiom layers, we obtain a possibility to see some hidden mysteries of thought".
Stone: "Mathematics has a vague and mysterious connection with the physical world, namely this connection is the fact that the process of thought takes place into our mind".
Construction of the above hierarchy of mathematical ideas is important for our spiritual development since such development needs an understanding of external information. Such information moves from the outside toward the interior. The understanding is a personal creative process and it sends in an opposite way.
Lakatos asked: "If mathematics is founded on intuition, why must we go further and further?
Let us take a risk to answer: "We must go further and further in order to construct a stable channel from our lower mind (intellect) to our higher mind (intuition, i.e. the soul).
When we understand this purpose and realize it (every time, when we have understood (for self) a mathematical idea, we have became a more clever person; possibly, some fiber is constructed in the channel from our lower mind (intellect) to our higher mind (intuition or soul)) then we receive a right to declare about "the unreasonable effectiveness of mathematics" (E. Wigner: "How can mathematics be so effective in science") -- and for our mind also!
Every mathematician makes it into his domain, on his level, and millions make it, when they understand mathematical ideas. For this goal it is necessary for mathematicians to construct the subtle structures of mathematical ideas for different mathematics domains including the domains for initial mathematical learning.
A. A. Bailey said (see [4]): "A development cycle has been over. At last, man, as a thinking and feeling being, possibly, arrived to a realizing of essence of his instrument, by what he must work... Whitner leads his mind that is perfected by step-by-step? .... Some more beatiful and uncovering large possibilities something will arise from our mind if it will be correctly realized and used. We can see this new power -- intuition in modern homo sapiens...A new synthesis of mind and soul must be generated in thought on a level of highest intelligence and it must lead humanity from the fourth or humam realm to the spiritual or fifth realm in the end".
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.
REFERENCES
1. A. A. Bailey. A Treatise on Cosmic Fire. Lucis Press Ltd., London, England, 1973.
2. A. A. Bailey. A Treatise on the Seven Rays: Vol. I,II. Esoteric Psychology. Lucis Press Ltd., London, England, 1973.
3. V. P. Polesskii. The Hidden Structures of Thought and Mathematics. Manuscript (in Russian), 1997.
4. A. A. Bailey. From Intellect to Intuition. Lucis Press Ltd., London, England, 1973.
5. M. Kline. Mathematics: the Loss of Certainty. New York. Oxford Univ. Press. , 1980
6. J.Dieudonne. The Difficult birth of Mathematical Structure 1840-1940. Scientific culture in the contemporary world. p.7-23, Milan: Scienta.
7. A. Heyting. Intuitionism. An Introduction. North-Holland Publishing Company, Amsterdam, 1956.
8. S. MacLane. Mathematics: Form and Function. Springer Verlag. New York, Berlin, Heidelberg, Tokyo. 1986.
9. N.Bourbaki. The Architecture of Mathematics. Amer.Math.Month.,57, p.221-232, 1950.
10. J. Dieudonne. Shur functions and group representation, Austerike. 87-88, p. 7-19, 1981.
11. V. P. Polesskii. Three questions about initial concepts of mathematics: What are they? What structure have they? How do they arise? Manuscript (in English), 1997. http://www.spircom.org/russia.htm