V. Tz. Penev, N. G. Zidarov, B. P. Zidarova
 

Central Laboratory of Mineralogy and Crystallography,
Bulgarian Academy of Sciences
Acad. Georgi Bonchev Str., bl. 107, 1113 Sofia, BULGARIA
Fax: (+359 2) 9797 056; Phone: (+359 2) 97 97 055
E-mail: vpenev@clmc.bas.bg; nzidarov@interbgc.com; bzidarova interbgc.com
Web site: http://www.clmc.bas.bg

ABSTRACT

Based on analysis of the evolutionary stages of the development of different sciences the problem for the geometrization of the language of mineralogy is grounded in a general form as the main evolutionary task, to be solved in the forthcoming qualitatively new evolutionary stage of the development of mineralogy. The way for solving this problem is outlined by formulating a system of four basic tasks to be solved during the next stage of the development of the language of mineralogy. As a result of the logical analysis of the foundations of the language of mineralogy, the system of categories "structure", "composition" and "construction" is correctly logically defined. The relations with these categories with the cognitive and the classical structural paradigms and the notions internal energy of a thermodynamic system, order, symmetry, etc., are analyzed. The conceptual scheme for entirely mathematical spatial representation of arbitrary chemical (including mineralogical) structures, both ordered and disordered, is briefly represented. The main obtained geometrical results are represented, analized and interpreted in the terms of mathematical functions. The use of the developed original conceptual scheme for entirely mathematical coordinate representation of arbitrary chemical and mineral structures is demonstrated by the examples of fluorite and pyrite.

In the present work, we are presenting the basic conceptual, logical and mathematical results obtained till now and concerned with defining and solving the problem of geometrization of the language of mineralogy.

GROUNDING OF THE PROBLEM

Before discussing the main results obtained in the framework of the project "Geometrization of the language of mineralogy", we are presenting some preliminary general remarks. Thus, we are trying to create at least partially the cognitive context needed and at the same time to ground the necessity for solving this problem. It is worth noting that such a cognitive context is missing in both mineralogical and natural sciences literature. So:

I. According to Blanshe [1], the development of every scientific branch or science passes through four consecutive stages: descriptive, inductive, deductive, and axiomatic. Their peculiarities can be summarized as follows [2]: during the first two, the descriptive and the inductive stages, the accent is over the collection of empirical knowledge. During the next, higher, deductive stage, the accent shifts on the organization of the already collected (and continuing to be collected) empirical knowledge in the form of a deductive theory (or a set of deductive theories), i.e. on the development of such deductive formulations of a given scientific branch or science, which are based on a small number of properly chosen basic statements (in different scientific branches or sciences these basic statements are called differently, - principles, postulates, axioms, etc.). During the highest, axiomatic, stage of development one accentuates on the strict logical analysis of the deductive formulations of a given scientific branch or science, developed during the preceding stage in order to reformulate them in an axiomatic form. It should be noted that the boundaries between the stages in the development of a given scientific branch or science, are to a great extent conditional. This is especially valid for the boundary between the deductive and axiomatic stages. It is worth noting that the main "tools" during these both stages of the development of a science, are the logical analysis and the mathematical modeling.

A nice illustration of the above-given scheme are the stages the development of elementary (Euclidian) geometry had passed through, namely, the descriptive and inductive stages (till Euclid, 3^th century B.C.), the deductive stage (with stronger and stronger axiomatic features) since Euclid till Lobachevski (more than 22 centuries!), the axiomatic stage since Lobachevski till now. Indeed, it is known from the history of mathematics that Euclid's "Elements" systematizes in fact the collected till then empirical geometrical knowledge in the form of a logically orderly deductive theory. At the same time, in "Foundations of Geometry" by D. Hilbert (occurring some twenty-three centuries after Euclid) one can find an already really strict axiomatic formulation of the preceding deductive formulations of elementary geometry. The same stages are evident in the development of classical mechanics: an inductive stage since antiquity till Newton, a deductive stage (with stronger axiomatic features) since Newton till now (lasting still more than three centuries). Similar is the situation for almost all branches of physics they either are still in the inductive stage of their development or relatively recently (as compared to the Euclidian geometrics) have entered the deductive stage.

One can conclude from the aforesaid that:

Conclusion 1. It is impossible to develop deductive formulations (and, even less axiomatic formulations) of scientific branches unless clarifying, at least principally, their logical structure and defining in a correct logical manner the sets of their basic notions and relationships, as well as unless developing entirely mathematical and unique representations of these notion and relationships.

Conclusion 2. The processes of developing deductive and axiomatic formulations of a given scientific language are quite long term lasting and, at the same time, mutually interconnected.

Conclusion 3. The language of mineralogy (and, consequently, the mineralogy itself) is recently still at the end of the inductive and the beginning of the deductive stages of its development.

II. The analysis of evolution [3] of such well-developed sciences as geometry and physics makes it possible to find the basic laws in the development of natural science knowledge. One of them is that our knowledge on particular groups of phenomena had been developed and is continuing to develop towards creating of a complex of such axiomatic formulations of the different sets of logically consistent, non-contradictory and veracious statements of the respective science, that are characterized by a wide use of entirely mathematical spatial concepts and models. This regularity is referred to as geometrization of a given scientific language. Examples of a terminated process of geometrization are the formulations of diverse geometries, of a quite advanced process the formulations of classical and quantum mechanics, thermodynamics, electrodynamics, theory of relativity, etc., of a just beginning process the reformulation of the language of chemistry. We accept that the above said [4, 5, 6, 7] regularity governs the development of mineralogy as well, which, like chemistry, is now in an earlier evolutionary stage as compared to geometry and physics.

III. The main "tool" of each science or scientific branch is its language.

ІV. In the language of mineralogy one uses first of all the languages of chemistry and crystallography. That is why the solution of the problem of geometrization of this language is closely related to the same problem in the languages of chemistry and crystallography. At the same time the process of geometrization of the language of crystallography had started long ago and is well progressed. That is why the geometrization of the language of mineralogy turns out to depend directly on the solution of the same problem in chemistry. On the other hand the problem of geometrization of this scientific language has already been solved in principle [3]. This fact together with "The concept for existence of qualitatively different evolutionary stages in the development of different scientific branches and sciences" [3, 8] allowed us [4, 5] to formulate and to ground the problem of geometrization of the language of mineralogy and to outline the way for solving it.

MAIN RESULTS

OUTLINING THE WAY FOR SOLVING THE PROBLEM

By the preliminary comments given above, we formulated, at least partially, the main conceptual context needed and at the same time, grounded in a general way the necessity of solving the problem of geometrization of the language of mineralogy. Now, we shall outline the pathway of solving this problem [4, 5, 6].

It is clear from the definition of the notion geometrization that to solve the problem of geometrization of the language of mineralogy means to solve the following system of four closely related problems: 1) to clarify the logical structure of this language in order to formulate correctly the mineralogical notions and relations; 2) to construct an entirely mathematical spatial representations for the latter; 3) to formulate a system of basic mineralogical statements, playing the role of axiomatics in mineralogy; 4) to verify logical of these mineralogical statements which are not axioms, i.e. to deduct them from the formulated mineralogical axioms.

RESULTS FROM THE LOGICAL ANALYSIS

І. The basis for solving of the first problem is the logically correct definition of the triad of cognitive categories structure, composition and construction. These categories are defined by us through the following system of three interconnected axiomatic statements [2, 9]:

Definition A. By the term structure of a given complex object in an exactly specified energy state we denote this unity of a particular compositon and a particular construction which characterizes uniquely the object in this energy state.

Definition B. By the term composition of a given complex object in an exactly specified energy state we denote this set of relatively independent particular parts which characterizes uniquely the object in this energy state.

Definition C. By the term construction of a given complex object in an exactly specified energy state we denote this particular mutual disposition of relatively independent parts, composing the object in the usual three-dimensional physical space, which characterizes uniquely the object in this energy state.

In these definitions energy state of a given complex object is the particular value of the internal energy of this object, considered as a thermodynamic system (TDS).

From the logical point of view this system of definitions is in fact an integral part of the axiomatics of a series of natural sciences because it defines explicitly, uniquely, non-contradictorily and logically consentient, the triad of cognitive categories.

It should be emphasized that the such defined triad of categories structure, composition and construction: 1) is valid for arbitrary complex objects physical, chemical, mineralogical, etc.; 2) does not include any requirements for the existence of ordering in the physical space V_Е(3), i.e. does not include any restrictions to the possible mutual dispositions in V_Е(3) of the particular relatively independent parts, which construct the corresponding structure; 3) is directly connected to the notion internal energy of the object considered as a thermodynamic system (TDS). To be more precise, the specific structure of every complex object depends on the particular value of its internal energy (i.e. on its specific energy state).

From the above comments one can draw the following conclusions [2, 9]:

A. The meaning of such defined triad of categories is much larger than that used so far in natural sciences. Indeed, the removal of the concept for ordering in the physical space V_Е(3) from the logical matter of the category structure enlarges its range to such an extent that it becomes applicable to all kinds of ideal and real physical, chemical, mineralogical, etc. objects, both entirely ordered and partially or entirely disordered (i.e. to all kinds of solids, liquids, gases, plasmas, atoms, mono- and polyatomyc ions, molecules, etc.).

B. On binding the meaning of these categories with the notion internal energy of TDS one can interpret every particular structure (physical, chemical, mineralogical, etc.) as a specific manifestation of an exactly defined amount of internal energy. In addition, one can see from the definitions that: 1) the total amount of internal energy in any particular structure reveals always in two differing aspects, namely, a part of its is in the form of a specific composition, while the rest is in the form of a specific construction; 2) any particular structure is an exactly specified relationship between the corresponding amounts of internal energy "accumulated" in the form of specific composition and specific construction; 3) the different types of symmetries of the construction into V_Е(3) can be interpreted as different forms of "accumulation" of the corresponding amounts of internal energy in the physical space V_Е(3); 4) the change of the internal energy state (i.e. of the specific amount of internal energy) leads to an exactly specified change in the corresponding structure, and, depending on the change in the amount of internal energy, the changes are only in the construction, or only in the composition, or in both. That is why, in studying a complex object much attention is paid to the investigation of the relationship between its internal energy and its specific structure in the corresponding energy state. On its turn, its becomes clear from the above conclusions that in each structure the different types of defects in the composition and the construction play, figuratively said, the role of a complex system of interconnected "gates" ("sluices") through which the energy can flow both inwards (i.e. to flow in the considered TDS) and outwards (i.e. to flow out of the considered TDS).

II. In continuing the investigation of the the relationship between the common for all natural sciences categorial and notional (i.e., epistemological) basis and the fundamentals of the language of mineralogy the category material objects (matter) and the notions solids, liquids, and gasses are defined in a logically correct manner. It is shown that from a logical point of view [2, 9]:

• The notion mineral is a concretization of the cognitive categories structure, matter, volume, and order.

• Each material object (including each mineralogical object) is a chemical structure (i. e. a unity of chemical composition and construction in the ordinary physical space V_M(3)). So, in the language of mineralogy for the first time originates the necessity of a constitutional union of both main approaches for describing objects - the geometric one used for describing crystal morphology and construction of minerals and the substantial one used for describing the chemical composition of minerals.

ІІІ. The general (cognitive) structural paradigm which underlying in the basis of the scientific cognition is formulated and logically analyzed [2, 9]. Through concretizing of the meaning of categories structure, composition and construction the classical structural paradigm, which is in the basis of several natural sciences, is obtained from the general structural paradigm. The logical relationships between these particular forms of the classical structural paradigm are analyzed, which are in the fundamentals of chemistry, crystal chemistry and crystallography, and, consequently in the fundamentals of mineralogy. It is shown that: a) from the classical structural paradigm through adding the relatively weak logical requirement "the composition to be chemical" one obtains the chemical structural paradigm; b) from the chemical structural paradigm through adding the much stronger logical requirement for a partial or total periodicity of the construction in the ordinary physical space V_E(3) one obtains the crystal chemical structural paradigm; c) if in the crystal chemical structural paradigm we reject the logical requirement "the composition to be chemical" and accept that the role of the composition in the considered structure is simply played by mathematical point in V_E(3) one obtains the crystallographic structural paradigm.

RESULTS FROM THE MATHEMATICAL MODELING

ІV. The base for solution the second problem in the case of mineral structures is the conceptual scheme for entirely mathematical spatial representation of arbitrary chemical structures [3, 10]. This scheme is based on the original method developed for constructing entirely mathematical spatial representations of all species¹ of simple chemical objects, both free and bounded. In brief, the meaning of this method is as follows: The different species of simple chemical objects are mathematically represented uniquely through the corresponding set of points in the newly defined Mendeleev's space V_M(3) with respect to Mendeleev's coordinate system K_M=Ozrj, where: O is the origin of K_M; z is the number of neutrons, and r is the number of protons in the nuclei of the presented species of simple chemical objects; where t is the oxidation number of the represented species of simple chemical objects, and n (n=1,...,32) is the number of that column in the corresponding starting 32-columna tabular Periodic system, which contains the represented species of simple chemical objects when they are in electrically neutral state.

The described method allows us to represent entirely mathematically and uniquely the composition of any chemical structure (and thus, of any mineralogical object) by giving the coordinates of the corresponding set of points in V_M(3) with respect to K_M. On the other hand, the construction of any structure (chemical, mineralogical, etc.), both entirely ordered and partially or entirely disordered, is represented mathematically uniquely by giving the coordinates of the corresponding set of points in the physical space V_E(3) with respect to an arbitrary coordinate system K_E(3). Consequently, in order to represent entirely mathematically and uniquely an arbitrary chemical or mineralogical structure, one has to:

1) Construct the Cartesian product V_S(6) = V_M(3) x V_E(3) of the Mendeleev's space V_E(3) (in which the chemical composition is represented mathematically) and the physical space V_E(3) (in which the construction of the considered structure is represented mathematically). The constructed six-dimensional space V_S(6) will be referred to as the space of the chemical structures.

2) Construct in V_S(6) one six-dimensional coordinate system of the chemical structures K_S(6)=(O_S;z,r,j;u,v,w), which is the Cartesian product K_S(6) = K_M(3) x K_E(3) of the Mendeleev's coordinate system K_M=(O;z,r,j) and an arbitrary coordinate system K_E=(O_E;u,v,w) in the physical space V_E(3), such that there is no simple chemical objects in its origin O_E.

It is easily seen that every chemical and mineralogical structure(both entirely ordered and partially or entirely disordered) is represented entirely mathematically and uniquely with respect to K_S(6) in V_S(6) by giving the coordinates (z,r,j;u,v,w) of the corresponding set of six-dimensional points. The first three coordinates (z,r,j) of the points of this set represent the uniquely the chemical composition of the considered structure with respect to K_M(3) in V_M(3), while the second three coordinates (u,v,w) of these this points represent its chemical construction with respect to K_E(3) in V_E(3). In this way, we sketched the conceptual scheme for entirely mathematical spatial representation of arbitrary chemical structures (and consequently, also of arbitrary mineralogical objects) and demonstrated in practice that the problem of geometrization of the language of mineralogy is solvable.

MAIN GEOMETRICAL RESULTS

V. Based on the previously defined [10, 11] notions free chemical point, bonded chemical point and chemical graph, presenting entirely geometrical images in V_S(6) of free simple chemical objects, bonded simple chemical objects and complex chemical objects, respectively, we developed in a general form entirely geometrical images of such basic mineralogical, crystal-chemistry and crystallographic notions as entirely ordered chemical structure (i.e. crystal lattice, a solid with long-range ordering), partially ordered chemical structure (i.e. a fluid, a solid with short-range ordering), entirely disordered chemical structure (i.e. a gas, an amorphous solid) unit cell, etc.

It is shown that [11]:

• The notions free chemical point, bonded chemical point, and chemical graph cover the all variety of chemical (including mineralogical) objects and are sufficient for the entire mathematical representation of their arbitrary combinations, both totally order and partially and totally disordered.

• Every simple chemical object can be represented entirely geometrically through defining the coordinates of the corresponding chemical point in the space of chemical structures V_S(6) and every complex chemical (and mineralogical) object can be represented entirely mathematically through defining the corresponding chemical graph in V_S(6).

• The structure of every complex chemical (including mineralogical) object is represented entirely geometrically through defining in V_S(6) of the corresponding chemical graph, representing the given complex object.

• The composition of every chemical (inc. mineralogical) object is represented entirely geometrically through defining in V_S(6) of the set of the apexes of the corresponding chemical graph representing the given complex object, i.e. through defining the corresponding set of bonded chemical points.

• The construction of every complex chemical and mineralogical object is represented entirely geometrically through defining in V_S(6) of the set of edges of the corresponding chemical graph, representing the given chemical object.

• The role of figures in the space of chemical structures V_S(6) is played by the chemical points (both free or bonded) and chemical graphs, while that of transformations in this space is played by the well known geometrical transformations of the physical space V_E(3) and the motions in the Mendeleev's space V_M(6).

• Every entirely ordered chemical structure (i.e. both composition and construction of every crystal lattice) is represented entirely mathematically and uniquely in V_S(6) through the corresponding periodic chemical graph.

• For each periodic chemical graph (i.e. for each crystal lattice) there exists such a minimum chemical graph (named elementary graph or elementary cell), through which by translations along the three directions in the ordinary three-dimensional physical space V_E(6) one obtains the given periodic chemical graph representing the corresponding entirely ordered chemical structure.

• The set of chemical structures includes the subsets of the objects with natural and such with synthetic origin. In the language of mineralogy the entirely ordered natural objects of earth inorganic origin and with definite composition is referred to as minerals. That is why every periodic chemical graph, representing mineral will be referred hereafter to as mineral graph.

It is clear from the aforesaid that [7]:

• From a geometrical point of view, mineralogy is a science for the mineral graphs in the space of the chemical structures V_S(6) and for their possible transformations.

• In a further mathematical reformulation of the language of mineralogy, an important role will be played by both the combinatorial trend of graphs theory and the mathematical apparatus developed for describing colored graphs.

INTERPRETATION OF THE MODELS IN THE TERMS OF MATHEMATICAL FUNCTIONS

VІ. It is shown that [6, 7]:

• Each chemical structure (including each mineralogical object) can be represented entirely mathematically and uniquely in the space of chemical structures V_S(6) through a corresponding simple discrete function defined in the physical space V_E(3) and having values in the Mendeleev's space V_M(3).

• All chemical structures (and consequently all mineralogical objects) which are of a periodic construction can be represented entirely mathematically and uniquely in V_S(6) through the corresponding periodic discrete simple irreversible functions.

It becomes clear from the last two statements that:

• Mineralogy, just as chemistry, is dealing with the description and study of one exactly defined subset of the set all possible functions in V_S(6). This is the subset of these simple discrete irreversible functions, which describe the various chemical structures (both totally ordered and partially or totally disordered).

• Descriptive mineralogy is concerned with the description and study of one exactly specified subset of the set of all discrete simple and irreversible functions. This is the subset of the periodic discrete simple irreversible functions in V_S(6), which represent entirely mathematically and uniquely the corresponding entirely ordered mineralogical objects.

The classification of minerals is in fact a classification of such periodic discrete simple irreversible functions in V_S(6), which are of a mineralogical meaning.

EXAMPLES

In IV we outlined the conceptual scheme of an entirely mathematical representation of arbitrary chemical (inc. mineral) structures. The usage of this scheme was demonstrated by us on the examples of rutile [2], wurtzite [7], markazite [12], perovskite [12], and sulvanite [12]. In the present works, we do the same on the examples of fluorite and pyrite (Tables 1 and 2).

In these tables the following abbreviations and symbols are used: SCO – Simple Chemical Object; SSCO – Species of Simple Chemical Objects;  stands for the mathematical image with respect to K_M(3) in Mendeleev's space V_M(3) of the corresponding SSCO, obtained using the second generalized mathematical model [3, 10]; N_n in the symbol  stands for the number of neutrons, where N_p for the number of protons (i.e.the sequential number in the Periodic system) in the nuclei of the considered SSCO, h stands for the oxidation state of the SSCO, and k (k=1,...,32) is the number of this column in the corresponding initial form T₂ of the Periodic system to which the electro-neutral atoms of the considered SSCO belong.

The Cartesian coordinates (u, v, w) with respect to K_E(3) in the physical space V_E(3) of the simple chemical objects building the elementary cell of fluorite are as those in http://cst-www.nrl.navy.mil/lattice/struk.xmol/c1.pos, and of the pyrite are as those in http://cst-www.nrl.navy.mil/lattice/struk.xmol/c2.pos. On its turn, the coordinates (z, r, j) of the corresponding SSCO with respect to K_M(3) in Mendeleev's space V_M(3) are obtained using the following rules (see IV): z = N_n; r = N_p and j = (k - h)(2p/32), where h is the oxidation state of the considered SSCO, while k (k=1,...,32) is the number of this column in the corresponding initial table form T₂ of the Periodic system to which the electro-neutral atoms of the considered SSCO belong [3, 10].

CONCLUSIONS

From the results obtained, it becomes clear that:

• The problem of the geometrization of the language of mineralogy is solvable.

• Solving this problem is quite important as it will raise substantially the cognition effectivity of the language of mineralogy, i.e. of its main "tool".

• The total solution of this problem is enormous in size and necessitates the prolonged efforts of numerous researchers.

REFERENCES:

^[1] Blanshe, R. 1965. L' axiomatique. Paris, Press universitaires de France.

^[2] Пенев В., Н. Зидаров, Б. Зидарова. 2004. Логически анализ и математическо формализиране на основите на езика на минералогията предпоставка за неговото геометризиране. София, Год. на СУ "Св. Кл. Охридски", ГГФ, кн. 1-геология, 2005а, 97, 85-96.

^[3] Penev V. Geometrization of the fundamentals of chemistry. 1997, Riva Publ. House, Sofia, 224 pp. (in Bulgarian with an extended summary in English)

^[4] Пенев В., Н. Зидаров. 1999. Геометризация языка минералогии: Новая эволюционная стадия ее развития. В сборник: Матер. ІІ Межд. Минерал. Конференции „История и философия минероллогии", Октябр 4-8, Сыктивкар, Россия, 26-29. (in Russian and English)

^[5] Penev, V., N. Zidarov, B. Zidarova. 2002. Geometrization of the language of mineralogy: І. Formulation of the problem and outlining the way of its solution. - Compt. rend. Acad. bulg. Sci., 55, 5, 47-50.

^[6] Пенев, В., Н. Зидаров, Б. Зидарова. 2004. Геометризирането на езика на минералогията предстоящ качествено нов етап в неговата еволюция. - „МИНЕРОГЕНЕЗИС - 2004", Научна сесия в чест на 90-годишния юбилей на акад. Иван Костов, СУ „Св. Кл. Охридски", София, януари 22-23 , 50-51.

^[7] Пенев, В., Н. Зидаров, Б. Зидарова. 2005. Геометризиране на езика на минералогията. Юбилеен сборник 10 години Централна лаборатория по минералогия и кристалография "акад. Иван Костов" към БАН, Академично изд-во "Марин Дринов", София, 123-129.

^[8] Penev, V. 2002. "Geometrization of the language of chemistry", official site ot the scientific research project - http://www.clmc.bas.bg/staff/V_Penev/gcl/

^[9] Penev, V., N. Zidarov, B. Zidarova. 2004. Geometrization of the language of mineralogy: ІІ. Formulation and logical analysis of the structural paradigm underlie the mineralogy. - Compt. rend. Acad. bulg. Sci., 57, 7, 59-64

^[10] Penev, V., L. Konstantinov, M. Marinov. Logical structure of the fundamentals of chemistry, conceptual schemes of its geometrization and spatial mathematical models of the sets of different species of simple chemical objects. Academic Open Internet Journal, vol. 2 (2000), Part 2: Chemistry - http://www.acadjournal.com/2000/v2/part2/p3/

^[11] Penev, V., L. Konstantinov. Geometrical images of the simple and complex chemical objects in the space of chemical structures V_M(3). Investigation of the connections between the languages of chemistry and geometry. Academic Open Internet Journal, vol. 11 (2004), Part 2: Chemistry - http://www.acadjournal.com/2004/v11/part2/p6/

^[12] Penev, V., N. Zidarov, B. Zidarova. 2004. Geometrization of the language of mineralogy: III. Entirely mathematical and unique coordinate representation of mineral structures. - Compt. rend. Acad. bulg. Sci., 57, 7, 65-70

NOTES

¹ Species in this study means the major subdivision of a genus or subgenus, regarded as the basic category of chemical classification (see for example Random House Webster's meanings of this term.)

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