Philosophical and Methodological Aspects of Discrete Mathematical Chemistry as a New Field of Knowledge in Theoretical Chemistry in its Logical and Historical Context

Chechetkina Irina Igorevna PhD in Chemistry Associate professor, Department of Philosophy and History of Science, Kazan National Research Technological University 420097, Russia, respublika Tatarstan, g. Kazan', ul. Dostoevskogo, 74 A, of. 1 iralena@mail.ru Other publications by this author

In the philosophy of science, the study of methodological problems of the mathematization of science is an urgent strategy, since it allows us to study the genesis and development of science more fully, consider obstacles in its path and develop a new understanding of symbolic reality. A number of the author's works have been devoted to this issue in the field of chemical sciences ^[1-4]. Nevertheless, the interaction between chemical sciences and discrete mathematics remains still unexplored, which leads to uncertain methodological boundaries of new mathematical chemical sciences (mathematical chemistry, discrete mathematical chemistry, chemoinformatics and digital chemistry) arising in various interdisciplinary fields of knowledge, and an ambiguous understanding of their subject of study. Therefore, in order to study in detail the process of mathematization of chemistry with the help of discrete mathematics, it is necessary to turn to its history, depending on both internal scientific factors of development and social ones. The latter circumstance leaves an imprint on the peculiar methodology of these sciences.    

Discrete mathematical chemistry is a new direction in theoretical chemistry, it developed at the junction of various fields of chemistry, discrete mathematics, regression analysis and programming at the end of the XX century, but has deep traditions dating back to the middle of the XIX century. Then the English mathematician A. Cayley for the first time presented a mathematical model of the structure of a molecule in the form of a graph, where the valence of the atom acted as the degree of the vertex of the graph.

The further development of discrete mathematical chemistry at the beginning of the XX century was associated with the attempts of Sylvester, Clifford and Gordon to develop models of molecules based on the theory of the classical theory of the structure of molecules and the mathematical theory of invariants of binary forms. In these models, each chemical bond corresponded to objects of discrete mathematics – components of two–dimensional vectors and discrete representations of chemistry - individual atoms occupying different positions in the molecule ^[5].

However, at that time, the discrete analogy between chemistry and mathematics was met with hostility by mathematicians who believed that chemistry could hardly benefit from this branch of algebra. For example, the works of A.A. Balandin in the field of matrix algebra and its application in theoretical chemistry have been ignored for a long time.

The situation changed a little by the end of the first third of the XX century. in quantum chemistry, then chemists began to apply graph theory to solve particular problems of chemistry in the field of chemical bond theory: the representation of molecular structures in resonance theory and in the method of molecular orbitals of H?ckel. But the methods of discrete mathematics were not widely used at that time.

Nevertheless, chemists continued their work in this new field of knowledge. And in 1947 G. Wiener discovered the first topological index of a chemical graph, which was associated with molecular branching and was called by him the "path number". It was related to the number of vertices in the graph and the distances between them. It was an arithmetic number obtained by adding the shortest distances between carbon atoms in a graph, and it correlated with the boiling points of alkanes ^[6] and many other physical properties. In the future, other researchers began to use such a scheme for calculating topological indices and searching for correlations with the physicochemical properties of molecules on their basis, and it is still used in various fields of chemical knowledge.

The next stage in the development of discrete mathematical chemistry was associated, as Restrepo and Villaveses ^[7] believe, with the social conditions in the Eastern Bloc countries of Western Europe in the second half of the XX century. They believe that at this time most theoretical chemists began to calculate the exact numerical values of the energies of electronic interactions in atoms, and these calculations required large computer capacities.  Western theoretical chemists had such computers, but there were no such computers in the Eastern Bloc countries (Romania, Bulgaria), but theoretical chemists in these countries had the best mathematical training in the field of quantum chemistry. They turned to another field of knowledge – discrete mathematics, which then did not require special expensive computer costs.

Methods of discrete mathematics contributed to the formalization of chemical knowledge, with the help of these means, such properties of molecules as chirality and aromaticity (1970 – 1972) were mathematically substantiated by Ruch and Randich, then in 1990 the concept of mathematical formalization of the chemical structure was created by Randich ^[8], in 1994 new allotropic forms of carbon – fullerenes were predicted and nanotubes by Balaban ^[9], and in 2012 Restrepo gave an interpretation of D.I. Mendeleev's periodic law as a topological space.

It should be noted that, starting from 2000 and to the present day, a lot of reference books on molecular descriptors are being produced, which are being developed by theoretical chemists working in the field of discrete mathematical chemistry ^[10].

Another powerful impetus for the further development of discrete mathematical chemistry was the appearance of digital computers, their appearance in the 70s of the XX century was associated with the task of optimizing computers. They used the graph theory of discrete mathematics to translate chemical information into numbers (molecular descriptors), so the field of discrete mathematical chemistry began to develop even more. Digital computers worked with discrete steps and stored information in discrete bits. It was this circumstance that allowed us to further advance the introduction of discrete mathematics into the field of theoretical and experimental chemistry. Since that time, the stage of automation of spectral data, planning of a chemical experiment, creation of a database of chemical compounds, development of nomenclature and computer modeling of new chemicals with specified target properties and search for optimal ways of chemical synthesis begins.

Among the domestic researchers in the field of computer modeling of molecular structures and chemical synthesis using methods of discrete mathematics and combinatorics, the works of N. S. Zefirov can be distinguished. Also noteworthy are the works of V.I. Sokolov in the field of stereochemistry, who interprets the concepts of chirality, conformation and configuration, measurement of molecules using non-numerical methods of mathematics: algebra, topology and graph theory. He shows that organic chemistry and biochemistry are linked together using chemical algebra or topology in such a direction of theoretical chemistry as stereochemistry ^[11].

N.S. Zefirov ^[12] identifies three ways in which the expansion of discrete mathematics, in particular graph theory, into chemistry takes place: the first way is connected with the classical theory of structure, which has isolated, on the basis of experimental data, the structural fragments of the molecule responsible for the physicochemical properties of molecules, the second way is connected with quantumchemical methods of calculating molecules and the third way is the development of computer technology and the degree of its accessibility for chemists.  

Currently, there is a rapid development of discrete mathematical chemistry on the border with such areas as computer engineering and computer science, which leads to the emergence of new disciplines – chemoinformatics and digital chemistry. 

Questions arise about the boundaries of these disciplines and the possibility of using discrete methods and machine algorithms in them to solve various problems of chemistry. The methodological boundaries between these sciences remain blurred, since all these sciences solve interrelated problems using mathematical modeling methods and the use of computers. Let's try to divide these sciences according to the specifics of mathematical modeling, which will make it possible to characterize the subject of discrete mathematical chemistry.

The subject area of modern discrete mathematical chemistry has not yet been sufficiently defined. Sometimes it is considered as a part of mathematical chemistry or referred to the field of chemical informatics, since its mathematical methods and computational algorithms allow solving various problems of chemistry. The range of chemical problems considered in discrete mathematical chemistry is constantly growing: the concept of chemical bonding, isomerism, directions of reaction flow is being clarified, the reaction mechanism is being studied, chemical classification issues are being considered, structure–property, structure–activity relationships are being described, new molecular descriptors are being searched, and much more. Mathematical chemistry and chemical informatics solve approximately the same theoretical problems, only at a more abstract level of mathematical modeling. The practical tasks of these new chemical mathematical sciences are also the same – the search for new useful properties of chemical compounds, the identification of toxicity and biological activity, the production of new drugs.

There are a number of features of discrete mathematical chemistry that distinguish it from other fields of knowledge:

1) the introduction of discrete mathematics goes directly into chemistry without the participation of an intermediary in this process – physics, which distinguishes it from mathematical chemistry, which uses physics as a conductor of mathematics and the whole arsenal of new mathematical tools developed for 25 areas of theoretical chemistry: differential equations, partial differential equations, group theory, Lie algebras, combinatorics, graph theory, theory of partially ordered sets, linear algebra and matrix theory, probability theory and statistics, number theory, algebraic and combinatorial geometry, topology, functional analysis, von Neumann algebras, Hilbert spaces. The arsenal of mathematical tools of discrete mathematics in chemistry is much more modest: it includes theoretical computer science, group theory, formal logic, set theory, combinatorics, graph theory, probability theory, number theory, algebra, calculus of finite differences, geometry and topology. Nevertheless, it is a powerful computing tool that penetrates not only chemistry, but also the sciences bordering on it: in all areas of mathematical chemistry, chemoinformatics, computational chemistry, digital chemistry, biochemistry and pharmacology, environmental sciences, crystallography, automation and control systems ^[13].

2) discrete mathematics is not only a tool for formalizing chemistry, its application demonstrates mathematical thinking in chemistry, which mathematician Weil spoke about as linking variables in a function ^[14]. So, in discrete mathematical chemistry, the variables can be fragments of any molecular structure, and any physicochemical property of a molecule can act as a function.

3) the degree of idealization in mathematical modeling in discrete chemistry is significantly less compared to chemoinformatics and digital chemistry.  The degree of idealization is associated with the construction of various mathematical models based on different levels of detail of the chemical structure of molecules and the abstractness of models ^[15]. So, we can distinguish the simplest level of modeling – arithmetic, when a molecule is described by a gross formula, which contains information about which atoms and in what quantity are present in the molecule. It can be used to create databases on chemical compounds.

The next level of modeling is topological, where a molecule is described by a structural formula and a mathematical model corresponds to it – a multigraph (chemical graph), the vertices of which are "colored" with symbols of chemical elements, and the edges of the graph mean interactions between atoms. A higher level of detail and idealized representations of the model is a molecular graph, which is written as numbers in matrix form. Chemical graphs and multigraphs are an area of application of discrete mathematics. Finally, an even more idealized level of model representations follows – the stage of various mathematical descriptors as invariants of molecular graphs that describe the chemical structures and electronic structure of molecules in the form of numbers, which are also studied using discrete mathematics. These modeling levels, ranging from topological to individual descriptors, relate to discrete mathematical chemistry.

The whole field of descriptors (fragmentary, physico-chemical, quantum-chemical, molecular field descriptors and many other descriptors) is covered by chemoinformatics, which uses them to build statistical models "structure–property" and "structure – activity". The highest stage of idealized model representations is the conceptual domain of chemical space. The chemical space is a generalized mathematical model of all available and potential properties of chemical compounds, and individual "structure–property" or "structure–activity" models are maps of this space to descriptors describing molecular graphs ^[16].

Digital chemistry is searching for new effective ways of moving through the chemical space ^[17], for this it develops algorithms for generating chemical structures in accordance with the classical theory of chemical structure, and then translates the "chemical" language into the language of "numbers", using graph theory as the language of discrete mathematical chemistry.

Now it is possible to determine the place of discrete mathematical chemistry, chemoinformatics and digital chemistry in accordance with the hierarchical system of mathematical models. Discrete mathematical chemistry deals with traditional issues of isomerism, considers the structure–property relationship, for this it uses models of multigraphs, molecular graphs and part of mathematical descriptors to describe and predict individual structure–property relationships. These tasks form its subject. Discrete mathematical chemistry "translates" chemical compounds into molecular graphs and descriptors. To do this, it can represent its objects – molecules in various ways, for example, from the point of view of the classical theory of chemical structure, molecules can be represented as "atoms" and "bonds" or as "nuclear–electronic systems" in quantum chemistry. Then the set of molecules is translated into a set of numbers using matrix algebra and a molecular descriptor is calculated. This is her method.

 Chemoinformatics, using computer technologies and methods of machine learning and artificial intelligence, transforms chemical structural data into information using the mathematical apparatus of discrete mathematical chemistry, but at the same time it treats molecules differently, they represent objects of chemical space ^[18].

Digital chemistry is the automation and implementation of discrete mathematical chemistry in computer form. Digital chemistry can solve various tasks: to search for new compounds with programmable properties, or, for example, to develop automation of chemical synthesis, in which, with the help of technological developments in the field of machine and software, chemical structural properties are translated into the language of numbers. To do this, a connection is created between the ontological structures of chemistry or a chemical process and the equipment for conducting reactions – a chemical automaton. The translation of a chemical language describing molecular structures into a programming language is carried out using graph theory, previously the graph code can be compiled into byte code ^[19-20].

Discrete mathematical chemistry is an independent field of knowledge that arose as a result of the integration of methods of non-numerical mathematics and various fields of chemical knowledge. She gradually stood out from various fields of chemical sciences, as evidenced by her history. It has its own subject – the study of structure–property relations, for this it uses mathematical modeling based on the translation of molecular graphs into the language of numbers (molecular descriptors).

New sciences, such as chemoinformatics and digital chemistry, have emerged due to the formalization of knowledge using discrete mathematical chemistry. The latter is a tool, a means of cognition and the language of these sciences, which have begun to develop intensively in recent decades due to the introduction of mathematical representations of discrete mathematical chemistry into them. However, the fascination with discrete mathematical chemistry and mathematical algorithms of various batch programs without knowledge of the theoretical and experimental foundations of chemistry leads to meaningless results, the construction of fantastic and exotic structures in chemistry. Therefore, there should be a union of mathematicians, theoretical chemists and experimental chemists working in this field of knowledge.