Types of mathematical models used in historical research. Comments by Slavko Mathematical and Statistical Methods in Historical Research

Features of the rationalistic direction: mathematical methods are used where a digital characteristic is needed for the object under study + the desire to streamline a large amount of digital material. As a result of this approach, there is a fixation of the need to construct verbal expressions that fix the relationship to large, quantitatively filled facts.

The free direction is more voluntarily, implies the presence of elements of novelty.

The use of these two directions has led to the formation of two approaches in modern historiography:

1) specifically historical;

2) globalist.

Specifically historical approach: the accuracy of the mathematical method is assumed. Figures more than other data capture accuracy. + Accuracy of concepts, they have a special role. Developing…
information theory = supposed to be analyzed through the prism of mathematical methods.

The globalist approach at this stage falls short of the status of a theory of history. Causes:

1) within the framework of this approach, there is a rather careless attitude to historical facts;

2) within the framework of this approach, there is a simplification of understanding of complex phenomena;

3) weak connection with the modern philosophy of history (?).

Conclusion: both directions are part of historical science, but mathematical methods are effective only if their boundaries are clearly defined. With regard to these two directions, the specifically historical method is more representative.

The French Revolution of the 18th century in modern Russian historiography

At this stage, post-Soviet Russian historiography is increasingly taking shape. The starting point is the concept of Manfred, in which the Marxist methodology is clearly traced. French Revolution = change of social formations. This approach gave a universal model for assessing revolutions (including the Russian one) that took place at different times in different countries. + The relevance of studying the French Revolution was determined.

Since the second half of the 80s. a departure from the traditional approach is planned, because rethinking of the October Revolution begins. => A.V. Ado is the first to suggest moving away from the simplistic approach in papers from the 1990s. In 2000, the largest researcher Chudinov, on the basis of these articles, stated that there had been a paradigm shift in the study. By 2000, there was a gradual departure from the ideological cliché. + Leadership in determining directions from Manfred passed to his students. But at the same time, Chudinov said that, despite the paradigm shift, the departure from the Marxist approach is ineffective. Smirnov, 2 years later, in his article marked a departure from the ideological cliche, but, in his opinion, the ideological cliché did not disappear, but became different: now it is determined not by the ruling regime, but by the media. This approach has existed for the past decade.

Another aspect that determines the perception of the Great French Revolution is the concept. The key concept, which is perceived ambiguously, is the term feudalism. In particular, there is a position that expresses doubts about the adequacy of the use of this concept in relation to the then existing order. Gurevich (school of annals): “seigneurial” is a more adequate concept than “feudal”. As a result, in the 2000s the idea that it is necessary to agree on concepts was declared. It was proposed to correct the term feudalism in relation to the situation in France before the revolution: to separate 2 terms - seigneurial and feudal. It was suggested that in relation to the political structure of France, the concept of feudalism can be used, since. it refers to the socio-economic formation. As for other, social, economic relationships, it was proposed to use the term senior. => As a result, the use of terms in such a context can lead to different assessments regarding feudalism and seigneury. For historians, this approach turned out to be acceptable, because. in this way it was possible to understand the reason for the difference in conclusions => problems => recognition that the French Revolution, as a phenomenon, has not been studied enough. The available research is fragmentary. Terms about the essence of the feudal system turn out to be unproven. In addition, there is a term about the lack of prospects in this matter. (Prospects were determined by the urgency of the problems (Smirnov) + there is a decrease in interest in socio-economic issues). In fact, in Russian historiography, the last book that fixes the problem of feudalism in the French countryside was Ado's book "Peasants and the Great French Revolution" in 1987. Since the mid-2000s. around 2005, the term "seniorial system" is being more and more approved.

The concept of absolutism. In European historical science in the 20th century. rejection of this concept. But in our historiography this tendency was not reflected. They did not abandon this concept in Russian historiography, but they began to define various facets of this concept. In 2005, in the French yearbook, it was indicated that this term can be understood as power unlimited by law, which is opposed by power based on the law. Absolutism in the complete sense, i.e. an attempt to make this concept as a cultural phenomenon.

The concept of the bourgeoisie.

Thus, in Russian historiography there is a variety of interpretations, which is associated with the lack of a unified theory => the perception of the French Revolution is moving towards Western understanding. There are no generalizing works, because Russian historiography has not reached the level of generalizations. The work of Revunenkov in 2002 is, in fact, a reprint with clarification of his work of the 70s. But there are a lot of works devoted to specific narrow problems. Those. material is being accumulated. The rejection of the Marxist interpretation does not mean the rejection of the use of the works of Marxist historians. The unwillingness of many historians to be drawn into the discussion is noticeable. The current situation in the perception of the French Revolution as a whole reflects modern Russian society: there is a process of reassessment of the revolution as a phenomenon.

The requirements of the state educational standard (SES) in the specialty - history STUDENT: Knows how to organize his work on a scientific basis, owns the methods of collecting, storing and processing information used in his professional activities, Able, given the current state of science and changing social practice, to reassess the accumulated experience and the ability to acquire new knowledge. He is capable of project activities in the professional field based on a systematic approach, is able to build and use models to describe and predict various phenomena, and carry out their qualitative and quantitative analysis.

Requirements of the state educational standard (SES) in the specialty - history (continued) Able to set a goal and formulate tasks related to the implementation of professional functions, knows how to use the methods of the sciences he studies to solve them. Owns general and private methods in the professional field. Able to plan their own activities, navigate in specialized literature Possesses in-depth knowledge in the field of professional specialization, owns modern methodology and methods for solving professional problems Able to form their own research programs in the field of professional specialization.

Principles of construction of the course "Mathematical Methods in Historical Research" The course "Mathematical Methods in Historical Research" is an integral part of the integral methodological training of a history student. This follows from a systematic understanding of the subject of the methodology of historical science, which includes: 1) the doctrine of the ways of understanding history associated with social methodology, the philosophy of history, and the study of historical theories; 2) the doctrine of the methods of obtaining historical knowledge - the methodology of historical knowledge, closely related to the historiography of historical science; 3) teachings about the methods of historical research - the methodology of historical research; 4) teachings about the system of historical methods - substantiation, generalizations, descriptions, explanations of the nature of general historical and particular scientific methods.

The principles of building the course "Mathematical Methods in Historical Research" This follows from a systematic understanding of the subject of the methodology of historical science, which includes: 1) the doctrine of the ways of understanding history associated with social methodology, the philosophy of history, the study of historical theories; 2) the doctrine of the methods of obtaining historical knowledge - the methodology of historical knowledge, closely related to the historiography of historical science; 3) teachings about the methods of historical research - the methodology of historical research; 4) teachings about the system of historical methods - substantiation, generalizations, descriptions, explanations of the nature of general historical and particular scientific methods.

Course Objectives The student must know and master: the conceptual apparatus of a specific methodology of historical research; be able to analyze scientific literature related to the use of mathematical methods in historical research. The student should be able to: navigate modern methods of historical research; it is reasonable to use specific methods for solving research problems in the course work and in the subsequent final qualifying work; determine the cognitive capabilities of certain methods for solving specific research problems.

Organization of the course Course ……………………………………………… Semester ………………………………………… Total class hours ……....…… Lectures ……………………..………… Seminars … Independent work Midterm control: 50 points in total, including: “The structure of term paper” test (March) -5 points + points for work on practical (5) Review of a scientific article (April) -10 points + points for work in practical classes (10) Essay on the topic “Mathematization of history: pros and cons” (May) -10 points + points for work in practical classes (5) + + points for work in practical classes (5) Final control: Pass -50 points

Thematic plan of the course History as a science, history as a reality The structure of historical research Methodology and methods of scientific research in historical science Characteristics of the main methods of historical research Mathematization of historical research Formalization and measurement of historical phenomena Modeling of historical phenomena and processes Methods of grouping statistical data

Basic literature Teaching aids Akhtyamov A.M. Mathematics for sociologists and economists: Proc. allowance. - M .: FIZMATLIT, Belova E.B., Borodkin L.I., Garskova I.M., Izmesteva D.S., Lazarev V.V. Historical informatics. M., Borishpolets K.P. Methods of political research. Tutorial. M., Borodkin L.I. Multivariate statistical analysis in historical research. M., Kovalchenko I.D. Methods of historical research. M., 1987, Quantitative methods in historical research. M., Kuznetsov I.N. Scientific research. Methodology for conducting and designing. M

Basic literature Teaching aids Lavrinenko V.N., Pushilova L.M. Study of socio-historical and political processes. Tutorial. M., Mazur L.N. Methods of historical research. Yekaterinburg, Mathematical Encyclopedic Dictionary. M., Methods of sociological research. Tutorial. / Under the editorship of Dobrenkov V.I., Kravchenko A.I. M., 2006 Nezhnova N.V., Smirnov Yu.P. Application of mathematical methods in historical research. Cheboksary., Fedorova N.A. Mathematical methods in historical research. Lecture course. Kazan, Kazan University Library Fedorov-Davydov G.A. Statistical methods in archeology. M., Formalized-statistical methods in archeology. Kyiv, Yadov V.A. Strategy of sociological research. Description, Explanation, understanding of social reality

Further Reading Henri L., Blum A. Methods of Analysis in Historical Demography. M., Kolomiytsev V.F. Methodology of history. M., Mannheim D., Rich R. Political science. Research methods. M., Mironov B.N. History in numbers. Mathematics in historical research. Moscow, Mathematical Methods in Historical-Economic and Historical-Cultural Researches. M., Mathematical methods in research on socio-economic history. M., Mathematical methods and computers in historical research. M., Mathematical methods in socio-economic and archaeological research. M., Parfenov I.D. Methodology of historical science. Saratov, Tosh D. Striving for Truth or How to Master the Skill of a Historian. M., 2002.

Teaching aids Mathematical methods in historical research. Training and metodology complex. - Izhevsk, Electronic version in the local network of UdGU Methodological dictionary of a student of history. Comp. O.M. Melnikov. Izhevsk, Volkov Yu.G. How to write a diploma, term paper, essay. Rostov-on-Don, Vorontsov G.A. Written work at the university. Rostov-on-Don Morozov V.E. Culture of written scientific speech. M., 2007.

Internet resources for the course of the Laboratory of Historical and Political Informatics of the Perm State Research University, : histnet.psu.ru. histnet.psu.ru Bulletin of the Association "Historian and Computer": Library of Electronic Resources of the Faculty of History of Moscow State University http: //

Topic 1. History as a science, History as a reality (2 hours) History as a reality. Official history. Counterhistory. History as a collective and individual memory of society. pseudoscience. Quascience. The specificity of the past as an object of knowledge. Separation of historical knowledge. History as a science. Scientific knowledge as a kind of human cognitive activity. Object and subject of historical science. Social functions of historical science.

Literature on the topic 1. Barg M.A. Historian-individual-society // Modern and recent history Bernal J. Science in the history of society. M., Gening V.F. Object and subject of science in archeology. Kyiv, Kelle V.Zh., Kovalzon M.Ya. Theory history (Problems of the theory of the historical process). M., Langlois Sh., Segnobos Sh. Introduction to the study of history. SPb., Legler V.A. Science, quasi-science, pseudoscience // Questions of Philosophy Methodological problems of history. Minsk Mogilnitsky B.G. On the nature of historical knowledge. Tomsk, 1978.

Literature on the topic 1. Mogilnitsky B.G. Introduction to the methodology of history. M., Rakitov A.I. historical knowledge. M., Rozov N.S. Philosophy and theory of history. M., 2003. Repina L.P., Zvereva V.V., Paramonova M.Yu. History of historical knowledge. Tutorial. M., 2003, Rumyantseva M.F. Theory of history. M., Ferro M. How the story is told to children in different countries of the world. M., Philosophy and methodology of science. In 2 vols. M., 1994.

Types of historical knowledge 1. Institutional (official history) Dominates society Expresses and legitimizes politics How the complex of historical ideas evolves Constantly changes the system of references The system of sources is strictly hierarchical: the main sources belong to the ideologists of the regime, laws, avoids personal sources Adapts to current politics

Types of historical knowledge. 4. History as a science. The specificity of social cognition in the natural sciences is that the subject of cognition is always outside the scope of the scientific phenomenon; in history: both the subject and the object belong to one whole - history Qualitative incompleteness of the process of development of history The object of history does not exist in reality in the sense in which reality is considered in natural science (“The past is not recoverable in any of its phases” T. Heirdahl)

Features of Science Universality – i.e. scientific knowledge is subject to all spheres of being. Fragmentation - science does not study being as a whole (philosophy), but various fragments of reality. Therefore, science is divided into separate disciplines. Every science has its own object and subject

L.I. Borodkin

(Chapter from textbook)

Mathematical models

in historical research

One of the developing and debatable areas of quantitative history of the 90s. is the mathematical modeling of historical processes. One of the evidence of this is the discussion about the methodological problems of modeling in history, which unfolded on the pages of the journal New and Contemporary History in 1997 1 . This discussion was attended by 15 historians from six countries in Europe and America.

Many models can be found in the literature. These are explanatory and descriptive (descriptive) models, theoretical and empirical, algebraic and qualitative, general and partial, a-priori and a-posteriori models, dynamic and static, extended and limited, simulation and experimental, deterministic and stochastic, semantic and syntactic, not to mention the other types of models you might encounter. The function of models can be research and heuristic, reducing and simplifying, explaining or managing, and in general - formalizing the study. Often models are used to bridge the gap between theory and practice.

A huge number of works are devoted to modeling problems, in which dozens and hundreds of definitions of the concept of "model", classifications of models, types of mathematical modeling are introduced. The term "model" in philosophical literature refers to "some really existing or mentally represented system, which, replacing and displaying in cognitive processes another original system, is with it in relation to similarity (similarity), due to which the study of the model allows you to get new information about the original ". This definition contains the genetic connection of modeling with the theory of similarity, the principle of analogy. Another aspect of modeling is reflected in the definition of the methodologist M. Wartofsky: "The model is the best intermediary between the theoretical language of science and the common sense of the researcher."

As regards mathematical models and the possibilities of their use by historians, this will be discussed in this chapter.

Methodological problems of the application of mathematical methods and models in historical research are devoted to a large number of works 1 , however, these problems are considered most thoroughly in the monograph by Acad. I.D. Kovalchenko 2 . The focus of this chapter is on the methodological and methodological problems that arise when considering the possibilities and limits of the application of mathematical models in historical research. The analysis of these problems requires preliminary consideration of more general aspects related to the regularities and stages of the process of mathematization of social knowledge. It is this broader context that is necessary to understand the specifics of mathematical modeling. historical processes.

11.1. Mathematical methods and models in social sciences:
patterns, specifics and stages of application

The process of introducing mathematical methods into the research practice of the social sciences and the humanities (called the mathematization of social knowledge) is multifaceted and contains the features of both integration and differentiation of modern science. The application of mathematical methods in historical research has a certain specificity in comparison, for example, with a similar process in sociological or economic research. At the same time, this process has certain common features with the process of mathematization of the natural sciences. Let us briefly consider some of the methodological problems associated with the application of mathematical methods in the social sciences and humanities and which are essential for our further discussion of the issues of constructing mathematical models of historical processes and phenomena.

The most general in methodological terms is the problem of explaining the fundamental possibility of using mathematics in various fields of knowledge. Discussing this problem, the famous mathematician, acad. B.V. Gnedenko writes about "the agonizing question that many generations of mathematicians and philosophers have asked themselves: how can science, seemingly without direct connections with physics, biology, economics, be successfully applied to all these areas of knowledge?" 1 . This question is all the more relevant because the concepts of mathematics and conclusions from them, which are introduced and constructed without obvious visible connections with the problems, concepts and tasks of various disciplines, are increasingly being used in them and contribute to more accurate knowledge.

The main "customers" for the development of mathematics today are, along with the natural sciences, the humanities and social disciplines, which put forward tasks that are poorly formalized within the framework of traditional mathematics 2 . This is an essentially new stage in the development of mathematics, given that during the history of mankind the real world has three times given powerful impulses to the development of mathematics 3 . The first time - in ancient times, when the needs of counting and land use gave rise to arithmetic and geometry. Mathematics received a second strong impulse in the 16th-17th centuries, when the problems of mechanics and physics led to the formation of differential and integral calculus. Mathematics receives a third powerful impulse from the real world today: these are the sciences about man, "large systems" of various types (including social ones), problems of information. "There can be no doubt," notes G.E. Shilov, "that the 'structuralization' of new areas of mathematics that are being formed under the influence of this impulse will require mathematicians many years and decades of hard work" 4 .

In this regard, the point of view of the outstanding modern mathematician J. von Neumann is also of interest: "The decisive phase of the application of mathematics to physics - the creation of the science of mechanics by Newton - could hardly be separated from the discovery of differential calculus. ... Importance social phenomena, the richness and multiplicity of their manifestations are at least equal to the physical ones. Therefore, one must expect - or fear - that mathematical discoveries of the same rank as differential calculus will be required in order to make a decisive revolution in this area" 1 .

The impact of the current stage of the scientific and technological revolution with its important social component has significantly changed the traditional idea of ​​mathematics as a "computational" science. One of the main directions in the development of mathematics today is the study of the qualitative aspects of objects and processes. Mathematics of the twentieth century is a qualitative theory of differential equations, topology, mathematical logic, game theory, theory of fuzzy sets, graph theory and a number of other sections, "which do not operate with numbers themselves, but study the relationship between concepts and images" 2 .

An important methodological problem of the mathematization of social knowledge is to determine the degree of universality of mathematical methods and models, the possibility of transferring methods used in one field of science to another. In this regard, one should, in particular, consider the question of whether special mathematical methods are needed for research in the social sciences and the humanities, or one can get by with the methods that arose in the process of mathematization of the natural sciences.

The basis for considering this range of issues is created by the unity of the methodological structure of social and natural science knowledge, which is found in the following main points: description and generalization of facts; establishment of logical and formal connections, deduction of laws; building an idealized model adapted to the facts; explanation and prediction of phenomena 3 .

The sciences of nature and society carry out a constant exchange of methods: the social sciences and the humanities increasingly involve mathematical and experimental methods, the natural sciences - individualizing methods, a systematic approach, etc.

It is essential that the use of mathematical models makes it possible to establish the generality of the processes studied by various branches of knowledge. However, the unity of the world, the commonality of the basic principles of knowledge of nature and society does not at all reduce the specificity of social phenomena. Thus, most of the mathematical models created in the process of development of physics and other natural sciences will hardly be able to find application in the social sciences and humanities. This follows from the obvious methodological position that it is the specificity, the internal nature of the phenomenon or process under study that should determine the approach to constructing the corresponding mathematical model. For this reason, the apparatus of many sections of mathematics is not used in the social sciences and humanities. The methods of mathematical statistics based on the results of probability theory 1 have received the greatest distribution in these disciplines. An explanation of this situation will require consideration of the question of the regularities and stages of the process of introducing mathematical methods in any branch of science.

The experience of mathematization of scientific knowledge indicates the presence of three stages (they are also called forms of mathematization) in this process. The first stage consists in "numerical expression of the studied reality in order to reveal the quantitative measure and the limits of the corresponding qualities" 2 ; for this purpose, mathematical and statistical processing of empirical data is carried out, a quantitative formulation of qualitatively established facts and generalizations is proposed. The second stage consists in the development of mathematical models of phenomena and processes in the area of ​​science under consideration (this is the level of particular theoretical schemes); it reflects the main form of mathematization of scientific knowledge. The third stage is the use of the mathematical apparatus for the construction and analysis of specific scientific theories (combining particular constructions into a fundamental theoretical scheme, the transition from model to theory), i.e. formalization of the main results of scientific knowledge itself 3 .

In the context of our consideration, it becomes necessary to at least very briefly touch upon the question - how is the concept defined in modern science "mathematical model"? As a rule, it is about a system of mathematical relationships describing the process or phenomenon being studied; in a general sense, such a model is a set of symbolic objects and relations between them. As G.I. Ruzavin, "until now, in specific applications of mathematics, most often they deal with the analysis of quantities and the relationships between them. These relationships are described using equations and systems of equations" 1, due to which a mathematical model is usually considered as a system of equations in which specific quantities are replaced by mathematical concepts, constant and variable quantities, and functions. As a rule, differential, integral and algebraic equations are used for this. The resulting system of equations, together with the known data needed to solve it, is called a mathematical model. 2 . However, the development of the latest branches of mathematics related to the analysis of non-numerical structures, the experience of their use in social and humanitarian research have shown that the framework of ideas about the language of mathematical models should be expanded, and then a mathematical model can be defined as any mathematical structure "in which its objects, as well as relations between objects, can be interpreted in various ways (although from a practical point of view, a mathematical model expressed in terms of equations is the most important type of model)" 3 .

While in the "exact" sciences all three forms of mathematization are used (which gives grounds to speak of the "incomprehensible effectiveness" of mathematics in natural sciences), the "descriptive" sciences mainly use only the first of these forms. Although, of course, in the totality of the social and human sciences, this process has certain differences. Economic research is leading here, in which the first two stages of mathematization have been firmly mastered (in particular, a number of effective mathematical economic models have been built, the authors of which have been awarded Nobel Prizes), there is a movement to the third stage 5 .

Assessing the current situation with the "lag" in general of social knowledge in terms of the degree of penetration of exact methods into them, some representatives of the natural sciences explain this by a number of reasons of a subjective nature. More justified is another point of view, based on the fact that the exact sciences study relatively simple forms of motion of matter. “Isn’t it because this “lag” arose,” writes a well-known probabilistic mathematician, “that people involved in the humanities were, perhaps,“ stupider ”engaged in exact ones? By no means! It’s just that the phenomena that make up the subject of the humanities are immeasurably more complicated those that are involved in exact ones. They are much more difficult to formalize. For each of this kind of phenomena, the range of reasons on which it depends is much wider ... And yet, in a number of cases, we are simply forced to build mathematical models here too. If not exact, then approximate. If not for an unambiguous answer to the question, then for orientation in the phenomenon" 1 . As G.I. Ruzavin, in most human sciences, which are traditionally considered inaccurate, the object of study is so complex that it is much more difficult to formalize and mathematize. Therefore, the desire to consider exact natural science as an ideal of scientific knowledge ignores the specifics of research in other sciences, the qualitative difference in the object of their study, the irreducibility of higher forms of movement to low ones 2 .

This already contains an approach to resolving the question of whether the results obtained with the help of mathematical methods in a particular area of ​​social knowledge correspond to those standards, criteria that are accepted in the "exact" sciences? On the one hand, social and natural sciences use a set of scientific criteria based on the same epistemological principles. The main requirements for the scientific method can be reduced to the following: objectivity, facticity, completeness of description, interpretability, testability, logical rigor, reliability, etc. 3 .

On the other hand, research activities within mathematical the standard of scientificity is primarily the knowledge of the logically possible; natural science the standard is focused on obtaining results that are effective for practical, substantive activities; social and humanitarian the standard of scientific knowledge "is oriented, in addition, to obtaining socially significant results consistent with the goals, basic values ​​of the socio-historical subject" 1 . Without pretending here to analyze the complex problem of the correlation of scientific standards, we note only the obvious irreducibility of the process of historical knowledge to purely logical or mathematical procedures. A comparison of the actual processes of mathematization of various areas of social knowledge reveals significant differences in the nature of these processes, which stem primarily from the specifics of the nature of knowledge in various social sciences. It seems that discussions about the limits of penetration of mathematical methods in the social sciences and humanities 2 cannot be fruitful without identifying types social knowledge.

A.M. Korshunov and V.V. Mantatov distinguish three types of social knowledge: socio-philosophical, socio-economic And humanitarian knowledge 3 . These types of knowledge can complement each other even within the same science. An example of such a connection is historical science, which gives a description of social events in all their specificity and individuality, spiritual uniqueness, but at the same time based on the laws of development, primarily economic ones. As noted by these authors, socio-economic knowledge approaches in its type the knowledge of the natural sciences 4 . That is why mathematical methods of cognition find effective application in the studies of socio-economic processes. An important condition for theorization of social knowledge, A.M. Korshunov and V.V. Mantatov, "is the development of a specialized language that opens up the possibility of constructing and operating with idealized models of reality. The construction of such a language is mainly associated with the use of the categorical apparatus of the corresponding scientific discipline, as well as the formal-sign means of mathematics and logic" 5 .

V.Zh. Kelle and M.Ya. Kovalzon, discussing the same problem, distinguishes two types of social knowledge 6 . One of them is similar to natural science and can be associated with the use of mathematical methods, but in all cases it involves such a description of social processes in which attention is focused on "the objective beginning of society, objective laws and determinants." For lack of a better term, this type of knowledge is called by the authors sociological 1 . Another type of knowledge is social and humanitarian or simply humanitarian. Within its framework, methods of scientific analysis and individualized description of the spiritual side of human life are developed. These types of social knowledge differ from each other primarily in that, in accordance with their cognitive capabilities, they reflect various aspects of reality, complementing each other. Since the boundaries between these types of knowledge are mobile and relative, they can be combined within the framework of one science (an example of this kind is given by story). The methodological significance of the proposed typology lies in the fact that it provides an approach to resolving the "eternal dispute between the humanities and their opponents on the question of what scientific knowledge about society should and can be - or only passed through the" mathematical filter ", strict, formalized," accurate", or purely humanitarian, revealing the "human", spiritual side of socio-cultural reality, not claiming to be accurate and fundamentally different in nature from natural knowledge" 2 . Recognizing the existence of various types of scientific social knowledge, we thereby remove the indicated problem of the dichotomy of scientific knowledge and transfer the conversation to another plane - studying the specifics of various types of social knowledge, their cognitive potential and, accordingly, the possibilities of their formalization and modeling.

The second aspect of social knowledge, influencing the process of its mathematization, is determined by the maturity of the relevant scientific field, the presence of an established conceptual apparatus that allows one to establish the most important concepts, hypotheses and laws at a qualitative level 3 . "It is based on such a qualitative analysis of the objects and processes under study that one can introduce comparative and quantitative concepts, express the found generalizations and established patterns in the exact language of mathematics" 4 , thus obtaining an effective analysis tool in this scientific field. In this regard, it seems to us that the point of view of Acad. N.N. Moiseev, who believes that "fundamentally non-mathematizable" disciplines do not exist at all. Another thing is the degree of mathematization and the stage in the evolution of a scientific discipline at which mathematization begins to work" 1 .

The noted factors and features of the process of mathematization of social knowledge also manifested themselves in the experience of applying mathematical methods and models in historical research, which at the same time have certain specifics. Let us consider here a number of methodological and methodological aspects of this process, which in recent years have become the focus of attention of historians who use the methods of mathematical modeling in concrete historical research.

11.2. Mathematical models of historical processes:
specificity, levels, typology

Having mastered almost the entire arsenal of traditional mathematical and statistical methods during the first decade of its development (including descriptive statistics, sampling method, time series analysis, correlation analysis, etc.), domestic cliometrics in the second half of the 1970s switched to the active use of multivariate methods. statistical analysis ("tops" of applied mathematical statistics). To date, most of the work related to the use of mathematical methods in historical research is based on the statistical processing of data from historical sources; these works, in accordance with the periodization discussed above, should be attributed to the first stage of the mathematization of scientific research. At this stage, the solution of many topical problems of historical science 2 was promoted.

However, the improvement of the methodology of historical research in the 1980s created the prerequisites for the transition to the second stage of mathematization - the construction of mathematical models of historical processes and phenomena. As will be shown in this paper, there are various approaches to the classification of such models.

The problem of modeling historical processes and phenomena has a pronounced specificity. The rationale for this specificity is contained in the works of I.D. Kovalchenko, which characterized the essence and goals of modeling, proposed a typology of models of historical processes and phenomena, including reflective-measuring And imitation models 1 . Highlighting two stages of modeling (essential-content and formal-quantitative), I.D. Kovalchenko notes that quantitative modeling consists in a formalized expression of a qualitative model by means of various mathematical means 2 . The role of these tools differs significantly in the construction of reflective-measuring and simulation-prognostic (more precisely, retro-prognostic) models.

Models of the first type characterize the studied reality invariantly, such as it was in reality. Measurement modeling is based, as a rule, on the identification and analysis of statistical relationships in the system of indicators characterizing the object under study. Here we are talking about checking the essential-content model using the methods of mathematical statistics. The role of mathematics in this case is reduced to the statistical processing of empirical material.

Much less tested in the practice of domestic cliometric studies are mathematical models, the use of which is not limited to the processing of source data. The purpose of such models may be to reconstruct the missing data on the dynamics of the process under study over a certain time interval; analysis of alternatives of historical development; theoretical study of the possible behavior of the studied phenomenon (or class of phenomena) according to the constructed mathematical model. Models of this type can be classified as imitation And analytical 3 .

As is known, in the study of modern socio-economic processes, simulation and prognostic models that, replacing the object of knowledge, acting as its analogue, allow you to simulate, artificially reproduce options for its functioning and development. Thus, they serve as an effective tool for solving numerous problems related to forecasting, management, planning, etc.

Obviously, when studying the past, when the researcher is dealing with an already accomplished reality, simulation modeling has its own specifics compared to imitation of the subsequent development of the current reality. The experience accumulated in domestic and foreign historiography allows us to distinguish two types of simulation models: imitation-counterfactual And imitation-alternative models of historical processes 1 .

The problems of counterfactual modeling, associated with the arbitrary reshaping of historical reality, do not at all mean the impossibility of using “non-reflective” modeling in historical research. Moreover, by the mid-1990s this direction was marked by the Nobel Prize, which was received by famous American cliometrists - Robert Vogel and Douglass North. The text of the justification for the decision of the Nobel Committee noted, in particular: "R. Vogel and D. North were pioneers in the direction of economic history, which was called the "new economic history" or cliometrics, i.e. the direction of research that combines economic theory, quantitative methods, hypothesis testing, counterfactual modeling" 2 .

For us, however, more important is the possibility of using mathematical models in the study alternatives historical development. The problem of alternativeness is given a lot of attention in the works of historians-methodologists of the second half of the 1990s. A. Ya. Gurevich 3 considers this problem as one of the main ones at the present stage of development of historical research. Alternativeness in history is one of the main aspects of the analysis of historical patterns in the works of BG Mogilnitsky 4 .

Models can be an effective tool for exploring alternative historical situations. Modeling one or another of the possible outcomes will allow a deeper understanding of the real course of historical development and the objective meaning and significance of the struggle of social forces for one or another variant of this development 1 . The imitation of an alternative historical situation and the calculation of the values ​​of indicators of interest to the researcher should be based on certain, to some extent probable and legitimate assumptions. Justifying these assumptions is critical. In simulation-alternative models that characterize, although counterfactual, but objectively possible states of an object, the model parameters are determined on the basis of data characterizing the real states of the system under study.

Speaking about the need to develop new methods and models that "capture the specifics of historical phenomena", K.V. Khvostova comes to the conclusion that “a detailed quantitative analysis of local-temporal socio-economic and political trends... would lead to a more thorough formulation of the problem of alternatives to historical development. would answer the question about the probability of further functioning, which the interrupted trend had, and thus about the random or regular nature of the factors that caused the cessation of its development” 2 .

Nizhny Novgorod State University N.I. Lobachevsky National Research University Educational, scientific and innovative complex "Social and humanitarian sphere and high technologies: theory and practice of interaction" Main educational program Main educational program 030600.62 "History", general profile qualification (degree) bachelor Educational and methodological complex in the discipline "Mathematical methods in historical research” Negin A.E., Mironos A.A. MATHEMATICAL METHODS IN HISTORICAL RESEARCH Electronic teaching aid Activity 1.2. Improving educational technologies, strengthening the material and technical base of the educational process Nizhny Novgorod 2012 MATHEMATICAL METHODS IN HISTORICAL RESEARCH. ., Negin A.E., Mironos A.A. Electronic teaching aid. - Nizhny Novgorod: Nizhny Novgorod State University, 2012. - 31 p. The teaching aid deals with the use of mathematical statistics methods in historical research, as well as the use of mathematical modeling tools for the reconstruction of historical events and processes. The use of mathematical methods in historical research is illustrated by specific examples of the analysis of source complexes carried out in the study of key problems of Russian history. The manual contains information about the structure of the course, a list of control questions and recommended literature for self-study. The electronic educational and methodical manual is intended for UNN students studying in the direction of preparation 030600.62 "History", studying the course "Mathematical Methods in Historical Research". 2 TABLE OF CONTENTS page Introduction. 4 Section 1. Methods of mathematical statistics in historical research 5 1.1. The specifics of the application of mathematical methods in history. 5 "Mathematization" of historical knowledge: possibilities and limitations 1.2. Sampling method 9 1.3. Cluster analysis method 12 1.4. Correlation, regression and factor analysis 16 Section 2. Modeling in historical research 22 2.1. Types of mathematical models used in historical research 22 2.2. Mathematical methods in classical and experimental archeology 25 2.3. Problems of historical modeling. Cliodynamics in reconstruction of the past and forecasts of the future 2.4. Modeling by means of fractal geometry 30 The structure and content of the discipline 34 "Mathematical methods in historical research" Questions for preparing for the exam 38 Recommended reading 39 3 Introduction. The development of historical science, as well as other areas of scientific knowledge, is closely associated with the development of new technologies that expand cognitive capabilities. In modern conditions, the main resources are concentrated in the field of computer technology. It is in this area that promising opportunities are concentrated for improving the methodological tools of historical science. The computer creates fundamentally new conditions for the work of a historian with a source: it makes it possible to process huge amounts of data, multidimensional analysis, and even modeling of historical processes and events. Modern software tools also impose new requirements on the researcher himself: often freeing him from the need for detailed knowledge of the technology of working with data, their “manual processing”, they make him take a much closer look at the formal-logical component of research activity. The use of computer technologies in historical research entails the mathematization of historical knowledge, provides a basis for a wider application of interdisciplinary approaches, thanks to which it has become possible to obtain more accurate data about the past and verify the existing theoretical developments of previous generations of historians. The significance of mathematical methods is multifaceted; at the same time, they act as a powerful tool in the research arsenal, and as a "communicative resource" that provides the possibility of interdisciplinary synthesis. The third-generation educational standard put into effect in the field of study "History" imposes increased requirements on the level of knowledge and competencies of future graduates of historical faculties in the field of using information technologies and mathematical methods in historical research. A modern bachelor of history should be able to use "basic knowledge in the field of the fundamentals of computer science, elements of natural science and mathematical knowledge" in his professional activities. The course "Mathematical Methods in Historical Research" occupies a leading place in their development. A necessary part of the educational process within the framework of this course is familiarization with the existing experience in the application of computer technologies and mathematical methods in specific works of modern historians and the acquisition of practical skills in applying this or that method, taking into account the experience of today's classical studies in this field. The material summarized within the framework of this educational and methodological manual is intended to help students master the experience gained by historical science in applying mathematical methods in solving problems of historical reconstruction. 4 SECTION 1. METHODS OF MATHEMATICAL STATISTICS IN HISTORICAL RESEARCH 1.1. The specifics of the application of mathematical methods in history. "Mathematization" of historical knowledge: opportunities and limitations In the social and human sciences, which study the patterns of existence and development of human society and the individual, the traditional arrays of information, which are usually used quantitative methods, are the so-called. "statistical sources" - population records, fiscal and cadastral data, etc. The second group, in relation to which quantitative methods are also actively used, is “mass sources” - arrays of documents of the same type in structure and composition of the information contained in them (for example, periodicals). Such information can be easily formalized and, therefore, reduced to a quantitative value with subsequent statistical processing. But one should not, however, think that statistical methods can only be used to analyze statistical sources, which in their original form are digital material. Statistics methods are also suitable for working with non-quantitative information, because they always deal with aggregates, groups, i.e. mass material, and not with individual cases, objects, individuals. Therefore, when describing a set of data, a statistical calculation is possible and, consequently, the use of statistical methods. Thus, the mathematization of historical information is a much more diverse and large-scale phenomenon, which has not only an explicit expression in the form of attracting and processing data containing proper quantitative information in the narrow sense. The introduction of the processing of statistical data using mathematical methods in historical research and in the auxiliary historical disciplines accompanying them began as early as the 19th century. It was then that the growing source base of both written and archaeological sources required processing, systematization and verification using elements of mathematical knowledge. A peculiar direction that ultimately allows historical information to be brought to a certain quantitative embodiment and, thus, processed by mathematical means, is the use of experimental methods in history and archeology. In the middle of the 19th century, thanks to the efforts of Napoleon III, the birth and formation of the so-called military archeology and reconstruction took place. He purposefully financed excavations in Alesia, with his support, the first attempt was made to reconstruct an ancient rowing vessel - a trireme and a medieval throwing machine - a trebuchet. In these experiments on the reconstruction of ancient technology, for the first time, the massive application of mathematical methods in the study of the development of 5 ancient technologies was noted. During the second half of the 19th and early 20th centuries, a whole series of experiments based on mathematical calculations followed, which aimed at restoring and testing working models of Greek and Roman siege equipment and throwing machines. Thus, the sportsman and philanthropist R. Payne-Gallway reconstructed the Roman one-arm machine - the onager, rather vaguely described by Ammianus Marcellinus. This large onager managed to launch a stone ball weighing 3.6 kg at a distance of 450 meters! At the beginning of the 20th century, the initiative passed to German researchers. Major E. Schramm, in collaboration with classical scholars and with the support of Kaiser Wilhelm II, built twelve examples of ancient throwing machines. After the grandiose work done by E. Schramm, no new reconstruction attempts were made over the next sixty years, until later new archaeological finds appeared that clarified many details. Concerning the problems of using statistical methods in research on ancient history, one should mention, for example, the calculations of J. Le Boeck, presented by him in his books The Third August Legion and The Roman Army of the Early Empire1. He, for example, compared the African and Spanish legions, in which the ratio of Italians and local natives was completely different. In spite of this, the Latin cognomina were in the majority: 96 to 4 for Africa and 94 to 6 for Spain. He notes that, in general, Greek names among legionnaires are extremely rare and their bearers can be divided into 3 categories: those who really came from the East, soldiers from the "camp" (there is no consensus on the origin of the term origo castris) and those who lived during the reign of Hadrian (as you know, an Hellenophile). In Africa, where most of the time only one legion, the III Augusti, was stationed, one can trace the changes in the ethnic composition from documents, especially numerous for the 2nd century BC. and the era of the North. As a result of his calculations, J. Le Boeck came to the conclusion that the 1st century is the century of the Italians and Gauls. At the beginning of the II century. AD Africans begin to join the legion (and some of them did this already in the 1st century), but there are still fewer of them than the Bithynians, people from the Lower Danube and especially the Syrians after the Parthian campaigns of the same Trajan. At the end of the II century. the percentage changes in the opposite direction - Africans predominate, first of all the natives of the Maghreb, and then Numidia. At the beginning of the III century. the share of "foreigners" remained stable. The legion, disbanded between 238 and 253, was restored, perhaps through the recruitment of natives; but in the middle of the III century. it was already lost the custom to indicate the origin of the recruit. The successful introduction of statistics into the studied documents on medieval and modern history was carried out by historians who worked within the framework of the so-called Annales school, which arose on the basis of the journal of the same name in 1929. Representatives of the Annales school sought to comprehensively consider historical material, as part of the creation of the so-called "total history" (histoire totale). The first attempt at such an embodiment of this ideal of all-encompassing history is attributed to F. Braudel, the leader of professional French historians in the mid-twentieth century. In his work 1 Le Bohec Y. La Troisième Légion Auguste. Paris, 1989; Le Boeck J. The Roman army of the Early Empire / Per. from fr. M. N. Chelintseva. - M., 2001. 6 "The Mediterranean and the Mediterranean world in the era of Philip II" (1947) all aspects of this huge topic were covered vividly and in detail: physical geography and demography, economic and social life, political structures and politics of Philip II and his rivals in the Mediterranean. According to Braudel, in the study of history, mathematical modeling should be applied as widely as possible and a genuine "social mathematics" should be developed. The historians of the Annales school were the first to turn to a new type of local history. The power of this approach of "local total history" was demonstrated by another already mentioned French historian E. Leroy Ladurie in his works "Peasants of Languedoc" (1966) and "Montaillou" (1978). These studies were limited to a single village over several generations. Methodological developments close to the Annales school were used in their research by the well-known Russian medievalist historian Yu. L. Bessmertny (1923-2000). So, in his book "Life and Death in the Middle Ages" on the material of the history of France in the 9th-18th centuries. Yu. L. Bessmertny analyzed the forms of marriage and family, traced the change in views on the role of women in the life of medieval society, spoke about the attitude towards childhood and old age, about “self-preserving” behavior in different social strata, reproduced medieval ideas about illness and death. The author examines the change in the most important demographic parameters - marriage, fertility, mortality, natural population growth. Already in the late 50s. cliometrics arises and develops (cliometry - English. Cliometrics is a direction in historical science that involves the systematic use of mathematical methods. A close, actually synonymous concept is “quantitative history”, understood as historical knowledge obtained using mathematical methods in historical research. The name of this direction is made on behalf of Clio, the muse of history and heroic poetry in Greek mythology. Cliometrics is an interdisciplinary field, originally associated with the application of econometric methods and models in economic history research. The term cliometrics first appeared in print in December 1960 in an article by J. Hughes, L. Davis and S. Reuter "Aspects of Quantitative Research in Economic History". However, a surge of interest in such studies, often referred to as the "cliometric revolution", is associated with the 1960s. A special role in the development of this direction (cliometric approaches to the study of economic history) was played by the American journal "Journal of Economic History", whose editors in the 1960s. became Douglas North and William Parker - supporters of the cliometric approach. During the same period, cliometric conferences began to be held regularly in the United States. American researchers, relying on cliometric methods, successfully studied the role of railway construction in the development of industrialization and development processes, US agriculture in the 19th century, the economic efficiency of slave labor in the American economy, etc. In 1993, Robert Fogel and Douglas North received the Nobel Prize in Economics for their cycle of work in the field of cliometrics. The decision of the Nobel Committee notes that the 7th Prize was awarded "for the development of new approaches in research in economic history, based on the application of economic theory and quantitative methods to explain economic and institutional changes." Since the 1970s The cliometric approach is beginning to be actively used in studies on economic history in Great Britain, the Scandinavian countries, Spain, Belgium, Holland, and other countries. In a broader sense, the use of quantitative methods in historical research (quantitative history) became widespread in Germany (the Center for Historical and Social Research of the University of Cologne plays the main role here) and the USSR (Russia), where the “cliometric school” began to take shape in the 1970s. last century. The formation of quantitative history was accompanied by a large number of scientific conferences, publications, and the appearance of periodicals, such as, for example, "Historical Methods" (since 1967. , since 1978 - "Historical Methods Newsletter") in the USA, "Computer and the Humanities" (since 1966), "Historische Sozialforschung" (since 1976 - "Historical Social Research") in Europe. This direction was aimed at a qualitative transition to understanding history as a developed science (science), systematically applying not only methods and models, but also theories of related sciences. Representatives of the "Annales school" experienced a strong influence of quantitative ideas. E. Le Roy Ladurie's polemically pointed statement is well known: "History that is not quantifiable cannot claim to be considered scientific." In the USSR, Moscow State University became the center of research on quantitative history. M.V. Lomonosov, where, in the 1970s - 1980s, a community of scientists was formed who used mathematical methods and computers in historical research. Academician I.D. Kovalchenko became the undisputed leader of the new direction. Since 1979, the All-Union Seminar "Quantitative Methods in Historical Research" (L. V. Milov, L. I. Borodkin and others) has been operating on the basis of the Faculty of History of Moscow State University. For almost half a century of active development of the “quantitative methodology” of history, we can talk about a significant internal evolution of both the scientific direction itself (beginning with cliometric approaches to the study of economic history), and the emergence on its basis of related areas - in particular, actively developing in the last two decades historical informatics, which has become an interdisciplinary field that develops theoretical and applied problems of using information technologies in historical research and education. However, all these interdisciplinary areas are connected by a common basic approach - the mathematization of historical knowledge. Is not it. Borodkin, considering the history of the emergence and development of historical informatics, singles out two periods that are significantly different in their content: the first is the era of "large" computers (the beginning of the 1960s - the end of the 1980s) and the second is the "microcomputer revolution" (the end of 1980s - mid 1990s). To date, we can talk about three successive stages of the mathematization of historical science: 1) mathematical and statistical processing of empirical data and the quantitative formulation of qualitatively established facts and generalizations, including traditional mathematical and statistical methods (descriptive statistics, sampling method, time series analysis, correlation analysis) ; methods of multivariate 8 statistical analysis; 2) development of mathematical models of phenomena and processes in some area of ​​science; 3) the use of mathematical apparatus for the construction and analysis of a general scientific theory. According to L.I. Borodkin, the third stage in history has not yet been used at all, the second is under active development. Already at the end of the 20th century, as a kind of reaction to attempts to establish “scientism” in historical research, “neo-antipositivist” concepts appeared that denied the possibility of scientific knowledge not only of the past, but also of the present. From this point of view, the effectiveness of the use of mathematical methods in history is denied and it is proposed to return to the position of artistic, poetic and metaphorical methods of its understanding and description, in which the historian still seems to be more of a storyteller than a researcher. The obvious limitations pointed out by "skeptics" regarding the use of quantitative methods in historical research are related to the lack of direct observation, subject-object correlation, multifactorial manifestations and the corresponding multidimensionality of the study, as well as the weak uniformity of the information used. At the same time, of course, new methods of historical research based on the use of mathematical data processing tools have made it possible to revise a number of already known problems at a different level of generalization, as well as to set and solve fundamentally new, major problems of studying the historical past. 1.2. Sampling Often, historians have at their disposal a large array of sources and data that they are not able to fully process. This applies, first of all, to research on Modern and Contemporary history. On the other hand, the deeper you have to look into the depths of centuries, the less information you can operate on. In both of these cases, it is useful to use the so-called sampling method, the essence of which is to replace a continuous survey of massive homogeneous objects with a partial study of them. At the same time, a part of the elements, called a sample, is selected from the general population, and the results of processing sample data are eventually generalized to the entire population. Only a representative sample that correctly reflects the properties of the general population can serve as the basis for characterizing the entire population. This is achieved by random selection of elements of the general population, in which all its elements have equal chances of being included in the sample. The application of this method is equally suitable for studying various phenomena and processes of our time, and for processing data from previously conducted selective statistical studies, such as censuses. In addition, the sampling method also finds application in the processing of data from natural samples, from which only fragmentary data remain. So, quite often, such partially preserved data include act materials, documents of current office work and reporting. Depending on how the selection of population elements in the sample is carried out, there are several types of sample surveys in which the selection can be random, mechanical, typical and serial. Random selection is a selection in which all elements of the general population have an equal opportunity to be selected, for example, using lots or a table of random numbers. The draw method is used if the number of elements of the entire population under study is small. With a large amount of data, random selection by lottery becomes difficult. More suitable, in the case of a large amount of data being processed, is the method of using a table of random numbers. The selection method using a table of random numbers can be seen in the following example. Assume that the population consists of 900 items, and the intended sample size is 20 units. In this case, from the table of random numbers, numbers not exceeding 900 should be selected until the required 20 numbers are dialed. The numbers written out should be considered as serial numbers of the elements of the general population that fell into the sample. For very large populations, it is better to use mechanical selection. So, when forming a 10% sample, only one of every ten elements is selected, and the entire set is conditionally divided into equal parts of 10 elements. Then, from the top ten, an element is randomly selected (for example, by drawing lots). The remaining elements of the sample are determined by the specified selection proportion N by the number of the first selected element. Another type of directional selection is typical selection, when the population is divided into groups that are qualitatively homogeneous. Only after that, within each group, a random selection is made. Although this is a more complex method, it gives more accurate results. Serial selection is a type of random or mechanical selection carried out for enlarged elements of the initial population, which is divided into groups (series) during the analysis. The above sampling methods do not exhaust all types of selection used in practice2. As an example of the application of the sampling method in historiography, let us consider in more detail the analysis of the movement of grain prices in Russia in the 18th century carried out by Russian researchers3. The task was to determine the average prices for bread in individual provinces, regions and in Russia as a whole for each year of the 18th century, as well as to identify the dynamics of grain prices over the century. However, in the course of the study, it became clear that it would not be possible to compile tables with a continuous series of prices, since the data in various archives were only partially preserved. For example, data for 1708 were available for only 36 counties of the country. Only for the periods from 1744 to 1773 and from 1796 to 1801 data on most Russian cities have been preserved. In this regard, it was decided 2 For the most complete acquaintance with the various types of selection, we advise you to refer to the book: Pite F. Sampling in censuses and surveys. M., 1965. 3 Mironov B.N. Grain prices in Russia for two centuries (XVIII-XIX centuries). L., 1985. 10

used in historical research.

The basis of the computational experiment is mathematical modeling.Mathematical model- a system of equations (differential, integral and algebraic), in which specific quantities are replaced by constant and variable quantities, functions.

The purpose of modeling is replacement of the real object of study by its model, which must be investigated, transferring conclusions to the object.

As in any other experiment, a number of general stages can be distinguished in mathematical modeling.

At the initial stage a mathematical model is built for the object under study. Then a computational algorithm is developed (in the form of a set of chains of algebraic formulas and logical conditions). At the third stage the development of a computer program for the implementation of the algorithm is carried out, and then the actual calculations are carried out on the computer. Finally, at the final stage, the calculation results are processed, which are subjected to a comprehensive analysis.

There are many models in the literature: explanatory and descriptive (descriptive), theoretical and empirical, algebraic and qualitative, general and partial, a-priori and a-posteriori models, dynamic and static, extended and limited, simulation and experimental, deterministic and stochastic, semantic and syntactic.

The use of mathematical methods in historical research has certain specifics.

Most of the works related to the use of mathematical methods in historical research use the statistical processing of data from historical sources. But in the 1980s there was an improvement in the methodology of historical research, which made it possible to proceed to the second stage - the construction of mathematical models of historical processes and phenomena.

In the works of I.D. Kovalchenko proposed a typology of models of historical processes and phenomena, including reflective-measuring And imitation models 8 . The researcher singles out two stages of modeling (essential-content and formal-quantitative), noting that quantitative modeling consists in a formalized expression of a qualitative model by means of various mathematical means 9 .

Reflective-measuring models represent the studied reality as it was in reality, revealing and analyzing statistical relationships in the system of indicators characterizing the object under study. The purpose of simulation models is to reconstruct the missing data on the dynamics of the process under study over a certain time interval. Here it is possible to analyze the alternatives of historical development and theoretical study of the behavior of the studied phenomenon (or class of phenomena) according to the constructed mathematical model. There are two types of simulation models: imitation-counterfactual And imitation-alternative models of historical processes.

Usually counterfactual modeling is associated with the arbitrary reshaping of historical reality, but, on the other hand, it can be an effective tool for exploring alternative historical situations. This is where analytical and simulation models come into play. The former are characterized by the recording of the processes of functioning of the system under consideration in the form of functional relations (equations). simulation The models reproduce the studied process itself in its functioning in time. At the same time, elementary phenomena are simulated with the preservation of their logical structure and sequence of flow in time. With the help of a modeling algorithm, based on the initial data on the initial state of the process (input information) and its parameters, it is possible to obtain information about the states of the process at each subsequent step. The advantage of simulation models compared to analytical ones is that they provide the possibility of modeling very complex processes (with a large number of variables, nonlinear dependencies, feedbacks) that are not amenable to analytical research. The main disadvantage of simulation modeling is the fact that the resulting solution (the dynamics of the process being modeled) is always of a particular nature, responding to fixed values ​​of the system parameters, input information and initial conditions.

Considerable attention in modeling is attracted by the problems verification models of historical and social processes; at the same time, for many mathematical and simulation models, the parameters are fixed a priori, while in statistical models the parameters are estimated from the data that verifies this model.

The decision on the use of mathematical, statistical or simulation modeling to build a theory depends on the nature and volume of the available initial data.

Table 1

Comparison of Three Dynamics Modeling Approaches 10