# Introduction. History and Mathematics

For centuries there have been disputes over whether mathematics could and should be applied to the humanities. On the one hand, most scientists believe that an academic discipline only acquires the status of a "normal science" when it systematically applies mathematical and other formal methods. From this scientific perspective many fields of historical research look "underdeveloped". On the other hand, in recent decades, postmodernism and other fashionable intellectual currents have questioned the status of history as a science, even though positivism had seemed to provide historians with a sound scientific basis for developing rigorous methods for searching for, analyzing and validating historical data. Debates over the interpretation of narratives and discourses have tended to delineate and thus limit historical research to the role of a type of artistic fiction. We believe that the development and application of mathematical methods for the purpose of analyzing and modeling historical dynamics could effectively counteract attempts to eliminate the scientific and objective character of historical research.

Of course, the famous typology proposed by Wilhelm Windelband (1848--1915) and Heinrich Rickert (1863--1936) retains its value. These two scholars proposed to divide the academic research into *nomothetic* and *idiographic* branches. According to them, sciences that seek to discover general laws for indefinitely repeatable events and processes are *nomothetic*; in contrast, disciplines which are *idiographic* describe what is particular and non-recurrent. Rickert considered history an idiographic science, as, according to him, its aim was not to generalize but to analyze specific historical processes and events; historical studies should deal with the particular, so selection of facts is inevitable and such selection cannot but rest on value-assumptions (see, *e.g.*, Rickert 1986). However, it is hardly useful to divide the sciences so dichotomously and declare that a field of science only belongs to one or the other camp. Nothing is gained by building a "Berlin Wall" between the sciences and humanities.

One of the main aims of our almanac is to counteract the absolutist approach to separating the humanities and sciences. Its goal is not to unite artificially all the sciences and humanities, it is rather to contribute to the building of bridges between them. It does not make sense to deny the differences in the subject and methods between natural sciences and the humanities. But this does not mean that cooperation between these disciplines is not possible.

When we named this almanac *History and Mathematics*, we meant both history and mathematics in the wide sense of these words; its main subject is the study of socio-historical processes by using a wide range of mathematical methods.

The first attempts to apply mathematics to the analysis of phenomena that seemed to be rather far from the "World of Numbers" were undertaken long ago. Developments in both mathematics and various scientific disciplines led to the application of mathematical methods to the study of more and more fields. But the effect was not just in one direction -- from mathematics to other sciences; developments in the other sciences also spurred the branches of mathematics to develop new mathematical methods for dealing with new types of research data and questions. One may suppose that new branches of mathematics might develop in the future in order to solve various new problems in mathematical history. For example, Georgy Malinetsky (Малинецкий 1996: 102) suggests that the development of mathematical history might result in the development of an original new mathematical apparatus, as this has already happened with respect to mathematical economics and psychology.

Rapid computerization both poses additional problems and creates new opportunities for the application of mathematical methods to the various fields of the humanities. Presently, we have a considerable number of publications on the development of electronic historical databases; there also exist various computerized information systems aimed at the storage, ordering, processing, and analysis of historical data (see, *e.g.*, Boonstra, Breure, and Doorn 2004; Borodkin, Thaller, and Turner 1995; Denley 1994; McCrank 2002; Speck 1994; Thaller 1989, 1992; Woollard 1999). One can observe a wide application of mathematical statistics, methods of multidimensional clustering and classification, including those based on the fuzzy sets' theory, to historical research. Computer technologies are widely used in archaeology, *e.g.* to visualize three-dimensional models of excavations, or to analyze stratigraphic data; they are applied to the study of historical sources (*e.g.*, for text identification, or information processing); they are also widely used in the classroom on the basis of network and multimedia software systems, and so on. A very promising direction is the application of geo-information technologies in historical research; for example, these technologies have made it possible to visualize spatial models through the integration of various datasets. On the other hand, the development of the general systems theory, cybernetics, and complexity studies has made it possible to identify a number of new aspects of socio-historical phenomena that can be formalized and studied with the application of mathematical methods.

It appears necessary to emphasize that the role of mathematics in the study of human history should not be restricted to pure "cliometrics", that is to the development of computer databases, systematization and quantification of historical data, as well as to the mathematical analysis of these databases. Mathematics can also be used for the development of theoretical interpretations of human history, for historical modeling and forecasting. For example, in Russia this process started in the 1980s with the development of mathematical models of concrete historical processes (see, *e.g.*, Иванилов, Огарышев, Павловский 1993). In recent years we observe the development of a number of more general mathematical models that describe non-linear dynamics of agrarian and industrial societies, processes of social self-organization (Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Nefedov 2004; Podlazov 2004; Tsirel 2004; Korotayev, Malkov, and Khaltourina 2006a, 2006b; Korotayev and Khaltourina 2006).

Of course, the development of the mathematical models of historical processes confronts a number of problems that are connected with the following points:

*etc.*

However, notwithstanding all the above mentioned difficulties, interest in the mathematical modeling of historical processes is constantly growing.

This first issue of the *History & Mathematics* almanac in English^ is devoted to the application of mathematical methods to the analysis and modeling of the global historical processes.

Within this almanac two groups of articles can be singled out. The first deals with the periodization and modeling of global development (articles by *Leonid Grinin*, *Andrey Korotayev*, *Akop P.Nazaretyan*, and *Arno Tausch*). The second applies mathematical methods to reconstructing the deep history of global development (articles by *Michael L.Burton*, *A.Kimball Romney*, *Carmella C.Moore*, and *Natalia L.Komarova*).

Within the first group, two articles (by *Leonid Grinin* and *Andrey Korotayev*) apply mathematical methods to problems concerning the macro-periodization of the history of the World System (Korotayev's article) and the world historical process as a whole (Grinin's article). *Akop P.Nazaretyan* studies the long-term evolution of social violence and the social means used to limit it, while also touching upon problems of periodization. The authors of this almanac study the problems associated with periodization from different perspectives (*i.e.*, technological and economic in Grinin's article; demographic and macrosocial in Korotayev's; cultural-psychological and technological in Nazaretyan's). However, all agree that the mathematical modeling of historical macroprocesses suggests a fresh approach to the periodization issue.

Tausch presents new quantitative insights on the dynamics of the contemporary development and, with a complete database for 140 nations, proceeds to an analysis of the determinants of economic growth and ecological and social development in these nations, which allows him to make a number of interesing forecasts. Articles by Grinin, Korotayev, and Nazaretyan also contain a number of important forecasts.

Ken Boulding (1970) was not only joking when he maintained that the interpretation of history is a dangerous business. There are grounds to hope that the application of mathematical methods may diminish this "danger". In any case, the application of these methods allows researchers to perform deep historical reconstruction more effectively than when purely qualitative methods are used. We believe that this conclusion is confirmed by the two articles previously identified as belonging to the second group.

Both of these articles deal with the evolution of human languages, but from very different perspectives. In the first of these articles, Komarova proposes a mathematical model of the formation and development of human language. In the second article, Burton, Romney, and Moore apply statistical analysis to both linguistic and cross-cultural anthropological data in order to reconstruct some important features of the long-term evolution of human social organization.

Finally, we would like to state that there is one additional positive point in the application of mathematical methods by the specialists in the humanities. The point is that they contribute to their scientific self-discipline, by providing means for rigorous verification/falsification of their hypotheses. The introduction of these methods could diminish the current extreme polysemanticism of the terminology used in the humanities. Rudolf Carnap (1966) noted long ago that even in physics the use of words from natural languages (like *law*) leads to problems for accurate monosemantic expression of one's ideas. However, physicists have come to develop and use effective conventions for the meaning of basic terms. In the humanities the basic terms still have numerous contradictory definitions. We believe that the introduction of mathematical methods to the field can somehow alleviate this problem due to the univocacy of mathematical language.

The articles published in this almanac have been prepared on the basis of papers submitted for the presentation at the 1^{st} International Conference "History and Mathematics" (Russian State University for the Humanities, Moscow, December 20--22, 2006).