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:
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 1st International Conference "History and Mathematics" (Russian State University for the Humanities, Moscow, December 20--22, 2006).
Michael L.Burton is Professor of Anthropology at the University of California, Irvine. Current research interests include the origin of human systems of gender and family relationships; the histories of language families; the origins and development of food production systems, current transformations in systems of food production, consumption, and exchange; and the use of quantitative methods in the field of anthropology. Publications include "Sexual Division of Labor in Agriculture" (American Anthropologist 1984, with Douglas R.White), "Regions Based on Social Structure" (Current Anthropology 1996, with John Whiting, Carmella Moore, and A.K.Romney), and "Language and Region Codes for the Standard Cross-Cultural Sample" (Cross-Cultural Research 1999). Email: MLBurton@UCI.edu.
Leonid Grinin is a senior research fellow of the Volgograd Center for Social Research, a vice-editor of the journals History and Modernity and Philosophy and Society, and a co-editor of the Social Evolution & History. Current research interests include the long-term trends in the evolution of technologies and their influences on sociocultural evolution, periodization of history, and long-term development of the political systems. He is author of over 100 scholarly publications, including such books as "Philosophy, Sociology, and Theory of History", "Productive Forces and Historical Process", Formations and Civilizations", "State and Historical Process". Among his more important journal articles are "The Early State and Its Analogues" (Social Evolution & History 1: 131--176), "Democracy and Early State" (Social Evolution & History 3: 93--149), "Early State and Democracy" (in The Early State, Its Alternatives and Analogues, pp.419--463. Volgograd: Uchitel), and "The Early State and Its Analogues: A Comparative Analysis" (in The Early State, Its Alternatives and Analogues, pp.88--136. Volgograd: Uchitel). Email: LGrinin@mail.ru.
Natalia Komarova is an Associate Professor of Mathematics, University of California, Irvine, USA. Her research interests lie at the interface between the mathematical and life sciences. The two main topics are mathematical biology (modeling of initiation and development of cancer, biophysics and virus dynamics) and mathematical models of language evolution (with elements of learning theory and historical dynamics). Natalia Komarova received the 2002 Prize for Promise and a 2005 Sloan Fellowship. She co-authored the book Computational Biology of Cancer: Lecture notes and mathematical modeling (with D.Wodarz; Singapore: World Scientific Publishing, 2005) and published over 50 scientific articles, including "Oscillations in population sizes -- from ecology to history" (Structure and Dynamics, 1, 2006), "The Evolution of Altruism: from Game Theory to Human Language" (Spiritual Information: 100 perspectives, West Conshohocken, PA: Templeton Foundation Press, 2005, co-authored), "Replicator-Mutator Equation, Universality Property and Population Dynamics of Learning" (Journal of Theoretical Biology, 230, 2004), "Language Dynamics in Finite Populations" (Journal of Theoretical Biology, 221, 2003, co-authored), "Language, Learning and Evolution" (Language Evolution: The States of the Art, Oxford: Oxford University Press, 2003, co-authored), "Evolution of Universal Grammar" (Science, 291, 2001, co-authored). Email: Komarova@UCI.edu.
Andrey Korotayev is Director and Professor of the "Anthropology of the East" Center, Russian State University for the Humanities, Moscow, as well as Senior Research Fellow of the Institute for Oriental Studies and the Institute for African Studies of the Russian Academy of Sciences. He is a laureate of the Russian Science Support Foundation Award in "The Best Economists of the Russian Academy of Sciences" nomination (2006). He is author of over 250 scholarly publications, including Ancient Yemen (Oxford University Press, 1995), Pre-Islamic Yemen (Harrassowitz Verlag, 1996), Social Evolution (Nauka, 2003), World Religions and Social Evolution of the Old World Oikumene Civilizations: a Cross-Cultural Perspective (Mellen, 2004), Origins of Islam (OGI, 2006, with Vladimir Klimenko and Dmitri Proussakov). Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth (URSS, 2006, with Artemy Malkov and Daria Khaltourina), Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends (URSS, 2006, with Artemy Malkov and Daria Khaltourina), and Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends and Africa (URSS, 2006, with Daria Khaltourina). Email: AKorotayev@mail.ru.
Carmella C.Moore is Associate Research Anthropologist at the University of California, Irvine. Her research has been in the fields of medical anthropology, psychological anthropology, and cross-cultural research methods. Her publications include The Psychology of Cultural Experience (2001 with Holly F.Matthews), "Material Culture, Geographic Propinquity, and Linguistic Affiliation on the North Coast of New Guinea" (1994 with A.K.Romney) "Regions Based on Social Structure" (Current Anthropology 1996, with Michael L.Burton, John Whiting, and A.K.Romney), "The Contribution of Medical Anthropology to the Comparative Study of Culture" (Medical Anthropology Quarterly 2001, with Arthur J.Rubel), "The Universality of the Semantic Structure of Emotion Terms" (American Anthropologist 1999, with A.K.Romney, Ti-Lien Hsia, and Craig D.Rusch), and "Methods for the Study of Inter-and intra-cultural variability" (American Anthropologist 1999, with A.K.Romney). Email: CCMoore@Uci.edu.
Akop P.Nazartyan is Senior Research Fellow of the Oriental Institute, Russian Academy of Sciences and Professor of the Moscow State University. He is author of over 250 scholarly publications, including Aggression, Morals and the Crises in World Cultural development. (Synergetics of historical progress) (Moscow: Nasledie, 1996, in Russian), Aggressive Crowds, Mass Panic, and Rumors. Lectures in Social and Political Psychology (St.Petersburg: Piter, 2003, in Russian), Civilization Crises within the Context of Universal History. Self-Organization, Psychology, and Forecasts (Moscow: Mir, 2004, in Russian), "Fear of the dead as a factor in social self-organization" (Journal for the Theory of Social Behaviour 35 : 155--169), and "Western and Russian traditions of Big History" (Journal for General Philosophy of Science 36: 63--80). Email: ANazaret@yandex.ru.
A.Kimball Romney is Research Professor of Anthropology at the University of California, Irvine and a member of the National Academy of Sciences of the United States. He has made important research contributions to the cross-cultural study of socialization, to cognitive anthropology, to the use application of multidimensional scaling models to anthropology, to the study of social networks, and to the field of cross-cultural research. His books include The Mixtecans of Juxtlahuaca, Mexico (1965), Systematic Data Collection (1988 with Susan Weller), and Metric Scaling (1990, with Susan Weller). Important articles include "Cognitive Aspects of English kin Terms (American Anthropologist 1964, with Roy G.D' Andrade), "Sibling Terminology and Cross-sex Behavior" (American Anthropologist 1967 with Sara B.Nerlove), "Culture as Consensus" (American Anthropologist 1986 with Susan Weller and William Batchelder), "Material Culture, Geographic Propinquity, and Linguistic Affiliation on the North Coast of New Guinea" (Current Anthropology 1994 with Carmella Moore), "Regions Based on Social Structure" (Current Anthropology 1996, with Michael L.Burton, John Whiting, and Carmella Moore), and "Methods for the Study of Inter-and intra-cultural variability" (American Anthropologist 1999 with Carmella C.Moore). His current research is primarily focused on the study of color perception. Email: AKRomney@UCI.edu.
Arno Tausch is in his academic function Adjunct Professor (Universitaetsdozent) of Political Science at Innsbruck University, Department of Political Science. In his academic career, he was also Associate Visiting Professor, Department of Political Science, University of Hawaii at Manoa, and Guest Researcher, International Institute for Comparative Social Research, Science Center, West Berlin, upon invitation by the late Karl Wolfgang Deutsch, Stanfield Professor of International Peace at Harvard University. He served as an Austrian diplomat abroad and was Counselor for Labor and Migration at the Austrian Embassy in Warsaw and he is now Ministerial Counselor in the Department of European and International Affairs at the Ministry for Social Security, Generations and Consumer Protection in Vienna. His research program is focused on world systems studies, development and dependency studies, European studies, and quantitative peace research. He authored or co-authored 10 books in English, 5 books in German, and over 110 printed or electronic scholarly and current affairs publications in 6 languages (English, Finnish, French, German, Russian and Spanish) for over 40 journals and/or publishing institutions. Email: Arno.Tausch@bmsg.gv.at.