Andrey Korotayev

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History & Mathematics: Processes and Models of Global Dynamics 2010 5 –12 




History & Mathematics:  

Processes and Models of  

Global Dynamics



Leonid Grinin, Andrey Korotayev, Peter Herrmann  


Multidisciplinarity is one of the salient features of contemporary science. This 

seems to be congruent with the globalization process as the globalized world 

will need a "global" science that is able to integrate and to unite various fields 

in order to solve fundamental problems. It may be said that, in some sense, 

the History & Mathematics almanac is "genetically" interdisciplinary as it was 

initially designed as a means to contribute to the construction of a bridge be-

tween the humanities, social, natural, and mathematical sciences (see the Intro-

duction to its first Russian issue [Гринин,  Коротаев,  Малков 2006: 4–11]). 

That time this very combination of words – History and Mathematics – might 

have looked a bit artificial. However, it gradually becomes habitual; what is 

more, it becomes to be recognized as quite an organic and important scientific 

phenomenon. This appears to be supported by the point that the recent two 

years have evidenced the publication of eight issues of the History & Mathe-

matics almanac in Russian and two issues in English.


 Various conferences in 

this direction are held now quite regularly, and, what is especially promising, 

they bring together representatives of very diverse fields of human knowledge. 

One of the most recent conferences of this kind was held in December 2009 in 

the Institute of History and Archaeology (Ekaterinburg, Russia). The confer-

ence has confirmed the existence of a critical mass of researchers within the 

world science that apply mathematical and quantitative methods to the study of 

history. Against this background the current discussions on the establishment 

of the Mathematical History academic journal do not appear coincidental.  

The present issue is the third collective monograph in the series started by 

almanacs  History & Mathematics: Analysing and Modeling Global Develop-

ment (Grinin, de Munck, and Korotayev 2006) and History & Mathematics: 

Historical Dynamics and Development of Complex Societies (Turchin, Grinin,  

de Munck, and Korotayev 2006). As one can see, every issue has its own subti-

tle. This issue is not an exception. Its subtitle is Processes and Models of 

Global Dynamics.  



 See Bibliography at the end of this Introduction.  


The notion of "process" in one of its most wide-spread meanings denotes 

a certain sequence of states or phases in the change/development of something. 

The more ordered this sequence, the deeper its understanding, the higher  

the probability of the respective process to be described mathematically in 

a successful way, to model it (and, frequently, to use it in practical activities). 

That is why the constant and profound interest of our almanac in various proc-

esses is quite explicable (especially as regards major long-term processes). It is 

evidenced in particular by the subtitles of the first and the second English al-

manacs (see above).  

Ashby (1958) notices that the class of systems is enormously wide, the class 

of processes can be well compared to the one of systems. However, within the 

present almanac we are naturally dealing first of all with social and historical 

processes. The ideas that social life is somehow connected with certain proc-

esses appeared already in antiquity – for example, ideas of constant regression 

(e.g., from the Golden to Iron Age), or ideas of cyclical processes. The first 

more or less scientific historical theories were connected with the analysis of 

such cyclical process – here one can recollect theories of Polybius, and later 

Ibn Khaldūn (see, e.g., Ibn Khaldūn 1958; Turchin 2003; Korotayev and 

Khaltourina 2006; Гринин 2010), Machiavelli, or Vico (see, e.g.,  Гринин 

2010). Theories of progress that appeared in the 18


 century and flourished in 

the 19


 century were also based on an idea of some naturally determined proc-

ess, a process of constant and endless enhancement (see, e.g., Turgot 1795 

[1766]; Condorcet 1970 [1795]). However, the first social processes, to whose 

study mathematical methods started to be applied, were economic and demo-

graphic ones (see, e.g., Борисов 2005).  

The representation of a process in a form of a model implies a rather pro-

found understanding of its nature, the possibility to identify regularities de-

scribing its course and (sometimes) to forecast it. That is why it is quite natural 

that models are present in every issue of History & Mathematics.  

Note that the notion of "model" is used to denote a rather wide class of phe-

nomena (Wartofsky 1979; Новик,  Садовский 1988: 450). Models (and their 

presentation and analysis belong to the main directions of our almanac) are fre-

quently interdisciplinary by their own nature – not least as they are still clearly 

linked to action, in the meaning of an active character of their elements. 

Mathematical models constitute just a subset of a wider set of models, but mathe-

matical models are also extremely diverse. And our almanac is open to any of 

such models. Yet, we still prefer explanatory models that are capable of inden-

tifying causes of a particular phenomenology. In addition, our almanac is based 

on the trends of modern post-non-classical science (see, e.g.,  Степин 2000); 

we try to present such models that take into account both regular processes and 

such processes that go beyond the framework of repeating phenomena, such 

models that describe non-linear processes, chaos, phase transitions, stochastic 

dynamics, etc.  

Leonid Grinin, Andrey Korotayev, Peter Herrmann 

It was in very distant epochs that philosophers and thinkers tried to embrace 

the whole universe with a single idea; however, we feel that there is still a need 

for conceptually and unifyingly organizing our knowledge. As justly noticed by 

Erwin Schrödinger (1944), it has become almost impossible for a single mind to 

master more than one small specialized part of the science, but someone should 

still risk to attempt to synthesize facts and theories. The use of models for such 

generalizations facilitates significantly the respective task, as it helps to achieve 

(as is formulated by Hermann Haken [2006]) an enormous degree of the com-

pression of information. Thus, we have tried to combine in the subtitle of 

the present issue of the almanac the most important aspects of our interests.  

The contributions to the present issue study processes within very diverse 

spheres of social life. However, the name of our almanac is History & Mathe-

matics, and it appears reasonable to consider the possible quantitative basis for 

an apparently unique chain of events, as history is frequently viewed. The fin-

est shades of sounds and light can be reduced at a certain level to unified mate-

rial-energetic substances/structures; similarly, historical events, processes, and 

phenomena that are apparently entirely different with respect to their time-

scales, uniqueness, novelty, and significance have some common foundations 

(and what is important, those foundations may sometimes be quantified). We 

believe that such a common foundation of sociohistorical phenomena is consti-

tuted by the temporal nature of any historical events (see Гринин,  Коротаев 

2008 for more detail). But this means also that the historical time perspective is 

one of at least medium, more likely long-term scales (see Braudel's longue 

durée). Note that temporal vectors have the same units of measurement and are 

characterized by unidirectionality, whereas the latter gives to social change a 

character of process, as any process is a directional current of changes. Thus, 

we speak about some types of directionality in history, as the time itself is di-

rectional; in addition, historical processes are usually ordered by some causal 

logic, various positive or negative feedback loops of the first, second, or higher 

orders,  etc. Due to such reasons, history is studied more and more just as a 

process, or, to be more exact, as a system of various processes, within which 

one can, for example, detect "waves", or "cycles" with various periods ranging 

from a few years to hundreds, or even thousands of years (see, e.g., the contri-

bution of Grinin, Korotayev, and Malkov to this volume, or Grinin 2006a, 

2006b, 2007a, 2007b; Korotayev and Khaltourina 2006; Korotayev, Malkov, 

and Khaltourina 2006; Korotayev and Tsirel 2010).  

We believe that scales, durations, degrees of orderliness/stochasticity of re-

spective changes, their prevalence, the degree to which they are known to us, 

etc. determine to a considerable extent our conclusions about the very nature of 

historical development – whether it should be considered as deterministic, or 

stochastic; linear, or non-linear; cyclical; liable to bifurcations or so on. 

The more regularities can be found in various event series, the easier it is to de-

tect fundamental similarities in various historical and social processes.  


This almanac considers processes of various durations and with diverse 

characteristics. We have grouped them into three sections.  


*   *   * 


The first section (titled Analyses of the World Systems and Global Proc-

esses) starts with Tony Harper's article "The Trajectory of the World System 

over the Last 5000 Years" where he quantitatively delineates the non-random 

trajectory of the World System over the last 5000 years. A mathematical model 

is used to characterize the relationship between maximum urban area size and 

the total population of the World System at century intervals and is predicated 

on urban areas having a Pareto distribution. The trajectory exhibits two distinct 

characteristics, that of periods of oscillation punctuated by periods of continu-

ous, directed change. At any century the position of the World System can be 

represented by the log-transform of F = αC – γ, and it can be shown that 

changes in this position are brought about by changes in either lnα or lnC



Also, it is shown that the trajectory is most affected by changes in the exponent, γ

Further, the World System trajectory also exhibits cyclical behavior. Other 

characteristics of the trajectory are also investigated.  

Christopher Chase-Dunn, Richard Niemeyer, Alexis Alvarez, Hiroko Inoue, 

Kirk Lawrence, and James Love present their article "Cycles of Rise and Fall, 

Upsweeps and Collapses: Changes in the scale of settlements and polities since 

the Bronze Age". This paper uses estimates of the sizes of settlements and poli-

ties to examine patterns that need to be understood in order to explain the 

growing scale of human socio-cultural institutions. All systems of interacting 

polities oscillate between relatively greater and lesser centralization as rela-

tively large polities rise and fall. This is true of systems of chiefdoms, states, 

empires and the modern system of the rise and fall of hegemonic core states. 

But there has also been a long-term trend in which polities have increased in 

population and territorial size since the Stone Age and the total number of polities 

has decreased. These trends have been somewhat masked in recent centuries be-

cause the processes of decolonization and the emergence of nation-states out of 

older tributary empires have increased the number of smaller polities. But the 

general trend toward larger polities can be seen in the transition from smaller to 

larger hegemonic core states (from the Dutch to the British and to the United 

States), and in the emergence of international political organizations and an ex-

panded and active global civil society that participates in contemporary world 


The Lisbon performance of the countries of the European Union is analyzed 

from a long-term, Kornai structural perspective by Arno Tausch, Almas Hesh-

mati, and Chemen S. J. Bajalan in their article "On the Multivariate Analysis of 

the ‘Lisbon Process’".  The international team of authors presents in a simple 

form the mathematical methods used in this essay. Then, they analyze Lisbon 

indicator performance by factor analytical means. Tausch, Heshmati, and Ba-

Leonid Grinin, Andrey Korotayev, Peter Herrmann 

jalan conclude that only a Schumpeterian vision of capitalism as a process of 

"creative destruction" (or rather "destructive creation"?) can explain these con-

tradictions, which they empirically reveal in this analysis, and which beset 

the "Lisbon process" from the very beginning. This factor analysis tells us that 

a majority of the kernel Lisbon indicators go indeed hand in hand with high com-

parative price levels; high transport costs; high greenhouse gas emissions; low 

business investment rates; and low youth educational attainment rates. The au-

thors conclude that in reality we are facing four underlying and contradictory 

processes including a Lisbon productivity factor; high eco-social exclusion; the 

employment performance; and the neo-liberal European model. 


*   *   * 


The second section (titled The Models of Economic and Demographic Proc-

esses) starts with an article by Leonid Grinin, Andrey Korotayev, and Sergey 

Malkov "A Mathematical Model of Juglar Cycles and the Current Global Cri-

sis". The article presents a verbal and mathematical model of medium-term 

business cycles (with a characteristic period of 7–11 years) known as Juglar 

cycles. The model takes into account a number of approaches to the analysis of 

such cycles; in the meantime it also takes into account some of the authors' own 

generalizations and additions that are important for understanding the internal 

logic of the cycle, its variability and its peculiarities in the present-time condi-

tions. The authors argue that the most important cause of cyclical crises stems 

from strong structural disproportions that develop during economic booms. 

These are not only disproportions between different economic sectors, but also 

disproportions between different societal subsystems; at present these are also 

disproportions within the World System as a whole.  

The proposed model of business cycle is based on its subdivision into four 


– recovery phase (which could be subdivided into the start sub-phase and 

the acceleration sub-phase); 

– upswing/prosperity/expansion phase (which could be subdivided into 

the growth sub-phase and the boom/overheating sub-phase); 

– recession phase (within which one may single out the crash/bust/acute cri-

sis sub-phase and the downswing sub-phase);  

– depression/stagnation phase (which could be subdivided into the stabiliza-

tion sub-phase and the breakthrough sub-phase).  

The article provides a detailed qualitative description of macroeconomic 

dynamics at all the phases; it specifies driving forces of cyclical dynamics and 

the causes of transition from one phase to another (including psychological 

causes); a special attention is paid to the turning point from the peak of overheat-

ing to the acute crisis, as well as to the turning point from the downswing to re-




The proposed mathematical model of Juglar cycle takes into account the 

following effects that are typical for the market economy:  


positive feedbacks between various economic processes;  


presence of a certain inertia, time lags in reactions of the economic sub-

system to the change in conditions; 


amplification by the financial subsystem of positive feedbacks and time 

lags in the economic subsystem; 


excessive reaction to changing conditions during the acute crisis sub-


The authors suggest that the current crisis turns out to be rather similar to 

classical Juglar crises; however, there is also a significant difference, as the cur-

rent crisis occurs at a truly global scale. Yet, due to this truly global scale of the 

current crisis, the possibilities of regulation with the national state's measures 

have turned out to be ineffective, whereas the suprastate regulation of financial 

processes hardly exists. It is shown that all these have led to the reproduction of 

the current crisis according to a classical Juglar scenario. 

Michael Golosovsky presents an article titled "Hyperbolic Growth of the Hu-

man Population of the Earth: Analysis of existing models". This work focuses 

on 1) demographic problems arising from the growing human population of 

the Earth and 2) the quantitative estimates of the future growth of the Earth's 

population. The author discusses the existing models of the global human 

population growth using a popular presentation level and without appealing to 

so-phisticated mathematical language. Instead of proposing a new mathemati-

cal model of the population growth, Golosovsky advances a new perspective 

for the mathematical modeling: phase transitions which are well-know in phys-

ics. In particular, he demonstrates that the world's demographic transition is ac-

tually a phase transition that has been affecting all aspects of our life. 

"A Dynamic Model of Historical Economies" by Lucy Badalian and Victor 

Krivorotov presents the concept of history as domestication of sequential geo-

climatic zones, with clear boundaries, unique domesticated animals/plants, 

a dominant energy source. A zone's social institutions form a system of feeding 

chains uniquely adapted to its conditions. The respective mathematical model 

presents historical development as a fundamentally nonlinear process – imbal-

ances start a chain of events. In this context, globalization presents a compensa-

tion for exhausted resources of the initial zone, especially, energy/food. How-

ever, the entry of periphery-suppliers is hardly seamless. Their substantial dif-

ferences lead to unique local adaptations. Wars or conflicts, such as the 1870–

1871 French-Prussian war or the ongoing Middle Eastern conflicts, signal of 

rising tensions in the search of new solutions. Below, historical examples show-

case the wavelike process of imbalance generation – development in new places 

of signals of the exhaustion of the growth potential in the older zone. This proc-

ess periodically pushes toward domesticating the next geoclimatic zone. 

Leonid Grinin, Andrey Korotayev, Peter Herrmann 


*   *   * 


The third section (titled Сultural Dimensions) consists of an article "The Self is 

not Culture: toward a unified theory of self, identity and culture" by Victor C. 

de Munck who proposes a new theory of culture that focuses on cultural models 

and their relationship to the self and identity. The purpose of this theory is to 

explicate the linkages between what goes on in the mind of individuals with 

a notion of cultures as shared and distributed knowledge systems. The author 

builds on earlier theories of cultural models by bringing the "self" back as an 

agent that triggers the use of cultural models. He argues that the function of 

the self is to give "I-ness" to various identities. Identities in turn are formed 

through the historical interactions in particular contexts. These interactions are 

mediated through cultural models. The self activates identities that are con-

structed and evoked by different kinds of context which we refer to as eco-

niches for identities. Drawing on Simmel's work on the emergent socio-

psychological properties of small groups, Victor C. de Munck posits that there 

are three different superordinate categories of identity which are referred to as 

ego-niches. There are many different kinds of basic level identities within each 

superordinate category and these draw on a menu of cultural models which are, 

in turn, used to construct contingent schemas which are then used to generate 

behavior. The author's theory proposes a necessary synthesis of sociocultural 

and psychological processes to develop a theory of the relationship between 

cognition and action. 



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l'esprit humain. Paris: Vrin.  

Grinin, L. E. 2006a. Periodization of History: A Theoretic-Mathematical Analysis. History 

and Mathematics: Analyzing and Modeling Global Development / Ed. by L. E. Grinin, 

V. de Munck, and A. V. Korotayev, pp. 10–38. Moscow: KomKniga/URSS.  

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