Conflict, Complexity and Mathematical Social Science: Volume 15

Peace is as old as mankind. The gospel of peace and harmony is inherent in all religions and expressed through philosophy, literature, art, and music. However, in contemporary academic disciplines, peace has been addressed differently. In sociology, individual and group conflicts that also involve psychological conflicts have been a matter of discussion for years. For the last two or three decades, international, interregional, and intraregional conflicts have received a prominent place. The study of conflict in economics is a recent phenomenon. In management, industrial relations and labor relations cover this topic through contract negotiations, mediation, and arbitration. Extensive studies are now available in many applied areas such as energy, water resources, and the environment.

Set theory provides a foundational approach to mathematics, and mathematics provides an abstract way of looking at social reality. The first section presents some of the elementary concepts of set theory. The second section presents a variety of examples of social reality and shows how the abstract features of reality can be modelled by set theory. The third section shows how set theory can provide a way of looking at the accounts of social reality presented in humanities disciplines such as history and literature. The fourth section briefly indicates how set theory and the concept of a structure provide a foundational approach to mathematics. The fifth section looks at the debates surrounding realism and, albeit warily, espouses mathematical social science realism.

In this section we shall give a brief account of the elementary concepts of set theory – elementary in the sense that they are concepts which students might meet in the first week of a mathematics foundation course at a university; and elementary also in the sense that everything else in mathematics is dependent on them. The concepts are at once both extremely simple and extremely abstract.

In his discussion of early lexical development in children Barrett (1995) notes that utterances can be of the following types: an expression of an internal state, a response to a specific context, a social-pragmatic utterance and a referential utterance. A referential utterance can be thought of in the following way: the thing referred to, a mental representation, a word representation and a word sound. An utterance may refer to an object, an action, an attribute or an event. Some utterances are used as the names of classes of objects while other utterances are used as the proper names of individual objects. Looking at this in abstract we might say that, in the early years of childhood, language is used to refer to elements, sets, functions and relations – in other words to the mathematical structures which were discussed in Chapter 2. Of course although early language is used to refer to mathematical objects, the character of the language itself takes the form of ordinary language.

Consider now the set of all possible events that can occur in a given context. There is a distinction between an elementary event and a compound event. The set of elementary events is exhaustive, exclusive and elementary: the elementary events cover all the possible events; no two of them can occur at the same time; and all other events are constituted by compounds of these. Denoting the set of all elementary events by E, the set of all (possibly compound) events is the power set of E, S ^E. The set of events, S ^E, consists of pairs of events: for each event e there is its complementary event not-e; and for the event not-e there is its complementary event not-(not-e)=e. In any given world only one event of any complementary pair can occur.

Are the statements which we make true?…and, if they are true, how do we know that they are true? To address these two questions let us first note that this book is interested in two types of truth: mathematical truth and scientific truth. Mathematical truth applies to statements within a mathematical theory. A statement is true within a theory either if it is one of the axioms of the theory or if it can be deduced from the axioms of the theory (see Chapter 3). Scientific truth applies to statements about the real world. A statement about the real world is true if it corresponds to what happens in the real world. A theory about the real world is true if all of its statements correspond to what happens in the real world. Given a mathematical theory which is consistent (i.e. true within itself) or a specific statement which is true within the theory, we can enquire whether or not the theory or statement is true in relation to reality.

According to Kolm (1998, p. 3), social ethics addresses the question ‘what should be done in society?’ The topic of justice constitutes a very large part of social ethics although other virtues are also important. Kolm distinguishes between macro-justice and micro-justice. For the former, Kolm proposes ‘a combination of the three rationales of rights and duties about capacities: process-freedom, partial income equalisation by efficient means, and the satisfaction of basic needs and the alleviation of deep suffering’. Sen (1992, pp. ix, 21–22, 150) argues that ‘a common characteristic of virtually all the approaches to the ethics of social arrangements that have stood the test of time is to want equality of something – something that has an important place in the particular theory’. For example, even libertarian thinkers such as Nozick who are perceived as being anti-egalitarian place importance on people having liberty and hence that equality of liberties is important. Sen's own capability approach ‘has something to offer both to the evaluation of well-being and to the assessment of freedom’.

I am interested in a set X of entities and refer to this as a multiple-entity system unless the set contains just a single entity. Each entity can be characterised either by a single attribute or by many attributes. In general, then we have a system of n entities with m attributes, giving nm attributes in all. A model of a system usually focuses on the variables associated with the attributes. So a model for a unitary entity with nm attributes, a model for a system of nm entities each with just one attribute and a model for a system of n entities with m attributes may be all formally identical with one another.

The subject matter of psychology can be illustrated by addressing the questions, ‘who am I?’, ‘what do I spend my time doing?’ and ‘reflecting on my life, where have I come from and where am I going?’. I am a human being and also more generally an animal. I am a biological entity and also a psychological entity. I have an identity and a personality. I am an individual and also a member of a social system. I am similar to other human beings but also different. In some respects I am normal and in other respects abnormal.

In his soliloquy, Hamlet reasons about the choice of whether to end his life or not. Our own experience of choice contains many such instances of choice as the outcome of a reasoning process. This aspect of choice is not discussed in this book although I would look to Chapter 3 on mathematics, logic, artificial intelligence and ordinary language to provide a route into investigating this aspect.

The Wikipedia (2008) entry for mathematical sociology cites four books with ‘mathematical sociology’ in the title: Coleman (1964), Fararo (1973), Leik and Meeker (1975) and Bonacich (2008). Fararo (1973, pp. 764–766) provides a guide to the literature in mathematical sociology covering journals, bibliographies, reviews and expository essays, readers, texts, original monographs and research papers. Many of the references are either broader than mathematical sociology, for example, concerning the behavioural sciences in general, or narrower, dealing with a particular topic within sociology, or concerning a related field such as social psychology. Three classical original monographs are identified: Dodd (1942), Zipf (1949) and Rashevsky (1951). Included in a second generation of monographs is Coleman's (1964) ‘An Introduction to Mathematical Sociology’. Could it be that this is the first use of the phrase ‘mathematical sociology’?

DefinitionsA game consists of a set of players I, a pure-strategy space S, a value space V and a value function f from the strategy space to the value space.

The aim of this section is to develop a model of the linkage between the macro-dynamics of price and the micro-dynamics of individual buyers and sellers, drawing on classical micro-economic theory (Jehle & Reny, 2001). Jehle and Reny's book is in three parts. The first part discusses economic agents, namely consumers and firms. The second part discusses markets, in other words what happens when the economic agents interact; and welfare, namely the social value of the outcome. The third part is on strategic behaviour, covering game theory, information economics and auctions and mechanism design. Here, the emphasis will be on the core concepts rather than on the mathematical details. In emphasising the core concepts, attention will be drawn to the fact that these core concepts have a much wider range than simply micro-economics.

Taken all together, how would you say things are these days – would you say that you are very happy, pretty happy or not very happy? (USA General Social Surveys Question 157)On the whole, are you very satisfied, fairly satisfied, or not at all satisfied with the life you lead? (Eurobarometer Survey Series)

Our physical universe is 1.5×10¹⁰ years old. It began with the Big Bang. There is some debate about what happened in the first tenth of a second! The first 3×10⁵ years were radiation dominated. Since then it has been matter dominated. (This in accordance with the first law of thermodynamics which states that total mass-energy is conserved.) The universe has continuously expanded in space and in the future either this may continue, or expansion may stabilise at a fixed size or the universe may contract in the Big Crunch (depending on the spatial curvature). At a certain scale the universe is spatially isotropic and homogeneous. Its trajectory exhibits increasing entropy in accordance with the second law of thermodynamics. These statements are in accordance with certain models and empirical data: distant galaxies are receding from us at a velocity proportional to their distance; there is greater spatial uniformity at greater distances from us; there is uniform presence in space of radiation with a temperature of 2.7K; etc.

The mathematical science approach to the study of social affairs has been much debated not least among scholars of international relations. Wight (2002, p. 37) reviews the current debate – discussing the views of Michael Nicholson and Steve Smith quite extensively – and comments:all of this adds up to a very confused picture in terms of the philosophy of science. IR has struggled to incorporate an increasingly diverse set of positions into its theoretical landscape. In general, the discipline has attempted to maintain an unsophisticated and outdated two-category framework based on the science/anti-science issue.…Currently there are three continuums that the discipline seems to consider line up in opposition to each other. The first of these is the explaining/understanding divide (Hollis & Smith, 1990). The second is the positivism/post-positivism divide (Lapid, 1989; Sylvester, 1993). The third is Keohane's distinction between rationalism and reflectivism (Keohane, 1989). The newly emerging constructivism claims ‘the middle ground’ in between. (Adler, 1997; Price & Reus-Smit, 1998; Wendt, 1999)