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Topos Modeling of Social Conflict: Theory and Methods
Abstract
Category theory and, more specifically, topos categories provide a more expressive type of mathematical modeling and, thereby, open the door to social models that are both rigorous and expressive. The present analysis draws upon four views of topos categories to construct a rich model of a logic based on recognition theory. The topos initially provides support for: (1) set theory (and classical logic), (2) topological regions dualism. The resulting structures support: (3) a local intuitionist logic that can vary by situated circumstances and actor types, and (4) the specification of regions that are defined by finely-differentiated classifiers. These tools construct an integrated topos category of social recognition that supports diverse forms of local logic. The latter are then explored regarding their contributions to a mathematical model of social conflict. Categorial analysis of historical patterns, as well as a complementary simulation model, is used to illustrate the advantages of such an approach.
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The application of computational models to historical and policy-oriented issues highlights the extent to which advances in the theory of social action are needed. Social science theory, while insightful, is qualitative, fragmented and redundant, and often lacks a determinate analytical framework. The present paper seeks to strengthen social theory as a means of making theory-based analysis a more viable alternative. More specifically, such an analysis undertakes 1) to synthesize existing theory in order to articulate it at a higher level of abstraction, and 2) to use categorical mathematics as a means of accomplishing this synthesis. The paper concludes with a discussion of how such theoretical advances can contribute to more effective social models.
Computational social science (CSS) has created innovative forms of modeling and analysis. However, now CSS appears to be approaching a barrier in the form of a complexity ceiling that imposes constraints throughout the modeling lifecycle, from design to validation. Given the precedent of the physical sciences, a 'natural' solution is to reformulate the domain at ever-higher levels of mathematical abstraction. In this pattern, an appropriate mathematical formalism is a prerequisite to establishing a foundation for CSS. The present analysis makes the case that category theory can provide an effective mathematical framework capable of representing social complexities.
This article presents a new theory of war that is grounded in the insights of Clausewitz on the social nature of conflict. Clausewitz had argued that war is a political process; he therefore distinguished between ‘war’—understood in political terms—and warfare—understood as fighting. He then created a typology covering a spectrum of war ranging from total to limited, the political stakes of a conflict determining where it would fall on the spectrum. I develop and modify this basic framework by arguing that the social organization of the actors has a determining role in predicting the stakes of war. I then show how this framework helps us understand some key problems in the political science literature on war and conflict. I attempt to show two main things: (1) that there are different types of wars (and that these differences are not necessarily related to the standing of the actors, i.e. the presence or absence of sovereignty); and (2) that how war and warfare are related is more complicated than previously understood and that this has implications for the political science literature on order, conflict and violence.
Exploration and exploitation have emerged as the twin concepts underpinning organizational adaptation research, yet some central issues related to them remain ambiguous. We address four related questions here: What do exploration and exploitation mean? Are they two ends of a continuum or orthogonal to each other? How should organizations achieve balance between exploration and exploitation-via ambidexterity or punctuated equilibrium? Finally, must all organizations strive for a balance, or is specialization in exploitation or exploration sometimes sufficient for long-run success? We summarize the contributions of the work in this special research forum and highlight important directions for future research.
Randall Collins has introduced a simplified model of emotional dynamics in which emotional energy, heightened and focused by interaction rituals, serves as a common denominator for social exchange: a generic form of currency, except that it is active in a far broader range of social transactions. While the scope of this theory is attractive, the specifics of the model remain unconvincing.
After a critical assessment of the currency theory of emotion, a field model of emotion is introduced that adds expressiveness by locating emotional valence within its cognitive context, thereby creating an integrated orientation field. The result is a model which claims less in the way of motivational specificity, but is more satisfactory in modeling the dynamic interaction between cognitive and emotional orientations at both individual and social levels.
It is useful to be able to define a two-dimensional point-set surface determined by a point cloud. One popular definition is Levin's MLS surface. This surface is defined on a domain which is a three-dimensional subset of R 3 , a narrow region around the input point cloud. If we were to extend the definition outside the domain, we would produce components of the surface which are far from the point cloud. This is important in practice, since when moving points onto the MLS surface, we need to begin with an initial guess which is within the domain. We visualize the domain in two dimensions, and explain why it is so narrow. We also consider two MLS variants which can be defined on a wider domain without producing spurious surface components. One is efficient and works well except near sharp corners. The other is computationally expensive but seems to work well everywhere.
A Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory. After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various 'refinements' of the fundamental concepts of category and functor.
This chapter discusses the problems and pitfalls of introducing new technology into an organization. In particular, it focuses upon the problems which can arise when introducing CASE tools and the methods and approaches which accompany them into a new environment.
This book argues that Clausewitz developed a wide-ranging political theory of war by reflecting not only on the success, but also on the limitations and the failure of Napoleon's method of waging war. Successes, failures, and defeats of Napoleon forced Clausewitz to reflect on questions that went beyond purely military matters and led him to develop a political theory of war. Analyses of three paradigmatic campaigns - which are symbolized by Jena, Moscow, and Waterloo - are placed at the center of understanding surrounding Clausewitz's On War and it's inherent problems, like the relation of absolute and real war, theory and practice, the primacy of politics, the enigma of the first chapter, and the dialectics of attack and defense. Clausewitz's analysis of these war campaigns are the cornerstones of On War and redounded to the 'wondrous trinity', which is the basis for a general theory of war and which is quite different from the so-called trinitarian war.
This book sheds light on Clausewitz's purpose in writing On War, and the methodology he employed. While many scholars agree that Clausewitz's work is frequently misunderstood, almost none have explored his methodology to see whether it might enhance our understanding of his concepts. This book lays out Clausewitz's purpose and methodology in a brisk and straightforward style. It then uses that as a basis for understanding his contributions to the ever growing body of knowledge of war. The specific contributions this study addresses are Clausewitz's theories concerning the nature of war, the relationship between war and politics, and several of the major principles of strategy he examined. These theories and principles lie at the heart of the current debates over the nature of contemporary conflict. They also underpin much of the instruction that prepares military and civilian leaders for their roles in the development and execution of military strategy. This book is organized into three parts. The first provides students with background information concerning Clausewitz's purpose and method. The second lays out his theories regarding the nature of war, his ideas concerning the relationship between war and policy, and the complementary balance between friction and military genius. The final part examines his concept of strategy, and several of his strategic principles, particularly the centre of gravity, and reveals how they relate to contemporary war. Together, these themes represent the core of what professional military curricula usually cover of Clausewitz.
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Is functional analysis inherently static? An answer to this question requires a precise definition of such analysis. A functional system is defined as several variables which compensate for each other's variation, thus maintaining some property of the system. Four ways in which change is incorporated in functional systems are specified and some of the major arguments which attempt to demonstrate the static nature of functional analysis are discussed and rejected. Finally, two examples of functional analysis of change are presented: E. R. Leach's analysis of changing political systems in Burma, and Parsons and Smelser's analysis of the differentiation of ownership from control in the American economy.
This paper tries to show that both functionalism and the Hegelian-Marxian dialectic present one-sided, but complementary and reconcilable, views of society. The basic postulates of functionalism and the dialectic reveal strengths and limitations which suggest a reformulation of both approaches in minimal form. Basically, the problem is to arrive at a theory of society that achieves an adequate balance between stability and the various sources of endogenous and exogenous change, between consensus and conflict, and between equilibrium and disequilibrium. After examining Dahrendorf's efforts in that direction, we conclude that convergence and overlap between the two theories show promise of a fruitful synthesis.
Introduction.- Philosophical Motivations.- Kochen-Specker Theorem.- Introducing Category Theory.- Functors.- The Category of Functors.- Topos.- Topos of Presheaves.- Topos Analogue of the State Space.- Topos Analogue of Propositions.- Topos Analogues of States.- Truth Values.- Quantity Value Object and Physical Quantities.- Sheaves.- Probabilities in Topos Quantum theory.- Group Action in Topos Quantum Theory.- Topos History Quantum Theory.- Normal Operators.- KMS States.- Future Research.- Topos and Logic.- Worked out Examples.
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A solution is proposed to several long-standing problems in the theory of rational social action: emotional or altruistic behavior that escapes cost/benefit calculations; the lack of a common metric among different spheres of action; and naturalistic evidence that choice in real-life situations involves little calculation. Emotional, symbolic, and value-oriented behavior is determined by a social mechanism, the dynamics of interaction rituals (IRs). Because IRs vary in the amount of solidarity they provide, and in their costs of participating, there is a market for ritual participation that shapes the distribution of individual behavior IRs generate a variable level of emotional energy (EE) in each individual over time, and EE operates as the common denominator in terms of which choices are made among alternative courses of action. Individuals apportion their investments in work and in ritual participation to maximize their overall flow of EE. The economy of participating in interaction rituals shapes individual motivation for participating in the economy of material goods and services. Microsituational cognition is determined by the EE and the cognitive symbols generated by IRs, bringing about the tendency to narrow the range of alternatives that are consciously focused upon in choice situations. Nevertheless, the aggregation of microsituations is subject to interactional markets, which gives individuals a rational trajectory in the medium-run drift of behavior.
Theories of social problems and deviant behavior are normative; they interpret facts within the context of ideological assumptions about the nature of man and society. Normative theories can be classified as variants of two ideal types- order and conflict theories. As an exercise in the use of these models, American sociological approaches to the Negro question are examined with the following conclusions: (1) they are similar to politically liberal interpretations; (2) both sociological and liberal approaches to the race question represent conservative variants of order theory.
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In this paper we present a categorization method for systems research and show a categorical treatment of general systems using this method. This method yields a unified framework to categorize systems classes and their associated systemic concepts.The basic idea of the method is that, first, the structure of a object of concern is represented by a diagram and then the functor category is defined from the diagram category to a category which represents associated properties of the object and finally the desired categorization is given by a subcategory of the functor category.This paper categorizes several important classes of general systems by this method and expresses relationships between the classes by functors. Then, the concept of state and that of a state space representation are discussed categorically as an important application of the method. The results of the paper shows that this method can be a basis of the categorical theory of general systems.
Proposes a model of social systems based on general systems research, cybernetics, and information and communication theory. Contains reviews and critiques of important sociological approaches. Harvard Book List (edited) 1971 #677 (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Examines the correlation between the exploration of new possibilities and the exploitation of old certainties in organizational learning. Also discusses the difficulty in balancing resource management between gaining new information about alternatives to improve future returns (i.e., exploration) and using information currently available to improve present returns (i.e., exploitation). Two models which evaluate the formation and use of knowledge in organizations are developed. The first is a model of mutual learning in a closed system having fixed organizational membership and stability. The second is a model which considers the ways in which competitive advantage is affected by knowledge accumulation. The analysis indicates that the choice to rapidly develop exploitation over exploration might be effective in the short term, but is potentially detrimental to the firm in the long term. (SFL)
The dominant strategy among game theorists is to pose a problem narrowly, formalize that structure, and then pursue analytical
solutions. This strategy has achieved a number of stylized insights, but has not produced nuanced game-theoretic solutions
to larger and more complex issues such as extended international historical conflicts, or the detailed assessment of variegated
policy alternatives. In order to model more complex historical and policy-oriented processes, it has been proposed that a
broader computational approach to game theory that has the potential to capture richer forms of social dynamics be used, namely
the ‘multigame.’ In the multigame approach there are multiple games each of which is open, prototypical, implicit, reciprocal,
positional, variegated and historical. When later implemented, the multigame approach will offer the potential to rigorously
model complex international historical conflicts and variegated policy alternatives that, heretofore, typically required qualitative
analysis.
The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis.
Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories.
Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts.
Contents Tutorial • Applications • Further Reading
A simple theory of power relations is developed in an effort to resolve some of the ambiguities surrounding "power," "authority," "legitimacy," and power "structures," through bringing them together in a coherent scheme. After defining a reciprocal power-dependence relation, attention is focused upon properties of balance and "balancing operations" in such relations. The theory dictates exactly four generic types of balancing process, and discussion of these leads directly into processes of group formation, including the emergence of group norms, role structure and status hierarchy, all presented as the outcome of balancing tendencies in power relations. Within the framework of this theory, authority appears quite naturally to be legitimized power, vested in roles, and "legitimation" is seen as a special case of the coalition process through which norms and role-prescriptions are formed. Finally, through treating both persons and groups as actors in a power-network (two or more connected power-dependence relations) the door is opened for meaningful analysis of complex power structures. Brief reference is made to findings from two experiments pertaining to hypotheses advanced in this theory.
Translated from the German and edited by Edward A. Shils and Henry A. Finch.