Conflict, Complexity and Mathematical Social Science

287 Pages·2010·3.142 MB·English

by  Gordon Burt, Gordon Burt

CONFLICT, COMPLEXITY
AND MATHEMATICAL
SOCIAL SCIENCE
Contributions to
Conflict Management, Peace Economics
and Development Volume 15
SERIES EDITOR
MANAS CHATTERJI
BOOKS IN THE SERIES
Eurasia: A New Peace Agenda, edited by M. D. Intriligator, A. I. Nikitin and
M. Tehranian
Military Missions and Their Implications Reconsidered: The Aftermath of September 11th,
edited by G. Caforio and G. Kümmel
Managing Conflict in Economic Convergence of Regions in Greater Europe, by F. Carluer
Cultural Differences between the Military and Parent Society in Democratic Countries,
edited by G. Caforio
Conflict and Peace in South Asia, edited by M. Chatterji and B. M. Jain
War, Peace, and Security, edited by J. Fontanel and M. Chatterji
Armed Forces and Conflict Resolution, edited by G. Caforio, G. Kümmel and
B. Purkayastha
Regional Development and Conflict Management: A Case for Brazil, by R. Bar-El
Crisis, Complexity and Conflict, by I. J. Azis
Putting Teeth in the Tiger: Improving the Effectiveness of Arms Embargoes, edited by
M. Brzoska and G. A. Lopez
Peace Science: Theory and Cases, by P. Gangopadhyay and M. Chatterji
Advances in Military Sociology: Essays in Honor of Charles C. Moskos (Two Volume Set),
edited by Dr Giuseppe Caforio
Arms and Conflict in the Middle East, by Riad A. Attar
Economics of War and Peace: Economic, Legal, and Political Perspectives edited by
Benjamin E. Goldsmith and Jurgen Brauer
Contributions to
Conflict Management, Peace Economics
and Development Volume 15
CONFLICT, COMPLEXITY
AND MATHEMATICAL
SOCIAL SCIENCE
GORDON BURT
The Open University, UK
United Kingdom – North America – Japan
India – Malaysia – China
Emerald Group Publishing Limited
Howard House, Wagon Lane, Bingley BD16 1WA, UK
First edition 2010
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ISBN: 978-1-84950-972-5
ISSN: 1572-8323 (Series)
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CONTENTS
FOREWORD xiii
ACKNOWLEDGMENTS xv
CHAPTER 1 INTRODUCTION AND OVERVIEW 1
Overview of the Chapters 3
Set theory and social reality 3
Mathematics, logic, artificial intelligence and ordinary language 3
Possibility and probability: Value, conflict and choice 4
Theory, evidence and reality: The mean and median
ideals of competing groups 5
Social design, ethics and the amount of value 6
Change, multiple-entity systems and complexity 6
Mathematical psychology 7
Models of choice 8
Mathematical sociology 8
Mathematical political science and game theory 9
The mathematical economics of social participation: Complexity 10
Life and history: The speculative pursuit of value 10
World history: The growth and distribution dynamics
of power, truth and value 11
Debating the mathematical science approach to international
relations 12
CHAPTER 2 SET THEORY AND SOCIAL REALITY 13
The Elementary Concepts of Set Theory 13
Set Theory Models of Social Reality 15
Set Theory Models of History and Literature 19
Local history 21
Models of history: England in the fourteenth century 21
Set Theory, Structures and Mathematics 24
Reality and Realism 24
v
vi CONTENTS
CHAPTER 3 MATHEMATICS, LOGIC, ARTIFICIAL
INTELLIGENCE AND ORDINARY LANGUAGE 29
Ordinary Language 30
Early lexical development in children 30
Ordinary language and non-mathematical discourse 31
Truth-intending and non-truth-intending discourses 32
Meaning and precisionymeaninglessness and vagueness 33
Linguistics: The study of ordinary language 34
Mathematical Inquiry and Completed Mathematical Knowledge 35
The process of mathematical inquiry: An elementary
illustration 36
Completed mathematical knowledge: Concepts, propositions
and contexts 37
Completed mathematical knowledge: Proofs 39
Mathematical Logic 41
Propositional calculus 42
Predicate calculus 44
The relationship between a language and a reality 45
Artificial Intelligence 47
CHAPTER 4 POSSIBILITY AND PROBABILITY:
VALUE, CONFLICT AND CHOICE 51
Possibility and Probability 52
The set of all possibilities and the set of all propositions 52
Probability theory 54
Value, Conflict and Choice 55
Value and choice 55
Multidimensional value: The probability of consensus or
conflict 56
Social choice and social welfare 58
The condorcet majority principle, the de borda ranking
principle and the welfare principle 58
Arrow’s list of principles and his general impossibility theorem 60
The probability of things going wrong 60
Success Equals Power Plus Luck 61
The probability of success and power 62
Success versus decisiveness: Three voting rules for the european
council of ministers 63
Contents vii
CHAPTER 5 THEORY, EVIDENCE AND REALITY: THE
MEAN AND MEDIAN IDEALS OF COMPETING GROUPS 67
Theory, Evidence and Reality 68
A priori probability and empirical probability 70
Error and approximation 71
Evidence about reality 73
Continuous Spaces and Single-peaked or Euclidean Preferences 74
The median ideal and the mean ideal 74
Thedistributionofidealpoints:Representativedemocracyassocial
choice 76
The weighted mean ideal and power 77
The Median Ideals of Competing Groups 79
The legislative median and partisan policy 80
How does the overall median relate to the component medians? 82
CHAPTER 6 SOCIAL DESIGN, ETHICS AND THE
AMOUNT OF VALUE 87
Ethics and the Utilitarian Social Welfare Function 88
The Amount of Value 89
The individual and social value of objects and attributes 90
Limitations on the value of a social design 91
Social Value Functions on a Continuous Space 92
Single-peaked functions 93
Distance functions (symmetric about the peak) 94
The modulus function 94
The quadratic function 95
A Practical Application 98
Ten dimensions of educational design 98
One dimension 99
The results 99
The average student: What does the mean function look like? 100
Differences between students 101
Three different students 101
Students have different ideals 101
What do the individual utility functions look like? 102
Implications for educational design 103
Limitations on the mean value 103
The social tension in the best option 103
viii CONTENTS
How else might we define ‘best’? 104
To increase satisfaction, increase choice 104
CHAPTER 7 CHANGE, MULTIPLE-ENTITY
SYSTEMS AND COMPLEXITY 107
Single-entity, Multiple-entity and Composite Systems 108
Information Systems 109
A process as a simple sequence 110
Symbol processing and symbol processors 111
Generative processes 111
Behavioural Systems 112
The trajectories of a single entity in one-dimensional space 112
The trajectory of multiple interacting entities 114
Multiple entities with non-identical probabilities 116
Empirical evidence of within-group heterogeneity of
propensities 118
The most probable trajectory of a macro parameter 120
A toy example 121
Quantitative sociodynamics 122
CHAPTER 8 MATHEMATICAL PSYCHOLOGY 123
Psychology 124
Mathematical Psychology 124
Theory, Evidence and Reality 127
Measurement theory 127
Mathematics, methodology and statistics 129
Model selection 129
Models of individuals and models of sets of individuals 129
Models of Life and Models of Single Steps in Life 130
Perception and psychophysics 132
Perception of the duration of time 133
Weber’s law 133
The perception of colour 134
Imitation: Response reproduction 134
Psychometrics: Individuals, situations and events 134
Multi-dimensional individuals and situations 136
Naturalistic and controlled allocation of individuals to
situations 137
Contents ix
CHAPTER 9 MODELS OF INDIVIDUAL CHOICE 139
The Nature of the Choice Process 140
Choice as the outcome of a reasoning process 140
Choice as the outcome of a mathematical reasoning process 140
Inexact reasoning, heuristics and error 141
Choice: The establishment of the options and their value 141
Classical Models of Choice 143
Static, single-unit models 143
The yes-or-no threshold model 144
The probabilistic threshold model 144
The variable experience model 145
The three-state threshold model (one stimulus) 145
The three-state threshold model (two stimuli) 147
Signal detection theory 148
Hypothesis testing 151
Bayesian hypothesis testing 153
Statistical decision theory 154
More than two options: Classification into three classes 155
Static, multi-unit models 156
Dynamic, single-unit models 156
A threshold model of procrastination 156
Stochastic general recognition theory 157
Dynamic, multi-unit models 158
Limitations to the Value of Individual Choice 159
CHAPTER 10 MATHEMATICAL SOCIOLOGY 161
Background to Mathematical Sociology 162
People Participation in Social Activities 163
Life: A trajectory through social structure 163
Family and kinship 164
Leisure and holidays 166
Shopping 167
Work and the organisation: Rules or choice? 167
The classical economic versus behavioural theories
of the firm 167
Rationality, rules, religion or randomness? 168
A social process 169
Travel 169