Cover 1
Half-title 3
Title 5
Copyright 6
Contents 7
Preface 11
1 Introduction 13
1.1. WHAT\u2019S IN THE BOOK? 13
1.2. SOME EXAMPLES OF FORMAL AND INFORMAL THINKING 14
1.3. A BIT OF HISTORY 16
1.4. HOW BIG IS THE EARTH? ERATOSTHENES\u2019 SOLUTION 17
1.5. A CRITIQUE OF ERATOSTHENES 24
1.6. APPLICATIONS OF MATHEMATICS TO SOCIAL AND BEHAVIORAL ISSUES 26
1.7. STATISTICS 28
2 Applying Probability Theory to Problems in Sociology and Psychology 30
2.1. INTRODUCTION 30
2.2. DEFINING PROBABILITY AND PROBABILITY MEASURES 31
2.3. HOW CLOSELY CONNECTED ARE WE? 35
2.4. CONSCIOUS AND UNCONSCIOUS MEMORIES 39
2.5. SOME FINAL COMMENTS 44
APPENDIX 2A. THE BASIS FOR KOLMOGOROV\u2019S AXIOMS 44
APPENDIX 2B. SOME IMPORTANT PROPERTIES OF PROBABILITY MEASURES 45
Joint, Conditional, and Complementary Probability 45
Probability Distributions 48
Standard Scores 50
The Normal (Gaussian) Distribution 51
3 From Physics to Perception 54
3.1. THE PSYCHOPHYSICAL PROBLEM 54
3.2. WEBER\u2019S LAW 56
3.3. FECHNER\u2019S LAW 59
3.4. STEVENS\u2019S SCALING TECHNIQUE: DERIVING THE PSYCHOPHYSICAL FUNCTION FROM MAGNITUDE ESTIMATION 65
3.5. JUDGING COMPLEX OBJECTS 73
3.6. A COMMENT ON MEASUREMENT 77
4 When Systems Evolve over Time 79
4.1. SYSTEMS OF VARIABLES 79
4.2. DIFFERENCES AND DIFFERENTIATION 80
Derivatives and Difference Equations 81
4.3. EXPONENTIAL GROWTH AND DECAY 82
4.4. NUMERICAL ANALYSIS: THE TRANSMISSION OF JOKES AND COLDS 88
Exploring the Model 90
4.5. QUESTIONS ABOUT MODELING 93
Challenge Problem: How Should an Influenza Vaccine Be Distributed? 97
4.6. GRAPHICAL ANALYSIS: THE EVOLUTION OF WAR AND PEACE 98
Historical Note 98
Richardson\u2019s Hostility Model (Expanded) 99
Acceptable Levels of Armament 102
The Burden of Arms 102
Challenge Problem: The General Conditions for System Stability 106
4.7. MAKING LOVE, NOT WAR: THE GOTTMAN-MURRAY MODEL OF MARITAL INTERACTIONS 108
Challenge Question: What Happens When Polyanna Meets the Grinch? 113
4.8. CONCLUDING COMMENTS ON MODELING SIMPLE SYSTEMS 113
APPENDIX 4A. A PROOF OF THE EXPONENTIAL GROWTH EQUATION 115
5 Non-linear and Chaotic Systems 116
5.1. CONTINUOUS CHANGE AND SUDDEN JUMPS 116
5.2. THE LOTKA-VOLTERRA MODEL OF PREDATOR AND PREY INTERACTIONS 118
Challenge Questions 123
5.3. THE LOGISTIC EQUATION: INTRODUCTION AND BEHAVIOR WHEN K<1 123
5.4. NON-ZERO ASYMPTOTES AND CYCLES AS K INCREASES 128
5.5. CHAOS 133
5.6. CHAOS AND NETWORK MODELS 135
5.7. CLOSING COMMENTS ON CHAOS 142
6 Defining Rationality 144
6.1. AXIOMATIC REASONING 144
6.2. DECISION MAKING UNDER RISK 145
6.3. THE CONCEPT OF UTILITY 147
6.4. VON NEUMANN AND MORGENSTERN\u2019S AXIOMATIC APPROACH TO DECISION MAKING 151
6.5. THE UTILITY OF MONEY 155
6.6. A SUMMARY OF THE ARGUMENT 160
6.7. PSYCHOLOGICAL RESEARCH ON DECISION MAKING 163
The Allais Paradox 164
Prospect Theory 165
Some Disturbing Observations from the Field 169
6.8. THE PROBLEM OF VOTING 170
6.9. DEFINITION AND NOTATION 173
6.10. ARROW\u2019S AXIOMS: THE RESTRICTIONS ON SOCIAL WELFARE FUNCTIONS 174
6.11. ILLUSTRATION OF THE DEFINITIONS AND CONCEPTS FOR THE THREE-PERSON SOCIETY 176
6.12. A PROOF OF ARROW\u2019S THEOREM 178
6.13. COMMENTARY ON THE IMPLICATIONS OF ARROW\u2019S THEOREM 185
6.14. SUMMARY COMMENTS AND QUESTIONS ABOUT AXIOMATIC REASONING 186
7 How to Evaluate Evidence 188
7.1. THE LEGACY OF REVEREND BAYES 188
7.2. BAYES\u2019 THEOREM 190
7.3. SOME NUMERICAL EXAMPLES 192
Further Reasoning about Silver Blaze 193
Medical Diagnosis 194
7.4. CALCULATING THE ODDS 196
7.5. SOME EXAMPLES OF SIGNAL DETECTION 197
7.6. A MATHEMATICAL FORMULATION OF THE SIGNAL DETECTION PROBLEM 199
Definitions and Assumptions 200
The Basic Signal Detection Situation 202
7.7. THE DECISION ANALYST\u2019S PROBLEM 203
Step 1 \u2013 Identifying the Criterion 203
Step 2 \u2013 Analyzing the Receiver Operating Characteristic 206
7.8. A NUMERICAL EXAMPLE OF ROC ANALYSIS 211
7.9. ESTABLISHING A CRITERION 215
7.10. EXAMPLES 219
Example: The Attorney\u2019s Problem 219
Example: Detecting Potential Spies 220
Example: Diagnostic Radiology 222
7.11. FOUR CHALLENGE PROBLEMS 225
Challenge Problem 1: Racial Profiling 225
Challenge Problem 2: Profiling in Homeland Security Issues 226
Challenge Problem 3: A Last Look at Homeland Security 227
Challenge Problem 4: A Return to Mathematics 227
8 Multidimensional Scaling 228
8.1. THE BASIC IDEA 228
8.2. STEPS AND TECHNIQUE 231
8.3. EXTENSIONS TO NON-GEOMETRIC DATA 234
8.4. EXTENDING THE IDEA TO CONCEPTUAL CLASSES 235
8.5. GENERALIZATIONS OF SEMANTIC SPACE MODELS 239
8.6. QUALIFICATIONS ON THE SEMANTIC SPACE MODEL 241
Challenge Problem 242
9 The Mathematical Models Behind Psychological Testing 243
9.1. INTRODUCTION 243
9.2. A BRIEF REVIEW OF CORRELATION AND COVARIANCE 246
9.3. PREDICTING ONE VARIABLE FROM ANOTHER: LINEAR REGRESSION 252
9.4. THE SINGLE FACTOR MODEL: THE CASE OF GENERAL INTELLIGENCE 256
9.5. MULTIFACTOR THEORIES OF INTELLIGENCE AND PERSONALITY 261
9.6. GEOMETRIC AND GRAPHIC INTERPRETATIONS 266
9.7. WHAT SORT OF RESULTS ARE OBTAINED? 267
APPENDIX 9A. A MATRIX ALGEBRA PRESENTATION OF FACTOR ANALYSIS 268
10 How to Know You Asked a Good Question 271
10.1. THE PROBLEM 271
10.2. AN ILLUSTRATIVE CASE: VOCABULARY TESTING 272
10.3. THE BASICS OF ITEM RESPONSE THEORY 274
10.4. STANDARDIZATION: ESTIMATING ITEM AND PERSON PARAMETERS SIMULTANEOUSLY 277
10.5. THE APPLICATION PHASE: ADAPTIVE TESTING 279
10.6. MORE COMPLICATED IRT MODELS 281
10.7. MATHEMATICS MEETS THE SOCIAL WORLD: MATHEMATICAL ISSUES AND SOCIAL RELEVANCE 284
APPENDIX 10A. THE ADAPTIVE TESTING ALGORITHM 286
APPENDIX 10B. AN EXERCISE IN ADAPTIVE TESTING 287
11 The Construction of Complexity 289
11.1. SOME GRAND THEMES 289
11.2. THE PROBLEM OF COMPLEXITY 290
11.3. CELLULAR AUTOMATA CAN CREATE COMPLICATED CONSTRUCTIONS 293
11.4. IS CAPITALISM INHERENTLY UNFAIR? RECONSTRUCTING A SIMPLE MARKET ECONOMY 295
11.5. RESIDENTIAL SEGREGATION, GENOCIDE, AND THE USEFULNESS OF THE POLICE 301
11.6. IS THIS A NEW KIND OF SCIENCE? 306
12 Connectionism 309
12.1. THE BRAIN AND THE MIND 309
12.2. COMPUTATION AT THE NEURAL LEVEL 311
12.3. COMPUTATIONS AT THE NETWORK LEVEL 315
12.4. A PHILOSOPHICAL ASIDE 319
12.5. CONNECTIONIST ARCHITECTURES 321
12.6. SIMULATING A PHENOMENON IN VISUAL RECOGNITION: THE INTERACTIVE ACTIVATION MODEL 323
12.7. AN ARTIFICIAL INTELLIGENCE APPROACH TO LEARNING 325
12.8. A BIOLOGICAL APPROACH TO LEARNING: THE HEBBIAN ALGORITHM 331
12.9. THE AUTO-ASSOCIATOR 333
12.10. A FINAL WORD 336
13 L\u2019Envoi 337
References 340
Index of Names 345
Index of Subjects 349