Mathematics behavior

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