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Cognitive Constructivism and the Epistemic Significance Of Cognitive Constructivism and the Epistemic Significance of Sharp Statistical Hypotheses in Natural Sciences Julio Michael Stern IME-USP Institute of Mathematics and Statistics of the University of S~aoPaulo Version 2.31, May 01, 2013. arXiv:1006.5471v8 [stat.OT] 26 Oct 2020 2 3 A` Marisa e a nossos filhos, Rafael, Ana Carolina, e Deborah. 4 5 \Remanso de rio largo, viola da solid~ao: Quando vou p'ra dar batalha, convido meu cora¸c~ao." Gentle backwater of wide river, fiddle to solitude: When going to do battle, I invite my heart. Jo~aoGuimar~aesRosa (1908-1967). Grande Sert~ao,Veredas. \Sert~ao´eonde o homem tem de ter a dura nuca e a m~aoquadrada. (Onde quem manda ´eforte, com astucia e com cilada.) Mas onde ´ebobice a qualquer resposta, ´ea´ıque a pergunta se pergunta." \A gente vive repetido, o repetido... Digo: o real n~aoest´ana sa´ıdanem na chegada: ele se disp~oempara a gente ´eno meio da travessia." Sertao is where a man's might must prevail, where he has to be strong, smart and wise. But there, where any answer is wrong, there is where the question asks itself. We live repeating the reapeated... I say: the real is neither at the departure nor at the arrival: It presents itself to us at the middle of the journey. 6 Contents Preface 13 Cognitive Constructivism . 14 Basic Tools for the (Home) Works . 16 Acknowledgements and Final Remarks . 16 1 Eigen-Solutions and Sharp Statistical Hypotheses 19 1.1 Introduction . 19 1.2 Autopoiesis and Eigen-Solutions . 20 1.3 Functional Differentiation . 24 1.4 Eigensolutions and Scientific Hypotheses . 26 1.5 Sharp Statistical Hypotheses . 27 1.6 Semantic Degradation . 34 1.7 Competing Sharp Hypotheses . 35 1.8 Final Remarks . 37 1.8.1 Noether and de Finetti Theorems . 38 1.8.2 Compositionality . 38 2 Language and the Self-Reference Paradox 41 2.1 Introduction . 41 2.2 Eigen-solutions and Language . 44 2.3 The Languages of Science . 47 2.4 The Self-Reference Paradox . 50 2.5 Objective Idealism and Pragmatism . 52 2.6 The Philosophy of C.S.Peirce . 54 7 8 CONTENTS 2.7 Final Remarks . 62 3 Decoupling, Randomization, Sparsity, and Objective Inference 65 3.1 Introduction . 65 3.2 The Decoupling Principle . 67 3.3 Covariance Structure Models . 70 3.4 Simpson's Paradox and the Control of Confounding Variables . 73 3.5 C.S.Peirce and Randomization . 76 3.6 Bayesian Analysis of Randomization . 78 3.7 Randomization, Epistemic Considerations . 80 3.8 Final Remarks . 83 4 Metaphor and Metaphysics: The Subjective Side of Science 85 4.1 Introduction . 85 4.2 Statistics is Prediction. Is that all there is? . 87 4.3 Rhetoric and Self-Fulfilling Prophecies . 88 4.4 Language, Metaphor and Insight . 91 4.5 Constructive Ontology and Metaphysics . 95 4.6 Necessary and Best Worlds . 102 4.7 Efficient and Final Causes . 104 4.8 Modern Metaphysics . 107 4.9 Averaging over All Possible Worlds . 111 4.10 Hypothetical versus Factual Models . 114 4.11 Magic, Miracles and Final Remarks . 117 5 Complex Structures, Modularity, and Stochastic Evolution 121 5.1 Introduction . 121 5.2 The Ergodic Path: One for All . 122 5.2.1 Block Partitions . 123 5.2.2 Simulated Annealing . 125 5.2.3 Heuristic Acceleration . 128 5.3 The Way of Sex: All for One . 129 CONTENTS 9 5.3.1 Functional Trees . 130 5.3.2 Genetic Programming . 133 5.3.3 Schemata and Parallelism . 135 5.4 Simple Life: Small is Beautiful . 136 5.4.1 Overfitting and Regularization . 137 5.4.2 Building Blocks and Modularity . 140 5.5 Evolution of Theories . 143 5.5.1 Nested, Mixture, Separate and Series-Parallel Models . 144 5.5.2 Complementary Models . 148 5.6 Varieties of Probability . 151 5.6.1 Heisenberg's Uncertainty Principle . 151 5.6.2 Schr¨odinger's Wave Equation . 154 5.6.3 Classic and Quantum Probability . 156 5.7 Theories of Evolution . 159 5.7.1 Systemic View and Probabilistic Causation . 160 5.7.2 Modularity Requires Quantization . 164 5.7.3 Quantization Entails Objective Probability . 168 5.8 Final Remarks . 170 6 The Living and Intelligent Universe 171 6.1 The Origami Example . 173 6.2 Autopoietic Control, Precision, Stability . 175 6.2.1 Organic Morphogenesis: (Un)Folding Symmetries . 175 6.3 Object Orientation and Code Reuse . 177 6.3.1 Doing, Listening and Answering . 178 6.3.2 Mnemes, Memes, Mimes, and all that. 182 6.4 Hypercyclic Bootstrapping . 183 6.4.1 Bootstrapping Paradoxes . 185 6.5 Squaring the Cycle . 188 6.6 Emergence and Asymptotics . 190 6.7 Constructive Ontologies . 193 10 CONTENTS 6.8 Distinctions and Probability . 196 6.8.1 Spencer-Brown, Probability and Statistics . 198 6.8.2 Overcoming Dilemmas and Conceptual Dichotomies . 200 6.9 Final Remarks and Future Research . 202 Epilog 205 References 209 A FBST Review 245 A.1 Introduction . 245 A.2 Bayesian Statistical Models . 246 A.3 The Epistemic e-values . 248 A.4 Reference, Invariance and Consistency . 251 A.5 Loss Functions . 254 A.6 Belief Calculi and Support Structures . 255 A.7 Sensitivity and Inconsistency . 257 A.8 Complex Models and Compositionality . 260 B Binomial, Dirichlet, Poisson and Related Distributions 263 B.1 Introduction and Notation . 263 B.2 The Bernoulli Process . 264 B.3 Multinomial Distribution . 269 B.4 Multivariate Hypergeometric Distribution . 271 B.5 Dirichlet Distribution . 273 B.6 Dirichlet-Multinomial . 277 B.7 Dirichlet of the Second Kind . 280 B.8 Examples . 281 B.9 Functional Characterizations . 285 B.10 Final Remarks . 287 C Model Miscellanea 289 CONTENTS 11 C.1 Contingency Table Models . 290 C.2 Weibull Wearout Model . 291 C.3 The Normal-Wishart Distribution . 294 C.4 Structural Models . 296 C.4.1 Mean and Covariance Structure . 297 C.4.2 Numerical Optimization . 298 C.5 Factor Analysis . 299 C.5.1 EM Algorithm . 300 C.5.2 Orthogonal and Oblique Rotations . 301 C.5.3 Frobenius Norm and Projections . 302 C.5.4 Sparsity Optimization . 303 C.6 Mixture Models . 305 C.6.1 Dirichlet-Normal-Wishart Mixtures . 306 C.6.2 Gibbs Sampling and Integration . 308 C.6.3 EM Algorithm for ML and MAP Estimation . ..
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