The Reflection of Life: Functional Entailment and Imminence in Relational Biology (IFSR International Series in Systems Science and Systems Engineering Book 29)

Cover......Page 1 Le reflet de la vie......Page 8 Prerequisites......Page 12 Contents......Page 14 E.2 What Is Life?......Page 18 E.3 The Modelling Relation......Page 19 E.4 Biology Extends Physics......Page 20 E.5 Relational Biology......Page 21 E.6 Mapping and Its Relational Diagram......Page 22 E.7 Efficient Cause......Page 23 E.8 Compositions......Page 24 E.9 Impredicativity......Page 27 E.11 Biological Realization: Metabolism-Repair System......Page 28 E.12 Closure to Efficient Causation......Page 32 0.3 Definition......Page 34 0.6 Schröder-Bernstein Theorem......Page 35 0.10 Property......Page 36 0.15 Cardinality of the Power Set......Page 37 0.17 Indexed Family......Page 38 0.19 Indexed Partition......Page 39 0.21 Sequence......Page 40 0.22 Preorder......Page 41 0.23 Monotonic Sequence......Page 42 0.24 Subsequence......Page 43 0.25 Enumerating Sequence......Page 44 1.1 Definition......Page 45 1.5 Definition......Page 46 1.6 Notations......Page 47 1.8 Definition......Page 48 1.10 The Category Set......Page 49 1.14 Lemma......Page 50 1.18 The Power Set Functor......Page 51 1.19 Theorem......Page 52 1.21 Theorem......Page 53 1.24 Hardy’s Idea of a Mapping......Page 54 1.25 Well-Posed Problem......Page 55 1.26 Examples of Multi-valued Mappings......Page 56 Definition A......Page 58 2.2 Definition......Page 59 2.4 Definition......Page 60 2.5 The Constant Empty-Set-Valued Mapping......Page 61 2.7 Definition......Page 62 2.9 Definition......Page 63 2.11 Theorem......Page 64 2.14 Lemma......Page 65 Inverse Images......Page 66 2.16 Definition......Page 67 2.18 Corollary......Page 69 2.20 First Combination......Page 70 2.23 Theorem......Page 71 2.24 Lemma......Page 72 2.27 Theorem......Page 73 2.32 Theorem......Page 74 3.1 Four Causes in a Mapping......Page 75 3.3 Material Connections......Page 76 3.4 Definition......Page 77 3.5 Lemma......Page 78 3.9 Theorem......Page 79 3.11 The Category Rel......Page 80 3.12 Inverse Functor......Page 81 3.14 Definition......Page 82 3.16 Lemma......Page 83 3.18 Definition......Page 84 3.19 Idempotence......Page 85 3.21 Power Submonoid......Page 86 3.22 Example......Page 87 3.23 Operation Closure......Page 88 3.24 Closure Operator......Page 90 3.26 Lemma......Page 91 3.28 Lemma......Page 92 3.29 Commutation......Page 93 3.30 Eightfold Partition......Page 94 3.31 Partial Order......Page 95 3.34 Directed Graph......Page 96 3.36 Reachability......Page 97 3.37 Undirected Graph......Page 98 3.40 Connectedness......Page 99 4.1 The Symmetric Closure......Page 101 4.3 Definition......Page 102 4.4 Definition......Page 103 4.6 Self-Independence......Page 104 4.8 Theorem......Page 105 4.9 Theorem......Page 106 4.11 Definition......Page 107 4.13 Theorem......Page 108 4.15 Examples......Page 109 4.16 Theorem......Page 112 4.17 Another Example......Page 113 4.18 Counting Maximal Independent Sets......Page 114 5.1 Cartesian Product......Page 115 5.3 Sur-/In-/Bijective Mappings......Page 116 5.5 Sequential Composition......Page 117 5.7 Various Special Mappings......Page 118 5.8 Illustration......Page 119 5.10 The Poset of Relations......Page 120 5.11 Union and Intersection......Page 121 5.13 Square Product......Page 122 5.14 Illustration......Page 123 5.15 Various Special Mappings......Page 125 5.17 The Simplest (M,R)-Network......Page 127 5.19 ‘Inverse’......Page 128 5.21 Powers......Page 130 5.23 Reachability......Page 132 5.24 Connectedness......Page 133 5.25 Eightfold Partition......Page 134 5.26 Infinite Matrix......Page 135 6.1 The Logic of Entailment......Page 137 6.2 Infer Versus Imply......Page 138 6.4 Axioms of Relational Biology......Page 139 6.6 Arrow Differentiation......Page 140 Definition......Page 141 Definition......Page 143 6.9 Partitioning of......Page 144 6.10 Composition as Mapping......Page 145 6.11 Composition as Relay......Page 146 6.12 Exponential......Page 147 6.13 Hierarchical Composition......Page 148 6.14 Functional Entailment......Page 149 7.2 Theorem......Page 150 7.5 (M,R)-System Maps......Page 151 7.6 Unfolding......Page 152 7.7 Process Closure......Page 153 7.9 Definition......Page 154 7.10 Definition......Page 155 7.13 That Which Functionally Entails......Page 156 7.14 Definition......Page 157 7.15 Range of a Mapping......Page 158 7.16 Imminence......Page 159 7.18 Coproduct Decomposition......Page 160 7.20 The Symmetric Closure Imm±......Page 163 7.24 Theorem......Page 164 8.1 Fiat......Page 165 8.5 Lemma......Page 166 8.11 Random Variable......Page 167 8.15 Variance......Page 168 8.17 Properties......Page 169 8.19 Chebyshev’s Inequality......Page 170 8.23 Theorem......Page 171 8.26 Incompleteness as Metaphor......Page 172 8.27 Whence Clef......Page 173 8.29 Postulate of Biopoiesis......Page 174 8.31 Relational Biology......Page 175 9.1 Imminence Mapping of a Model......Page 176 9.3 Functional Closure......Page 177 9.6 Specificity......Page 179 9.9 Sequential Imminence......Page 180 9.10 Square Product Imminence......Page 182 9.11 Iterated Inverse......Page 183 10.1 Six Modes of Nodal Connection......Page 186 10.2 Common Causes......Page 187 10.3 Resolutions......Page 188 Entailment Between Systems......Page 190 10.4 Sequential Interaction......Page 191 10.5 Metabolism Bundle......Page 192 10.6 Hierarchical Interaction......Page 194 10.7 Range Partition......Page 195 10.8 Unfolding (M,R)-Network......Page 197 10.9 Unfolding (M,R)-Systems......Page 199 11.2 Effect......Page 201 11.4 Necessity......Page 202 11.5 An Example......Page 203 11.7 Modes of Relational Interaction......Page 204 11.8 Metabolism Symbiosis......Page 207 11.9 Repair Symbiosis......Page 209 11.10 Bacterial and Fungal Infection......Page 210 11.12 Ubiquity of Symbiosis......Page 212 12.2 Interacting Entailment Networks......Page 214 12.3 Relational Pathophysiology......Page 215 12.4 Material Entailment as Metabolic Interaction......Page 216 12.5 Alternate Material Entailments......Page 218 12.6 Functional Entailment of Metabolism......Page 219 12.7 Stability......Page 220 12.8 Metabolic Entailment of Repair......Page 221 12.9 Alternate Entailments of Replication......Page 223 12.11 Mutagenesis......Page 224 12.12 Relational Evolution......Page 225 12.13 Synthetic Life......Page 226 12.14 Trial and Error......Page 227 12.15 Anthropic Principle......Page 228 13.1 Suicidal Chreod......Page 230 13.3 Replicator......Page 231 13.4 Empty Imminence......Page 232 13.5 Definition......Page 233 13.7 Clef Closure......Page 234 13.8 Origin of Viruses......Page 236 13.9 Functional Entailment on Three Hierarchical Levels......Page 237 13.10 Prion Infection......Page 238 13.11 Pseudographs......Page 240 14.1 Three Remedies......Page 242 14.5 Theorem......Page 243 14.7 Alteration......Page 244 14.8 Image Factorization......Page 245 14.9 Isomorphic Therapy......Page 246 14.10 Reversal Therapy......Page 247 14.11 Epigenetic Perturbation......Page 248 14.12 Isomorphism Iterated......Page 249 14.13 Relational Oncology......Page 250 14.15 Disease Management......Page 251 14.16 Anamnesis......Page 252 14.18 Credo......Page 253 Acknowledgments......Page 254 Bibliography......Page 255 Index......Page 257 ​​​A. H. Louie's The Reflection of Life: Functional Entailment and Imminence in Relational Biology is a continuation of the exploratory journey in relational biology which began with his 2009 monograph More Than Life Itself: A Synthetic Continuation in Relational Biology. The theme of his first book was 'What is life?'; the theme of this sequel is "How do two life forms interact?" Biology is a subject concerned with organization of relations. Relational biology is the approach that advocates 'function dictates structure", rather than 'structure implies function'. It is mathematics decoded into biological realizations. The book demonstrates some of the powers of the approach of relational biology, and illustrates how pertinent problems in biology can be better addressed this way. In the first volume the theory was developed by using partially ordered sets, lattices, simulations, models, Aristotle's four causes, graphs, categories, simple and complex systems, anticipatory systems, and metabolism-repair [(M,R)-] systems. Here in the second volume, these tools are expanded to employ set-valued mappings, adjacency matrices, random graphs, and interacting entailment networks. The theory of set-valued mappings culminates in the imminence mapping, which equips the further investigation of functional entailment in complex relational networks. Imminence in (M,R)-networks that model living systems addresses the topics of biogenesis and natural selection. Interacting (M,R)-networks with mutually entailing processes serve as models in the study of symbiosis and pathophysiology. The formalism also provides a natural framework for a relational theory of virology and oncology. This book will serve researchers and graduate students in mathematics and biology. A.H. Louie's The Reflection of Life: Functional Entailment and Imminence in Relational Biology is a continuation of the exploratory journey in relational biology which began with his 2009 monograph More Than Life Itself: A Synthetic Continuation in Relational Biology. The theme of his first book was 'What is life?'; the theme of this sequel is "How do two life forms interact?" Biology is a subject concerned with organization of relations. Relational biology is the approach that advocates 'function dictates structure", rather than 'structure implies function'. It is mathematics decoded into biological realizations. The book demonstrates some of the powers of the approach of relational biology, and illustrates how pertinent problems in biology can be better addressed this way. In the first volume the theory was developed by using partially ordered sets, lattices, simulations, models, Aristotle's four causes, graphs, categories, simple and complex systems, anticipatory systems, and metabolism-repair [(M, R)-] systems. Here in the second volume, these tools are expanded to employ set-valued mappings, adjacency matrices, random graphs, and interacting entailment networks. The theory of set-valued mappings culminates in the imminence mapping, which equips the further investigation of functional entailment in complex relational networks. Imminence in (M, R)-networks that model living systems addresses the topics of biogenesis and natural selection. Interacting (M, R)-networks with mutually entailing processes serve as models in the study of symbiosis and pathophysiology. The formalism also provides a natural framework for a relational theory of virology and oncology. This book will serve researchers and graduate students in mathematics and biology