This book explains a certain way of appreciating “the world filled with non- linearity.” Its core is conceptual analysis and phenomenology, which is backed up by renormalization philosophy. The main target of the book is young people who have just started to appreciate the world seriously. The author wishes the book to be helpful also for those who have been observing the world, but who wish to appreciate it afresh from a different angle.

1 Looking at the Nonlinear World . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Characteristics of linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.2 Characteristics of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . 10

    1.3 Intrinsically nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    1.4 What do we mean by ‘appreciating the world’? . . . . . . . . . . . . . 16

    1.5 The structure of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Conceptual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    2.1 Starting with typical examples—chaos as an example . . . . . . . 39

    2.2 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    2.3 Characterizing chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    2.4 How to quantify ‘history’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    2.5 How to quantify information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    2.6 Measure-theoretical dynamical systems . . . . . . . . . . . . . . . . . . . . 76

    2.7 How to quantify chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    2.8 Preparation for characterizing randomness . . . . . . . . . . . . . . . . . 91

    2.9 What is computation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

    2.10 Turing machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    2.11 Characterizing randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    2.12 Understanding the essence of chaos . . . . . . . . . . . . . . . . . . . . . . . 105

    2.13 Is the characterization of randomness satisfactory? . . . . . . . . . . 109

    2.14 How is `complexity’ understood? . . . . . . . . . . . . . . . . . . . . . . . . . 110

3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    3.1 What is phenomenology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    3.2 Phenomenology too universal to be recognized . . . . . . . . . . . . . 133

    3.3 How to obtain phenomenology—relation to renormalization . . 139

    3.4 Two approaches to renormalization . . . . . . . . . . . . . . . . . . . . . . 144

    3.5 ABC of renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    3.6 Longtime behavior and renormalization: a simple example . . . 161

    3.7 Resonance and renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

    3.8 How reliable is the renormalization group result? . . . . . . . . . . . 173

    3.9 Proto-renormalization group approach to system reduction . . 176

    3.10 Statistics seen from the renormalization group point of view . 183

4 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    4.1 What is a model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

    4.2 Correspondence between models and reality . . . . . . . . . . . . . . . . 195

    4.3 Models as tools of description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

    4.4 Models as tools of deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

    4.5 Examples of modeling—examples of abduction— . . . . . . . . . . . 207

    4.6 What characterizes good models? . . . . . . . . . . . . . . . . . . . . . . . . . 217

    4.7 By-products of modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5 Toward Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

    5.1 Meaning and value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

    5.2 Pasteur process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

    5.3 Fundamental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

    5.4 What do `fundamental conditions’ imply? . . . . . . . . . . . . . . . . . 255

    5.5 How can we approach complex systems? . . . . . . . . . . . . . . . . . . . 261

    5.6 Is there a `theory of biological systems’? . . . . . . . . . . . . . . . . . . . 264

    5.7 How do fundamental conditions evolve? . . . . . . . . . . . . . . . . . . . 269

    5.8 How do systems complexify? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

    5.9 Integration step and its logical consequence . . . . . . . . . . . . . . . . 275

    5.10 `Lessons’ we learn from complex systems . . . . . . . . . . . . . . . . . . 281

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .