Mathematics and the Natural Sciences: The Physical Singularity of Life (Advances in Computer Science and Engineering: Texts, 7)

This book identifies the organizing concepts of physical and biological phenomena by an analysis of the foundations of mathematics and physics. Our aim is to propose a dialog between different conceptual universes and thus to provide a unification of phenomena. The role of 'order" and symmetries in the foundations of mathematics is linked to the main invariants and principles, among them the geodesic principle (a consequence of symmetries), which govern and confer unity to various physical theories. Moreover, an attempt is made to understand causal structures, a central element of physical intelligibility, in terms of both symmetries and symmetry breakings. A distinction between the principles of (conceptual) construction and of proofs, both in physics and in mathematics, guides most of the work. The importance of mathematical tools is also highlighted to clarify differences in the models for physics and biology that are proposed by continuous and discrete mathematics, such as computational simulations. Since biology is particularly complex and not as well understood at a theoretical level, we propose a "unification by concepts" which in any case should precede mathematization. This constitutes an outline for unification also based on highlighting conceptual differences, complex points of passage and technical irreducibilities of one field to another. Indeed, we suppose here a very common monist point of view, namely the view that living objects are "big bags of molecules". The main question though is to understand which "theory" can help better understand these bags of molecules. They are, indeed, rather "singular", from the physical point of view. Technically, we express this singularity through the concept of "extended criticality", which provides a logical extension of the critical transitions that are known in physics. The presentation is mostly kept at an informal and conceptual level. Contents 14 Preface 6 Chapter 1 Mathematical Concepts and Physical Objects 19 Introduction 19 1.1 On the Foundations of Mathematics. A First Inquiry 25 1.1.1 Terminological issues? 25 1.1.2 The genesis of mathematical structures and of their relationships – a few conceptual analogies 28 1.1.3 Formalization, calculation, meaning, subjectivity 31 1.1.4 Between cognition and history: Towards new structures of intelligibility 35 1.2 Mathematical Concepts: A Constructive Approach 37 1.2.1 Genealogies of concepts 37 1.2.2 The “transcendent” in physics and in mathematics 41 1.2.3 Laws, structures, and foundations 49 1.2.4 Subject and objectivity 55 1.2.5 From intuitionism to a renewed constructivism 58 1.3 Regarding Mathematical Concepts and Physical Objects 62 1.3.1 “Friction” and the determination of physical objects 63 1.3.2 The absolute and the relative in mathematics and in physics 65 1.3.3 On the two functions of language within the process of objectification and the construction of mathematical models in physics 66 1.3.4 From the relativity to reference universes to that of these universes themselves as generators of physical invariances 69 1.3.5 Physical causality and mathematical symmetry 70 1.3.6 Towards the “cognitive subject” 73 Chapter 2 Incompleteness and Indetermination in Mathematics and Physics 75 Introduction 75 2.1 The Cognitive Foundations of Mathematics: Human Gestures in Proofs and Mathematical Incompleteness of Formalisms 76 2.1.1 Introduction 76 2.1.2 Machines, body, and rationality 77 2.1.3 Ameba, motivity, and signification 79 2.1.4 The abstract and the symbolic; the rigor 80 2.1.5 From the Platonist response to action and gesture 83 2.1.6 Intuition, gestures, and the numeric line 87 2.1.7 Mathematical incompleteness of formalisms 91 2.1.8 Iterations and closures on the horizon 93 2.1.9 Intuition 96 2.1.10 Body gestures and the “cogito” 100 2.1.11 Summary and conclusion of part 2.1 101 2.2 Incompleteness, Uncertainty, and Infinity: Differences and Similarities Between Physics and Mathematics 103 2.2.1 Completeness/incompleteness in physical theories 103 2.2.2 Finite/infinite in mathematics and physics 111 Chapter 3 Space and Time from Physics to Biology 119 3.1 An Introduction to the Space and Time of Modern Physics 121 3.1.1 Taking leave of Laplace 121 3.1.2 Three types of physical theory: Relativity, quantum physics, and the theory of critical transitions in dynamical systems 123 3.1.3 Some epistemological remarks 129 3.2 Towards Biology: Space and Time in the “Field” of Living Systems 131 3.2.1 The time of life 131 3.2.2 More on Biological time 133 3.2.3 Dynamics of the self-constitution of living systems 138 3.2.4 Morphogenesis 142 3.2.5 Information and geometric structure 146 3.3 Spatiotemporal Determination and Biology 150 3.3.1 Biological aspects 150 3.3.2 Space: Laws of scaling and of critical behavior. The geometry of biological functions 151 3.3.3 Three types of time 154 3.3.4 Epistemological and mathematical aspects 157 3.3.5 Some philosophy, to conclude 161 Chapter 4 Invariances, Symmetries, and Symmetry Breakings 167 4.1 A Major Structuring Principle of Physics: The Geodesic Principle 167 4.1.1 The physico-mathematical conceptual frame 169 4.2 On the Role of Symmetries and of Their Breakings: From Description to Determination 176 4.2.1 Symmetries, symmetry breaking, and logic 176 4.2.2 Symmetries, symmetry breaking, and determination of physical reality 179 4.3 Invariance and Variability in Biology 183 4.3.1 A few abstract invariances in biology: Homology, analogy, allometry 183 4.3.2 Comments regarding the relationships between invariances and the conditions of possibility for life 187 4.4 About the Possible Recategorizations of the Notions of Space and Time under the Current State of the Natural Sciences 193 Chapter 5 Causes and Symmetries: The Continuum and the Discrete in Mathematical Modeling 199 Introduction 199 5.1 Causal Structures and Symmetries, in Physics 200 5.1.1 Symmetries as starting point for intelligibility 204 5.1.2 Time and causality in physics 205 5.1.3 Symmetry breaking and fabrics of interaction 208 5.2 From the Continuum to the Discrete 213 5.2.1 Computer science and the philosophy of arithmetic 214 5.2.2 Laplace, digital rounding, and iteration 216 5.2.3 Iteration and prediction 219 5.2.4 Rules and the algorithm 221 5.3 Causalities in Biology 228 5.3.1 Basic representation 229 5.3.2 On contingent finality 233 5.3.3 “Causal” dynamics: Development, maturity, aging, death 234 5.3.4 Invariants of causal reduction in biology 236 5.3.5 A few comments and comparisons with physics 238 5.4 Synthesis and Conclusion 238 Chapter 6 Extended Criticality: The Physical Singularity of Life Phenomena 243 Introduction 243 6.1 On Singularities and Criticality in Physics 245 6.1.1 From gas to crystal 245 6.1.2 From the local to the global 247 6.1.3 Phase transitions in self-organized criticality and “order for free” 249 6.2 Life as “Extended Critical Situation” 254 6.2.1 Extended critical situations: General approaches 258 6.2.2 The extended critical situation: A few precisions and complements 260 6.2.3 More on the relations to autopoiesis 262 6.2.4 Summary of the characteristics of the extended critical situation 263 6.3 Integration, Regulation, and Causal Regimes 264 6.4 Phase Spaces and Their Trajectories 268 6.5 Another View on Stability and Variability 273 6.5.1 Biolons as attractors and individual trajectories 273 Chapter 7 Randomness and Determination in the Interplay between the Continuum and the Discrete 277 Introduction 277 7.1 Deterministic Chaos and Mathematical Randomness: The Case of Classical Physics 280 7.2 The Objectivity of Quantum Randomness 283 7.2.1 Separability vs non-separability 285 7.2.2 Possible objections 287 7.2.3 Final remarks on quantum randomness 291 7.3 Determination and Continuous Mathematics 292 7.4 Conclusion: Towards Computability 296 Chapter 8 Conclusion: Unification and Separation of Theories, or the Importance of Negative Results 299 8.1 Foundational Analysis and Knowledge Construction 299 8.2 The Importance of Negative Results 303 8.2.1 Changing frames 307 8.3 Vitalism and Non-Realism 310 8.4 End and Opening . . . 315 Bibliography 317 Index 331 The Book Aims At The Identification Of The Organising Concepts Of Some Physical And Biological Phenomena, By Means Of An Analysis Of The Foundations Of Mathematics And Of Physics. This Is Done In The Perspective Of Unifying Phenomena, Of Bringing Different Conceptual Universes Into Dialog. The Analysis Of The Role Of “order” And Of Symmetries In The Foundations Of Mathematics Is Linked To The Main Invariants And Principles, Among Which The Geodesic Principle (a Consequence Of Symmetries), Which Govern And Confer Unity To The Various Physical Theories. Moreover, We Attempt To Understand Causal Structures, A Central Element Of Physical Intelligibility, In Terms Of Symmetries And Their Breakings. The Importance Of The Mathematical Tool Is Also Highlighted, Enabling Us To Grasp The Differences In The Models For Physics And Biology Which Are Proposed By Continuous And Discrete Mathematics, Such As Computational Simulations. A Distinction Between Principles Of (conceptual) Construction And Principles Of Proofs, Both In Physics And In Mathematics, Guides This Part Of The Work.as For Biology, Being Particularly Difficult And Not As Thoroughly Examined At A Theoretical Level, We Propose A “unification By Concepts”, An Attempt Which Should Always Precede Mathematisation. This Constitutes An Outline For Unification Also Basing Itself Upon The Highlighting Of Conceptual Differences, Of Complex Points Of Passage, Of Technical Irreducibilities Of One Field To Another. Indeed, A Monist Point Of View Such As Ours Should Not Make Us Blind: We, The Living Objects, Are Surely Just Big Bags Of Molecules Or, At Least, This Is Our Main Metaphysical Assumption. The Point Though Is: Which Theory Can Help Us To Better Understand These Bags Of Molecules, As They Are, Indeed, Rather “singular”, From The Physical Point Of View. Technically, This Singularity Is Expressed By The Notion Of “extended Criticality”, A Notion That Logically Extends The Pointwise Critical Transitions In Physics.