A. H. Louie

This rare publication continues an exploratory journey in relational biology, a study of biology in terms of the organization of networked connections in living systems. It builds on the author’s two earlier monographs which looked at the epistemology of life and the ontogeny of life. Here the emphasis is on the intangibility of life, that the real nature of living systems is conveyed not by their tangible material basis but by their intangible inherent processes. Relational biology is the approach that hails ‘function dictates structure’; it is mathematics decoded into biological realizations. Therefore, the work begins with a concise introduction to category theory, equiping the reader with the mathematical metalanguage of relation biology. The book is organized around three parts: Part I is a comprehensive study of the most important functor in relational biology, the power set functor. The author lays the set-theoretic foundations of the functorial connections in relational biology, exploring relations, mappings, and set-valued mappings. In Part II, Natural Law receives a new mathematical formulation founded on two axioms: ‘Everything is a set.’ and ‘Every process is a set-valued mapping.’ The reader sees how Metabolism–Repair networks, equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems. Part III expounds the various shades of invertibility in general, and the inversion of encoding to decoding in particular. A plethora of mathematical and biological examples illustrate the category-theoretic concepts of equivalence and adjunction. This book's algebraic approach to biological models will appeal to researchers and graduate students in mathematics, biology, and the philosophy of science.

This rare publication continues an exploratory journey in relational biology, a study of biology in terms of the organization of networked connections in living systems. It builds on the author’s two earlier monographs which looked at the epistemology of life and the ontogeny of life. Here the emphasis is on the intangibility of life, that the real nature of living systems is conveyed not by their tangible material basis but by their intangible inherent processes. Relational biology is the approach that hails ‘function dictates structure’; it is mathematics decoded into biological realizations. Therefore, the work begins with a concise introduction to category theory, equiping the reader with the mathematical metalanguage of relation biology. The book is organized around three parts: Part I is a comprehensive study of the most important functor in relational biology, the power set functor. The author lays the set-theoretic foundations of the functorial connections in relational biology, exploring relations, mappings, and set-valued mappings. In Part II, Natural Law receives a new mathematical formulation founded on two axioms: ‘Everything is a set.’ and ‘Every process is a set-valued mapping.’ The reader sees how Metabolism–Repair networks, equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems. Part III expounds the various shades of invertibility in general, and the inversion of encoding to decoding in particular. A plethora of mathematical and biological examples illustrate the category-theoretic concepts of equivalence and adjunction. This book's algebraic approach to biological models will appeal to researchers and graduate students in mathematics, biology, and the philosophy of science.

???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.

A. H. Louie's More Than Life Itself is an exploratory journey in relational biology, a study of life in terms of the organization of entailment relations in living systems. This book represents a synergy of the mathematical theories of categories, lattices, and modelling, and the result is a synthetic biology that provides a characterization of life. Biology extends physics. Life is not a specialization of mechanism, but an expansive generalization of it. Organisms and machines share some common features, but organisms are not machines. Life is defined by a relational closure that places it beyond the reach of physicochemical and mechanistic dogma, outside the reductionistic universe, and into the realm of impredicativity. Function dictates structure. Complexity brings forth living beings.