The New Evolutionary Paradigm

Keynote Volume

Ervin Laszlo

• Publisher: Routledge
• Year: 2019
• Print ISBN: 9780367339227
• eBook ISBN: 9781000477931
• Edition: 1
• Topic: Biological Sciences
• Subtopic: Evolution
• Index: Biological Sciences

Originally published in 1991, The New Evolutionary Paradigm provides an innovative and cross disciplinary look at evolution. While Darwin's theory of evolution was originally restricted to the life sciences, in recent years the same principles have been applied successfully to historical, social and natural sciences. The papers included in The New Evolutionary Paradigm analyse the facts, observations, and accumulated data from the significance of a general evolution theory cannot be overemphasised; a new understanding of the cosmos and man's relationship to it could lead to the systemization of the irreversible change that takes place in society and nature. This book will appeal to scientists, sociologists and those interested in transdisciplinary evolution theories.

CHAPTER 1

Complex Dynamical Systems Theory: Historical Origins, Contemporary Applications

RALPH ABRAHAM

Editor’s Introduction: In this opening chapter Ralph Abraham, a pioneer of the mathematical modeling and simulation of complex systems, traces the origins of the latest mathematical theories applicable to the systems that emerge and evolve in various domains, in nature as well as in society.

A concise and definitive account, the chapter serves as a general introduction to the mathematical underpinnings of general evolution theory, comprehensible to the layman as well as to the specialist. A number of the basic concepts that appear throughout this volume are here described and defined, including the concept of dynamical system itself, and the modeling and simulation of its various stable and unstable states with the help of static, periodic and chaotic attractors and the processes known as bifurcation.

Aware of the practical potential of the mathematical simulation of real-world systems as well as of the mushrooming problems that face contemporary people and societies, Abraham stresses the urgency and importance of researching and developing applications of the mathematical models of complex systems in the human and social sciences.

Introduction

Since the last glaciation, we have extensive records of some ten thousand years of the struggles of the human species for survival within the ecosystems of Terra. We have coextensive records of the evolution of consciousness, wisdom, intelligence, arts, sciences, and technology. The mutual interactions between these two levels of history have been critical to the survival of our colony up to the present moment, and will

Dedicated to Erich Jantsch (1929–1980)

continue to be critical, as we face the challenges to come. In this essay, we examine the cognitive strategies entwined in the historical records of the sciences, and propose an extrapolation for the near future which may be essential for our survival: the mathematical acceleration of social theory.

We will begin with a brief history of the role of mathematics in the development of the sciences since Newton, from the viewpoint of modeling and simulation. Then, we will outline three case studies: dynamics, physiology, and sociology. Finally, we propose an inexpensive project for the accelerated development of a large-scale model of our emerging planetary society, suitable for high-speed simulation by existing supercomputers.

The motivation of this essay, and the proposed project, is the challenge of meeting the oncoming evolutionary crisis, and surmounting it, through a timely increase of our understanding of complex systems and their transformations. For we feel that this increase in understanding will come soon, or never.

Historical Interactions Between Mathematics and the Sciences

Mathematics is not a science, nor is science mathematical. The applications of mathematics to the sciences involves, in fact, a relatively small part of our mathematical activity, and an even smaller smaller part of our scientific efforts. Yet historically, this interaction has been particularly important in the development of each. This is particularly true since Huyghens, Newton and Leibniz, who were primarily responsible for the cognitive style which dominates scientific theory today.

Applied mathematics, as we may call the interaction between mathematics and the sciences, has two aspects: modeling and simulation. Modeling denotes the creative activity of building a mathematical model for a given phenomenon, or experimental domain. It may involve any branch of mathematics in the architecture, construction, testing, and evaluation of a model. Simulation, on the other hand, denotes the operation of an existing mathematical model for purposes of prediction, or study, of the target system. The computer revolution has changed the dominant method of simulation from classical analysis to numerical computation and graphical presentation. We wish now to focus on the modeling aspect of applied mathematics, which was called mechanics in ancient Greece.

According to this mechanical paradigm, our cognitive strategy in technical matters is mechanical. That is, we understand complex phenomena by constructing models, rather than by verbal, symbolic, or other representations. Models may be physical machines (such as orreries or planetaria), pictorial representations (such as photographs) or mathematical models (symbolically represented, as in F = ma). The relationship between the model system and the real target system is a conventional (fictitious) one, and need not be an ideal analogy in order to be cognitively useful. Many different models of the same target system (a spectrum of models) may be used at once, to advance understanding. In fact, this may actually be understanding. We call this the mechanistic approach to science.

This approach differs from that of dogmatic science, in which the model comes, over time, to be identified with the target system. For example, a traditional physicist may assume that the electrostatic potential of Maxwell’s model has an actual existence in the phenomenal universe.

Accepting the mechanistic approach, let us review the role of the modeling aspect of applied mathematics in the history of the sciences since Newton.

Throughout the period 1680–1930, there was a growing list of spectacularly good models for physical phenomena. These have become, with surprisingly little evolution since their original creation, the cornerstones of mathematical physics: dynamics of particles and continua, electrodynamics, gravitational theory, thermodynamics, statistical mechanics, quantum theory, and so on. In each case, history follows the same pattern: experimental evidence mounts, cognitive strategies form and dissolve, data are increasingly numerical, models become increasingly mathematical, and so on. Eventually, someone has a revelation or intuitive leap, and theory emerges in a new simplicity of understanding, clothed in a splendid model (Maxwell’s equations, Einstein’s tensor, etc.) which stands as an ideal model for a long time. In this pattern of punctuated evolution in the sciences, the mathematical models play a key role in the formative stages and cognitive strategies, through interaction with the experimental and theoretical developments. This common pattern is a central point in this essay, and can be learned in detail from a single case study. An ideal case is d’Alembert’s wave equation for the vibrating string, which established the dominant modeling style of mathematical physics in 1752.

We will now go on to consider three other cases, one each from the physical, biological, and social sciences.

Theoretical Dynamics

The word mechanics meant model-making to the ancient Greeks, as we have noted above, while the word dynamics referred to the medicinal power of plants. In the context of the physical sciences, these two words have become synonymous, and denote the science of force, mass and motion begun by Galileo. From the point of view of mechanics (model-making), the history of mechanics (dynamics of particles and continua) provides outstanding examples of the role of models in the creation of theory. It is very instructive to study them in detail, but here we will be satisfied with a brief listing.

In 1560 or so, Galileo made use of real (physical) models (marbles, inclined boards, leaning tower of Pisa, and so on) to elucidate the basic principles of motion. After creating the calculus in 1665, Newton used it to make mathematical models for the same phenomena in 1685. From the study of these models grew classical analysis, one of the main branches of mathematics. The goal of analysis was to obtain predictions (that is, simulated data) from the models (differential equations) by symbolic integration (that is, from explicit functions).

In 1865, James Thompson invented the first mechanical analog computer for the simulation of these same models, providing a second simulation strategy. In the 1920’s, Van der Pol began using electronic analog computers for modeling and simulation, and these became fast enough to compete with classical analysis as a practical method. Later, during World War II, they became fast enough to be used as bombsites, simulating trajectories according to Newton’s model. Shortly thereafter, digital computers replaced them as the simulation strategy of choice for most dynamic models.

The models created by Newton (coupled systems of nonlinear differential equations) are basic to all the simulations which followed, whether by classical analysis or analog or digital computation.

Simple Dynamical Schemes

An outstanding problem of theoretical dynamics is the stability of the solar system. In 1885, Poincaré showed that Newton’s methods of classical analysis were inadequate to resolve this fundamental problem. He went on to establish totally new mathematical methods for the study of dynamical systems. These were geometric, rather than analytic, and gave rise to new branches of mathematics such as differential topology. The new methods, applied to systems of ordinary differential equations, are now known as dynamical systems theory (or qualitative nonlinear dynamics). They have provided a synthesis of all the outstanding models of the physical sciences into a single modeling strategy.¹

A dynamical system is based upon a state space, or geometrical model for the virtual states of the target system. Each point of the state space represents a single, instantaneous state, perhaps through some number of observable parameters. The dynamic is a infinitesimal rule of evolution: each state is characterized by a unique evolutionary tendency, described by a velocity vector.

The behavior of these mathematical systems is well-known, through three centuries of experimental and theoretical findings. A given initial state evolves along a unique trajectory. After a temporary phase, the transient response, this trajectory approaches asymptotically to a limit set called an attractor, and a dynamical equilibrium is attained. These occur in three flavors: static, periodic, and chaotic.

Static attractors, also called rest points, have been extensively applied since the time of Newton. A system under the influence of a static attractor approaches the final destination and slows to a halt.

Periodic attractors, also called oscillations, have dominated dynamics for the past century. A system approaching an oscillation will behave more and more like a perfect oscillation as time goes on.

Chaotic attractors, also called strange attractors, are newly discovered, and provide for an understanding of many kinds of aperiodic behavior. Much is now known to be signal, which was previously considered to be noise.

In a given dynamical system, there are usually several attractors. As each initial state will evolve to one of them, the state space may be decomposed into sets sharing the same final fate, which are called basins. The basins are divided by separatrices. The state space, with the attractors, basins, and separatrices drawn upon it, is called the portrait of the dynamical system. This portrait comprises the full understanding of the dynamical behavior of the model, at least as far as long-run prediction is concerned.

Most useful models contain adjustable constants, or control parameters, which may be used to adjust the dynamic on the fixed state space. Such a model is called a dynamical scheme. As the controls change, so does the portrait. The response diagram of the scheme is a graph showing the dependence of the portrait upon the control parameters. The response diagram is the master map which gives this kind of model great power in applications. Points in this diagram where the portrait changes in a particularly significant way are called bifurcations.

Catastrophe theory has provided excellent pedagogic examples of response diagrams for various schemes, establishing their importance as graphic representations in many scientific disciplines. Further, it demonstrates the usefulness of mathematical theory in these applications, as the theory excludes many bifurcations which might otherwise be expected.^2,3

For our present purposes it suffices to observe that dynamical schemes unify all the best-known models of the physical sciences within a single modeling strategy.

Theoretical Physiology

After the introduction of dynamical systems theory by Poincaré in 1882, and the maturation of mathematical physics f...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Half Title
• Original Title Page
• Original Copyright Page
• Dedication
• Table of Contents
• Preface to the Series
• Preface
• Foreword
• Introduction
• Chapter 1 Complex Dynamical Systems Theory: Historical Origins, Contemporary Applications
• Chapter 2 Chaos and Transformation: Implications of Nonequilibrium Theory for Social Science and Society
• Chapter 3 Biological Evolution: The Changing Image
• Chapter 4 Evolution and Intelligence
• Chapter 5 Modelling Biological and Social Change: Dynamical Replicative Network Theory
• Chapter 6 Social Evolution: A Nonequilibrium Systems Model
• Chapter 7 Economic and Social Evolution: The Transformational Dynamics Approach
• Chapter 8 The Evolutionary Paradigm and Neoclassical Economics
• Chapter 9 Cultural Evolution: Social Shifts and Phase Changes
• Index