Waves are ubiquitous in nature. They have been studied in the last couple of decades in such diverse forms and varied fields that they may now be said to constitute a new discipline — the science of waves (Lighthill, 1978). This wide and varied interest in waves has been particularly helped by the appearance of that strange entity, the soliton. The wave adopts such diverse forms that it is difficult to present a precise unifying definition. However, we may agree that waves (or disturbances), in an otherwise quiet or uniformly moving medium, have propagation properties and therefore involve the variable time, and have distinct features such as crests and troughs which themselves move with definite speeds. It should, however, be noted that not all waves are oscillatory. Thus, shock waves and solitary waves are not oscillatory. Nevertheless, these are regarded as (nonlinear) entities of great physical importance.

Two major types of waves have been distinguished (Whitham, 1974). The first is called hyperbolic and requires the system of n governing partial differential equations to have n real characteristic directions and correspondingly n linearly independent left eigenvectors of the relevant matrix (Courant & Hilbert, 1962). The second type of waves, called dispersive, are categorised by a real dispersion relation connecting the frequency and wave number (Bhatnagar, 1979). These definitions are broadened suitably to apply to partial differential equations with variable coefficients as well as nonlinear ones.

Self-assembly in general

To know a pattern is to know its roots – to understand a pattern, we must understand its history. Configurational explanations are historical explanations of pattern topology. Formally, a configurational explanation is a topologically naive automaton – a logical adder – that combines a set of raw materials and a templet to produce a final pattern. When the raw materials are topologically naive, the templet must contain significant explicit form information and the final pattern is extensively templeted. In contrast, when the raw materials are themselves topologically knowledgeable and allow the assembly of only one or a very few different configurations, the templet need not provide much additional form information and the final pattern is self-assembling.

A key and a lock, a hand and a glove, and a three-pronged plug and an electrical socket are all self-assembling units – the underlying ‘universal assembly laws’ are restrictive, and they can be fitted together in only one way.

How can a scientist make sense of very complex systems? What logical abstractions can he use to dissect phenomena that cannot easily be reduced to simpler parts? Like all scientists, researchers faced with complex systems wish to construct rigorous scientific explanations; yet the particular phenomena that they have at hand are often forbiddingly tangled. Some complex phenomena are so heterogeneous that they appear almost random; but, surprisingly, they differ from the truly random in that they are faithfully reproducible. ‘Random’ means more than extremely heterogeneous – it also means unpredictable.

In most methods for analyzing random phenomena, these two ideas – heterogeneity and unpredictability – are inextricably intertwined. For this reason, the standard stochastic tools for explaining random phenomena often cannot be effectively applied to heterogeneous yet highly stereotyped (predictable) phenomena. The realm of the very heterogeneous is not the usual realm of classical physics. But, the task of scientifically explaining heterogeneous phenomena is the task of most other disciplines. It represents the daily challenges of biology, psychology, sociology, economics, meteorology, epidemiology, metallurgy and geology. Theoreticians in each of these disciplines continually face inescapable, and at times irreducible, heterogeneity. And, irreducible heterogeneity is tantamount to complexity.

This book presents the beginnings of one epistemological approach to the natural science of complex systems.

Scientific abstractions answer the question: ‘What?’ and, more specifically, they answer the question: ‘What is the exact form of pattern x in the real world?’ Scientific abstractions formally describe the elements and the relations in certain real world patterns; they do so in a manner that maps the abstract realm directly back to the real world; and, in broad terms, this is the meaning of ‘what something is’.

To be precise, however, there is a critical caveat to this simple characterization. Scientific abstractions are called ‘abstractions’ because they abstract certain features from the real world. Abstractions are selected representations; they are not full representations, and they are not the real world items themselves. These are important distinctions, for the only truly complete answer to ‘What is the form of pattern x in the real world?’ is a display of the actual pattern itself [J. Bronowski (1978) The Origins Of Knowledge And Imagination Yale Univ. Press, New Haven]. Any other answer, any rephrasing of pattern x, any reproduction, any metaphor – i.e., any abstraction – implies that we have been at work selectively viewing and, in fact, selectively structuring the real world pattern, and this is unavoidable.

The power of configurational explanations lies in their explicit separation of the precursor elements and the templets. To use them to best advantage, I suggest the following rules (which presume some consistent philosophical assumptions about inference, such as the set of five postulates proposed by Russell [R.E. Egner & L.E. Denonn (1961) ‘Non-demonstrative inference’ The Basic Writings Of Bertrand Russell Simon & Schuster, NY, pp. 655–66]):

Define the topology of the pattern

Choose a substantive pattern

For a configurational explanation to be predictive, the pattern that it explains should be coherent and relatively permanent. When a pattern is fragmented or ephemeral, formulating configurational explanations can be an empty exercise.

Science strives to reduce our experiences to symbols. Experiences are colorful, multi-faceted, and fuzzy along the edges; symbols are bland, one-dimensional, and precisely-bounded. Real world observations can be bulky and ill-shaped and can have both strong and tenuous ties with a myriad of other real world observations; abstractions are built of simple, smooth-surfaced elements, uncoupled from other constructs, and abstractions can easily be carried in one's pocket. Scientifically, we give up the shifting and elusive mystery of the world, but, in exchange, we gain the standardized and reproducible abstractions from which we can build precise determinate explanations.

The reduction of experience to useful symbols – the construction of scientific abstractions – is the essential scientific endeavor. In this sense, all of science is reductionism. At the same time, there is another sense to scientific reductionism. Once a real world phenomenon has been abstracted, the scientist attempts to create a scientific explanation of that abstraction, and this too is a type of reductionism. To reduce is to recreate, and reductionism is actually constructionism.

A scientific abstraction is a model of some real world phenomenon, and a scientific explanation is then a further transformation of that phenomenon, carried out largely in the abstract realm.

In A Child's Garden Of Verses, Robert Louis Stevenson wrote:

The world is certainly bountifully supplied with different things, and while these all appear new and wonderful to a child, they have largely become old and familiar to an adult. A scientist, on the other hand, dreams of a balance between the child world and the adult world. The scientist would like most things to be old and familiar while a few things still remain new and wonderful.

The scientific pursuit is turning the new and wonderful into the old and familiar: a scientist tries to make the world around him understandable.

In this work, I owe a special debt to two colleagues. Raymond J. Lasek, from whom I learned biology, always provided a thoughtful, challenging, and critical response as I presented him with various steps in my thinking out of these ideas. Ulf Grenander was my warm and generous host in the Division of Applied Mathematics at Brown University. His Pattern Theory is the language that I have borrowed in an attempt to formalize my ideas about templeting. A language is more than a tool, it structures one's thoughts; and Prof. Grenander's language is responsible for much of the organization of my ideas. Both of these scientists engaged me in innumerable hours of epistemological discussion, and I cannot help but to have adopted many of their good ideas. Some of these are now woven inextricably throughout this book.

I would also like to thank the Alfred P. Sloan Foundation and the Whitehall Foundation, which provided financial support for this work.

The title of Chapter 2 is borrowed whole from the book The Nature Of Explanation in which the author, Kenneth Craik, makes a strong case for the importance of determinate explanations.

The story that I present in this book is not as tidy as I had originally hoped, but I think that this is the nature of detailed abstractions that must be tethered to the real world.

Simple and complex are two ends of a spectrum. Simple things have few parts, the parts are organized in a homogeneous fashion, and the whole can be fairly easily grasped in one fell swoop. Complex things have many parts, the parts are organized heterogeneously, and it takes a concentrated effort to comprehend the whole. A musical scale is simple, a Bach fugue is complex. A color chart is simple, a Jackson Pollock painting is complex. The alphabet is simple, the Bible is complex.

At times, the simplicity or complexity of an item can be deceptive. We sit down to a steaming meal of clams, lobster tails, crab legs, chicken, and vegetables. ‘Delicious, but obviously complicated to make,’ we comment to our host. ‘Not at all complicated,’ says he. ‘Your simply steam all of the ingredients in a large pot for an hour. The meal cooks itself.’ On the other hand, the first course was quenelle de brochette – egg-shaped fish mousses, looking like tiny homogeneous puddings.

Abstractions are for pockets; they are miniatures of the world that we can carry around with us, that we can take out at our leisure and examine, and that we can tinker with. We can poke them and probe them and rearrange their parts. In essence, they are pocket toys.

Abstractions are pocket models of the world, and the scientific abstractions are a special class of these pocket models. For the scientist, abstractions must be useful models of the real world – the scientist would like to ensure that what he learns from tinkering with an abstraction will lead him to understand parts of the real world that he has not directly put his hands on. Scientific abstractions must have tiny portals that are windows into the unknown. For this to hold true, for a scientist's abstractions to enable him to see beyond his reference texts, he must be able to generalize from observations and experiments on the abstraction to observations and experiments in the real world. Thus, certain relations must exist between the abstraction and the real world phenomena and certain internal relations must also exist between the parts of the abstraction.