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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 1: The Foundations for a New Kind of Science
Section: An Outline of Basic Ideas
Page 861

Complexity and science. Ever since antiquity science has tended to see its main purpose as being the study of regularities - and this has meant that insofar as complexity is viewed as an absence of regularities, it has tended to be ignored or avoided. There have however been occasional discussions of various general aspects of complexity and what can account for them. Thus, for example, by 200 BC the Epicureans were discussing the idea that varied and complex forms in nature could be made up from arrangements of small numbers of types of elementary atoms in much the same way as varied and complex written texts are made up from small numbers of types of letters. And although its consequences were remarkably confused, the notion of a single underlying substance that could be transmuted into anything - living or not - was also a centerpiece of alchemy. Starting in the 1600s successes in physics and discoveries like the circulation of blood led to the idea that it should be possible to explain the operation of almost any natural system in essentially mechanical terms - leading for example René Descartes to claim in 1637 that we should one day be able to explain the operation of a tree just like we do a clock. But as mathematical methods developed, they seemed to apply mainly to physical systems, and not for example to biological ones. And indeed Immanuel Kant wrote in 1790 that "it is absurd to hope that another Newton will arise in the future who will make comprehensible to us the production of a blade of grass according to natural laws". In the late 1700s and early 1800s mathematical methods began to be used in economics and later in studying populations. And partly influenced by results from this, Charles Darwin in 1859 suggested natural selection as the basis for many phenomena in biology, including complexity. By the late 1800s advances in chemistry had established that biological systems were made of the same basic components as physical ones. But biology still continued to concentrate on very specific observations - with no serious theoretical discussion of anything as general as the phenomenon of complexity. In the 1800s statistics was increasingly viewed as providing a scientific approach to complex processes in practical social systems. And in the late 1800s statistical mechanics was then used as a basis for analyzing complex microscopic processes in physics. Most of the advances in physics in the late 1800s and early 1900s in effect avoided complexity by concentrating on properties and systems simple enough to be described by explicit mathematical formulas. And when other fields tried in the early and mid-1900s to imitate successes in physics, they too generally tended to concentrate on issues that seemed amenable to explicit mathematical formulas. Within mathematics itself - especially in number theory and the three-body problem - there were calculations that yielded results that seemed complex. But normally this complexity was viewed just as something to be overcome - either by looking at things in a different way, or by proving more powerful theorems - and not as something to be studied or even much commented on in its own right.

In the 1940s, however, successes in the analysis of logistical and electronic systems led to discussion of the idea that it might be possible to set up some sort of general approach to complex systems - especially biological and social ones. And by the late 1940s the cybernetics movement was becoming increasingly popular - with Norbert Wiener emphasizing feedback control and stochastic differential equations, and John von Neumann and others emphasizing systems based on networks of elements often modelled after neurons. There were spinoffs such as control theory and game theory, but little progress was made on core issues of complexity, and already by the mid-1950s what began to dominate were vague discussions involving fashionable issues in areas such as psychiatry and anthropology. There also emerged a tradition of robotics and artificial intelligence, and a few of the systems that were built or simulated did show some complexity of behavior (see page 879). But in most cases this was viewed just as something to be overcome in order to achieve the engineering objectives sought. Particularly in the 1960s there was discussion of complexity in large human organizations - especially in connection with the development of management science and the features of various forms of hierarchy - and there emerged what was called systems theory, which in practice typically involved simulating networks of differential equations, often representing relationships in flowcharts. Attempts were for example made at worldwide models, but by the 1970s their results - especially in economics - were being discredited. (Similar methods are nevertheless used today, especially in environmental modelling.)

With its strong emphasis on simple laws and measurements of numbers, physics has normally tended to define itself to avoid complexity. But from at least the 1940s, issues of complexity were nevertheless occasionally mentioned by physicists as important, most often in connection with fluid turbulence or features of nonlinear differential equations. Questions about pattern formation, particularly in biology and in relation to thermodynamics, led to a sequence of studies of reaction-diffusion equations, which by the 1970s were being presented as relevant to general issues of complexity, under names like self-organization, synergetics and dissipative structures. By the late 1970s the work of Benoit Mandelbrot on fractals provided an important example of a general approach to addressing a certain kind of complexity. And chaos theory - with its basis in the mathematics of dynamical systems theory - also began to become popular in the late 1970s, being discussed particularly in connection with fluid turbulence. In essentially all cases, however, the emphasis remained on trying to find some aspect of complex behavior that could be summarized by a single number or a traditional mathematical equation.

As discussed on pages 44-50, there were by the beginning of the 1980s various kinds of abstract systems whose rules were simple but which had nevertheless shown complex behavior, particularly in computer simulations. But usually this was considered largely a curiosity, and there was no particular sense that there might be a general phenomenon of complexity that could be of central interest, say in natural science. And indeed there remained an almost universal belief that to capture any complexity of real scientific relevance one must have a complex underlying model. My work on cellular automata in the early 1980s provided strong evidence, however, that complex behavior very much like what was seen in nature could in fact arise in a very general way from remarkably simple underlying rules. And starting around the mid-1980s it began to be not uncommon to hear the statement that complex behavior can arise from simple rules - though often there was great confusion about just what this was actually saying, and what, for example, should be considered complex behavior, or a simple rule.

That complexity could be identified as a coherent phenomenon that could be studied scientifically in its own right was something I began to emphasize around 1984. And having created the beginnings of what I considered to be the necessary intellectual structure, I started to try to develop an organizational structure to allow what I called complex systems research to spread. Some of what I did had fairly immediate effects, but much did not, and by late 1986 I had started building Mathematica and decided to pursue my own scientific interests in a more independent way (see page 20). By the late 1980s, however, there was widespread discussion of what was by then being called complexity theory. (I had avoided this name to prevent confusion with the largely unrelated field of computational complexity theory). And indeed many of the points I had made about the promise of the field were being enthusiastically repeated in popular accounts - and there were starting to be quite a number of new institutions devoted to the field. (A notable example was the Santa Fe Institute, whose orientation towards complexity seems to have been a quite direct consequence of my efforts.) But despite all this, no major new scientific developments were forthcoming - not least because there was a tremendous tendency to ignore the idea of simple underlying rules and of what I had discovered in cellular automata, and instead to set up computer simulations with rules far too complicated to allow them to be used in studying fundamental questions. And combined with a predilection for considering issues in the social and biological sciences that seem hard to pin down, this led to considerable skepticism among many scientists - with the result that by the mid-1990s the field was to some extent in retreat - though the statement that complexity is somehow an important and fundamental issue has continued to be emphasized especially in studies of ecological and business systems.

Watching the history of the field of complexity theory has made it particularly clear to me that without a major new intellectual structure complexity cannot realistically be studied in a meaningful scientific way. But it is now just such a structure that I believe I have finally been able to set up in this book.

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 861.
© 2002, Stephen Wolfram, LLC