An Essay on the Importance of Being Nonlinear

The fields of scientific knowledge¹ have been established through the painstaking effort of large numbers of dedicated investigators over long periods of time. As a scientist I am in awe of the present day rate of growth in information and in the growing number of practitioners of science. One can easily be discouraged by an inability to absorb all this information or to know all one’s colleagues, or even those engaged in areas of research directly related to one’s own. It appears that the modern explosion in scientific knowledge has far outstripped our collective ability to comprehend and the slower paced research of the 18th and 19th centuries has a strong appeal. This attraction results from a distortion of the true situation, however, and the feelings of isolation and impotence are not unique to the contemporary scientist. De Sola Price (1963) points out that during the past three hundred years:

“...every doubling of the population has produced at least three doublings in the number of scientists, so that the size of science is eight times what it was and the number of scientists per million population has multiplied by four.”

Many traits of humankind have been suggested as being characteristics of the species. It seems to me that the ability and/or desire to plan for future events must be one of the dominant traits of civilized people. Such planning has certainly contributed to the survival of the species and has perhaps led to a social being with a predisposition for anticipating, if not predicting the future. In a sociological context, forecasting is important for endeavors ranging from economic to military to recreational. As scientists we strive to understand the underlying processes and to correct deficiencies in both our understanding of these processes and in the limitations of forecasting from extant data. The study of the question of our fundamental limitations of forecasting has been termed “predictability.”

In the preceding section we have sketched some of the dominant styles of thought used in the understanding of complex events in natural philosophy. In our discussions we stressed the dependence of these schema on the notion of linearity. The motivation for these presentations was to uncover the often implicit and obscure dependence of these schema on linear ideas. We are now in a position to examine the limitations of some of these ideas in describing a number of familiar, but all too often ill-understood phenomena. We maintain that these processes can be understood by determining the proper nonlinear descriptions to replace the inadequate linear ones. As the title of this section indicates, we focus on the reasons why a nonlinear perspective is to be preferred over a linear one.

In the previous sections we have repeatedly indicated how linear concepts collapse under the weight of uninterpretable data and are replaced by nonlinear ones. Examples include the existence of non-integer exponents in the functional representation of data, long tails in the distribution functions, saturation phenomena and rhythmic behavior in open (biological) systems. Each has obliged us to seek a nonlinear representation of the underlying process. In each case we observed how a comfortable linear notion was inadequate to describe a particular phenomenon due to the existence of one or more nonlinear mechanisms and therefore had to be discarded. An example we considered is the amplification mechanism responsible for the log-normal distribution being replaced by a Pareto tail at the high end of the income distribution [cf. (§3.2)]. The log-normal distribution relies on a linear interpretation in terms of the logarithm of an individual’s income. The amplification mechanism at high incomes is nonlinear, however, and forces the replacement of the linear-based log-normal distribution with the nonlinear-based Pareto power law. The practical question remains: “How does one learn the techniques that are of value in one’s field without becoming burdened with perhaps interesting but nonetheless irrelevant mathematical baggage?”

The series of lectures on which this essay is based was developed for an audience with a heterogeneous scientific background which varied from music theory to psychiatry to organic chemistry. The intent of the series was to draw parallels between different disciplines and to perhaps uncover an underlying unity. Instead of this, we in fact uncovered the self-imposed limitations that scientific investigators have established in their efforts to make sense out of complex systems and processes. Starting with the first of the modern scientists (Newton) I have attempted to show how the evolution of a linear world view has dominated scientific thinking in natural philosophy. The efforts of Newton to understand mechanistic relations and those of Gauss to interpret large data sets, along with their contemporaries, have contributed to this limited and limiting view of the world. It is not my intent to in any way minimize the monumental contributions of these two great men to our understanding of the physical world. But I do wish to emphasize the influence that this style of thought has had outside the physical sciences. For example, it was this perspective that prompted Descartes to say:

“If we really knew all the parts of the seed of any particular animal, man, for example, we could deduce from that alone, by certain and mathematical reasoning, the shape and conformation of each of his limbs.”