A Dedicated Conservationist Embraces The Science of Complexity

James Case

SIAM News, Volume 27, Number 6, July 1994

The Quark and the Jaguar: Adventures in the Simple and the Complex
By Murray Gell-Mann
W.H. Freeman and Co., New York, 1994, xviii + 392 pages, $23.95

Early reviews of Murray Gell-Mann's first ever book have been surprisingly lukewarm. Representing the author's personal attempt to gain a unified perspective on the entire scientific enterprise, from elementary particle theory (the Quark) to the social and biological sciences (the Jaguar), the book identifies the problems Gell-Mann and his seemingly countless army of collaborators consider most important and assesses the progress humankind has made, or is likely to make, toward solving them.

Not since Poincaré has so accomplished a theoretician accepted so ambitious a challenge. Anyone would be beyond his depth in such an undertaking, and Gell-Mann surely knows that fault will be found with his treatment of certain topics. Minor embarrassment would be a small price to pay, however, to achieve what this book attempts. For Gell-Mann, unlike Poincaré, does not write as a disinterested philosopher of science. On the contrary, he is a dedicated conservationist, eager to achieve a sustainable equilibrium between human beings and their environment that will preserve the Earth's biodiversity. This book is his manifesto!

Keenly aware of the gaps in the argument he yearns to make on behalf of diversity and sustainability, Gell-Mann proposes a conceptual framework within which those gaps may be filled. The newborn science of complexity occupies a prominent place in his grand design. Because it is so new and ambitious, however, his remarks concerning that fledgling science are unavoidably futuristic in tone. Indeed, the passages relating to it may well be classified, along with the science fiction of masters like Jules Verne and H.G. Wells, under the heading of "informed speculation." Yet the emphasis is on the "informed," and the speculation is intended not to amaze and entertain, but to exhort and enlighten. There already exists, at least in Gell-Mann's mind, a science of complexity. As Gauss was the first to deploy complex numbers with complete confidence, so Gell-Mann is the first to embrace this new discipline without reservation. He does so in part because he foresees a day when mathematically provable generalizations will emerge from the welter of special results now being obtained via the simulations that have so far characterized the field.

Previous popular accounts of complexity theory, the work of ordinary scientists turned expositors, groped toward an understanding of what a science of the complex might someday look like and accomplish. Gell-Mann's, in comparison, is the work of an extraordinary scientist who has given years of thought to the matter. His unique insight is due, in part, to his long involvement with the Santa Fe Institute, which, created for the study of complexity, has afforded him an unparalleled opportunity to digest the concept of a "complex adaptive system" and the knowledge of such systems that has so far accumulated. The Santa Fe Institute, in turn, owes its existence mainly to the fact that Gell-Mann (and a very few others) began long ago to think about complex systems.

The book's main flaw, if indeed it is a flaw, is that it devotes too few pages to explaining what complex adaptive systems are and what is now known about them. Although Gell-Mann's book is nominally self-contained, I as a reviewer would have had little idea what it was about had I not already been familiar with complexity theory through the excellent popular accounts of its development by M. Mitchell Waldrop and Roger C. Lewin (reviewed in SIAM News, May 1993). Gell-Mann writes as if his audience were already familiar with the work being done at the Santa Fe Institute, at least in broad outline, and wished only to learn how best to carry on with it.

Gell-Mann launches his exposition by partially ordering the sciences as to which (if either) of any two of them is the more fundamental. Physics, for example, is more fundamental than chemistry, because all the laws of the latter can in principle be deduced from those of the former. For similar reasons, chemistry is more fundamental than biology, and biochemistry is more fundamental than psychology. He argues in addition that mathematics, to the extent that it is the study of logical systems that may or may not ever be found to occur in any empirical science, may be considered the most fundamental science of all. Although there is nothing particularly original in all this, it does help set the tone for the rest of the book.

There are at least two reasons, in Gell-Mann's opinion, for which the future is unpredictable. One has to do with the phenomenon of sensitive dependence on initial conditions, first documented by Poincare, which would make accurate long-term predictions a practical impossibility even if they were possible in theory. The second reason is that the quantum mechanical nature of the universe makes certain knowledge of the future impossible even in principle. Good bets are to be found, but sure things aren't.

Being based on the so-called "decoherence functional" $D(\cdot,\cdot)$, defined for any two "ensemble histories" $A$ and $B$, Gell-Mann's interpretation of quantum mechanical uncertainty goes far beyond the familiar Heisenberg principle. Its purpose is to avoid the dependence on human observers that rendered early explanations of the matter unsatisfactory. $D(\cdot, c\dot)$ is assumed to satisfy the functional equation

$D(A or B, A or B) = D(A, A) + D(B, B) + [D(A, B) + D(B, A)]$ (1)

and to assume values in the interval [0,1] whenever its arguments coincide. Thus, it is at least potentially true that $D(X, X) = Prob{X}$. If, in addition, $A$ and $B$ "decohere" in the sense that $D(A, B) + D(B, A)$ vanishes, (1) furnishes a consistency condition to be satisfied by the probabilities assigned various ensemble histories.

Histories decohere, it turns out, unless they are too much alike. So, by demonstrating that coherence is an equivalence relation among ensemble histories, it can be shown that probabilities cannot be assigned to individual histories. The domain of the functional chosen to perform the assignment must consist of equivalence classes of coherent histories. This fact alone places a severe limitation on what can be known about ensemble histories. In particular, electrons from a common source that reach a particular detector by passing through either of two narrow slits in an otherwise impermeable barrier follow paths belonging to a common equivalence class. They cannot, in consequence, be assigned separate probabilities. Even the assignment of probabilities to ensemble histories would be an exercise in futility, were it not for the fact that the need to distinguish between members of a single coherence class seldom arises.

As Gell-Mann considers mathematics to be the most fundamental of the sciences, so he finds information theory to be the most useful branch of mathematics. With its help he is able to forge, from a variety of individually inadequate definitions of complexity, a workable conception of that surprisingly elusive attribute. The theory also prompts him to formulate the concept of an IGUS (information gathering and utilizing system) that provides the needed link between the basic sciences like chemistry and physics and the less fundamental ones like economics and ecology, which relate to sustainable equilibria between human beings and their environment and the preservation of biodiversity. Lastly, it links the would-be science of complexity with all the others. Complexity, at least to Gell-Mann, is only slightly more fundamental than ecology, economics, and other "policy sciences," and it is destined to become a powerful tool in their hands.

Gell-Mann is particularly anxious to put to rest the misconception that complexity theory somehow contradicts the second law of thermodynamics. This he does by pointing out that (1) that fundamental law applies only to closed systems, (2) entropy can perfectly well decrease in one subsystem if the loss is offset by gains in others, and (3) any loss of entropy attributable to increasing complexity is no more than a subsystem loss. The laws of the less fundamental sciences must, at all times, remain consistent with those of the more fundamental ones!

The essential unity of science is a theme that recurs throughout the book. It occurs naturally in the author's memorable discourse on the various unified and grand unified theories of the natural forces and surfaces again and again in other contexts. He is particularly anxious to persuade his audience that there is both a need and a place in the overall scheme for a science of the complex.

The book's final chapters are devoted to the subjects of diversity and sustainability. In them, the author's voice changes from that of an elder statesman of science to one of unapologetic advocacy. There is a need, he says, for people with "the courage to take a crude look at the whole, in addition to studying the behavior of parts of a system in the traditional [more precise] way." He cites a comparatively modest effort called Project 2050, begun under the auspices of the World Resources Institute, the Brookings Institution, and the Santa Fe Institute. There is every indication that a transition to sustainability had best begin long before the year in question.

The biosphere is, rather obviously, an exceedingly complex adaptive system. The project organizers hope, nevertheless, that enlightening models can be developed soon enough to enable participants to identify a few of the more dire threats to be encountered in any transition to sustainability. Gell-Mann quotes an early Project 2050 document to the effect that: "We are all in a situation that resembles driving a fast vehicle at night over unknown terrain that is rough, full of gullies, with precipices not far off. Some kind of headlight, even a feeble and flickering one, may help avoid some of the worst disasters." He further suggests that even this seemingly modest objective will prove elusive unless the essential unity of the sciences (including that of complexity) on the one hand, and the primacy of humankind's common interests on the other, are duly recognized.

James Case is an independent consultant who lives in Baltimore.

Reprinted from SIAM NEWS
Volume 27-6, July 1994
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved.