Murray Gell-Mann
Santa Fe Institute, Santa Fe, New Mexico
I congratulate Mercedes Pascual and Stefan Rahmstorf on their winning the competition in the area of Global and Complex Systems, on their prospects for great achievements in the future, and on their excellent contributions to this volume. They were chosen for the McDonnell Centennial Fellowships because of the impressiveness of their application essays, the importance of their topics, their promise as individuals, and the difference the award would make to them.
The Foundation insisted that the application essays themselves play a very important part in the selection process. But the competition was international, and therefore special allowance was supposed to be made for those applicants who did not have English as a native language. It turned out, however, that many of the best-written essays in English were submitted by just such people, including the two recipients of these awards. Most of the native speakers of English who applied were Americans. Maybe American education at the primary and secondary level really does leave something to be desired.
Naturally, a number of other applications submitted in this competition were also quite impressive. As a member of the Advisory Committee, I wished that the Foundation had had greater resources so as to afford some additional expenditures. As usual in these cases, I felt it would have been nice to fund some more of the best applicants. But in addition, and that is perhaps not quite so usual, I felt that it would have
been wonderful to fund collaborations among various applicants. There were several cases in which the researchers who wrote to us were tackling complementary aspects of the same situations. For example, some of them were concerned with different parts of the huge field encompassing global climate and the behavior of the oceans. Others treated various aspects of ecology, including fires of natural and human origin, changes in the character of lakes, and so on. What an appealing idea to bring these various scientists together to supplement their ongoing individual research projects with collaborative work exploiting the strong links among their topics of study. Even the two winners might benefit from exploring the connections between their projects.
Our Advisory Committee spent a good deal of time choosing the phrase “Complex and Global Systems” and discussing its meaning. In addition, unlike the other advisory committees, we met for an extra day and held a seminar on our topic.
By global systems, we mean, of course, those that relate to the biosphere of our planet as a whole and to the life forms in it, including us human beings. Both of the chapters in this section have to do with global systems.
The word “complex” is a little trickier to interpret. Some people use the term “complex systems” to refer to anything composed of many parts (even if those parts are rather similar, like ions in a crystal). I believe that is an abuse of language unless the parts exhibit a great deal of diversity.
In my view, and that of many of my colleagues, something is complex if its regularities necessarily take a long time to describe. It then has what I call “effective complexity.” The plot of a novel is complex if there are many important characters, a number of subplots, frequent changes of scene, and so forth. A huge multinational corporation is complex if it has many subsidiaries selling different goods and services, with a variety of management styles suited to the various countries in which they operate. The United States tax code is complex. So is Japanese culture.
Note that effective complexity refers to the minimum description length for the regularities, not the random or incidental features. Randomness is not what is usually meant by complexity.
In my talks on simplicity and complexity, I often make use of three neckties to illustrate my ideas. One of those ties has a very simple pattern of stripes in three colors, repeated over and over again with the same spacings between the stripes. The next one, (an Ermenegildo Zegna tie, the same brand Monica is supposed to have given the President) has a more complex pattern—its regularities would take somewhat longer to describe. The last one, a hand-painted tie from Austin, Texas, has a very complex pattern, with its regularities requiring a very long description
indeed. In this discussion I restrict myself to the pattern, ignoring things like soup stains, wine stains, and so forth.
But the choice of features to study is to some extent subjective. To a dry cleaner, for example, the pattern of the tie may be incidental, while the character of the stains is a crucial regularity.
How does one separate the total information about something into a part describing regularities and a part describing random or incidental features? Much of the discussion in the chapters by Pascual and Rahmstorf is precisely about this sort of issue. Where is a practical place to draw the line between pattern or regularity and what is treated as random or incidental?
A very useful way to characterize the regularities of an entity is to imagine that it is embedded in a set of entities that display its regularities and differ in incidental or random features, the entity being described and the other, imaginary ones are assigned probabilities. The set is then what is called an “ensemble,” as, for example, in statistical mechanics. Thus we arrive at a two-part description of the entity: first the ensemble to which it belongs and then the address—within that ensemble—of the specific entity itself. That is similar to what we do on a computer when we utilize a basic program and then feed specific data into that program. The regularities are like the basic program and the incidental features are like the additional data.
Those of us who study the fundamental laws that govern the behavior of all matter in the universe seem to be finding that those laws are very simple. However, they are probabilistic rather than fully deterministic. The history of the universe is thus codetermined by those simple laws and an inconceivably long string of chance events —“accidents” that could turn out in different ways with various probabilities. In this way the history we experience is embedded in an ensemble, a branching tree of possible histories, with the accidents at the branchings. In a splendid short story by Jorge Luis Borges, someone has constructed a model of the branching histories of the universe in the form of a Garden of Forking Paths.
In the history we experience, when each chance event happens, only one of the possible outcomes is selected. Before it happens, then, each future accident is a source of unpredictability, of randomness. But once it occurs, a past accident may be an important source of regularity.
Some of those chance events produce a great deal of regularity, at least over a considerable stretch of space and time. I call those events “frozen accidents.” They are responsible for most of the effective complexity that we see around us.
Think of all the accidents that have produced the people who have contributed to this volume. Start with the little quantum fluctuation that
led to the formation of our galaxy. That fluctuation was presumably not very important on a cosmic scale, but it was very important indeed to anything in our galaxy.
Then there were the accidents that led to the formation of the solar system, to the structure of our own planet, to the origin of life here on Earth, to the particular course of biological evolution, to the characteristics of the human race. Consider also the accidents of sexual selection, of sperm meets egg, that led to the genomes of everyone here. And then the accidents of development, in the womb and in childhood, that produced the adult human beings.
An example from biology of what seems to be a frozen accident is the fact that right-handed sugars and left-handed amino acids play important roles while the corresponding mirror image molecules do not. Some theorists have tried to account for that asymmetry by utilizing the left-handed character of the weak interaction for matter as opposed to antimatter. However, in forty years no one has succeeded, as far as I know, in making such an explanation stick. It seems that the asymmetry must be an accident inherited from very early forms of life.
Among those few historians who are willing to consider contingent history (speculating about what would have happened if something had gone differently), it is fashionable to discuss an incident that occurred in 1889. Buffalo Bill’s Wild West Show was touring Europe and reached Berlin. One of the most important acts was, of course, that of Annie Oakley, the female sharpshooter. She would ask for a male volunteer from the audience who would light up a cigar that she would then shoot out of his mouth. Generally, there were no such volunteers and her husband, himself a famous marksman, would step forward instead. On this occasion, however, there was a volunteer, the Kaiser, Wilhelm II, who had succeeded to the throne the previous year upon the premature death of his father. He took out an expensive Havana cigar, removed the band, clipped off the end, and lit it, waiting for Annie to shoot. She was a bit worried, having drunk heavily the night before, but she took aim and fired. We know the result. But what if things had gone differently?
Work on complex systems involves a mix of general principles and of specific details that result from history, indeed from historical accident. Take the study of the oceans and global climate. The rotation of the Earth matters, along with its period—the day—the length of which has varied over time. The length of the year and the inclination of the equator relative to the plane of the Earth’s orbit also matter. So do the layouts of the oceans and the continents. (Those, too, were different in the past, of course.) There are specific ocean currents and specific wind systems, and
so on and so forth. That is what makes for complexity. In comparison, the basic physics of the air and the water is in general fairly simple.
However, another consideration enters besides effective complexity. How hard is it to calculate the consequences of the physical laws? How much computation is necessary to solve the equations to the needed accuracy? That is related to what is called logical depth.
Often it is not easy to tell whether we are dealing with effective complexity or with logical depth. When I was a graduate student fifty years ago we wondered what kind of dynamics underlay the structure of atomic nuclei. It appeared that the laws governing the energy levels of nuclei would turn out to be extremely complex. Today, however, we have the theory of quantum chromodynamics—the field theory of quarks and gluons—supplemented by quantum electrodynamics, which describes the electromagnetic interaction. We now believe that the combination of those simple theories yields the energy levels of nuclei to an excellent approximation, but the calculations are so elaborate that even today’s machines and techniques cannot handle them properly. If, as we believe, the theories are correct (and many of their predictions have been verified by experiment), the energy levels of nuclei have very little effective complexity, but a great deal of logical depth.
Just as apparent complexity may reflect either effective complexity or logical depth or both, so too can apparent randomness reflect either fundamental indeterminacy or an effect of ignorance. Even in the classical deterministic approximation, indeterminacy can arise as a consequence of coarse graining, which refers to the level of detail at which a system is being studied. Neither observation nor calculation can be carried out with perfect accuracy, and so there is necessarily some coarse graining. But the coarse-grained system is no longer deterministic, because the coarse-grained past does not determine the coarse-grained future, but only the probabilities of various coarse-grained futures. That effect is particularly striking when the system is nonlinear and exhibits the phenomenon known as chaos, in which variations in the tiniest details of the initial state can produce big changes later on.
In connection with chaos, I am reminded of an incident that took place on the European tour promoting my book, The Quark and the Jaguar. I stopped in Barcelona, where my Spanish editor Jorge Wagensberg was also the director of the Science Museum, which had a wonderful exhibit on chaos. It featured a nonlinear version of a pendulum. The visitor was supposed to grab hold of the bob, let it go from a certain position and with a certain velocity, and watch the motion, which was also recorded on a drum by a pen. Then the visitor was supposed to grab the bob again and try to repeat the operation, reproducing as closely as possible the original
position and velocity. No matter how carefully that was done, the subsequent motion came out quite different from what it was the first time, beautifully illustrating the meaning of chaos in science. While I was playing with the nonlinear pendulum, I noticed two men who were idling nearby, and I asked Jorge what they were doing there. “Oh,” he replied, “those are two Dutchmen waiting to take away the chaos”. It turned out that the exhibit was to be taken to Amsterdam as soon as I finished inspecting it. But I have often wondered whether many wealthy organizations wouldn’t pay huge fees to these Dutchmen who could take away chaos.
The coupled systems of oceans and climate discussed by Stefan Rahmstorf and the coupled systems treated by Mercedes Pascual clearly exhibit both effective complexity and logical depth. Even though the basic equations involved are not that complicated, the specific features of the planet contribute a great deal of effective complexity, and the calculations are very difficult. As a result, a great deal of ingenuity is required in setting up the approximations, especially in choosing the right level of coarse graining in time and space and other variables.
Essentially, one is constructing an approximate model. Indeed, we have learned from our winners a great deal about the fine art of modeling, in connection with the dynamics of cholera and of fisheries —both interacting with the physical and chemical properties of bodies of water—and in connection with the interplay of climate, vegetation, and ocean currents.
At the Santa Fe Institute, which I helped to start and where I now work, researchers from a great many disciplines come together to investigate theoretical problems connected with simplicity and complexity and, in many cases, with complex systems. They construct models of real complex systems. (Usually these are computer models, but sometimes they are analytical ones or a mixture of the two.) Often these models are highly simplified. But how does one compare simple approximate models of complex systems with observation? What can one claim on the basis of such a model? Because it is very approximate, we would be embarrassed to have it agree accurately with observation! At our institute, we have had many discussions about these questions.
One answer is that it is often possible to identify features of the real system that obey, to a reasonable degree of accuracy, phenomenological laws that can be traced all the way from that real system through a sequence of approximations to the simple model under consideration. Then one can claim for the model that it helps to explain at least those features and those phenomenological laws. (Scaling laws are an example.) In some cases, one can also get a feeling for the level of accuracy to be expected for predictions based on the model.
When complicating a model by adding details, parameters, or assumptions, one has to decide, of course, whether the additional predictability is worth the extra complication. At the ends of the spectrum of detail, absurd situations are encountered. Mercedes Pascual, in that beautiful quotation from Borges, refers to one of those ridiculous limits, in which the model is simply a faithful reproduction of the entire system. At the other end of the spectrum one finds models that are so vague and general that they refer, very approximately, to a generic planet and lack most of the real characteristics and history of our own Earth. It is necessary to strike a wise compromise and avoid approaching either limit.
Both chapters contain just this kind of reasoning. Incorporating descriptive scaling laws into dynamics, as Mercedes Pascual recommends, is a good example of what I am discussing here. Mercedes Pascual and Stefan Rahmstorf are facing up brilliantly to the challenge of finding suitable models, in ecology and in the system of climate and oceans, balancing considerations of complexity, accuracy, and computational difficulty. They are carefully weighing how much special information must be included and how much should be left out. This kind of work will constitute, in my opinion, one of the main foci of theoretical research in the early decades of the new century. The intricate dance of simplicity and complexity, regularity and randomness, defines one of the most important frontiers of science today, particularly when it is applied to the history and likely future of our planet and its life forms, including us human beings.