An Introduction





Peter Corning


Tuesday, February 19, 2008

Complexity is Just a Word!

What is complexity, asks author-journalist George Johnson in the science section of The New York Times a few years back? Below the headline, "Researchers on Complexity Ponder What It's All About," Johnson reports that there is still no agreed-upon definition, much less a theoretically-rigorous formalization, despite the fact that complexity is currently a "hot" research topic. Many books and innumerable scholarly papers have been published on the subject in the past few years, and there is even a journal, Complexity, devoted to this nascent science. Johnson quotes Dan Stein, chairman of the physics department at the University of Arizona: "Everybody talks about it. [But] in the absence of a good definition, complexity is pretty much in the eye of the beholder."

This is not to say that the researchers in this area have not been trying to define it. In the 1970s, Gregory Chaitin and Alexei Kolmogorov (independently) pioneered a mathematical measuring-rod that Chaitin called "algorithmic complexity" -- that is, the length of the shortest "recipe" for the complete reproduction of a mathematical treatment. The problem with this definition, as Chaitin concedes, is that random sequences are invariably more complex because in each case the recipe is as long as the whole thing being specified; it cannot be "compressed".

More recently, Charles Bennett has focused on the concept of "logical depth" -- the computational requirements for converting a recipe into a finished product. Though useful, it seems to be limited to processes in which there is a logical structure of some sort. It would seem to exclude the "booming, buzzing confusion" of the real world, where the internal logic may be problematical or only partially knowable -- say the immense number of context-specific chaotic interactions that are responsible for producing global weather "patterns", or the imponderable forces that will determine the future course of the evolutionary process itself.

A number of researchers, especially some of those who are associated with the Santa Fe Institute, believe that the key lies in the so-called "phase transitions" between highly ordered and highly disordered physical systems. An often-cited analogy is water, whose complex physical properties lie between the highly ordered state of ice crystals and the highly disordered movements of steam molecules. While the "Santa Fe Paradigm" may be useful, it also sets strict limits on what can be termed "complex". For instance, it seems to exclude the extremes associated with highly ordered or strictly random phenomena, even though there can be more or less complex patterns of order and more or less complex forms of disorder -- degrees of complexity that are not associated with phase transitions. (Indeed, random phenomena seem to be excluded by fiat from some definitions of complexity.)

To confuse matters further, a distinction must be made between what could be labeled "objective complexity" -- the "embedded" properties of a physical phenomenon and "subjective complexity" -- its "meaning" to a human observer. As Timothy Perper has observed (on-line communication), the equation w = f(z) is structurally simple, but it might have a universe of meaning depending upon how its terms are defined. Indeed, information theory is notorious for its reliance on quantitative, statistical measures and its blindness to meaning -- where much can be made of very few words. The telephone directory for a large metropolitan area contains many more words than a Shakespeare play, but is it more complex? Furthermore, as Elisabet Sahtouris has pointed out (on-line communication), the degree of complexity that we might impute to a phenomenon can depend upon our frame of reference for viewing it. If we adopt a broad, "ecological" perspective we will see many more factors, and relationships, at work than if we adopt a "physiological" perspective. When Howard Bloom (on-line communication) quotes the line "To see the World in a Grain of Sand..." from William Blake's famous poem, "Auguries of Innocence", it reminds us that even a simple object can denote a vast pattern of relationships, if we choose to see it that way. Accordingly, subjective complexity is a highly variable property of the phenomenal world.

Perhaps we need to go back to the semantic drawing-board. Complexity is, after all, a word -- a verbal construct, a mental image. Like the words "electron" or "snow" or "blue" or "tree", complexity is a shorthand tool for thinking and communicating about various aspects of the phenomenal world. Some words may be very narrow in scope. (Presumably all electrons are alike in their basic properties, although their behavior can vary greatly.) However, many other words may hold a potful of meaning. We often use the word "snow" in conversation without taking the trouble to differentiate among the many different kinds of snow, as serious skiers (and Inuit Eskimos) routinely do. Similarly, the English word "blue" refers to a broad band of hues in the color spectrum, and we must drape the word with various qualifiers, from navy blue to royal blue to robin's egg blue (and many more), to denote the subtle differences among them. So it is also, I believe, with the word "complexity"; it is used in many different ways and encompasses a great variety of phenomena. (Indeed, it seems that many theorists, to suit their own purposes, prefer not to define complexity too precisely.)

The "utility" of any word, whether broad or narrow in scope, is always a function of how much information it imparts to the user(s). Take the word "tree", for example. It tells you about certain fundamental properties that all trees have in common. But it does not tell you whether or not a given tree is deciduous, whether it is tall or short, or even whether it is living or dead. The same shortcoming applies also to the concept of "complexity". Although there may be some commonalities between a complex personality, a complex wine, a complex piece of music and a complex machine, the similarities are not obvious. Each is complex in a different way, and their complexities cannot be reduced to an all-purpose algorithm. Moreover, the differences among them are at least as important as any common properties.

What in fact does the word "complexity" connote. One of the leaders in the complexity field, Seth Lloyd of MIT, took the trouble to compile a list of some three dozen different ways in which the term is used in scientific discourse. Yet this exercise produced no blinding insight. When asked to define complexity, Lloyd told Johnson: "I can't define it for you, but I know it when I see it."

Rather than trying to define what complexity is, perhaps it would be more useful to identify the properties that are commonly associated with the term. I would suggest that complexity often (not always) implies the following attributes: (1) a complex phenomenon consists of many parts (or items, or units, or individuals); (2) there are many relationships/interactions among the parts; and (3) the parts produce combined effects (synergies) that are not easily predicted and may often be novel, unexpected, even surprising.

At the risk of inviting the wrath of the researchers in this field, I would argue that complexity per se is one of the less interesting properties of complex phenomena. The differences, and the unique combined properties (synergies) that arise in each case, are vastly more important than the commonalities. If someone does develop a grand, unifying definition-description of complexity, I predict that it will add very little to the tree of knowledge (pardon the pun). But that shouldn't deter us from trying; the very effort to do so will surely enrich our understanding.

Thought for the day: Complexity is a qualitative property that we apply to both apples and oranges -- to borrow a cliché. They are both fruits and grow on trees but also differ from each other in important ways. Despite the many fruitless attempts (pardon the pun) to develop a general definition for the term, perhaps its only universal trait is that it taxes the human mind.


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Peter Corning