You may have noticed that in previous sub-units I've been talking a lot about complexity in an informal way. But I haven't yet defined the term "complexity". That's for a reason. It turns out that "complexity" is hard to define. Or, more precisely, it has too many different definitions in different fields. So how do complex systems researchers measure the complexity of a system? Seth Lloyd's paper called "Measures of Complexity: A Nonexhaustive List" gives something like 42 different definitions or ways of measuring complexity, including Shannon information, algorithmic complexity, fractal dimension, thermodynamic depth, etc., etc. Is there a single comprehensive and useful definition of complexity? It's very doubtful. These different definitions are useful for measuring different aspects of systems. In this course, we'll talk in depth about two of these: Shannon information and fractal dimension; how they're used, and why they might be useful. We'll also discuss the general problem of defining and measuring complexity in the real world. Before I go on, I want to mention the ideas about complexity in one excellent, classic, and very prescient paper, called Science and Complexity, written by the American mathematician Warren Weaver in 1948. Weaver divided the problems of interest in science into three categories. The first category he called Problems of Simplicity. These are problems that involve just a few variables. Some examples might be relating pressure and temperature in thermodynamics; or, in electricity, relating current, resistance, and voltage; in population dynamics, relating population versus time. These are all problems that were dealt with in the 19th and early 20th centuries, in physics, chemistry, biology, and so on. Then Weaver goes on to a second category that he calls Problems of Disorganized Complexity. These are problems involving billions or trillions of variables. So one example would be understanding the laws of temperature and pressure, as arising from trillions of disorganized air molecules in a room or in the atmosphere. These are understood through taking averages over the large set of variables. When we look at understanding temperature, we don't look at the particular position and energy of every individual air molecule. Rather, we understand temperature as the average energy over the trillions of molecules. And the science of averages comes under the rubric of statistical mechanics, which deals with these kinds of problems. The key here is that we're assuming very little interaction among the variables. That's what allows us to take meaningful averages. In the case of the temperature of a gas, the whole is the sum, or equivalently, the average of the parts. Weaver's last category was the problems of organized complexity. These are the problems that include the examples I gave earlier, the problems of interest to complex systems researchers. These are problems that involve a moderate to large number of variables. But the key here is that, due to their strong nonlinear interactions, the variables cannot be meaningfully averaged. Again, we'll talk more precisely about what "nonlinear" means in the next unit. Weaver characterized these as "problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole". So this really gets at the notion of emergence. This "organic whole" refers to the emergent behavior of the system. In his paper, Weaver gives a beautiful list of questions as examples of problems of organized complexity. It's very striking to note that, even though Weaver's paper was published in 1948, all of these problems point to issues that are still open questions in complex systems science almost seven decades later. I'll just run through a few of his questions here. What makes an evening primrose open when it does? What is the description of aging in biochemical terms? What is a gene, and how does the original genetic constitution of a living organism express itself in the developed characteristics of the adult? On what does the price of wheat depend? How can currency be wisely and effectively stabilized? How can one explain the behavior pattern of an organized group of persons such as a labor union, or a group of manufacturers, or a racial minority? Weaver went on to say "These problems... are just too complicated to yield to the old 19th century techniques, which were so dramatically successful on two-, three-, or four-variable problems of simplicity. These new problems, moreover, cannot be handled with the statistical techniques so effective in describing average behavior in problems of disorganized complexity." And going even further, Weaver said, "These new problems - and the future of the world depends on many of them - require science to make a third great advance, an advance that must be even greater than the 19th-century conquest of problems of simplicity or the 20th-century victory over problems of disorganized complexity. Science must, over the next 50 years, learn to deal with these problems of organized complexity." Well, it's been almost 70 years since Weaver wrote this paper. And a major purpose of this course is to let you know how far we've gotten in dealing with problems of organized complexity, and what new tools has the science of complexity developed to deal with them. Even though we're not going to go deeply into formal definitions in this class, let's do a bit more to investigate the question, "What is a complex system?" which has many different possible answers. Let's go straight to the experts. The next sub-unit gives a sampling of answers to this from several of the best-known experts in the field. Note that, while there's a great deal of variation in the answers, there is some convergence, as well, on a definition.