Editor's comment

The above image of the Mandelbrot set is an example of how extreme mathematical complexity can arise from the iteration of very simple rules.

In an increasingly complex world new concepts and methods are needed to understand and deal with complex systems.


Reading on: Complexity

Golden, Richard Self-Organizing Systems: a resource for teachers 1997 [700 words] — on complexity

The concept of complexity is hard to define because it is a subjective idea. We tend to think that those things that are complicated as complex. But complicated things can become simple with understanding. Is their complexity then lost? The ordinary driver looks under the hood of a modern automobile and is bewildered by the intricacies. The experienced mechanic says, "Look! It’s simple." He tweaks a gizmo and the stalled engine comes to life. What is complex seems to depend, to some extent, on what the observer brings to the observation.

Recognizing complexity— 5 ways
1. Internal complexity
There are different ways in which a system can be considered to be complex. The first of them is when a great many independent parts within a system are interacting with each other in a great many different ways and many interactions are occurring simultaneously. This is internal complexity. The complexity arises from the organization and the interactions. A clock may have internal complexity and still produce a relatively simple action.

2. Behavioral complexity
Regardless of a system’s internal complexity, a system can be considered to be complex if its behavior is controlled by many variables or is unpredictable. Simple systems are predictable; they offer no surprises. If a system surprises us, acts in ways that are counterintuitive, we assume that complex functions are at work. Such a system has behavioral complexity. In mathematics, for example, the “simple” equation z —> z2 + c describes “the most complex mathematical object ever invented,” the Mandelbrot set. When the behaviors of the values generated by the iteration process, using the equation, are plotted on the computer screen they produce images that are unbelievably varied and unexpectedly beautiful. (The illustration above is a magnification of a tiny portion of the Mandelbrot set.)
The computer is to the study of complexity what the microscope was to microbiology. It makes the study of the field possible.

3. Structural complexity.
A system with many feedback loops is hard to understand because of all of the subtle and changing inputs and can therefore be considered to be complex. Such a system can be said to have structural complexity. The system is continuously modifying itself by changing the interactions among its variables. Consider the difference between a greenhouse and an ecosystem.

4. Informational content
This is the idea that complexity of a system is measured by the length of the shortest possible description of that system. In this view, if the most concise description that can be made of a system is lengthy the system is complex. If it is easily described it is simple. (For contrast with this view of complexity consider the Mandelbrot set mentioned above.)

5. Hierarchical depth or nested systems within systems
The more levels or systems a particular system contains, the more complex it is. The universe, containing all systems, is the most complex system we know. An atom used to be considered the most elementary (meaning simple) particle of matter. We now know that it is a complex structure made up of protons, neutrons and electrons. The atomic nucleus itself, rather than a ball of protons and neutrons, is now seen as a seething nest of quarks and gluons interacting in bewildering complexity.

If, in general, systems tend to increase in complexity, what might account for this? One hypothesis is that increased complexity expands the ability of a system to respond to a changing environment. This would enhance the system’s survival. Some organisms, notably human beings, expand their response to that of control over the environment. This concept involves the notion that systems “progress” by becoming more “advanced” and that means greater control of their surroundings and their lives. The philosophical discussion of the relationship between increased complexity and freedom of more complex organisms is based on the above ideas…

An individual system may exhibit many of the different kinds of complexity described above and may do so simultaneously. Researchers have begun to try to develop a science of complexity and move it away from the subjective frame of reference. This is an important task. In our increasingly technological world we are beginning to recognize our need to deal with complex systems. New concepts and methods are required.