DEFINITION OF COMPLEXITY
D. C. Mikulecky
Professor of Physiology
Medical College of Virginia Commonwealth Universityhttp://views.vcu.edu/~mikuleck/
1.0 Why is "What is complexity" a question not so easily answered?
For some time we have been being told that there is a "new science" called complexity. Universities and other research institutions have programs in "complexity research" and journals carry this word in their title. What is complexity? What does it mean to be "complex"? The dictionary does not help us here. There is more to the idea than what a dictionary definition suggests.
If we turn to science where words are carefully defined and have more precise meanings we find that, in this case, things are not much better and may even be worse! It is here where we need to look, nevertheless, since it is here where the answer does lie. One person who was quick to exploit the failure of the scientific community to get together on this concept was John Horgan. In his essay in his June 1995 Scientific American editorial entitled "From complexity to perplexity", he mentions 31 definitions of complexity and associates the concept with:
He also points out the lack of a "unified theory" of complexity.
It is also worth noting that nowhere in his essay does he mention the definition which will be given here and which, to my satisfaction, completely clears up the confusion.
Later, in his book, The End of Science, he adds some more fuel to the fire. There he develops a stance which places all discussion which does not fit the mold we shall refer to as the Newtonian Paradigm into a category called "ironic science" about which he says: "At its best, ironic science, like great art or philosophy, or, yes, literary criticism, induces wonder in us; it keeps us in awe before the mystery of the universe. But it can not achieve its goal of transcending the truth we already have. And it certainly cannot give us-in fact protects us from- The Answer, a truth so potent that it quenches our curiosity for all time. After all, science itself decrees that we humans must always be content with partial truths."
Horgan doesn't know it but he did a good job of giving validity to the concept called "complexity" here. In fact he also doesn't know that the very definition furnished explains most of what he is concerned about and clears it up. I can be sure that he doesn't know it because the only place he ever refers to Robert Rosen is in his book where he says: "…and Robert Rosen, a Canadian biologist who was at the workshop…" It seems clear he has little knowledge of Rosen's work if this is all he has to say about it.
This may seem silly, but the entire real world is complex! That's right! So why the problem? The answer lies in the nature of science. This is what Horgan was struggling with in his book.
What Horgan defends, as "strong science" is essentially what we will call "hard science". By hard science we mean the model for all science that is the embodiment of the so-called "scientific method". Since everyone probably thinks they know what this is, it is worth spelling it out in some detail, just so we know what I am really talking about.
Descartes is attributed with the popularization of the machine metaphor and he did it in a very interesting way. He saw the body as a biological machine and the mind as something apart from the body. This is called Cartesian Dualism and survives to this day as one approach to the so-called mind/body problem. What the machine metaphor did was to set the tone for modern science. It has lasted since that time. Descartes really did not know what a machine is, or if he did, he never told anyone. Ironically, not only do we not have a good definition of complexity, but we also lack one for a machine. The importance of this metaphor is in the intuitive concept of machine that almost everyone shares. A machine is built up from distinct parts and can be reduced to those parts without losing its machine-like character. We call this idea "Cartesian reductionism". We will see that this is not true for complex (real) systems except under very special circumstances. Cartesian reductionism does not work for complex systems; it only reduces them to simple mechanisms.
Newton gave us three laws of motion, which were intended to describe the motion of the planets. It turned out that these laws could be applied in a seemingly perfectly general way. This broader application has been the foundation of the modern scientific method and will be referred to here as the Newtonian Paradigm.
In the center of this paradigm is dynamics. Dynamics is the way the laws of motion get applied. The local description of the motion is formulated as a differential equation called an equation of motion. The equation of motion is manipulated by using the calculus (integrated) and results in a trajectory, which is an algebraic equation for calculating a particle's position as a function of time. Later, this was made somewhat more complicated by quantum mechanics, but the central philosophy was never changed.
The paradigm has been generalized from particle motion to all systems if we recognize that quantum mechanics is part of that generalization. When we look carefully at the subject matter of physics, we see that it is the application of the Newtonian Paradigm to the universe. This application then makes the world into simple mechanisms. That is to say that the subject matter of physics is the study of simple mechanisms. Note that in this context, "simple" means the opposite of complex, not the opposite of complicated.
The success of the Newtonian Paradigm cannot be ignored. Most of modern science and technology is the result of it. For that reason alone it is difficult to suggest that it has limits and to then make that suggestion stick. Not only does the paradigm have limits, but also those limits are what gave rise to a concept like complexity.
The world therefore divides naturally into those things that are simple and those things that are complex. The real world is made up of complex things. Therefore the world of simple mechanisms is a fictitious world created by science or, more specifically, by physics as the hard version of science. This is the world of the reductionist. It is modeled by the Newtonian Paradigm and simply needs sufficient experimentation to make it known to us. Those experiments involve reducing the system to its parts and then studying those parts in a context formulated according to dynamics.
The paragraph above is one of the most controversial things one can say about modern science. It flies in the face of all that everyone is taught. Yet, it really does no harm to those teachings. It simply puts them into perspective. What is needed is to see that is the picture of how science is done.
How is science done? It is a combination of using our senses to observe the world around us and then to use some mental activity to make sense out of that sensory information. The process is what we will call the modeling relation. If we call the world we are observing and/or trying to understand the Natural System and the events that make it change as we observe causality, then that represents our object of study. What we do in our minds is to encode the natural system into another system that is of our making or choosing which we can call a formal system. Once we have chosen a formal system, we can manipulate it in various ways with the objective of mimicking the causal change in the natural system. These manipulative changes in the formal system we will call implication. Finally, once we think we have an appropriate formal system and have found an implication that corresponds to the causal event in nature, we must decode from the formal system in order to check its success or failure in representing the causal event. The following diagram represents the modeling relation we have just described.
If all the parts of the diagram are in harmony, in other words if 1 = 2 + 3 + 4, we say that the diagram commutes and we have a model. A model of the world is the outcome of a successful application of the scientific method, but it can also arise in other, less formal ways. Whenever someone tries to make sense out of the world, they are trying to construct a successful modeling relation, or a model.
Now the definition of complexity is complete. The world, from which we single out some smaller part, the natural system, is converted into a formal system that our mind can manipulate and we have a model. The world is complex. The formal system we chose to try to capture it can only be partially successful. For years we were satisfied with the Newtonian Paradigm as the formal system, forgot about there even being and encoding and decoding, and gradually began to change the ontology so that the Newtonian Paradigm actually replaced or became the real world (at least as seen through the eyes of science). As we began to look more deeply into the world we came up with aspects that the Newtonian Paradigm failed to capture. Then we needed an explanation. Complexity was born! This easily can be formalized. It has very profound meaning.
Complexity is the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties. It requires that we find distinctly different ways of interacting with systems. Distinctly different in the sense that when we make successful models, the formal systems needed to describe each distinct aspect are NOT derivable from each other.
Now we have it! Rosen spent his life refining this idea. There is far, far more to it. We will spell all that out very carefully.