A Close Look at a New Science: Chaos as Science or Science in Chaos?

D. C. Mikulecky
Department of Physiology
Medical College of Virginia Commonwealth Univ.
Box 980551 MCV Station
Richmond, VA 23298-0551
U. S. A.

Introduction: The Many Faces of Chaos

The concept of chaos has emerged as a central topic in science in the last few decades. It has also emerged as popular topic in the lay press. Some authors have commented on historical roots to the concept which go back for centuries. The meaning of the word "chaos" has therefore become many faceted and the scientific use of the word has therefore become enmeshed in a rather complex web.

These problems surrounding the word chaos reflect a wider set of problems that science is facing. These problems all stem, in one way or another, from the fact that science has evolved in a way which cuts itself off in a very complete way from its philosophical foundations. Most of the new trends in science use this lack of philosophical grounding rather than try to remedy the problem. This becomes a more far reaching issue when the practice leads to the misinforming of the public.

Language about "revolution" the inadequacies of "reductionism" and related concepts abound in the new writings, especially those writings aimed directly at the layman. Meanwhile much of science goes about its business unaware and uninterested in these claims at revolution. This phenomenon raises the obvious question of whether or not the self proclaimed "revolution" is really a revolution or merely a passing fad. An alternative possibility is that the revolution is merely old wine in new bottles.

There is, therefore, a real need for a close look at the philosophical aspects of the new writings using chaos as their springboard. The related issue of "complexity" has been reviewed from this perspective recently [Mikulecky, 1995]. In that case, as well as in this review, the writings of one person are the only solid foundation available to build such a discussion. In his work, Robert Rosen has faced the philosophical problems squarely and has presented a thorough, systematic way of examining the way we do science. His conclusions preceded the more recent declarations that reductionism was inadequate. He built on that thesis and defined "complexity", "emergence", and other now popular terms using the same analysis that ultimately leads to a more rigorous definition of the distinction between machines and organisms. Central to this analysis of science and how it operates is the use of the modeling relation as a tool for seeing the way we use models, analogs, simulations, and metaphors. The distinction between these different ways of dealing with our understanding of the natural world is an important component to sorting out the different claims being made and trying to get estimates of their validity.

Chaos as a Mathematical Object

In its most abstract and formal manifestation, chaos is a mathematical object which results from the numerical solution of certain non-linear differential equations or, in one case, a difference equation. In the language to be developed here, this is chaos as an aspect of a particular formal system, namely non-linear dynamics. This aspect of a well known formal system needs to be evaluated carefully as a component to a model for the natural world. The popularity of the concept seems to come more from its value as a metaphor. There seem to be very few cases in which a modeling relation is even attempted, let alone established (see discussion of the modeling relation below). This is consistent with the increased reliance on computer simulation as a substitute for theoretical models. The deficiencies inherent in doing things this way are subtle, if only because we have no established criteria for evaluating our practices, especially since our capacity to simulate or otherwise deal with non-linear mathematical systems in a more or less empirical way using computers has been increasing at an extremely rapid rate.

Attempts to give chaos a rigorous, "scientific" definition have been anything but successful and the word seems doomed to become a poorly defined buzz-word in the same sense as the word "complexity" [Mikulecky, 1996]. The reason for this apparent loss of a term with great significance has some lessons behind it. Not the least of which is the fact that chaos, as a concept, has been around for centuries and only recently has become a part of the vocabulary of hard science [Briggs and Peat, 1985; Abraham, 1994]. As a mathematical object it is a bit more definable: " ...aperiodic bounded dynamics in a deterministic system with sensitive dependence on initial conditions." [Kaplan and Glass, 1995]. The question for science is how good a model is such an object for natural systems? (See the modeling relation below).

In the following brief review, it should become clear that the use of chaos, the mathematical object, as anything but a metaphor is fraught with difficulties which stem from the very nature of the formalism and which have no way of being eliminated. This leads to the ironic result that the very core of much of hard science is the very agent for making a softer, more qualitative approach inescapable! Thus, the dilemma that led Poincar‚ to come up with his qualitative approach has been resurrected [Barrrow-Green, 1997; Abraham and Marsden, 1978].

Does Chaos Have an Edge?

To compound the issue, the notion that things happen "at the edge of chaos" has now become a very fashionable concept. So much so that one of the originators of that piece of terminology, Stuart Kauffmann, [Kauffmann, 1993,1995] has become

"...a research scientist who is leading the charge to bring complexity theory to a realm of life not usually touched by theoretical biology."

says Joseph Marshall in a recent issue of Working Woman magazine [Marshall, 1997]. The edge of chaos notion will require some examination, but before that, it is interesting to note that the issue of chaos is dealt with in about a paragraph in this article:

"What can complexity theory do for business?......It is not to be confused with chaos theory, which was last year's trend. Chaos theory concentrated on the pieces-the millions of causes and effects that add up to an elaborate Rube Goldberg device.......".

The article goes on to explain how "complexity theory" has something important to say about how systems (businesses are systems for this discussion) can become "self-organized".

"...For him [Kauffmann], the key idea is a kind of neo-darwinism, by which it is not just the individuals in a system who determine their own fitness. somewhat miraculously, the system itself is trying to help, much as Adam Smith's "invisible hand" of capitalism helps distribute the greatest amount of happiness to the greatest number of people."

Later in the article, the "edge of chaos" gets some definition:

"...The catch phrase that complexity operates at the edge of chaos is a way of referring to the internal tumult that, almost despite itself, coheres into order."

Finally, after briefly mentioning others who write about "complexity theory" and it's applications in the business world, the following assessment is offered:

"It is perhaps a natural tendency of all scientific theory to be yanked violently out of context in an attempt to find its human relevance.Thus, Darwinism led to social Darwinism, and Plato gave us the platonic relationship. Certainly science can provide provocative new ways to look at the world, but it seems useful to distinguish theory from metaphor. As a rule of thumb, it may be safest to rely most heavily on the ideas that are hardest to understand. By that standard, kauffmann's version of complexity wins hands down; it is a complexity that is truly complex."

So we have a winner. Let us examine Kauffmann's writings before we proceed to examine the edge of chaos notion further, only to return to it later.

The science of butterfly flight

We need to look into Kauffmann's version of the new science for some definitions and for a perspective about what is actually new and different about his approach. The idea of what chaos is relative to complexity (in Kauffman's terminology, the edge of chaos) is brought forth in the discussion of dynamical systems and their description in diagrammatic form in pictures called "phase portraits". These diagrams are curves in a space whose coordinates are a characteristic set of dynamic variables which are followed over time to define the evolution of the system. Kauffmann tells us [Kauffmann 1993, pp175]

"The most natural language for describing the behavior of an integrated system is dynamical systems theory.... To be concrete suppose there are three chemicals reacting in a vessel. The rate of formation and disappearance of chemical depends on the concentrations of (1) those chemicals either forming it or influencing its formation and (2) those influencing its conversion to other chemicals. In addition, each of the three chemicals may be added to or removed from the vessel, or their concentrations may be held constant or caused to change in arbitrary ways by outside forces. The most natural representation of such a system is a three dimensional state space."

This would be fine if we did not have the accompanying account of the failure of reductionist thinking in these complex systems and the promise that we are being shown something new. A state space representation is fine for a dynamical system. The dynamical system is a formal system, it is well defined and certainly is tightly coupled to the idea of states. However, it is not only not the best model for a complex system, it is very inadequate, and finally contains a strongly contradictory notion. Here is where Kauffmann stops using models of natural systems and begins to use metaphors instead. The problem is that he never seems to notice that he is making the change. He is heading for an account of 30 years worth of computer simulations of things like Boolean networks so it is understandable that he would not want to move too far away from the reductionist mode, especially with respect to computer simulation.

In dynamic system's theory, the state of the system at any time is then represented by a point in the state space and as the system evolves this point will move tracing out a curve in the three-dimensional space. Such a curve is called a trajectory of the system and these trajectories are the solution to a set of differential equations called the system's equations of motion. This procedure for representing a system is a direct analogy of the way particle motion is handled in Newtonian dynamics.

Kauffman then cites theorems which assure us that trajectories never merge. They may, however converge on a single curve or even to a single point called an attractor and all the trajectories which converge on a given attractor sculpt out a kind of landscape feature called a basin of attraction. Most attractors for simple systems are isolated in their basins and the "landscape" is fairly uncomplicated with simple hills (seperatrices) dividing the basins of attraction from each other. Next the landscape can become more complicated involving toruses or other objects. Finally: "... In addition to these classes of attractors , strange, or "chaotic", attractors exist...." These strange attractors have some special properties:

  1. They can be highly structured and consist of many loops all very near each other. The result is that two trajectories can start out very close to each other and subsequently diverge to very different regions of the phase space. This is the property known as "sensitivity to initial conditions".
  2. The second feature is that they are highly structured and fill only a portion of the state space. They therefore have a low fractal dimension. This brings about the notion of an intrinsic "order in chaos.

The sensitivity to initial conditions feature leads Kauffmann among many others to make the following comment:

"...this sensitivity to initial conditions is amusingly called the butterfly effect. a butterfly in the Amazon might, in principle, ultimately alter the weather in Kansas."

This little example is so often repeated as an example of what chaos theory teaches that it seems irreverent to point out how ridiculous an idea it is! Yet it is precisely this kind of statement that exemplifies the problems that these models present to us. Some of the problems result from the context in which such discussions are held and others from the very origin of the ideas themselves. Let us look at some of them.

Chaos as a Model: The Modeling Relation.

The butterfly effect example is amusing to Kauffmann, but it would seem for the wrong reason. The little example assumes that Lorenz's model of weather patterns which was one of the first demonstrations of "chaos" in a dynamic system somehow can be extended to deal with global weather patterns and their interconnectedness. Secondly...if one butterfly can cause all that trouble, what do we do with the fact that there are millions of them?

Is this a frivolous criticism? I think not. It is at the heart of the issue if we consider non-linear dynamics as a model for real complex systems [Rosen, 1993]. To see this better we need to examine what we do when we chose a mathematical formalism as a model of some real world phenomenon. I find Rosen's modeling relation the best way of illustrating this [Rosen 1985]

The Modeling Relation:

The modeling relation is a formal description of the way we do science. It is, among other things, a description of the bringing together experiment and theory [Rosen, 1985]. It consists of two distinct systems, both which become incorporated into our thought process. The first, on the left in the diagram, is called the natural system. The second, on the right in our diagram, is called the formal system. Although there may be other variations, in general, the natural system enters our mind as a set of sensory inputs, a sort of "raw data" which we will call percepts. This set of percepts constitutes the "alphabet" for an activity of our mind which is very basic and which is the underlying reason for our belief that the world is something other than a set of events which have no relationship to each other. Thus at an early stage, our influence on the system we are manipulating in our thought process is inevitable! We automatically form relationships between the percepts and then see these relationships as data as well, something we observed out there. These relationships between percepts we will call linkages. It is the creation of such linkages which leads directly to our sense of causality in the universe. This is symbolized in the diagram by the arrow labeled 1. We have created elaborate methods for systematizing our interactions with our surroundings, the most elaborate of which is what has come to be called "the scientific method". One purpose of the scientific method is to create well defined sensory methods through measuring instruments which assign quantitative labels to our percepts and their linkages in order to minimize the effect of our status as observer. Thus we find it effective to create labels for the percepts which we designate as observables. A side-effect of this strong emphasis on "objectivity" is that things which are not readily quantified are often relegated to a status of not being "worthy" of becoming objects of scientific study. This results in notions like the distinction between "hard" and "soft" science. The formal system, on the other hand, is totally our creation, or, to those who believe that formal systems have independent existence and can be "discovered" like natural systems, they can be at least chosen by us to use in a given modeling relation.

The formal system is therefore the means by which we get to play god. It has a structure so similar to the way we handle percepts that it is very tempting to assume that we construct formal systems in a manner which mimics our handling of natural systems. The basic "units" of a formal system also consist of objects in parallel with the percepts in the natural system. We can see them as mathematical objects such as sets, dynamical systems, etc. Then as with percepts, there are mappings and relations within and between them.

Also, in the manner of the natural system, we then symbolize these objects with the members of and alphabet and construct the relations and mappings as axioms, production rules, and algorithms or programs. These latter manipulations allow us to mimic the causality we have associated with the orderliness of "natural law". We call these manipulations in the formal system implication represented by arrow 3 in the diagram.

The natural and formal systems are more or less self-contained. They have no necessary relationship to each other or anything else outside themselves. More pointedly, they have nothing in themselves to tell us how to relate them to each other. This is why the making of model can never be a totally "scientific" activity. It will always contain a strong component of "art". It is no accident that the computer has allowed a new dimension into epistemology by extending the realm of science beyond laboratory experimentation to a new kind of experimentation on purely formal systems.

The artistic steps in the modeling relation are symbolized by arrow 2, the act of associating a particular formal system or subsystem with the natural system which we call encoding. Then we perform an operation within the formal system designed to mimic a causal event in the natural system and must follow up by interpreting the result, the act of decoding symbolized by arrow 4. It is not hard to see that the ability to quantify aspect of both systems makes the encoding and decoding steps much easier.

The criteria for a model to result is that the diagram "commutes" in other words the process of encoding, implication, and decoding "matches" the causal event. Symbolically,

(1) = (2) + (3) + (4).

It should be clear that like the "artistic" nature of the encoding and decoding steps, the overall determination of whether or not the diagram commutes is also something not contained within any of the aspects of the model itself. This too must be supplied from outside the model itself, and can be the subject of considerable controversy.

Analog models and metaphors:

Once the modeling relation is understood it can easily be extended to help us understand much of epistemological activity and used to clear up some difficult problems in methodology. One illustration is the ability to encode and decode a number of different natural systems into a single formal system. This makes any of the natural systems a substitute for the formal system in the modeling relation. Examples of analog models are the entire subject of Network Thermodynamics [Mikulecky, 1993] and also the field of non-linear dynamics itself. In a very real any dynamic system can be an analog for any other.

If we see correspondence between a formal system and a natural system in such a way that it becomes possible to make predictions about the natural system without encoding it, we have only the top part to the modeling diagram and the relationship is now a metaphor rather than a model. The most famous and insidious of these is the "machine metaphor" which was spawned by Descartes, when he likened the living system to a machine.

The problem: Chaos as model or metaphor?

What then is wrong with non-linear dynamics and the resultant strange attractors, etc. as the formal system to use as models for things like weather and dripping faucets and turbulence in streams etc.?

The problem has many facets. The first of which is the very technical point that in order for a model to exist there must be and encoding of a natural system into some formalism. As we have seen, the formalism which may exhibit chaos as one form of implication is non-linear dynamics. Systems are not encoded into "chaos" and chaos is not, by itself, a formal system. The formal system called non-linear dynamics may exhibit chaos under the right circumstances, but this result potentially becomes a prediction of behavior in the natural system, not really a model of it. The next issue is more troublesome from a philosophical point of view. The encoding and decoding in a dynamic system model is usually accomplished by the writing and solving of differential equations and the comparison of these solutions with experimental data. The difficulty with chaos as a "prediction" is the very property that made this particular kind of solution so astounding in the first place. To see this we must also recognize the third problem, namely that the formal system giving rise to a chaos is a set of differential equations representing a system with no environment! The butterfly story makes this so clear. The equations' solution depends on one environmental interaction, the initial conditions. Then, as is always the case in dynamic systems theory, time runs along smoothly and all the trajectories are known for all time because the system is isolated from the rest of the world and has no further interactions with it! How is it possible to miss the significance of this feature of the formalism? The answer to this question lies in the history and philosophy of science. This flaw exists throughout the Newtonian Paradigm and has been with us for a long time. It is the basis for the reductionist method. It was never seen as a problem because, in response to it as a problem, systems were broken down into simpler systems until the problem was no longer so severe. When it came time to look at systems more holistically, the problem became manifest as "emergence" and "complexity". This aspect of the failure of dynamics to describe faithfully the interactive nature of a complex system is developed carefully by Rosen in the references cited as well as by Kampis [Kampis, 1991]. The entire environmental influence is lumped into the "sensitivity to initial conditions." The meaning of this still is devastating since the decoding of such an implication into the natural system has no obvious way to be carried out any longer! The situation makes a fit with data impossible. What remains is to simply move from model to metaphor and talk about the strong similarity. An example of a real system will make this clearer.

The double scroll attractor

The point that a test of the fit to data is no longer possible with chaotic systems is made in the description of a chaotic system's strange attractor in electronics, the "double scroll" attractor and related attractors [Matsumoto, Chua and Komuro, 1985]. Here the strange attractor can either be produced by computer simulation using a circuit simulation program called SPICE [Mikulecky, 1993] or it can actually be realized by a simple electronic circuit involving a linear resistor, two linear capacitors, a linear inductor, and a nonlinear resistor. Thus, a simple circuit with only one non-linear element is sufficient to produce a very exotic strange attractor.

What is especially interesting about this paper is the appearance of two notes and the language which leads into these notes:

"...The usage of the word "chaotic attractor" is of course not rigorous in this paper as well as in others, in the sense that its existence has not been proven mathematically.3 However, we have succeeded in providing a physical proof by designing and building a physical circuit whose equation of motion is modelled4 by [a set of three coupled ordinary differential equations, one of which is non-linear due to the non-linear resistor.]

Here are the two notes:

3) "It has been argued by many researchers that "chaotic attractors" observed by digital simulation are of questionable validity because chaotic systems are by nature extremely sensitive to local truncation and round-off errors."

4) "Of course, due to component tolerances, the physical circuit .... is not exactly modeled by [the set of differential equations] with the parameters specified..... However, the fact that this circuit exhibits a chaotic attractor on an oscilloscope shows that [the set of differential equations] is indeed a robust model."

This is not the first nor will it be the last time hard science has to use such caveats. The philosophical ramifications are rather interesting if they are viewed using the modeling relation. We have three objects involved here. We have the physical circuit, which is merely some hardware wired together (It is important to note that the "non-linear resistor" is realized by a fairly elaborate subcircuit involving an operational amplifier). Then we have the experiment which can be done on the circuit which involves measuring currents and/or voltages as observables in various parts of the circuit, and finally, we have the mathematical model, which is never directly solved by analytical mathematics but which is rather digitally simulated on a special simulation program.

The first disclaimer the authors make is that the existence of the chaotic attractor has not been proved mathematically. The fact that an actual circuit can be put together which gives measured values of the simulated parameters which strongly resemble the simulated behavior is found to be highly suggestive of a "physical proof" of the fact that the "equations of motion" of the system do generate the attractor.

This is an interesting dilemma. Normally, we would check out a mathematical model by trying to adjust simulation parameters and initial conditions to produce as near a super position between the measured values of the system's observables and the computed values from the equations or a simulation. In this case we are faced with an impossibility. The paper is mainly devoted to a detailed analysis of the attractor and it is clear at a glance that this is indeed a complicated geometric entity. It is described as follows:

"...Microscopically speaking, the two thin "rings" of the double-scroll attractor are made of infinitely many layers of points compressed into a thin sheet (think of infinitely many sheets of "lead" being hammered into one conglomerate sheet). Macroscopically, a good way of describing the "anatomy" of the above attractor would be a "double-scroll" structure since the two sheet-like objects are curled up together into spiral forms with infinitely many rotations-while maintaining some space between the two scrolls which gradually decreases, thus causing them to meet eventually at some limit point."

Harking back to the "butterfly effect" and the synonymous sensitivity to initial conditions mentioned above as one of the two defining characteristics of strange attractors, in this case the words "infinite" as applied to the number of layers of points which are "squeezed into a relatively small structure as well as the existence of "infinitely" many rotations of curling up gives the picture of the dilemma quite well. Given some data to "fit" to this model how could one ever be sure one had enough data to pick the correct curve among so many so close together? And since a wrong choice could easily send the system off on a divergent path on the model relative to the one the measurement should have been assigned to, the situation is definitely hopeless. But we have been here before in the course of scientific history and we managed to find a way out. In fact, the way out was furnished by Poincaré‚ when he discovered chaos many years ago. It is the so-called "neo-qualitative approach" [Abraham and Marsden, 1978]. This approach is characterized by the adoption of a global geometric point of view. An advantage of the model so obtained is that the full generality of the theory becomes apparent when "unnecessary" coordinates are suppressed. A second advantage is the replacement of analytical methods by differential-topological ones in the study of the phase portrait. A third aspect of this approach is the emergence of a new question- the question of structural stability, the question addressed directly by Rene Thom's "catastrophe theory". Notice that in every aspect of the approach, the detailed relation of parts to whole is being sacrificed to enable something to be salvaged.

In a very real way, the reductionist world-view is hoist in it's own petard. Start with deterministic equations, make a model which is so unrealistic as to assume that the world exists in parcels which can be dynamically described by isolating them totally from any environment, and then you are hit with strange attractors, butterfly effects, and are forced to forsake the analytical approach and look at the system more holistically anyway! What is more the irony of all this seems to be totally missed within the context of the discussions that go on in hard science.

Back to the edge of chaos?

Let us summarize what we have said up to this point. Chaos itself is not a formal system which can be used to model natural systems. It is, at best, a possible metaphor for some of their aspects. Now where does that leave the notion of another important region at the edge of chaos where everything happens? The idea that new laws of the universe can be found based on these ideas needs to be examined very carefully. The origin of the idea comes from studies of cellular automata by Wolfram and their further interpretation by Kauffmann and others. The crux of these studies involves a massive amount of computer simulation, usually in the form of cellular automata, but also in Boolean networks and other systems.

The issue of simulation

Simulation is a relation between formal systems. It might be viewed as the modeling relation with the natural system being replaced by a formal system. This then requires that the causal events on the left now be reinterpreted as implications in the same manner as those on the right. Then the goal becomes one of simulating the implication on the left with that on the right. This can easily be done by numerical methods in the case of dynamic systems, for example. It is less obvious, but also true, that the use of cellular automata, Boolean networks, Network Thermodynamic simulations using SPICE, and any other computer generated representation of some system of interest is merely the realization of one formal system by another. This alone never constitutes a model in the sense of the modeling relation. Furthermore, since the result of a simulation is usually compared with data, there is no decoding into the formalism on the left from the simulation, but this is circumvented and a direct comparison is made with the natural system. The idea is that if the simulation compares favorably with the data from the natural system, the formalism being simulated is then a good model for the natural system.

The simulator is therefore is common to both the natural system and the formal system being used as a candidate for its model. Hence, to the extent that the computer simulations now being used to study "complexity" are to be seen as actual models of natural systems and not merely metaphors, they must be carefully translated back into the formal system they represent and that formal system must be shown to encode from and decode into the natural system in question. To the extent that formal system is dynamics, the simulation has nothing in itself to say about a system's complexity. Dynamics is not by itself a way of modeling complexity. Complex systems are systems which transcend dynamics and require more for their modeling. Furthermore, much being done in computer simulation is at best the making of metaphors. In these cases, their value in modeling the natural world is extremely limited.

As long as these minimal requirements for correspondence between the natural world and formal systems are ignored, the confusion will continue to increase. The apparent advances due to computer simulation will then be seen to be stumbling blocks rather than advances. It is time for a theory of models to be taken seriously [Casti, 1994 ] for until then the major source of chaos may indeed be science itself!

Hard and Soft Science and theSokal Affair

At the risk of alienating both sides in this debate, it might be constructive to suggest that everyone read the "Praeludium" to Rosen's book Life Itself. As in any controversy of this magnitude, no one has a corner on truth and there is some difficulty with the positions taken by all the participants. (For anyone who has not yet heard of the Sokal Affair, it is a recent episode in the hard science vs soft science dialectic revolving an article by Physicist Alan Sokal in the Journal Social Text. The article was revealed by Sokal to be a hoax at the very time it was published. It is possible to see this affair as being related in some very telling ways to the issues raised in this review.

My comments about the article in Working Woman can easily be interpreted as critical of it's author's view of the role of chaos and certain versions of "complexity research". This is particularly true of the account of Kauffmann's version of complexity theory being the most acceptable to that author merely on the grounds of its own complexity. One might see that sentiment paraphrased as saying that we should give more credence to Kauffmann's version of complexity theory because it is more like hard science.

The hard scientists accuse the soft scientists of being sloppy, anti-intellectual, and other characteristics which all seem to boil down to the notion that methods and practices of hard science, though imperfect, are the best means we have at getting at the truth, all others being clearly inferior if they have any merit at all. The soft scientists in their criticism of hard science point to the obvious arrogance of that stance and the obvious consequences of blinding one's self to alternatives in the name of "objectivity". Rosen sees this issue as one of "...simplicity vs complexity." He claims that "...There is, as yet, no comprehensive investigation of the ideas I have sketched....they are too new. But it seems that such ideas, or ideas like them, are necessary in many ways."

Just what are these ideas? They revolve around an entire new notion of complexity. It is unfortunate that this is the very word that is used so much in the new science of chaos and its broader home as defined by Kauffmann and others. What is the difference? It is not in the sentiments expressed by either subgroup among the complexity theorists. Both groups are clear in their acknowledgement that complexity involves a holistic approach and can't be approximated by traditional reductionist methods.

The differences are more subtle. Most discussions among readers of rosen's work miss these differences unless they are carried on for some time and are allowed to dig very deep. Robert Rosen does not skirt the issue of epistemology in his work he confronts it straight on. For that reason, he moves into areas that hard science has forsaken for a very long time. The result of his efforts seems to clearly take us where we need to go (I am tempted to say "want to go", but after being dragged there kicking and screaming even though I was sympathetic, I know better). And here is the point. This is what the Sokal affair as well as the new revolution in complexity research have in common. The systematic division between hard and soft science and between science and philosophy has caused what appears to be a severe form of blindness and lack of understanding among those of us who willingly took the path away from a more global, interdisciplinary notion of what it means to know something. We are, without struggle and hard work, unable to hear or see any merit in these epistemological tangles. We are caught up in the very kind of self-referential knot that Rosen and others have warned us about. We won't look into epistemology because our epistemology has no place in it for looking into epistemology!

Rosen allows into his analysis of the issue of what complexity is necessary epistemolgical considerations which in turn direct his scientific pursuit which in turn raises new epistemological questions. One choice is to ignore such tangled thought processes and accept what the broader community offers as a science of chaos and complexity. Notice how easily what I am saying here could be confused with the way the evaluation of complexity approaches was carried out in Working Woman. The difference should be clear, however. I am recommending Rosen's approach because I am convinced of its correctness and in spite of its difficulty, not because of it.

In summary, it would seem that the normal outcome of this long standing dialectic ought to be a form of synthesis. It is hardly likely that one side or the other will win. It seems like the efforts of Robert Rosen are the first steps along such a synthetic route. It might be very productive to go further.


Abraham, R. (1994) Chaos, Gaia, Eros, Harper Collins, NY.

Abraham, R. and C. D. Shaw (1987) Dynamics: A Visual Introduction, in Self- Organizing Systems: The Emergence of Order, (E. Yates, ed.), Plenum, NY, pp 543-598.

Abraham, R. and J. E. Marsden (1978) Foundations of Mechanics, Benjamin/Cummings, Reading, MA. (Introduction pp xvii - xxii).

Barrow-Green, J. (1997) Poincaré‚ and the Three Body Problem, Am. Math. Soc., Providence, R. I.

Briggs, J. and F. D. Peat (1989) Turbulent Mirror, Harper & Row, N. Y.

Casti, J. L. (1994) Complexification Harper Perrenial, NY.

Glass, L. and M. C. Mackey (1988) From Clocks to Chaos: The Rhythms of Life, Princeton Univ. Press, Princeton, NJ.

Hayles, N. K. (1991) Introduction: Complex Dynamics in Literature and Science, in Chaos and Order: Complex Dynamics in Literature and Science, (N. K. Hayles, ed.), Univ. Chicago Press, Chicago, pp 1-36.

Gleick, J. (1987) Chaos: Making a New Science, Viking, NY.

Horgan, J. (1995) From Complexity to Perplexity, Scientific Am. June:104-109.

Kampis, G. (1991) , Self-Modifying Systems in Biology and Cognitive SciencePergamon, NY.

Kaplan, D. and L. Glass (1995) Understanding Nonlinear Dynamics, Springer- Verlag, NY.

Kauffmann, S. (1993) The Origins of Order: Self-organization and Selection in Evolution, Oxford, NY.

Kauffmann, S. (1995) At Home in the Universe: The Search for the laws of Self Organization and Complexity, Oxford, NY.

Lewin, R. (1992) Complexity: Life at the Edge of Chaos, Macmillan, NY.

Marshall, J. (1997) Creativity at the Edge of Chaos, Working Woman, Feb.:47- 53.

Matsumoto, T., L. Chua, and M. Komuro (1985) The Double Scroll, IEEE Trans. on Cir. and Sys. vol CAS-32:798-818.

Mikulecky, D. C. (1993) Applications of Network Thermodynamics to Problems in Biomedical Engineering, NYU Press, NY.

Mikulecky, D. C. (1996) Complexity, communication between cells, and identifying the functional components of living systems: some observations, Acta Biotheoretica 44:179-208.

Rosen, R. (1993) Some Random Thoughts About Chaos and Some Chaotic thoughts About Randomness, J. Biol. Sys. 1:19-26.

Rosen, R. (1985) Anticipatory Systems: Philosophical, Mathematical & Methodological Foundations, Pergamon, NY.

Waldrop, M. M. (1992) Complexity: The Emerging Science at the Edge of Chaos, Touchstone, NY.

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