Chaos and Philosophy

Philosophical Musings on the Implications of Chaos Theory on the Human Understanding of the Natural World
Steven Jackett, April 24, 1994

We humans seem to have a natural penchant or curiosity or passion for attempting to understand the natural world around us. We want to understand all the forces of nature in the universe and their interactions that are at work determining the things that we see happen. We want to write equations representing the interactive relationships of these natural forces and substitute in the initial condition values and calculate future predicted states of the system (i.e. the universe). We observe things that occur in the world surrounding us and we try to explain them with hypothesized theories that should be intuitive, certainly must be consistent with our observations of the real world, and are further subject to rigorous mathematical proof. Still however, the ever evasive epistemological questions remain: How should we go about the approach to knowing these things?, How can we be sure of the certainty of what we think we know?, and Are there things that simply cannot be known by the human mind?

Given to us by Rene Descartes in the Platonic tradition, the predominant Western scientific method of proof is called the Axiomatic or Geometric Method. This approach begins with assumed, not proven, not questioned axioms or postulates that are generally accepted by everyone because they are obviously true. A proof proceeds from hypothesis to conclusion using a set of agreed upon simple rules of logical reasoning that allow step by step procession from each statement, which implies the next by strict logic until the proof is complete. Every theory or proposition that a particular hypothesis implies a particular conclusion must be proven in this way, along with any others that it may depend on so that all knowledge follows from the original premises, assumptions, axioms, postulates, and hypotheses. This serves as an effective way of preserving the most amount of certainty possible in the accumulation of knowledge, as well to make sure our theories remain consistent (non-contradictory) with each other; it also stresses the dependence of each theory on all the others.

Future experience can and should also lead to subsequent refinements of the theory while always holding to the constant requirement that the theory match observations of the real world. These explanatory theories lead to collections of equations that together describe the behavior of natural systems in our universe over time, space, or whatever. In this way, we develop abstract mathematical models to represent the real world behavior that we observe in natural systems in our surrounding environment.

The degree of accuracy or preciseness to which these models describe the real behavior of the system depends directly proportional to the completeness and refinement of our understanding of the forces behind or causing the effects that we see. We must keep in mind at all times, however, that the model is limited to those determining factors that our current understanding led us to take into consideration, which quite likely is incomplete. Each determining factor in the system has a different magnitude of effect, or weight, or how strong the effect is; we probably consider only those factors whose effects are significant.

The classical or Newtonian approach that seems to have most influenced Western thought, implies that we can understand natures physical laws that govern natural effects to the extent that we can write a finite number of simple linear equations containing a finite number of variables or determining factors that govern the behavior of natural complex dynamical systems that we observe in our environment. This approach contends that we can then measure all the initial conditions to be input into the system of simultaneous equations that model the behavior of the natural system, resulting in precise, absolute predictions of any future state of the system in time or space or whatever. This is the classical deterministic model of our understanding which contains the concept of reductionism, saying that individual knowledge of all the parts of the system add up to the understanding of the system as a whole.

However, through history, our research and experiment has gradually shown us that things tend to be much more complicated than simple linear relationships, and that whole systems exhibit emergent properties that are not found in any of the parts, but result from their interactions. There is a completely extreme view that certain occurrences in nature, especially over small- scale time or distance, occur entirely at random. Randomness vs. classical determinism has become a popular polarized duality to speculate over. I see these two as the extreme limits at opposite ends of a symbolic, representative, and theoretical spectrum or scale upon which to measure the natural world against our ideas. Neither of these two ideal extreme cases actually occurs in the natural world, but the duality serves as a useful tool providing information for our understanding. However it must remain in mind at all times that this way of understanding has been arbitrarily imposed on the natural world by the human mind. There are infinitely many other scales or spectrums upon which to measure and gain information and understanding about the natural phenomena.

The approach taken in the human inquiry into the nature of nature certainly depends on language and therefore culture primarily. Different cultures and their languages contain entirely different thought patterns, logical processes, and associations. Each different approach or measuring instrument might give us new and different information and/or might disguise some other understanding. This also results in much difficulty communicating philosophical concepts across cultures. The superposition of information gained from several different approaches, discarding rare or infrequent contradictions, will provide the most comprehensive understanding. Although, there may be a certain paradox concerning our attempt to completely understand the workings of a system in which we are deeply enmeshed as an inter-related, interdependent, interacting part. Several philosophers in epistemology including David Hume, Immanuel Kant, and Charles Saunders Peirce have referred to the "limits of human understanding", saying that there are some things that we simply have no way of knowing.

Our current study of Chaos Theory is an important step in the gradual realization that things are considerably more complicated than a classical Newtonian deterministic interpretation, but contrary to what the word "chaos" might suggest, they are not completely random without any kind of underlying pattern or order for us to study. The word "chaos" was originally a misnomer because there really is underlying pattern and order for us to study, but it is complicated; we are changing the meaning of the word "chaos". The natural complex dynamical systems that we study are modeled mathematically by chaotic systems of multiple simultaneous non-linear integro- differential equations in multiple variables that exhibit certain properties of "chaos", including sensitive dependence on initial conditions. These abstract mathematical models of chaotic systems that we see in our natural world can consist of an infinite number of equations in an infinite number of variables or determining factors, each with an infinite number of terms including every exponent (degree or order) of each variable, every derivative, every integral, every function, etc., along with the appropriate coefficients for each term determined by the strength or weight of each terms particular effect.

Therefore to proceed with the classical deterministic approach, we must measure and start with an infinite number of initial conditions as well as an infinite number of subsequent conditions that must be input all along the way which also affect the outcome or future state of the system. These initial and subsequent conditions have also been determined by the same governing laws of nature relating forces, having started from previous initial and subsequent conditions in space and time. Furthermore, this infinite complexity of natural systems is yet again complicated by the infinite number of these systems that are all inter-connected, inter-related, interdependent, and interacting. Also, the way we define these systems and draw boundaries between them at different scales in both time and space is completely arbitrary and is imposed on the continuous range of the naturally occurring phenomena by the human mind or by cultural patterns of thought. Although there is usually an obvious way to draw those boundaries according to natural patterns that will provide the most accurate understanding.

These recent realizations of the complexity of the natural universe seem to eliminate the possibility of achieving absolute determinism at least in the classical sense of the word. However, a more inclusive interpretation of the word "determinism" can be more easily adhered to. Even though it must be humanly impossible to fully grasp and understand all of the relationships of forces that govern the behavior of natural phenomenological systems in the universe to the extent that we can write equations and calculate solutions, it is easily presumable that every observable result or outcome or condition has been completely determined by some natural interaction of forces, conditions, or relationships however many and however complicated those interactions may be. I believe that no occurrence is completely random, but certainly determined by some complex natural process. Randomness does not really exist in the natural universe, but only as an abstract mathematical model, an extreme or ideal case at one end of a spectrum for understanding completely imposed on reality by man.

These recent realizations made possible in part by the study of chaotic complex dynamical systems that model natural systems in the universe suggest that a shift in paradigm, or our approach to asking and answering questions regarding our understanding of the natural world, may be in order. As we gain further information and understanding, we sometimes realize that maybe a different question would have been more appropriate to have asked from the beginning. As another mechanism built-in to the natural system we are a part of to assure the most amount of certainty possible, by causing us to periodically question and possibly change the current paradigm. We gain insightful information and an accurate model by using an entirely different approach called feedback analysis or recursive loop iterations of the same equations with variable parameters where each next value depends on the previous one. This is opposed to substituting future values for variables representing time, space, or whatever into empirical equations to get future states or conditions of the system without iterating through all the recursive steps. This new approach makes intuitive sense because then knowledge and information processed within the system need not be across the entire range, but only very locally in time, space, or whatever. This type of mathematical model seems to more accurately represent the observable characteristics and abilities of the natural world. It remains ever important not to confuse the model for the real. Each cell in the leaf of a tree need only communicate with its local neighboring cells in space, and only over the very recent past and the very near future in time, to proceed with proper development consistent with the overall shape, size, appearance, and function of the leaf and the whole tree that works so well that we are so intimately familiar with. This feedback analysis approach still remains completely deterministic, but is extremely complex and requires much intense tedious work most suitable for computers not humans.

An important method for understanding the behavior of natural complex dynamical systems by feedback analysis modeling uses a graphical representation of the simplest of chaotic systems of equations. Much information and understanding regarding the behavior of the natural complex dynamical system can be seen by examining the graphical representation. Certain critical points on the graph correspond to parameter values at which bifurcation in periodic cycles of the behavior of the natural system occur.

These bifurcations compound themselves leading to chaotic unpredictable behavior of the system. Most important of these models is the general logistic equation whose graphical representations include the Mandelbrot Set and the infinite number of Julia Sets that play an integral role in the understanding of the universal behavior of natural complex dynamical systems whenever two or more factors interact in a non-linear way, resulting in attractors, repellors, fixed points, periodic and wandering cycles, etc. in the long-term iterative feedback analysis. These figures have boundaries with fractal dimensions which indicates the infinite degree of complexity that is at work in the natural world as illustrated in these graphical representations.

These recursive feedback equations that describe the behavior of natural systems exhibit chaotic behavior, like sensitive dependence on initial conditions, at certain parameter values. In his paper on complex systems, Ollar Stone Fuller pointed out to us that natural systems have a common tendency to operate very near the threshold parameter values that result in chaotic behavior. He suggests that this allows the most efficient processing of information between all the parts of the system over time and space that is necessary for proper maintenance and function of the system. I also find this idea intuitively sensible because this operating point near the edge of chaos is where the system will find the most freedom and ability to drastically change the overall direction, outcome, or state of the system itself with only minor changes in initial, current, or subsequent conditions due to the chaotic property of sensitive dependence on initial conditions. This freedom and ability of natural chaotic systems to change is integrally important in the Darwinian process of natural selection due to adaptation and mutation of species over time as part of the natural behavior of the universe, the largest system. The sensitive dependence on initial conditions characteristic of chaotic systems is similarly responsible and important for the systems' resilient ability to return to its steady-state or equilibrium operating point after an outside disturbance, perturbation, or excitement. I speculate that these properties or characteristics are also similarly involved with natural systems' inherent abilities to self-organize and self replicate. Over the long-term or large-scale time or distance, overall properties or characteristics of the system tend to outweigh the effects of short-term or small-scale causes.

The feedback analysis way of modeling nature also promises to at least partially describe the way that the DNA hold all the stored genetic information to replicate and self-organize the life of the organism (system) in the way we know works best, since the infinite complexity of the systems' overall information is contained within the recursive feedback equation and is revealed after many iterations over large-scale time and space. Classical deterministic equation modeling would require an infinite amount of space to store the information necessary to replicate and self-organize the complexity of the organism (system), and an infinite amount of time to calculate the outcome. Nature, of course, knows the outcome instantaneously, it is only we humans who have to calculate it. Recursive feedback analysis accommodates both of these infinite complexities completely within the genes of the DNA!