Posted by Hindol Datta
|Chaos is inherent in all compounded things. Strive on with diligence! –Buddha|
Scientific theories are characterized by the fact that they are open to refutation. To create a scientific model, there are three successive steps that one follows: observe the phenomenon, translate that into equations, and then solve the equations.
One of the early philosophers of science, Karl Popper (1902-1994) discussed this at great length in his book – The Logic of Scientific Discovery. He distinguishes scientific theories from metaphysical or mythological assertions. His main theses is that a scientific theory must be open to falsification: it has to be reproducible separately and yet one can gather data points that might refute the fundamental elements of theory. Developing a scientific theory in a manner that can be falsified by observations would result in new and more stable theories over time. Theories can be rejected in favor of a rival theory or a calibration of the theory in keeping with the new set of observations and outcomes that the theories posit. Until Popper’s time and even after, social sciences have tried to work on a framework that would allow the construction of models that would formulate some predictive laws that govern social dynamics. In his book, Poverty of Historicism, Popper maintained that such an endeavor is not fruitful since it does not take into consideration the myriad of minor elements that interact closely with one another in a meaningful way. Hence, he has touched indirectly on the concept of chaos and complexity and how it touches the scientific method. We will now journey into the past and through the present to understand the genesis of the theory and how it has been channelized by leading scientists and philosophers to decipher a framework for study society and nature.
As we have already discussed, one of the main pillars of Science is determinism: the probability of prediction. It holds that every event is determined by natural laws. Nothing can happen without an unbroken chain of causes that can be traced all the way back to an initial condition. The deterministic nature of science goes all the way back to Aristotelian times. Interestingly, Aristotle argued that there is some degree of indeterminism and he relegated this to chance or accidents. Chance is a character that makes its presence felt in every plot in the human and natural condition. Aristotle wrote that “we do not have knowledge of a thing until we have grasped its why, that is to say, its cause.” He goes on to illustrate his idea in greater detail – namely, that the final outcome that we see in a system is on account of four kinds of influencers: Matter, Form, Agent and Purpose.
Matter is what constitutes the outcome. For a chair it might be wood. For a statue, it might be marble. The outcome is determined by what constitutes the outcome.
Form refers to the shape of the outcome. Thus, a carpenter or a sculptor would have a pre-conceived notion of the shape of the outcome and they would design toward that artifact.
Agent refers to the efficient cause or the act of producing the outcome. Carpentry or masonry skills would be important to shape the final outcome.
Finally, the outcome itself must serve a purpose on its own. For a chair, it might be something to sit on, for a statue it might be something to be marveled at.
However, Aristotle also admits that luck and chance can play an important role that do not fit the causal framework in its own right. Some things do happen by chance or luck. Chance is a rare event, it is a random event and it is typically brought out by some purposeful action or by nature.
We had briefly discussed the Laplace demon and he summarized this wonderfully: “We ought then to consider the resent state of the universe as the effect of its previous state and as the cause of that which is to follow. An intelligence that, at a given instant, could comprehend all the forces by which nature is animated and the respective situation of the beings that make it up if moreover it were vast enough to submit these data to analysis, would encompass in the same formula the movements of the greatest bodies of the universe and those of the lightest atoms. For such an intelligence nothing would be uncertain, and the future, like the past, would be open to its eyes.” He thus admits to the fact that we lack the vast intelligence and we are forced to use probabilities in order to get a sense of understanding of dynamical systems.
It was Maxwell in his pivotal book “Matter and Motion” published in 1876 lay the groundwork of chaos theory.
“There is a maxim which is often quoted, that “the same causes will always produce the same effects.’ To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more than once, so that the causes and effects cannot be the same in all respects. There is another maxim which must not be confounded with that quoted at the beginning of this article, which asserts “That like causes produce like effects.” This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied: but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of the points cause a railway train to run into another instead of keeping its proper course.” What is interesting however in the above quote is that Maxwell seems to go with the notion that in a great many cases there is no sensitivity to initial conditions.
In the 1890’s Henri Poincare was the first exponent of chaos theory. He says “it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.” This was a far cry from the Newtonian world which sought order on how the solar system worked. Newton’s model was posted on the basis of the interaction between just two bodies. What would then happen if three bodies or N bodies were introduced into the model. This led to the rise of the Three Body Problem which led to Poincare embracing the notion that this problem could not be solved and can be tackled by approximate numerical techniques. Solving this resulted in solutions that were so tangled that is was difficult to not only draw them, it was near impossible to derive equations to fit the results. In addition, Poincare also discovered that if the three bodies started from slightly different initial positions, the orbits would trace out different paths. This led to Poincare forever being designated as the Father of Chaos Theory since he laid the groundwork on the most important element in chaos theory which is the sensitivity to initial dependence.
In the early 1960’s, the first true experimenter in chaos was a meteorologist named Edward Lorenz. He was working on a problem in weather prediction and he set up a system with twelve equations to model the weather. He set the initial conditions and the computer was left to predict what the weather might be. Upon revisiting this sequence later on, he inadvertently and by sheer accident, decided to run the sequence again in the middle and he noticed that the outcome was significantly different. The imminent question that followed was why the outcome was so different than the original. He traced this back to the initial condition wherein he noted that the initial input was different with respect to the decimal places. The system incorporated the all of the decimal places rather than the first three. (He had originally input the number .506 and he had concatenated the number from .506127). He would have expected that this thin variation in input would have created a sequence close to the original sequence but that was not to be: it was distinctly and hugely different. This effect became known as the Butterfly effect which is often substituted for Chaos Theory. Ian Stewart in his book, Does God Play Dice? The Mathematics of Chaos, describes this visually as follows:
“The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month’s time, a tornado that would have devastated the Indonesian cost doesn’t happen. Or maybe one that wasn’t going to happen, does.”
Lorenz thus argued that it would be impossible to predict the weather accurately. However, he reduced his experiment to fewer set of equations and took upon observations of how small change in initial conditions affect predictability of smaller systems. He found a parallel – namely, that changes in initial conditions tends to render the final outcome of a system to be inaccurate. As he looked at alternative systems, he found a strange pattern that emerged – namely, that the system always represented a double spiral – the system never settled down to a single point but they never repeated its trajectory. It was a path breaking discovery that led to further advancement in the science of chaos in later years.
Years later, Robert May investigated how this impacts population. He established an equation that reflected a population growth and initialized the equation with a parameter for growth rate value. (The growth rate was initialized to 2.7). May found that as he increased the parameter value, the population grew which was expected. However, once he passed the 3.0 growth value, he noticed that equation would not settle down to a single population but branch out to two different values over time. If he raised the initial value more, the bifurcation or branching of the population would be twice as much or four different values. If he continued to increase the parameter, the lines continue to double until chaos appeared and it became hard to make point predictions.
There was another innate discovery that occurred through the experiment. When one visually looks at the bifurcation, one tends to see similarity between the small and large branches. This self-similarity became an important part of the development of chaos theory.
Benoit Mandelbrot started to study this self-similarity pattern in chaos. He was an economist and he applied mathematical equations to predict fluctuations in cotton prices. He noted that particular price changes were not predictable but there were certain patterns that were repeated and the degree of variation in prices had remained largely constant. This is suggestive of the fact that one might, upon preliminary reading of chaos, arrive at the notion that if weather cannot be predictable, then how can we predict climate many years out. On the contrary, Mandelbrot’s experiments seem to suggest that short time horizons are difficult to predict that long time horizon impact since systems tend to settle into some patterns that is reflecting of smaller patterns across periods. This led to the development of the concept of fractal dimensions, namely that sub-systems develop a symmetry to a larger system.
Feigenbaum was a scientist who became interested in how quickly bifurcations occur. He discovered that regardless of the scale of the system, the came at a constant rate of 4.669. If you reduce or enlarge the scale by that constant, you would see the mechanics at work which would lead to an equivalence in self-similarity. He applied this to a number of models and the same scaling constant took effect. Feigenbaum had established, for the first time, a universal constant around chaos theory. This was important because finding a constant in the realm of chaos theory was suggestive of the fact that chaos was an ordered process, not a random one.
Sir James Lighthill gave a lecture and in that he made an astute observation –
“We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proved incorrect.”