Short navigation among chaos, complexity and chance.

The mathematics of chaos is far from mature and sufficiently developed to be applicable to large systems. Nevertheless it has successfully found its own way into the doctrines that form the evolving body of the so-called complexity theory. Historically, you could say that it is the true progenitor.

Apparently the two disciplines have little in common. The chaotic systems with known differential equations sets are only few, and all can be considered simple in the sense that they have few degrees of freedom - the famous Lorenz system has only three, for example. This is probably due to the areas of study in which the systems known in the literature have been developed, because the principles of chaos seem easily applicable for large systems with many degrees of freedom, or with several state variables. The analysis of the measurements authorize the conclusion that also those systems that are likely huge in terms of dimensions may show an intrinsic chaotic nature, at least under some conditions.

The theory of complexity originated from reflections around systems with many dimensions, or even infinite. The latter, however, is a specific case because of the mathematics and theoretical aspects that it brings with it: a system with infinite dimensions, or an infinite number of state variables, would need infinite information to be described. In fact, every variable of state is independent of all others, and a system with an endless quantity of variables could carry infinite information in it, and could exhibit an infinitely extended dynamic. Perhaps such a system would be a totally random one.

If a system of small dimensions (such as Lorenz's one, or another chaotic system known in the scientific literature) may have such complex, varied and uneven dynamics that it can even be mistaken for a random system, then imagine what can be surprisingly motley, in theory, the dynamics of a non-linear system with a large number of state variables, or with a state space with many dimensions. This fact can not surprise any longer, once we have already observed amazing behaviours in much smaller systems.

The surprise is, however, in the case of these systems that we can clearly call complex, when we observe an entirely ordered and regular behaviour, where we would have rather expected a turbulent disorder reigning indomitably. I should make examples here, I know: I will make it good in a future post. Russian Nobel prize for physics Lev Landau, who has done major and seminal studies on turbulence, attributed this phenomenology to the infinte sum of frequencies or modes in the fluid dynamics, thereby reflecting the idea that complex behaviors were to be generated by complex systems characterized by a large number, tending to infinity, of degrees of freedom. Today we know that idea was wrong, and that a complex dynamic may be generated by a simple system like the Lorenz's one, and that a complex system can show a surprisingly smooth dynamic.

If the theory of chaos is the study of disordered behavior (or complex) of small systems, the theory of complexity - assuming you can find defined boundaries - has focused mainly on the study of organized and ordered behavior of large systems (hence complex). No surprise then that a complex system can also have a complex dynamic. In my view, even a "simple" system showing chaotic behavior can be said complex, because a system and its dynamic have a very inherent link, and you can not keep the two things apart.

Strange attractions.

The most used way to graphically show the evolution of a dynamic system is to draw its trajectory in the phase space: a space with as many dimensions as state variables (or even degrees of freedom) in the system. Each point of this space represents a very specific state in which the system can come to find itself. The phase space of the pendulum - for instance - consists of the Cartesian coordinates system identified by two variables: the angle from the vertical and the angular speed of the pendulum.
Not all points of the infinite phase space (also known as state space) are possible points for the system: indeed, most of them are impossible points and the system will never reach them. Mathematically this means that those points (combination of status variables) are not a solution of the differential equations governing the dynamics.
If we consider all possible points of the system in a chronological order, we will find that it draws a trajectory line in the phase space, which looks like the flight of a fly. The study of these trajectories tells much about systems.
Generally speaking the development of a trajectory depends on the initial state and thus by the initial conditions.
Left free to evolve a system displays normally a short-term dynamic, temporary, known as transient stage, and a long-term dynamic conditions. The latter can be a null state (think of a spring that is discharged, or the swings of a pendulum at the end of its oscillations, when it doesn't receive additional forces) or a steady state. The current in a transformer is periodic indefinitely, for example.
The state we are interested to is the chaotic one. In this case, different initial conditions cause different trajectories, which never overlap nor cross both a given point. The chaotic trajectories are an infinite development of a curve in the phase space that do not ever repeat twice the same even for a tiny tract.
Should we draw it, we would see that this curve never intersects itself. If you think about it, this fact is closely related to the impredicibility of chaos: should an overlap ever occur periodicity would settle up, because from that moment on the curve will repeat itself, giving rise to a periodic state that allows easy predictions of the system dynamics.
The bizarre geometric figure drawn by this trajectory is called strange attractor: the name betrays the surprise of those who saw it for the very first time and did not know how else that rich dynamic could be called. The set of all states that give raise to trajectories evolving with this curious geometry is called the attractor's basin of attraction.

A chaotic system can produce more than one strange attractor, with basins of attraction hardly distinguishable from each other or even overlapped - such as riddled basins, which are fractal basins. In these cases, a dynamic system might jump from one to another attractor due to external causes and perturbations. My words fail to depict the complexity that can surface from a similar case.

But the question is not purely theoretical, rather it is very real and closely concerns the world we live in. An example might be the climate, which so often deserves the honor of the first columns in these years.

What is Chaos?

We have been hearing that deterministic chaos was found in this or that natural phenomenon for over 20 years now, since the moment we opened our eyes almost suddenly to a reality that we had decided not to observe: that of real non-linear phenomena (ie almost all real phenomena tout court).
Personally one of the findings that impressed me most is chaos in the human heartbeat: in a healthy individual dynamics of the heartbeat as you can measure by means of an electrocardiogram is chaotic, while a regular dynamic is often symptom of a disease. Even the electroencephalogram of a healthy person has shown the typical and unambiguous marks of chaos.
What does it mean that a natural phenomenon shows chaotic behaviour? How is that a natural system is chaotic?
If we know the equations of a system then we can rely on some tools such as mathematics analysis, computer simulation that allows us to study how the behaviour of the system changes with its parameters.
The onset of chaos is always preceded by well detectable mathematical phenomena, points of transition from regular to complex behaviour: I mean, for example, period doubling that happens at given intervals and that ultimately flows into true chaos. These paths can be viewed in the so called bifurcations diagrams.
An universal characteristic of chaos is that the convergence ratio in the sequence of bifurcations (the so-called route to chaos), which is observed as the control parameter varies, is almost constant and is constant as the parameter approaches the infinte. This universal ratio is called delta of Feigenbaum, and allows us to know exactly when the next bifurcation will happen in a system that evolves towards chaos. So if we see a constant ratio that is close to the Feigenbaum's delta in the intervals between one bifurcation and the next one, we have an important clue that system has the characteristics of chaos in itself and is evolving towards a chaotic dynamic as consequence of the change of one of its structural parameters (eg tension of a diode).
The calculation of the so-called Lyapunov's exponents yields the conclusion that two close trajectories of the system differ on average: the mathematical consequence of this fact is the system's chaoticism for some given parameters (for other parameters the same system may not be chaotic). This is of course possible only if you know the equations of the system, in which case you can draw the exponents of Lyapunov mathematically.
If we do not know the system's equations, then we must carry out a measurement of its dynamic based on a series of samples long enough. On this series of measures we can quantitatively estimate some parameters: these are the coefficients of Lyapunov, or the Kolmogorov's entropy, or fractal dimension, or the size of correlation. These parameters can tell us whether we have chaos.
Finally we can say chaos has well detectable traces, and its traces have been discovered in many natural systems, whether living or not.

This should unanmbiguously answer the question: what is chaos. Yet it tells as little of what chaos means.

Some philosophical aspects about chaos.

Pierre Simon Laplace was one of the mathematicians who mostly influenced on Western thought. He was probably the supreme representative of scientific determinism.
I imagine that by now many of us have come to realize how deeply the deterministic thought - born within the calssic post-newtonian science - influenced our civilization, our way of thinking and feeling.
He wrote in 1812:

An intellect which at a certain moment would know all forces that set
nature in motion, and all positions of all items of which nature is composed, if
this intellect were also vast enough to submit these data to analysis, it would
embrace in a single formula the movements of the greatest bodies of the universe
and those of the tiniest atom; for such an intellect nothing would be uncertain
and the future just like the past would be present before its eyes.

Looking back behind us it is not difficult to note that, starting from the end of the eighteenth century and throughout the twentieth century, the common perception of the future collective and individual has often been marked by an aura of inevitability, predetermination (scriptum est): the future seen as something that moves from starting state perfectly defined towards a destination just as perfectly determined a priori. The scientific causality has clashed with religious finality, overwhelming it.
Many scientific discoveries seemed to corroborate that conclusion (at least until the advent of quantum mechanics with its portfolio of probability and uncertainty), with the result that - if not consciously, at least subconsciously -- few have shown any skepticism.
It would be interesting to develop the theme of political, social and cultural relapses of this trend of the century just past, which still influences our way of thinking and acting.
The deterministic vision that leaves no escape to free will is broken up at the end of last century due to some simple and (moreover) deterministic systems - the chaotic ones - rushing into the scientific landscape: despite their determinism these systems show a behaviour entirely impredictable even for an Intelligence, such as the one imagined by Laplace, which had knowledge of all the physical data of universe at any given moment. This Intelligence, even if aware of these data with infinite precision, yet it could not have present all the future and all the past before its eyes, because it could not find the closed-form solution of the differential equations governing a chaotic system.
So there are namely deterministic systems that exhibit a behavior that is impredicibile or indeterminabile: what sounds an oxymoron is actually fully supported by mathematics.

"Therefore, even God must let [the dynamics of] these chaotic systems evolve to
see what will happen in the future. There is no shortcut to the prediction for
chaotic systems."

Robert C. Hilborn - Chaos and Nonlinear Dynamics.

Lorenz's Butterfly.

It was 1960 when Edward Lorenz developed a model for forecasting weather conditions at first sight rather simple. He must have considered it simple too.
The model could be written as a system of nonlinear differential equations, quite simple, which however could not find a solution in the closed form. In other words it was not possible to integrate its differential equations, so as to make evident time dependence of the three variables.
This explanation would have helped to write each of the three variables as a function of time (eg y = f (t)) - which is tantamount to find a closed-form solution for the system of equations, as the mathematicians love to say, and that would have helped get a punctual forecast of the weather, or at least a prediction consistent with the assumptions Lorenz made in the development of its model.
This approach was perfectly compatible with the physical thought that dominated almost the entire last century, that all phenomenologies might be considered within the mathematic determinism, becaus the observable reality is an expression of some stability form of the observed system. And I apologise for the extreme synthesis of these concepts.
Although closed-form solution cannot be found, the system developed by Lorenz can be solved numerically, with the help of computer. The geometric representation - in the space of phases - of the trajectories of state that emerges from these simulations is surprising, and is today the most famous example of a chaotic system. The trajectories never repeat, neither two of them will ever cross or overlap.
The system is unpredictable, because you can not write the integral functions, nonetheless it is stable and shows some regularity. A hidden order, impossible to imagine a priori.
The system is sensitive to initial conditions: the smallest variation of these can give rise to a dynamic evolution completely different from what you would have if the change had not been there.
Then the famous reflection of Edward Lorenz:

can the beat of wings of a butterfly in Brazil generate a hurricane in Texas?

The "simplicity" of the simple pendulum.

Anyone who has done a course in physics at secondary schools met the so-called "simple pendulum", one of the most commonly used examples in teaching.
Known since antiquity, it has inspired the fundamental insights of Galileo Galilei that later brought the great man in the formulation of the properties of isocronism, and opened the way for concepts of motion and momentum that are the foundation of classic mechanics.
Apart from these historical notes, since our first meeting with the pendulum what we notice most is its "simplicity": the motion of the pendulum can be described by means of a simple equation. Depending on certain initial conditions - angle, mass, length - you can predict all the future behaviour of the pendulum without possibility of errors. The behaviour is periodic in time, provided you can neglect some forces that have no direct relation with the mechanism itself, but are inevitable in a real pendulum: friction and resistance.
However, you can incorporate these components into the motion of the pendulum, which from ideal and linear becomes real (at least a little' more, because the whole mass is considered as condensed in a single point and concentrated in the lower summit of the shaft) and non-linear. The equation is significantly modified, but at first glance one could not say that the effects may disrupt our first understanding of the pendulum: it should remain a simple and absolutely predictable dynamic system.
I invite you to make direct experience of such pendulum playing with the simulator available on the website MyPhysicLab.com.
Depending on the choice of system parameters the behaviour of the pendulum becomes very similar to a totally random behavior, although it remains absolutely deterministic. This is a chaotic dynamics, where you can not predict the kinematic conditions of the system (e.g. speed, position, acceleration) even if initial conditions are well-known.
This is apparently in disagreement with the deterministic nature of the pendulum.
If one looks at the trajectories of state variables of the system they never repeat: this can be demonstrated mathematically. There is no periodicity for the oscillations of the pendulum. The trajectories will never overlap, infinitely. The state variables are no longer constrained to repeat the same routes, but free to experience an infinite variety of possible trajectories within the state space. The pendulum, from simple, has become something much more complex.

The need of a new Humanism.

The scientific revolution finds its historic place between the sixteenth and seventeenth century, and will be followed by the "Age of Enlightenment" - the eighteenth - when the scientific knowledge is finally consecrated. Today we struggle to represent how intricate and arduous must have been the path of ideas that led to the primacy of Sciene - in its modern meaning - on all the superstitions, occultisms, ermetisms, alchemy and magic beliefs of the Middle Ages.
We also know that we must add the obscurantist attitude and open hostility of the Catholic Church to all these -isms, which often degenerated into open and arbitrary violence that will never be sufficiently condemned.
I think, sometimes, that an opposite but equally degenerate situation became popular in our time: a blind trust in any information that is simply surrounded by an aura of scientific.
The rigor of the scientific method has said slowly against an ancient priestly conception of knowledge, and this has undoubtedly been one of the main achievements in human history.
Today, however, sometimes I think that faith in magic and the supernatural, which mankind has successfully strained to release, has been replaced by an equally fideistic credit to any assertion that can boast a relationship with science, albeit indirect.
The role of the distortion made by mass media is probably a remarkable one.
I have had a scientific education, then certainly I am not going to condemn scientific methods or undermine the achievements. However I often think that the ideal of "progress", and "scientific progress" in particular, must not lead to a society totally devoted to the achievement of such "progress" as if those achievements are the true nature of being human.
I believe today that there is a need for a new Humanism, bringing man back at the centre of society and replacing the faith in the technical and financial development with the faith in man, the search for progress in the quest for happiness and a worthy human condition. Science should always be instrumental to this condition. The ideal of "progress", with its obvious declinations of "development" (scientific, economic, etc.) and "growth" must be reviewed in the light of a new centrality of man.

An ugualitarian society: utopia?

Yes.
Every time a human society left a power position unoccupied, that position has been filled by some entity or person almost immediately. It looks like any human society self-organizes to fill the power lack. Jacobinism first and Communism later have made equality a revolutionary message that has deeply transformed their relevant societies, but the inequality, the injustice, the differences have taken punctually place in new organizations, transformed only at their surface level to reflect the new features of renewed society. These revolutions have transformed the society to re-establish old social networks still characterized by the presence of "hub" of various weights. A "hub" is still a center of power.
Then the centers of power can not remain uninhabited, because the society itself prevents the lack of power to persist. We should not be surprised to think of society as an organisation having its own awareness: any superior body has mechanisms that fix situations highly unfavourable for its survival. There are hundreds of examples one can draw from the living world. Dismantled a pre-existing "hub" due to some revolution, a new "hub" appears to re-interpret the role of the previous one. Indeed, it is likely that the new hub was already existing, and that the revolution is the manifestation of the struggle between the newcomer and the pre-existing one, between the "subversive" and "conservative".
The question then becomes independent of the nature of man, and its virtues and weaknesses: the true terms of the problem are to search in complex systems, "living" systems, adaptive systems, the populations of cooprating individuals. Those mechanisms could be common to many classes of these systems, perhaps there even exist universal characteristics, which human societies can never escape even they will want to.
If this is true, as I think, then a society in which all individuals are equal is a pure utopia, and the state of total equality of members is devoid of any utility. it would be rather harmful and is not desirable. While it is desirable to fully understand these mechanisms, to transfer this knowledge in political systems, so that they fit the "structural" requirements of human societies to the benefit of prosperity and welfare of individuals.
.

Think the thought.

I developed a natural distrust towards the new theories that come to the honours of the limelight, because some of those "mathematical certainty" or "scientific" ones I've seen in 35 years of life were suddenly flipped revealing their inconsistency.
I think it is a good thing. I am used to think of knowledge as a vital process, which is now fashionable to call "cognition". And life and mutation walk hand in hand, so I'm not surprised that what is today taken for safe and scientifically proven, tomorrow may be revised and radically reinterpreted in the light of a new theory. When this happens often downward compatibility is retained: the old certainty falls into the new one as its subset, but the general meaning and philosophical considerations that were first descendants radically change. Someone calls it "revolution" and wanted to see - not without reason - even in the history of science and scientific thought a succession of revolutions that has something to do with the theory of catastrophes.

This long introductory hat will certainly distracted attention from my unlikely reader, but helps me to create the conditions for writing this post.
My reflection today concerns the thought. Not the scientific thought, but the human thought.
I can not split the ontological part from the espistemological one: the nature of human thought is myself that is writing, and is paradoxically circular that the human thought attempts to describe itself. So we feel dizzy when we embark upon speeches of this kind. In addition, the tool one uses to know the nature of something is her/his thought. So the means by which I am going to explore the nature of thought is the thought itselt, which is the subject of the investigation. Recalling that who lead the investigation is the thought itself, which is also under investigation, the resulting picture has the same consistency of a drawing of MC Escher. Useless considerations so far, but amusing, and hopefully have the honour to give the idea of the hardship that everyone experience when philosophically pay attention to the very true essence of reality around us. Walking these steps one may have to conclude that an ultimate reality does not exist in order to keep consistent with her/his reasoning, and that everything that we know is the interior projection and processing of an individual perception, and that the representation of the world belongs to the sphere of subjectivity . So the world where I live is certainly different from the world where you live - if you take for granted that the range and uniqueness of sensory responses of my body are different from those of yours, in the words of Humberto Maturana and Francisco Varela.

Yet today a model that helps us to reflect on the nature of human thought is there.
The hypothesis that thought is an expression of property emerging from the very complex organization of neurons that make up the brain and nervous system It is now mature and experimentally supported by neuroscience. This complicated intercellular communications network covers the whole body and collects various kinds of stimuli coming from outside (vision, hearing, but also humidity, temperature, static electricity) and the inner (blood pressure, levels of carbon dioxide, adrenaline, etc.) and from pre-existing configurations that surface due to certain conditions (for example what they commonly call memories and memory).
The first property emerging from such an organization, perhaps the most primordial, is self-consciousness, or conscience. The sense of ego springs up as the contrast with what is perceived as outside us. The close relationship between the neuro-brain system and body extends automatically the self-awareness to the whole body.
The perception of the self is perhaps the most arcane expression of thought, and together the most fascinating and mysterious.
The set of activities that allow the self to differentiate itself from the rest of the world is called cognition, as differentiating it allows you to "know" the world. The cognition is a continuous activity that begins and ends with the individual, hence it is easy to conclude that is the individual self. Someone gave it the name "mind", whereby mind and cognition would have to be synonymous.
At this point it is not difficult to recognize what we call "thinking" as the sequence of different times (or evolution) in the dynamics of the mind.
A model now agreed of complex adaptive system, the one of neural networks, allows to sketch a model for human thought. Indeed, the thought in general, given that cognitive activity, under this light, is not the prerogative of the human species only but it belongs to all so-called higher animals, at least.
We live in an age when for the first time the epistemological and ontological efforts of philosophy found the support of scientific and mathematical theories arising from applied sciences and from the theory of dynamical systems, which have enabled the development of an interpretative model of thought and mind.
A simple model, which I hope will not be celebrated as a scientific truth on which basis re-enunciate the concept of life and of human existence. In fact it is from theories of complexity that we have learned - not without surprise - that diversity, variety, impredicibility and unrepeatability are the properties of certain classes of dynamical systems that we can not but consider "simple" when compared with living systems. Therefore we have no hurry to "universalize" conclusions that we can say valid only in restricted areas of research for the time being . But in the same way we are not afraid to reinterpret the most intimate part of ourselves - our ego, our thoughts, our selves - in agreement with new models that perhaps could lead us far away, and that certainly should not be rejected for ideological or religious prejudices , or for fear of violating a sacred concept of us, which tightly associates our egos and our souls.

Symmetry and chaos: the shape of snowflakes.

To respond briefly to the question that gives the title to his book, yet I quote Ian Stewart:

Equations in itself devoid of any symmetry sometimes give rise to a regular dynamic and sometimes to chaos. It was discovered that this is true even for the symmetrical equations. If you change the numeric coefficients of the equations, one can obtain a chaotic dynamic that obeys to symmetrical rules. A symmetrical chaos. What are the systems of this kind? The simplest answer you get from thinking of the geometric representation and the attractors. What we discover is that the attractors are chaotic (because dynamics are chaotic) and symmetrical (because rules are symmetrical).

Ian Stewart - What shape is a snowflake? (translated from Italian).

The geometry of chaos and the dynamics of fractals.

An enigmatic title, a phrase that represents a sort of two-faced palindrome, conceptually not literally, because it wants to highlight the duality between the fractals and the deterministic chaos. Once again I quote from a famous mathematician:

Following the mathematics practice and representing the dynamics in geometrical terms, one can well focus the disconcerting dual nature of chaos.
Associated with each dynamic system, there is its own geometric space, the phase space [...]. In chaotic systems the lines of flow (or state trajectories), head towards complex forms such attractors. [...] The geometry of these attractors combines the concept of chaos and the concept of fractal.

Ian Stewart - What shape is a snow flake? (translated from Italian).

The mathematics of chaos and fractal geometry are fundamental discoveries of the twentieth century, which allowed the review of basic concepts of many branches of science - not to say all the knowledge - much to be presented today as the ignition spark of a revolution epistemological still underway, whose significance is still difficult to understand.
Concepts such as chaos and fractal, abstruse to the most today, should become a basis for anyone who wants a key to our fascinating universe.

Deterministic chaos: the boundary between randomness and rules.

The statistical mechanics, which Boltzmann has made a vital contribution, made it possible to understand the processes underlying the behavior of systems formed by large amounts of elements - or particles - interacting, fundamental step for the development of entire fields of physics - thermodynamics, gas dynamics, plasma dynamics, quantum mechanics -. The price was a tribute to randomness.
No one today calls into question the great importance of probability theory to explain the physical mechanisms, but we are all aware of the contrast between the determinism of equations dynamics - often raised to the rank ambiguous and absurd "laws of nature" - and the mere estimate of Chance of random events. Heisenberg, Gödel, Prigogine: these distinguished scholars, and many others have clarified that statistics is an essential method of investigation to understand phenomena and processes of nature and life. However there remains a sense of discomfort, and - in some cases - the disagreement with the intuition when you try to read the nature only with these instruments.
The ideas arising from deterministic chaos come help us to find a convergence of concepts that might seem very distant from each other:

Chaos is apparent randomness with a case purely deterministic. It is behavior with no rules governed entirely by the rules. The chaos lives in the shadows zone between order and randomness. [...] In some ways, in chaos there is a genuine randomness. With some approximation, one can say that the rules of a chaotic system attach microscopic randomness of the initial conditions and magnify it making it evident in its large scale behaviour. The debate is made more difficult by a philosophical problem: does true randomness really exist?

Ian Stewart: What shape is a snowflake? (translated from Italian).

Forms and cognition: the "laws of nature."

For those who felt a certain embarrassment when hear that man has discovered the "laws of nature" and that they are nothing more than mathematical formulas, whose harsh aridity often appears intuitively inadequate to play all the symphonies of the universe. The human mind goes tirelessly searching for forms. To survive in a hostile world, we have developed a sensitivity to configurations, which we use to predict what will happen.
Sometimes [...] configurations actually exist in reality and reveal important truth about the universe. We call them laws of nature. And this is what science does, bringing to light the secret configurations that make the universe. For human beings using mathematics is the most effective way to think about configurations. We therefore believe that the laws of nature are mathematical laws.

Ian Stewart - What shape is a snowflake? (translated from Italian edition).

The linear original sin.

Therefore, rather than describe the phenomena in their full complexity, the equations of classical science deal with small fluctuations, surface waves, small changes in temperature and so on. [...] This habit rooted to the point that many equations were linearized the moment they were written, so that the complete non-linear descriptions not even appeared in the scientific manuals. Consequently, most scientists and engineers came to believe that practically all natural phenomena could be described by linear equations.

Fritjof Kapra - The Web of Life.