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.
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