The Birth of Complexity

We all know perfectly well that, according to the second law of thermodynamics, a spontaneous increase in order is impossible in an isolated system — only entropy can grow by itself (mathematically, entropy is the logarithm of the number of elementary states).

The evolution of living organisms "from simple to complex" is not subject to this law, because organisms always receive energy from outside and, like refrigerators, spend part of it on increasing their internal order. Meanwhile the total disorder of the whole system (organism + environment) grows. This is also well known.

Meanwhile, mathematicians have long been familiar with the so-called S-theorem of Yu. L. Klimontovich, which, to my regret, I only learned about yesterday. What follows is a passage from N. M. Borisov's article "Evolution, Chance, Entropy (on how Darwin and Clausius turned out to be right at the same time)":

Studying processes of self-organization, Yu. L. Klimontovich proved in 1983 the so-called S-theorem, which states that as a system moves away from equilibrium due to a change in a control parameter, the values of the system's entropy, normalized to a given value of mean energy, decrease. As a consequence, information ("negative entropy", or negentropy) is generated not "exclusively by a mind", as creationists believe, but by a combination of random changes in the state of the system followed by the necessary (selective) memorization of the results of those changes. Thus the concepts of variability, heredity and natural selection introduced by Darwin were described in mathematical language within non-equilibrium thermodynamics. Inspired by the success of nonlinear non-equilibrium thermodynamics, Eigen even called Darwin, with his idea of natural selection, "a great physicist" in his Nobel lecture.

That's wonderful! This theorem puts evolution on a solid mathematical foundation. But, like many other remarkable results, it won't reach the general public any time soon — and probably in an extremely distorted form. But that's how the world works; Stanisław Lem described this phenomenon beautifully in his novel His Master's Voice:

Nineteen years ago, together with the young anthropologist Max Thornop (who died tragically in a car accident), I published a paper in which I proved that there is a limit of complexity for all finite automata governed by a hedonistically oriented program (these include, in particular, all animals, man among them). This program is based on punishments and rewards, which are experienced as suffering and pleasure. My calculations show that if the number of elements of the regulating center (the brain) exceeds four billion, a population of such automata exhibits a gravitation toward the extreme poles of the program. Either of the limiting variants may then gain the upper hand — or, in more everyday language, sadism or masochism; consequently their emergence in the course of anthropogenesis was inevitable. Evolution "agreed" to this solution because it operates with statistical quantities: what matters to it is the preservation of the species, not the defective states, ailments and sufferings of individuals. As a designer, it chooses adaptation to circumstances rather than the attainment of perfection. I managed to prove that in any human population, given complete panmixia (free interbreeding), no more than ten percent of individuals will show sufficiently balanced, hedonistically regulated behavior, while the rest will deviate from the norm. Although even then I was considered one of the best mathematicians in the world, the influence of this work on anthropologists, ethnologists, biologists and philosophers was nil. For a long time I could not understand why. My work was not a hypothesis but a formal, and therefore irrefutable, proof that certain properties of man, over which legions of thinkers had racked their brains for centuries, are the result of a purely statistical fluctuation that cannot be avoided when designing automata or organisms. Later, using the excellent material collected by Thornop, I extended my proof to the emergence of group ethical norms. But that work, too, was completely ignored. Years later, after countless discussions with humanists, I understood: they did not acknowledge my discovery because it did not suit them. The style of thinking I represented was considered by them something like bad taste, because it left no room for rhetorical wrangling. It was tactless of me to draw conclusions about human nature by means of mathematics! At best my undertaking was called "curious". In essence, none of the humanists could accept that the great Mystery of Man, the enigmatic properties of his nature, could be derived from the general theory of automatic control. Of course, they did not say so directly. Nevertheless, my result was held against me. I had behaved like a bull in a china shop: that which anthropology and ethnography with their field studies, as well as the most profound philosophical analysis of "human nature", had failed at, that which neither neurophysiology nor ethology (the science of animal behavior) had managed to formulate as a meaningful problem, that which had remained a fat preserve of ever-fruitful metaphysics, of the psychology of the subconscious, of classical and linguistic psychoanalysis and God knows what other esoteric disciplines — I had tried to cut, like a Gordian knot, with my nine printed pages of proof. They had grown used to their high rank of Keepers of the Mystery, which they called the Reproduction of Archetypes, the Instinct of Life and Death, the Will to Self-Destruction, the Drive toward Non-Being — and I, crossing out these sacred rituals with some transformation groups and ergodic theorems, declare that the solution has been found! That is why I was treated with carefully concealed antipathy: some unceremonious layman had encroached upon the Riddle, had tried to cement up its ever-living springs, to seal the lips that found joy in asking endless questions; and since my proof could not be refuted, the only thing left was to pass it over in silence. No, it is not wounded vanity speaking in me. After all, I was praised to the skies — true, for other work, in the field of pure mathematics. The experience, however, was very instructive. We underestimate the inertia of thought in many branches of knowledge. Psychologically this is quite understandable. The resistance our mind offers to the statistical approach is far smaller in atomic physics than in anthropology. We readily accept a consistent, experimentally confirmed statistical model of the atomic nucleus. We do not ask: "Well, but how do atoms really behave?" — yet in the sciences of man such an approach does not satisfy us.

(Both quotations are translated from the Russian texts the original post used.)