From the Book of Changes to the Algebra of Conscience, through technical analysis

In his splendid book on Harmony Mathematics, Alexey Stakhov devotes a section to Vladimir Lefebvre’s mathematical theory of decision and strategic interaction, which, unlike other developments in this area such as the much-hyped game theory, includes an intrinsic behavioral and moral component [62].

Everything starts from the observation that if someone is asked to divide a pile of string beans into two piles of bad and good ones, oddly enough the average result is not 50%-50% as expected, but 62%-38% in favor of those considered good.

According to Lefebvre, what lies at the bottom of this asymmetric perception is a triple gradation of definable logical implications within a binary logic or Boolean algebra:

A [a0consciousness→a1reflection→a2intention]

consciousness being the primary field, reflection the secondary level and intention a reflection of second order.

Within this structure of human reflexion, the personal evaluation of behavior implies, by logical implication, a self-image:
A= a0a1a2 ; A=(a2→ a1)→ a0.

The subsequent truth table gives us an asymmetric sequence of 8 possible values, 5 being a positive estimate and 3 a negative one. Naturally, 8, 5 and 3 are adjacent numbers in the Fibonacci series.

The extraordinary thing about the logical framework of ethical cognition created by Lefebvre is that, inadvertently, it allows a completely modern interpretation of the Book of Changes without distorting its sapiential and moral nature. It is not something that we are going to prove here, but given the deep influence of Yi Jing in the history of Chinese culture, it would be of utmost interest to look for a rigorous correspondence between both views.

Undoubtedly, Lefebvre’s framework is more formalized, and more focused on the image that the individual agent has of himself, than on the situation or critical juncture, as is the case with the Yi Jing; in fact, we could say that Lefebvre’s formalization is a Boolean and measurable limit for interactions with a low number of individual agents, but even then the correspondence remains intact and can lead to much broader and impersonal degrees of involvement, and therefore, of understanding.


Stakhov devotes another section to Elliott’s technical wave analysis, a tool also based on the Fibonacci series. The theoretical scope of Elliott waves is limited at best, even if its fractal-type analysis can be applied discretionally. But here we want to point out a much deeper methodological issue.

Lefebvre’s model shows us that these series have, or at least admit, a reflexive component —in fact his work has been qualified as a reflexive theory of social psychology. And that reflective component is precisely what is most missing in Elliott’s model. There is also a reflexive theory of the economy and the market, with positive and negative feedback cycles; this theory is not without merit, if we compare it, for example, with the hypothesis of the efficient market, which, even if it were not false, would always be incapable of telling us anything. Moreover, the reflexive theory does not only affect prices, but also the fundamentals of the economy.

Economic reflexivity is a model of self-interaction, as it is also of self-image, and this decisive element in human affairs can never be dispensed with, even if it is impossible to quantify it. But what happens when self-interaction runs through the whole behavior of natural systems? Let us think, for example, of the case of the pulse analysis we have mentioned, a transparent model of self-interaction, which, as a bonus, tends towards golden ratios in rhythm and pressure. But it is clear that in social systems reflexivity passes through the knowledge of an external situation, while in an organic system reflexivity is a pure question of action.

How much room does one type of reflexivity leave for the other? How far can the perception of the markets be manipulated? Today we also see that central banks have an undisguised and unabated intervention in prices, injecting money discretionally and becoming guardians of the financial bubbles. Well, the analysis of the pulse, the transformation of its modalities, and the study of the limits of conscious manipulation through biofeedback are giving us a good match or at least some sort of reference for this elusive problem. And if we want more intensive methods, besides the quantitative implications that all this already have, we can also apply to monetary flows CPUs models like that of the triangular flow-efficiency-size feedback that we commented on earlier.


In The Algebra of Conscience, Lefebvre made a comparative analysis of the Western and Soviet ethical context within the framework of the Cold War confrontation [63]. The W-system of the United States believed that the compromise between good and evil was bad, and that the confrontation between good and evil was good. The S-system of the Soviet world, on the contrary, believed that the compromise between good and evil was good and the confrontation between good and evil was bad —what Lefebvre does not say, as it goes beyond his formal analysis, is that the desire for confrontation of the W system was not based on any moral idiosyncrasy but on the essential expectation of further expansion; just as the desire of the S system to avoid confrontation was based on the fear of disappearing.

At any rate, from the choices and implications between commitment and confrontation, four basic attitudes emerge, that of the saint, the hero, the philistine and the dissembler: the saint embraces suffering and guilt to the maximum; the philistine wants to diminish suffering but may feel acute guilt; the hero minimizes his guilt but not his suffering; and the dissembler minimizes suffering and guilt.

Maybe we could have created 8 types instead of 4, in consonance or dissonance with 8 other scenarios or natural tropisms, with two more third grade binary implications or a single sixth grade one. Logic and calculus serve to ascend this ladder and understanding begins where calculus ends. Thus, the continuous proportion would show us in a very direct way its role of mediator between the discrete and the continuous, the digital and the analogical.

In reality we have here an unsuspected Centaur, namely, a halfway point between binary formal logic and the dialectical logic as in Hegel. But the proof that this is something genuine, and not a mere theoretical construct, is that it has a built-in asymmetry that is a distinctive feature of Nature. We could call it an asymmetrical logic of implication.

It has been said that Lefebvre’s theory was used at the highest levels of negotiation during the collapse of the Soviet Union; and here we will not go into judging what positive or negative role it might have had, and for whom. In any case, its framework only covers two agents, and not the scenario or juncture that may dispose of them. Yi Jing’s framework can serve not only to see how I perceive myself in the world, but also to evaluate how the world perceives me; in the end, both are parallel illusions. One could say that asymmetric perception is part of the same reality; a double asymmetry better embraces the implications and the unnoticed axis of the dynamics of a situation.

Surely one could make a great parlor game out of these arguments. More interesting, however, is the study of moral or social conscience through the analysis of its implications. And even more interesting is the awareness of how the external situation, for which we are always looking for an image, interpenetrates with the changing image we make of ourselves, forming both a momentary map of our situation and conscience.

One can only guess that Lefebvre knows the text of the Yi Jing; however it is evident that the line of his reasoning is completely independent from the Chinese classic and he has arrived at his “positional system” of moral consciousness starting from the Western logical and mathematical tradition. If he had had just a glimpse of the Taijitu with the golden section embedded, the immediate association would have shine in his mind. In the coming years we may witness many new associations of this kind induced by the evolving environment.

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