From the Max. Em-Power Principle to the Maximum Ordinality Principle

From the Max. Em-Power Principle to the Maximum Ordinality Principle

This Section synthesizes the Logical Process of such a passage, articulated in four steps.

The latter are more widely illustrated in the four Abstracts recalled here below, which correspond to the papers presented at four subsequent Emergy Conferences, held in Gainesville (Florida, USA) in 2004, 2006, 2008 and 2010, respectively. However, for further details, the corresponding complete papers are also available to the Reader:

Gainesville 2004 – Differential Bases of Emergy Algebra


The well-known rules of Emergy Algebra, originally formulated in steady state conditions, are reconsidered and analyzed from a dynamic point of view. In such a sense the paper points out their corresponding differential bases. The latter, in turn, represent the preferential guide to recognize their most profound physical meaning.

However, for the sake of completeness, a possible generalization of the same rules from steady state to variable conditions is also considered.

The analysis is particularly focused on the three fundamental generative processes represented by co-production, inter-action, and feed-back, which are formally described (under dynamic conditions) by means of the Incipient Fractional Differential Calculus. In so doing, the mathematical method adopted succeeds in defining the output exceeding Quality of the mentioned processes by means of the corresponding Ordinality of their associated output Transformities.

Such a dynamic analysis enables us to show that the rules of Emergy Algebra proposed by Prof. Odum under steady state conditions have a well-founded dynamic physical nature, adequately described by the differential operators adopted. The analysis also shows that the originally conceived rules of Emergy Algebra continue to hold even when the dynamics of a process becomes extremely complicated.

Gainesville 2006 – Emergy Analysis as the First Ordinal Theory of Complex Systems


The linguistic-mathematical tools developed to give an appropriate mathematical formulation to the Maximum Em-Power Principle, together with the already shown validity of the Rules of Emergy Algebra under variable conditions, suggested a possible development of a General Ordinal Theory of Complex Systems. This can ideally be thought of as an appropriate “transposition” of theorems and concepts of the well-known “Systems Theory” (conceived in terms of traditional Differential Calculus) into a new Mathematical Theory of Ordinal Systems, where the latter are described and analyzed in terms of “Incipient” Differential Calculus.

The activity aimed at formally defining some basic concepts (such as “Ordinality”, “Information”, “Organization”, etc.) led us to recognize an unexpected “similarity” between the concept of “Information” adopted and Odum’s Transformity.

Further consequential developments of such an initial analogy led us to the conclusion that Emergy Analysis can be considered as being the First Ordinal Theory in the field of Thermodynamics of self-organizing Systems. This result definitely becomes evident when the basic cardinal quantities, which in Emergy Analysis are understood in an “ordinal sense”, assume their explicit and appropriate formal expressions corresponding to their Ordinal nature.

In particular, in such a context the Rules of Emergy Algebra can adherently be seen as specific Rules of “genesis and transfer of Ordinality”, when the latter is accounted for in terms of its associated “Ordinal Information”.

Gainesville 2008 – From Transformity to Ordinality, or better: From Generative Transformity to Ordinal Generativity


The paper starts by recalling the three Generative Rules of Emergy Algebra (Co-production, Inter-action, Feed-back) in order to show a possible “hiatus” between the concepts understood (by Prof. Odum) in terms of “verbal enunciations” and their corresponding “mathematical translations”, that is the formal symbology habitually adopted to express them.

On the basis of such a hypothesis, and in the perspective of generalizing the previous Rules under dynamic conditions (where such a supposed discrepancy would have surely become much more marked), a different concept of derivative was introduced (the so-called “incipient” derivative).

This allowed us to remove, from the very beginning, such an initial “in-harmony” in those definitions, in which radically new concepts are directly expressed by means of old mathematical symbolisms.

The concept of “incipient” derivative was the one which led us to the first “transition” indicated by the title: Transformity, in fact, can more effectively be replaced by the concept of Ordinality.

However, further research on the fundamental role played by Generative Transformity (with respect to Dissipative Transformity) suggested an even more generalized approach to the analysis of Self-Organizing Systems. This gave origin to the second successive “transition” (also synthesized in the title): Generative Transformity, in fact, can profitably be replaced by Ordinal Generativity.

Some Ostensive Examples, taken from Classical Mechanics, Relativistic Mechanics and Quantum Mechanics, will then clearly show how the latter approach is able to realize Odum’s “dream” concerning “the Unification of Science”, when based on a System perspective, Energy hierarchy and the fundamental concept of Emergy.1

Gainesville 2010 – The Maximum Ordinality Principle. A Harmonious Dissonance


The subtitle recalls a previous article written as a tribute to H.T. Odum’s lifetime work. In this article I already pointed out that Odum’s genial creativity, and in particular his famous Maximum Em-Power Principle, would “manifest its true relevance mainly in the future, and for many years to come. This is because the real and effective introduction of a renewed concept of Quality is able to transform any scientific aspect”, including the same Classical Thermodynamics.

In such a context, the Maximum Ordinality Principle is nothing but the re-proposition of the same Maximum Em-Power Principle, once deprived of any “residual” reference to traditional concepts of Classical Thermodynamics (such as Energy, Exergy and so on). Such a reformulation might thus appear, at a first glance, as being a sort of “dissonance” with respect to the previous formulations of the Maximum Em-Power Principle, both in steady state and dynamic conditions. Vice versa, in this way the real novelty of the Maximum Em-Power Principle emerges in a much clearer way, by contributing to give a more “harmonious” picture of the surrounding world.

The present reformulation, in fact, enables us to delineate the reference guide-lines for a real and complete re-foundation of Classical Thermodynamics, now properly understood as “a Thermodynamics of Quality”. This is because classical quantities (such as Energy, Exergy and so on) simply represent a mere cardinal reflex of Generative Processes persistently evolving toward the Maximum Ordinality.

Such a re-foundation, on the other hand, is already implicit in the same Rules of Emergy Algebra, which represent one of the most important contributions to modern Science over the last four centuries.


1 Odum H. T., 1995. Energy Systems and The Unification of Science. University Press of Colorado, pp. 365-372.