Solution to the “Three-Body Problem” in the Light of the Maximum Ordinality Principle, as a “Suggestion” for a Ri-Orientation of the Present Scientific Perspective in “Favor” of the “Irreducible Quality”

Corrado Giannantoni

Ente Nazionale per le Energie Alternative, Rome, Italy.

DOI: 10.4236/jamp.2023.111014                

Abstract

This paper presents the Solution to the “Three-body Problem” in the Light of the Maximum Ordinality Principle. In the first part, however, it starts with the Solution to the Solar System, made up of “11 Bodies”. This is because, in such a context, the “Three-body Problem” can be analyzed in its all descriptive possibilities. Nonetheless, the paper also presents the Solution to the “Three-body Problem” with reference to Systems totally independent from the Solar System, such as, for example, the “Triple Stars” and the “Triple Galaxies”. In this way, the paper offers a sufficiently complete framework concerning the Solution to the “Three-body Problem”, always in the Light of the Maximum Ordinality Principle, described in detail in Appendix A.

Keywords

Three-Body Problem, Light of the Maximum Ordinality Principle, Solution to the Solar System

1. Introduction: The “Three-Body Problem” and Poincaré’s Conclusions

As is well known, the “Three-body Problem”, of fundamental relevance in Mechanics and, in particular, in Celestial Mechanics, in its most general formulation is “intrinsically” unsolvable [1] . That is, it does not admit a closed-form solution when the latter is researched on the basis of Physical-Mathematical Methods strictly pertinent and formally adherent to the “Subjacent” Principles of its Formulation.

This, however, does not mean that the Problem could not result as being “solvable” by adopting “other” and “different” General Principles, which can be considered as being particularly apt to describe the Phenomenological Reality under consideration.

The paper, in fact, will show that the “Three-body Problem” is “perfectly solvable”, and always in “explicit form”, when the Problem is formulated according to the Maximum Ordinality Principle.

Before doing that, however, the Reader is invited to take a preliminary look at Appendix A, which presents the Physical, Logical and Formal presuppositions to get the aforementioned solution, in the Light of the Maximum Ordinality Principle, for any Self-Organizing System made up of an arbitrary number of Bodies.

2. General Formulation of the “Three-Body Problem” in Classical Mechanics

The “Three-body Problem”, in its most general version, can be formulated by applying the second Law of Classical Mechanics to each body

$m_1\frac{d^2{\stackrel{\to }{r}}_1}{d{t}^2}={\stackrel{\to }{F}}_{21}+{\stackrel{\to }{F}}_{31}$ (1.1)

$m_2\frac{d^2{\stackrel{\to }{r}}_2}{d{t}^2}={\stackrel{\to }{F}}_{12}+{\stackrel{\to }{F}}_{32}$ (1.2)

$m_3\frac{d^2{\stackrel{\to }{r}}_3}{d{t}^2}={\stackrel{\to }{F}}_{23}+{\stackrel{\to }{F}}_{13}$ (1.3)

where the generic vector ${\stackrel{\to }{F}}_{ji}$ represents the Newtonian gravitational action of body j on body i.

As far as the second fundamental Law of Classical Mechanics, pertaining to the moment of motion quantity, it is easy to recognize that the total moment of the System corresponds to the sum of the moments of the single bodies

$\stackrel{\to }{b}={\stackrel{\to }{b}}_1+{\stackrel{\to }{b}}_2+{\stackrel{\to }{b}}_3$ (2.1),

with the condition

$\frac{d\stackrel{\to }{b}}{dt}=0$ (2.2)

because any gravitational force ${\stackrel{\to }{F}}_{ji}$ is parallel to its proper action arm ${\stackrel{\to }{r}}_i$ .

Consequently, by considering that Equation (2.2) is directly integrable in the form $\stackrel{\to }{b}=cost$ (2.3), the Problem consists in the integration of 3 vector Equations (1.1), (1.2), (1.3), which correspond to 9 scalar differential equations of the second order. Each one, in turn, is reducible to 2 differential equations of the first order. That is, 18 scalar equations of the first order in all, through the position

$\frac{d{\stackrel{\to }{r}}_i}{dt}={\stackrel{\to }{v}}_i$ (per $i=1,2,3$ ) (3.1)

Consequently, the Problem is described by a system of 18 differential equations of the first order, with 9 unknown coordinates of position (3 for each body) and 9 components of velocity (3 for each body) which, at any time, describe the configuration of the System.

The Problem becomes rigorously defined only when its initial conditions are assigned, that is when position and velocity of each body are assigned at the initial time (according to Cauchy)

${\stackrel{\to }{r}}_i\left(0\right)={\stackrel{\to }{r}}_{i,0}$ , ${\stackrel{\to }{v}}_i\left(0\right)={\stackrel{\to }{v}}_{i,0}$ for $i=1,2,3$ (3.2)

In spite of such an appropriate formulation, the Problem does not admit any analytical solution, which is characterized by properties of continuity and derivability in correspondence to all the orders of derivation necessarily involved in the same Problem. In fact, it is possible to find only two first integrals of the motion. To quote the same Poincaré [2] : “…le problème de trois Corps n’admet pas d’autre intégrale uniforme que celle des force vives et des aires”. Where the concept of “integral” is not understood in the traditional sense of “solution”, but as a “function of the solutions” (ib.) structured in the form

$F_i\left[x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right)\right]=cost$ (4)

where $x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right)$ represent the generic unknowns of the considered problem.

At this stage, we can consider the Solution to the “Three-body Problem” in the Light of the Maximum Ordinality Principle.

3. Fundamental Presuppositions for the Solution to the “Three-Body Problem” in Its Most General Formulation

One of the Fundamental Presuppositions is that of starting from the Solution to the Solar System, understood as a “System of 11 bodies” (the Sun, 9 Planets, and the Asteroid Belt), analyzed in the Light of the Maximum Ordinality Principle.

Such a choice is not only due to the fact that, by itself, it already represents the Solution to the “Three-body Problem” at a more General Level, but especially because:

If the Problem is faced according to this more General Perspective, the “Three-body Problem” can be examined (and solved) according to different initial formulations of completely general character. In fact:

1) The Problem can be formulated by considering three distinct Planets (or bodies) inside the Solar System (according to all their possible “combinations”);

2) At the same time, each one of these “Three-body System” can also be considered as characterized by a specific and proper Habitat. For example, the “residual” part of the Solar System;

3) However, it is also possible to describe anyone of these “Three-body System” as it were an “isolated” System, even if each one is internal to the Solar System. Such a description, in actual fact, would correspond to the Basic Description assumed in Classical Mechanics, which does not consider the inter-action with the surrounding Planets;

4) This does not exclude that it is also possible to consider an arbitrary “Three-body System”, totally independent from the Solar System, situated in the Universe or, more simply, in our Galaxy.

It is evident that each one of the possibilities previously mentioned will correspond to “one sole”, “specific” e “distinct” Solution to the “Three-body Problem”.

So that, by comparing the various different Solutions between them, it will be possible to Ostend which one, among them, can properly be considered the Solution to the “Three-body Problem”. Or, even better, as we will see, it will be possible to recognize a more general meaning associated to all the Solutions, when they are considered all together.

4. The Solar System in the Light of the Maximum Ordinality Principle

The Maximum Ordinality Principle [3] is described in detail in Appendix A, with reference to a Self-Organizing System made up of an arbitrary number of Bodies. In such an appendix, there is not only such a General Formulation of the Principle, but also its Explicit Solution in Ordinal Terms, up to its Solution in Operative Terms, that can directly be implemented in an appropriate Simulator EQS (Emerging Quality Simulator).

If such a General Formulation of the Principle is then referred to the Solar System, understood as a Self-Organizing System made up of 11 Bodies, it is formulated as follows

${\left(\tilde{d}/\tilde{d}t\right)}_s^{\left(\widetilde{11}/\widetilde{11}\right)}\left\{e^{\tilde{\alpha }\left(t\right)}\right\}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (5)

which, as we will see, is able to describe the Evolution of the System in explicit terms.

The number 11, in fact, refers to the number of Bodies that form the Solar System, that is: The Sun + 9 Planets + The Asteroid Belt, while the symbol $\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ indicates that the Solar System, during its time Evolution, is always adherent to its origin and Habitat conditions.

It is then worth underlining that Equation (5) represents the Formulation of the Maximum Ordinality Principle with specific and exclusive reference to the Solar System. In fact:

- If Equation (3.2.8) in Appendix A is written with (m, n) = (11, 11), and k is assumed equal to 1;

- This means that we are describing a “non-living” System;

- In fact, the assumption of orders of derivation higher than k = 1, results as being more appropriate for the description of “Living Systems” and “Conscious Systems”;

- So that, after having assigned the origin and Habitat conditions according to Equation (5.4.1) in Appendix A;

- And after having integrated the corresponding “incipient” differential equations of the first order (5.4.2);

- It is possible to get the Explicit Solution which, by adopting an “internal representation” (for the reasons illustrated in Appendix A), it can be written in the following form

${\left\{\tilde{r}\right\}}_s=e^{\left\{\begin{array}{cccc}0& {\tilde{\alpha }}_{12}\left(t\right)& \cdots & {\tilde{\alpha }}_{1,11}\left(t\right)\\ {\tilde{\alpha }}_{21}\left(t\right)& 0& \cdots & {\tilde{\alpha }}_{2,11}\left(t\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\alpha }}_{11,1}\left(t\right)& {\tilde{\alpha }}_{11,2}\left(t\right)& \cdots & 0\end{array}\right\}}$ (6)

that corresponds to the fact that the Solar System is thought as being structured by “couples” of elements (distinct between them) and every couple is referred, by difference, to the corresponding element of the main diagonal.

In this way, apart from the elements of the main diagonal, all the other couples ${\tilde{\alpha }}_{ij}\left(t\right)$ of the Solar System, understood as a Self-Organizing and Ordinal System, satisfy the following Specularity Relationships

${\left\{{\tilde{\alpha }}_{ij}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left\{{\tilde{\alpha }}_{ji}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (7)

which represent a deeper concept with respect to the traditional concept of “symmetry” (in fact, among other aspects, the symbol “ $\stackrel{\ast }{=}$ ” indicates a simple “assignation” condition).

By starting from this preliminary assumption, such couples, on the basis of the Interior Diffusive Generativity of the System illustrated in Appendix A, will be “Organized” according to the so-called Harmony Relationships (Equation (5.6.5) in Appendix A), which in turn are structured, in particular, in terms of the Ordinal Roots of Unity

${\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j$ (8)

and they are here re-proposed for clearness of exposition

${\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j\circledR \left\{{\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\right\}$

for

$j=1,2,3,\cdots ,11-1$ (9)

Such Equations are characterized by the fact that they are based on one sole couple of reference, generally termed as ${\tilde{\alpha }}_{12}\left(t\right)$ . At the same time, they are also characterized by (11-1) Correlation Factors ${\tilde{\lambda }}_{1,j+1}$ (which generally account for particular Habitat conditions, and for the sake of generality they are supposed as being “not null”), while the (11-1) Ordinal Roots ${\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j$ of $\left\{\tilde{1}\right\}$ (whose specific sequence depends on the adopted couple of reference) transform the Explicit Solution (9) into an “Emerging Solution”. That is, a Solution whose “Information Content” is much higher than the “information content” that corresponds to the initial formulation of the problem [4] .

These represent the Fundamental Relationships that characterize the Solar System as a Self-Organizing System.

In fact, the expressions at the first member of Equations (9), which are defined by Assignation on the basis of the second member of the same, are exactly those that furnish the various “exponents” of the Ordinal Matrix (6), which thus describe the Relation Space of the Solar System and, contextually, the Proper Space of the same System.

This is exactly the reason why they characterize the irreducible “Excess” of Quality of the Solar System with respect to the various constitutive elements.

Obviously, all these aspects have their “reflex” on the “Harmony” of the “Topological” Configuration of the Solar System, which has always to be understood as the “Configuration of the Ordinal Space” of the same.

In fact, the development of the Ordinal Roots of Unity, as shown in detail in Appendix A2, leads to a Formal Representation, of “operative nature”, in which the Relation Space of any System is represented by the three coordinates $\left\{\tilde{\sigma },\tilde{\phi },\tilde{\vartheta }\right\}$ , of “generative nature”, understood as characteristic and specific of “non-Living” System.

In such a Formal Representation, of “operative nature”, for simplicity of exposition the variables ${\tilde{\Sigma }}_0,{\tilde{\Phi }}_0,{\tilde{\Theta }}_0$ synthetically represent the Ordinal coordinates $\left\{{\tilde{\sigma }}_{12}\left(t\right),{\tilde{\phi }}_{12}\left(t\right),{\tilde{\vartheta }}_{12}\left(t\right)\right\}$ , at a generic time t, that are relative to the reference couple, generally termed as “couple 12”, which, on the other hand, can be chosen arbitrarily.

So that, as a “Reflex” of the previous Ordo-Cardinal Relationships (9), the Formal Representation of “operative nature” will be given by the following Equations, which are illustrated in detail in Appendix A:

1) ${\tilde{\rho }}_{1j}\left(t\right)=A\cdot e^{{\tilde{S}}_l\left(t\right)}$ (10.0) where ${\tilde{S}}_l\left(t\right)={\psi }_{1,1}\cdot E_{l,1}\cdot \left[B_l\cdot {\tilde{\Sigma }}_0\left(t\right)-C_l\cdot \left({\tilde{\Phi }}_{0}t+{\tilde{\Theta }}_0\left(t\right)\right)\right]$ (10.1)

2) ${\tilde{\phi }}_{1j}\left(t\right)={\psi }_{1,2}\cdot E_{l,2}\cdot \left[B_l\cdot {\tilde{\Phi }}_0\left(t\right)+C_l\cdot {\tilde{\Sigma }}_0\left(t\right)\right]$ (10.2)

3) ${\theta }_{1j}\left(t\right)={\psi }_{1,3}\cdot E_{l,3}\cdot \left[B_l\cdot {\tilde{\Theta }}_0\left(t\right)+C_l\cdot {\tilde{\Sigma }}_0\left(t\right)+C_l\left({\tilde{\Phi }}_0\left(t\right)+{\tilde{\Theta }}_0\left(t\right)\right)\right]$ (10.3)

$E_l{}_{,i}=\frac{{\epsilon }_{l,i}+4\pi \cdot l}{N-1}$ $B_l=\cos \left(\sqrt{2}\cdot {\psi }_l\right)$ $C_l=D_l=\frac{1}{\sqrt{2}}\sin \left(\sqrt{2}\cdot {\psi }_l\right)$ (10.4)

where

${\psi }_l={\psi }_2\cdot \frac{{\epsilon }_2+2\pi \cdot l}{N-1}$ (10.5).

These are the equations that (as illustrated in Appendix A) describe in “operative terms”, and according to the Maximum Ordinality Principle, the time “Evolution” of any Self-Organizing System made up of N bodies.

In particular, if the number of bodies equals 11, they describe the time “Evolution” of the Solar System.

However, before considering such a general aspect in detail, it is worth considering the distribution of the Planets in the Solar System in stationary conditions, with reference to a prefixed time $t_0$ .

In such a case, the previous Relationships will describe the “Topological” distribution of the Planets in the “Proper” Space of the Solar System. In fact, by considering for example the “couple Sun Mercury” as “couple of reference”, we can obtain the Distribution represented in Table 1, in which the Solution to the Problem of the “Solar System made up of 11 bodies” is compared with the distribution of the Planets based on the semi-empirical Bode’s Law.

5. Comparison between the Distribution of the Planets of the Solar System according to the MOP and That Obtainable on the Basis of Bode’s Law

As an introduction, it is worth recalling the fundamental characteristics of Bode’s Law and its foreseen distribution of the Planets.

5.1. The Distribution of the Planets in the Solar System according to Bode’s Law

In 1778, Johann Bode, after having found a certain “regularity” in the sequence of the Planets of the Solar System, proposed a simple Law, of semi-empirical nature, to describe their topological configuration on the basis of their “distance” from the Sun.

It is a very simple Law, because, by assuming the “distance” Earth-Sun equal to 1, the distance of all the other Planets from the Sun is given by

$d=0.4+0.3\cdot 2^n$ per $n=-\infty ,0,1,2,3,\cdots$ (11)

that is, respectively: 0.4 (Mercury), 0.7 (Venus), 1 (Earth), …. etc. up to Saturn, the last Planet known at that time.

This simple Law has always aroused a particular interest, not only for its predictive capacity (even if with some discrepancy in the case of the Planets Uranus and Neptune), but for its descriptive capacity. And, even more, because it has never received a valid physical interpretation in the context of the Traditional Approach, both in Classical Mechanics and in General Relativity. This can be explained, in particular, because of the absence of the Solution to the “Three-body Problem”.

Table 1. Distribution of the planets in the solar system.

At this point, we can consider the data in Table 1, in which the “Exits” of the “Emerging Solutions” of the Maximum Ordinality Principle are compared with the results of Bode’s Law.

5.2. Basic Criteria to Read and Compare the Data in Table 1

Such Criteria result decisively important when comparing any solution that is a result of the Traditional Approach, which is obviously strictly “cardinal”, and any Solution that originates from an Ordinal Generative Approach such as the Maximum Ordinality Principle.

In the first case, in fact, they usually speak about “distances” between the various couples of the System analyzed. On the contrary, in adherence to the MOP, such “terminology” should be substantially modified. By adopting, for example, a more appropriate term, such as “Uniance”, instead of that of “distance”.

This is because the concept of “distance” tends more to divide, than to unify. In fact, the same etymology of the word (from Latin “dis-stant”) indicates that “one element stays here and the other one stays there” or, equivalently, “one is here and the other one is there”.

On the contrary, the term “Uniance” indicates that the two elements form “one sole thing” of Ordinal Nature, precisely because they are the Exit of the same Generative Process. So that, the term “Uniance” expresses an Ordinal concept, and not a mere cardinal concept, such as that of “distance”. Any “Uniance”, in fact, is characterized by its own Ordinality.

So that, while in the first Approach one can speck of the sole “distance” $\rho$ , independently from the angular variables $\phi$ e $\vartheta$ (because considered as being three independent variables), in the Ordinal Approach the same three “variables” represent one sole “Entity” (that is a Whole). This is the reason why they are represented as $\left\{\tilde{\rho },\tilde{\phi },\tilde{\vartheta }\right\}$ , that is internal to curly brackets, because in this way, it is also possible to contextually point out that all of them are understood as variables of Generative Nature (from which the tilde notation).

On the basis of such premises, we can start to examine Table 1 in more detail.

Let us start from the data of the second and third column in Table 1. From their comparison it is possible to recognize what previously said with reference to Bode’s Law as a semi-empirical Law with respect to the astronomical measures obtained according to the Traditional Approach.

Our attention, however, is prevailingly addressed to the “comparison” between the “experimental data” (Column 3) and the cardinal-reflexed values obtained on the basis of the Description according to the Maximum Ordinality Principle.

In this respect, in fact, it is worth underlining, once again, that:

1) They properly speak of cardinal-associated “reflexed” values, because the values of Columns 4 and 5 are never totally separable from their correspondent Ordinalities;

2) In fact, in order to be compared with the corresponding “experimental data”, their pertinent levels of Ordinality are “progressively” reduced;

3) Such a “progressive reduction”, however, always continues to reflect, even if in a less marked form, the Ordinalities of the Origin Formal Relationships in which they appear;

4) More precisely, this means that the values of Columns 4 and 5 are obtained by reducing (in the Formal Model adopted) the Ordinality of the reference couple “12”. In fact, instead of considering the Originary Binary-Duet Relationship

${\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (12)

this is implemented in the “reduced” form

$\left\{{\tilde{\alpha }}_{12}\left(t\right)\oplus {\tilde{\lambda }}_{12}\left(t\right)\right\}$ (12.1)

that is, in the absence of its correlative and specific Ordinality;

5) One of the main consequences is that, while in the Traditional Approach the “orbits” of the Planets are described as they were “ellipses”, Relation (12.1) on the contrary leads to a description of the orbits that (as we will see later on) only approximately they can be considered as being “ellipses”;

6) In addition, while the traditional astronomical measurements usually refer to the aphelion of the Planets (or, like above, to a mean value between perihelion and aphelion, always with reference to an elliptical orbit), the Ordinal measurements always refer to the effective configuration of the various Planets of the Solar System at any given time, that is, with reference to their effective “angular” position. And this precisely because, as already anticipated, the variable $\left\{\tilde{\rho },\tilde{\phi },\tilde{\vartheta }\right\}$ can never be separated, not even in their “reflexed” values;

7) To conclude these introductory considerations, which are apt at illustrating the “principal” differences between the experimental values (totally cardinal) and the “cardinal reflexed” values of the Ordinal Approach, it is worth noting that Column 4 refers to the Solar System as it were “isolated” from its natural Habitat, while Column 5 accounts for the contribution of our Galaxy as a surrounding Habitat.

6. “Evolutive” Characteristics of the Solar System

The Harmony Relationships (10.1), (10.2), (10.3), (10.4), (10.5), on the basis of the same methodology shown at the previous Paragraph 4, are evidently able to furnish the Description and the Distribution of the “Uniances” of the various Planets at any time t, but not only. They are able to do much more. In fact, they are also able to describe all the Fundamental “Evolutive” Characteristics of the Solar System. And this Aspect will result as being particular important when we will consider the specific Evolution of a “Three-body System” “internal” to the Solar System.

6.1. The Azimuthal Angular Distribution of the Planetary Orbital Planes with Respect to the Ecliptic

This phenomenon has never found a satisfactory explanation, neither in Classical Mechanics nor in General Relativity. The main reason fundamentally depends on the fact that, in the absence of any explicit solution to the “Three-body Problem”, it is impossible to evaluate, in such Disciplines, the exact influence between the reciprocal orbits of the Planets.

The various angles of the orbital planes, in fact, are distributed in a cone of a rather large width (20˚), which reduces to 10˚ only if the extreme Planets (Neptune and Pluto) are “excluded” (because the latter are usually considered as being rather “anomalous”). Such a Distribution, on the contrary, is clearly described by The Harmony Relationships (10.1), (10.2), (10.3), (10.4), and (10.5) and, as we will see, the foreseen angles are persistently the same even if we consider the Evolution of any specific “Three-body System” “internal” to the Solar System.

6.2. Precessions of the Planets

The Maximum Ordinality Principle, when adopted to describe the Solar System, is also able to describe another “Irreducible Excess”: The Precessions of the Planets.

General Relativity, in fact, which has given a preliminary answer to this phenomenology, assumes that their values are always “constant” in time and they can be evaluated in terms of a direct interaction between two sole celestial bodies, such as in the case of Sun and Mercury.

The Harmony Relationships (10.1), (10.2), (10.3), (10.4), and (10.5), on the contrary, show that the Precession of any Planet is not “constant” in time and its entity (in time) has to be considered in the context of the Solar System understood as a Whole.

As we will see, this is also true even if one considers the precessions of Three Planets, understood as a “Three-body System”, “integrated” in the Solar System.

Such phenomenon of the “Procession of the Planets”, in addition, is strictly related to another characteristic phenomenon of the Solar System: The “Rosette Motion” of the Planets.

6.3. The “Rosette Motion” of the Planets

The “Rosette motion” of the Planets is strictly related to the Procession of the Planets because the Harmony Relationships (10.1), (10.2), (10.3), (10.4), and (10.5) reveal that the various Planets cannot be everywhere.

In fact, precisely because of the subsistence of those Harmony Relationships, they must be characterized by specific and reciprocal Relationships of “Uniance” with respect to all the other Planets. Relationships which, however (and it is worth recalling once again) are not the result of any form of “necessity”, because they originate from Assignation Relationships of Generative Nature.

Consequently, if we consider, for example, the “Rosette” described by Mercury with respect to the Sun (which, on the other hand, is identical to that described by the Sun if observed by Mercury), and we imagine other two Rosettes, that described by Venus (with respect to the Sun (e vice versa) and that of Mercury respect to Venus (e vice versa), these Rosetta motions are, at any time, harmoniously related between them as indicate by Harmony Relationship (9) and their corresponding “operative” version.

In addition, the “Rosetta Motions” just recalled are also able to show a real “novelty” with respects the present Astronomical knowledges.

6.4. The Azimuthal Angular Variations in Time of the Planetary Orbital Planes with Respect to the Ecliptic

The previous exposition indicates that the same orbital planes of the Planets have not the same perennial azimuthal angular distribution. In other terms, they have not a “constant” lying in time, as it is generally assumed in Classical Mechanics and in General Relativity. This is because, on the other hand, they are usually measured with respect to the ecliptic. But the same plane of the ecliptic is not properly constant in time.

The General Structure of the Harmony Relationships (10.1), (10.2), (10.3), (10.4), and (10.5), in fact, is valid for all the Bodies belonging to the Solar System. Obviously, apart from the specific values pertaining to each single Body.

This means that the abovementioned Harmony Relationships are not “directly” and “exclusively” referable to each single Body as such, but, as already pointed out, they are the Reflex of a much Wider “Harmony”, that “transcends” them, because the “Harmony” is directly referable to the Solar System as a Whole.

As a consequence of the previous considerations, such a Concept of Harmony continues to be valid not only with reference to single Bodies inside the Solar System, but also with reference to groups of Bodies. Such as, for example, any “Three-body System” Internal to the Solar System.

7. The “Three-Body Problem” Described, Analyzed, and Solved according to Different General Perspectives

The previous description of the Solar System, understood as an “11-body System”, is evidently already indicative, by itself, of the possibility of obtaining the Solution to the “Three-body Problem”.

In fact, in Classical Mechanics the Solar System is not solvable, by itself, in explicit terms, in particular because the insolvability of the “Three-Body Problem”. This means that if the Solar System admits a solution in explicit terms, this fact proves in favor of the Solution to the “Three-body Problem”. But not only. In fact, the Solution to the Solar System as an “11-Body System”, because of its very General Description, offers the possibility of describing, analyzing and solving the “Three-body Problem” according to several Different Perspectives, always in the context of the Solar System, as it will be shown in detail in the next paragraphs.

7.1. Three Distinct Planets (or, Alternatively, the Sun and Two Planets) Understood as “Part” of the Solar System

In this case, it is possible to consider each one of the $\frac{11!}{3!\left(11-3\right)!}$ combinations of 11 bodies, considered in groups of 3.

The correlative solution to the Problem can directly be obtained from the Harmony Relationships (9), which, as already anticipated, are always valid with reference to any “arbitrary couple”, conventionally termed as “12”.

Consequently, after having conventionally denominated as 1, 2, 3, the three considered Bodies, and having properly defined the correlative sequence of the Ordinal Roots of Unity with specific reference to the selected “couple 12”, the first member of Relationships (9) (which is defined, by assignation, in adherence to the second member) will furnish the Ordo-cardinal Coordinates of the couple “12” e “13”, as here indicated

${\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j\circledR \left\{{\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\right\}$

for

$j=1,2$ (13)

and, “by difference”, also the Ordo-cardinal Coordinates of the couple “23”, that is:

${\left\{{\tilde{\alpha }}_{2,3}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{2,3}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (14)

Consequently, by starting from the Harmony Relationships (13) and (14), the correlative Relations (10.1), (10.2), (10.3), (10.4), (10.5) will furnish, correspondently, the cardinal-reflexed values of “operative nature” of the three variables $\left\{\tilde{\rho },\tilde{\phi },\tilde{\vartheta }\right\}$ pertaining to the “three couples” of the considered Bodies.

In this respect, it is important to observe, and also to underline, that although the Ordinal-cardinal Relationships (13) and (14) and their corresponding Relationships of “operative nature”, specifically refer to the considered Three Couples of Bodies, the Harmony that they manifest in reality is “part” of a more General Harmony, which concerns the entire Solar System understood as Whole. Consequently, and at the same time, the Three Couples of Bodies faithfully “reflect” the general “Evolutive” Characteristics of the Solar System previously described. That is: The distributions of the “Uniances” at any time, the angular distribution of the orbital planes, the Precessions of the Planets, the “Rosette motions” of the single Planets, and the angular variations in time of the planetary orbital planes with respect to the Ecliptic.

7.2. Three Distinct Planets, as in the Previous Case, with the Difference That Now the Solar System Is Thought as Being “Part” of Our Galaxy

In this case, the Solution to any “Three-body Problem” substantially present the same Structure of the previous case, however with the difference that now the corresponding ${\tilde{\alpha }}_{ij}\left(t\right)$ that characterize the single Couple of Bodies will result slightly modified, as a consequence of the different initial configuration of the Solar System, because in this case, the Solar System is represented with its pertinent “Habitat”. However, by taking into consideration that it is impossible to simulate the entire Galaxy, the latter will be simulated as an “Incorporated Habitat”, exactly as already done in [5] , from which we have taken that data of Column 5 in Table 1, denominated “System + Habitat”.

Nonetheless, even in this case, the Ordo-cardinal Relationships that describe the “Evolution” of the Three considered Bodies continue to manifest a Harmony which is always “part” of that more General Harmony which characterizes the Entire Solar System, although it is now characterized by its “Incorporated” Habitat. This is because the Solar System, according to the Maximum Ordinality Principle, always tends to maximize its Ordinality and that of its correlative Habitat.

This evidently means that, apart from some more or less marked specific differences with respect to the previous case, the various “Three-body Systems” now considered will always “reflect”, by themselves, the specific “Evolutive” Characteristics of the Solar Systems, although characterized by its Habitat. That is: The correlative distributions of the “Uniances” at any time, the angular distribution of the orbital planes, the Precessions of the Planets, the “Rosette motions” of the single Planets, and the angular variations in time of the planetary orbital planes with respect to the Ecliptic.

7.3. Three Distinct Planets (Such as in Case 7.1) with the Difference That Now They Are Considered “Isolated” by Remain Part of the Solar System

In analogy to the more general case of the Solar System made up of 11 Bodies, the Problem can be now formulated as follows

${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left(\tilde{3}/\tilde{3}\right)}\left\{e^{\tilde{\alpha }\left(t\right)}\right\}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (15)

and the solutions can be obtained by means of the same procedure followed in the case of the Solar System as a Whole.

Now, however, the Solution pertaining the Three Couple of Bodies will be different with respect to the corresponding Solutions when the Three Couple of Bodies are considered as being “part” of the Entire Solar System.

Consequently, the Solution now obtained to the Problem will show different values with respect to the general “Evolutive” Characteristics of the Solar System, such as the distributions of the “Uniances” at any time, the angular distribution of the orbital planes, the Precessions of the Planets, the “Rosette motions” of the single Planets, and the angular variations in time of the planetary orbital planes with respect to the Ecliptic.

7.4. Three Distinct Planets (Such as in the Previous Case 7.3) with the Difference That They Are Now Considered Characterized by an “Incorporated” Habitat

It is obviously also possible to consider the Thee Couple of Bodies, of the previous case, as being characterized by an “Incorporated” Habitat, which, for example, could possibly simulate the “remaining part” of the Solar System, initially not considered in the formulation of the Problem.

Also in this case, however, the corresponding Solutions will result substantially different from those of Case 7.1. This is because the “Incorporated” Habitat cannot properly simulate the Harmonious Properties of the Solar System when it is considered as a Whole.

7.5. Solution to the “Three-Body Problem” When the Bodies Are “External” to the Solar System

As a preliminary example, we can consider that of the so called “Triple Stars”.

7.5.1. The “Triple Stars”

This is a case relatively “frequent”. Let us think, for example, of the North Star.

This “Three-body System”, completely external to the Solar System, can evidently be described as Self-Organizing System as it were “completely isolated”. This hypothesis, on the other hand, is precisely that which corresponds to the “basic hypothesis” of the “Three-body Problem” in Classical Mechanics.

In such a case, the Description and the Solution to the Problem is perfectly analogue to that previously shown with reference to the Solar System made up of 11 Bodies and completely “isolated”. The sole difference is that now the Fundamental Equation of the Maximum Ordinality Principle will be written with reference to a “Three-body System” (like in the case of previous Paragraph 7.3)

${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left(\tilde{3}/\tilde{3}\right)}\left\{e^{\tilde{\alpha }\left(t\right)}\right\}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (16)

The corresponding Solution can be then obtained with modalities substantially analogue to those previously shown in the case of the Solar System made up of 11 Bodies. The fundamental difference will consist in the fact that now the Solution will present a different number of Harmony Relationships and, at the same time, they will be characterized by a specific different Structure.

In fact, they will present correlative variations in the Ordinal Roots of Unity, which now are of the Ordinality 1/(3-1)), as well as, specific variations in those coefficients which are pertaining to the Relationships of “operative nature”. In particular, the corresponding periodicity factors.

In all cases, it is important to underline that this modality of Description and Solution is not circumscribed to the sole Celestial Mechanics, with specific reference to Three Bodies “totally isolated”.

In fact, it can also be adopted, for example, in Quantum Mechanics. This is possible on the basis of the considerations and reflections that will presented later on, in a successive paragraph, specifically devoted to the Relationships between “Forces” and “Diffusive Generativity”.

In the case of the considered System of “Triple Stars”, it may also happen that it might result as being very near to other Stars, or even, very near to a Galaxy. In such a case the “Three-body System” can be thought and described as being characterized by a proper Habitat. In this sense, it is like to consider, equivalently, the System of “Triple Stars” as it were described in the context of the Entire Universe, whose “effects”, however, would be simulated in the form of an “Incorporated” Habitat, in a perfect analogy with the previous considered case of the Solar System.

7.5.2. The “Triple Galaxies”

This case is substantially more general than the previous one.

We can think, for example, of the “Triplet of Lion”, also known as Group M66.

This is a little group of Galaxies, at a “distance” of about 35 million light years from Earth, in the Constellation of Lion, made up of Three “Spiral” Galaxies (M66, M65 and NGC 3628).

Also, in this case, the Solution to the “Three-body Problem” is not dissimilar from the previous considered case, even if it is much more improbable to identify a potential “Habitat” to be incorporated.

Consequently, the Description of the System will formally be that of an “Isolated” System, although always “part” of the Universe. This is an aspect that will be reconsidered in more detail in the last paragraph, devoted to a possible Over-Conclusion.

8. Conclusions

On the basis of what was previously shown, it is possible to assert that the Formulation of the “Three-body Problem”, as it is considered in Classical Mechanics (see Part 2), does not represent the most general Formulation of the Problem.

Mainly because:

1) Apart from the consideration of the “effects” of the “forces” between the three Bodies, it does not consider the “effects” of the presence of a correlative “Habitat”;

2) For example, the Formulation does not consider that the Three Bodies could be Three Planets of the Solar System, and consequently, it does not consider the correlative “effects” of the “forces” due to all the other Planets;

3) Even less, it does not foresee the possibility of considering an “Incorporated” Habitat;

4) This is also true, in particular, when the System of the Three Bodies is totally “external” to the Solar System. In such a case the Mathematical Formulation “circumscribes” the Description to the sole Three Bodies, even when in some cases (let us think of the “Triple Stars” and the “Triple Galaxies”), it could be more appropriate to consider their “Evolution” in the presence of a correlative Habitat.

In addition, that is apart from the previous considerations, there is an Aspect of Particular Relevance that has to be mentioned with reference to all the previous Solutions pertaining to any considered Formulation of the “Three-body Problem”. That is:

8.1. The Relationship between “Forces” and “Diffusive Generativity”

In the context of Classical Mechanics, the relationships between the various Bodies are described in terms of “forces”, understood as “efficient causes”, of mechanical nature, to describe the “Three-body System” according to the Mathematical Formulation given in Paragraph 2.

In the Ordinal Context, on the contrary, the General Tendency of the Systems to the Maximum Ordinality, and the correlative Relationality between the Bodies, is “Guided” by the Diffusive Generativity of the System understood as a Whole.

As a consequence, it is precisely such a Diffusive Generativity that “first” generates the “Binary-Duet” Couples between the various Bodies, and then establishes between them the Harmony Relationships, in order to successively re-organize their Ordo-cardinalities in terms of Ordinal Roots of Unity.

This in fact is precisely the main characteristic that qualifies a System as a “Self-Organizing” System.

These considerations, however, also lead to highlighting another important Aspect, strictly related to the previous one.

8.2. All the Proposed Solutions to the “Three-Body Problem” Result as Being of “A-Functional Nature”

What was previously said with reference to the Properties of the “Diffusive Generativity” represents, at the same time, one of the fundamental reasons for the fact that all the proposed Solutions to the “Three-body Problem” are not of “functional nature”, as, on the contrary, they are researched for (even if “inexistent”) on the basis of the Formulation of the Problem in Classical Mechanics.

This is because, as it is possible to clearly recognize from Equations (10.1), (10.2), (10.3), (10.4), and (10.5), these give the Solution to the Problem in operative terms, which however is structured in such a form that, by starting from the “specific” Ordo-Cardinalities of the reference couple “12”, they realize a sort of “Recompositio ad Unum”, with a completely different result.

On the other hand, this is a specific consequence of the fact that all the various Solutions to the “Three-body Problem”, previously presented, have been obtained with reference to the Maximum Ordinality Principle, and not with reference to the Formulation of the Problem as usually considered in Classical Mechanics.

This contextually leads to some other Aspects of particular Relevance concerning the proposed Solutions.

8.3. Aspects of Particular “Relevance” of the Previous Solutions to the “Tree-Body Problem”

The Solutions to the “Three-body Problem” previously obtained show that each System “Evolves” in its “Proper Space” and its “Proper Time”, and, its “Topological Evolution” has its specific “reflex” in terms of “Uniances” between the various parts of the System, and not in mere terms of “distances”.

These two Aspects, however, which are specific and characteristic of all the Solutions to the Maximum Ordinality Principle, will be more properly dealt with in Appendix A, precisely because the latter is specifically devoted to the general properties of the Maximum Ordinality Principle and its explicit Solutions.

At the same time, there is another aspect that is worth clearly pointing out with the reference to the Solutions to the “Three-body Problem”. It is an Aspect which, although equally characteristic of all the Solutions to the Maximum Ordinality Principle, and for this reason, it is equally mentioned in Appendix A, in the case of the “Three-body Problem”, it plays a particular “Role”, that can be the basis of a “Relevant Over-Conclusion”.

9. Over Conclusion: The “Cosmological Problem” in the Light of the “Three-Body Problem”

The previous Solutions to the “Three-body Problem” have pointed out that such a Problem does not admit a unique and exclusive Solution. In fact, any Solution “involves” the consideration of an appropriate and specific Habitat.

This is true not only for a “Three-Planet System” internal to the Solar System, which at least suggests, as its proper Habitat, the consideration of an “Incorporated” residual Solar System.

It is also true for the same Solar System, even considered as a Whole. Because it is part of our Galaxy.

And even in the case of the same Galaxy, which, in turn, can be considered as being a part, together with other Galaxies, of the Entire Universe.

Such a Progressive Ascendency of the Description reveals that any Solution can always be considered as being not less than.

Such a consideration, if extended to the Entire Universe, leads to consider the latter as being a “Self-Organizing System”, whose “Quality” represents an “Irreducible Excess”.

Consequently, all the “Emerging Solutions” previously shown with reference to the various “Three-body Problem”, should be considered as being “part” of One Sole Emerging Solution, directly referable to the “Entire Universe”.

In this sense, the “Three-body Problem”, as previously analyzed in the Light of the Maximum Ordinality Principle, and characterized by the absence of one sole and unique Solution, suggests that it could consequently more properly be considered in the Light of a “Cosmological Perspective”.

In fact, when considered from such a specific Perspective, it shows a more direct Relationship with the Entire Universe, understood as a “Self-Organizing System”. And this is an Aspect that, in the case of the “Three-body Problem”, “appears” with a much more marked “Evidence” than in the case of other “Self-Organizing Systems” in Nature.

This is also the reason why the Solutions to the “Three-body Problem can also represent a “Suggestion”, an “Invitation”, a “Proposal” for a Ri-Orientation of the Present Scientific Perspective in “Favor” of the “Irreducible Excess” of Quality, which “reveals” in the Phenomena of the Universe, in particular when such an “Irreducible Quality” is described” in the Light of the Maximum Ordinality Principle.

Appendix A. Physical, Logical and Formal Presuppositions for the Solution to the “Three-Body Problem”: The Maximum Ordinality Principle, from the “Incipient” Derivative to EQS Simulator

Appendix A, articulated in three parts, presents a synthesis of the developments concerning the Maximum Ordinality Principle, with reference to a Self-Organizing System made up of an arbitrary number N of Bodies.

All the various developments, in adherence to what is indicated in the title, have been illustrated in the various papers presented at the Biennial Emergy Conferences (University of Florida) from 1999 to 2020.

In this respect, Appendix A will also present two aspects of particular Relevance:

1) The Process of Genesis of The Harmony Relationships and

2) The Process of Genesis of The Ordinal Roots of Unity.

The Maximum Ordinality Principle [3] is nothing but the reformulation of the Maximum Em-Power Principle [6] [7] [8] , given however in a more general form by means of a new concept of derivative, the “incipient” derivative, whose mathematical definition has already been presented in [9] - [15] .

In this way, both Emergy and Transformity are replaced by the concept of Ordinality. This is the reason why the Principle was renamed as the Maximum Ordinality Principle.

Consequently, on the basis of the Mathematical Formulation of the Maximum Ordinality Principle [3] and, in particular, its adoption as “One Sole Reference Principle” [16] , we can now present, in more details, the radically New Perspective that such a Principle offers to Modern Science. That is: “Every System is a Self-Organizing System”.

In order to give a clear presentation of the fundamental differences between such a New Perspective with respect to the Traditional Scientific Approach, Appendix A will start from the consideration of a synoptic picture of the basic characteristics of the two mentioned Scientific Approaches (see Table 1), successively analyzed and compared, in more detail, in the context of Appendix A.

1. Fundamental Characteristics of the Two Scientific Approaches

In this respect, it is worth starting from recalling that Self-Organizing Systems and their “emerging properties” began to be studied by L. Boltzmann toward the end of XIX century [17] . Several other Authors (e.g. A. Lotka [18] [19] [20] ) dealt with such a theme. However, Self-Organizing Systems received the most significant contribution by H.T. Odum (1955), with the genial introduction of a more appropriate formal language.

The consequential faithful developments of Odum’s approach have led us to the formulation of a unique general Principle, the Maximum Ordinality Principle (MOP), which is able to describe, by itself, the behavior of any given System as a Self-Organizing System: both “non-living” Systems, “living” Systems and “thinking” Systems too (e.g. Human Systems).

Such a conclusion then results as being deeply different from that of Modern Science, which, from Newton on, is persistently orientated at describing any known system as it were a “mechanism”.

Appendix A, after having synthetically recalled the formulation of the MOP and after having pointed out its corresponding descriptive advantages, will focus on the intrinsic new perspective offered by the MOP, especially in thinking, decision making and acting, with respect to the Traditional Approach. In particular, with reference to any form of relationship between Man and his surrounding Environment.

In particular, and with reference to this fundamental aspect, the basic differences between the two afore-mentioned perspectives will be brought out by comparing, on the one hand, “side effects” (related to the Traditional Approach) and, on the other hand, the “Emerging Exits” (specifically pertaining to the New Approach).

Let us then consider first the Traditional Approach that characterizes Modern Science.

1.1. The Traditional Scientific Approach

Modern Science is characterized by a persistent and progressively ascendancy toward ever more general Physical Laws and Principles.

However, before any formulation of a single hypothesis or a physical theory, Modern Science (let us say, from Newton on) adopts three fundamental pre-suppositions (see Table A1): the causality principle (also termed as “efficient causality”), classical logic (also termed as “necessary logic”), and functional relationships (between the various parts of any System analyzed).

Table A1. Synoptic comparison between the basic presuppositions of the two differential formal languages and their main corresponding fundamental characteristics.

On the basis of such fundamental presuppositions, and only after having developed a strictly conform consequential formal language (that is the Traditional Differential Calculus (TDC)), Modern Science progressively ascends toward ever more general Physical Laws and Principles:

1) From Phenomenological Laws (e.g. Kepler’s Laws); 2) to Physical Laws specific of each Discipline (e.g. Newton’s Laws, Maxwell’s Equations, etc.); 3) up to the three well-known Thermodynamic Principles.

Such a progressive development has given origin to a hierarchy of a multiplicity of quantitative Physical Laws and Principles, in particular as a consequence of the first basic presupposition: the causality principle. This Principle, in fact, has led Modern Science to introduce “different causes” in different Disciplines. The Principle of causality, in fact, tends to “sub-divide” the entire phenomenology (at present known) in different “branches”, precisely because, on the basis of such a presupposition, it leads Scientists to research for the most “appropriate causes” pertaining each specific set of phenomena each time considered.

In this way, Modern Science persistently propends to show that: “Every System is a mechanism”.

Such a conclusion, however, although confirmed by experimental results, can be considered as being valid only from an operative point of view, but not from an absolute point of view. This is because “necessary logic” (second basic presupposition) does not admit any form of “perfect induction” (see Popper’s Falsification Principle).

In fact, as synthetically illustrated in Table A1, in the strict contest of “necessary logic”:

1) After having formulated a single or more hypotheses (such as in the case of a Theory);

2) After having formalized them in an appropriate formal language (faithfully conform to the three above-mentioned basic presuppositions);

3) After having drawn the consequential conclusions;

4) And after having also obtained experimental confirmations of the previous formal conclusions;

5) It is impossible, in any case whatsoever, to assert the uniqueness of the inverse process. That is: it is impossible to show that the hypotheses adopted are the sole and unique hypotheses capable to explain those experimental results.

This is precisely because of the absence, in “necessary” logic, of any form of perfect induction.

In fact, only in the presence of a perfect induction it would be possible to assure the uniqueness of the inverse process and, thus, to transform the adopted hypotheses into an absolute perspective.

This means that Modern Science, precisely because based on necessary logic, should always be “open” to recognize that there always exist many other possible Approaches (in principle infinitive) capable to interpret the same experimental results.

At this stage, after having synthetically recalled the basic characteristics of Modern Science, we can analyze in more detail the fundamental properties of the New Perspective, synthetically indicated in parallel (for a better comparison) in the right side of Table A1.

1.2. “Emerging Quality” of Self-Organizing Systems and Adoption of New Mental Categories

After having synthetically recalled the basic characteristics of Modern Science and its corresponding formal language, we can now analyze the fundamental properties of a New Scientific Perspective, which leads to the introduction of a new Formal Language, the Incipient Differential Calculus (IDC). As anticipated, the fundamental properties we are referring to are synthetically indicated in parallel (for a better comparison) in the right side in Table A1.

Such a New Scientific Perspective is based on the phenomenological “Emerging Quality” of Self-Organizing Systems [21] . This represents the fundamental aspect that leads to the adoption of the corresponding new mental categories (shown in Table A1).

The expression “Emerging Quality of Self-Organizing Systems” refers to the fact that Self-Organizing Systems always show an unexpected “excess” with respect to their phenomenological premises. So that they usually say: “The Whole is much more than its parts”.

Such an “excess” can be termed as Quality (with a capital Q) because it cannot be understood as being a simple “property” of a given phenomenon. This is because it is never reducible to its phenomenological premises in terms of traditional mental categories: efficient causality, logical necessity, functional relationships.

This evidently suggests a radically new gnoseological perspective, which corresponds to recognize that: “There are processes, in Nature, which cannot be considered as being pure mechanisms”.

This also leads, in adherence, to the adoption of “new mental categories”1 and, correspondently, to the development of a completely new formal language, so that the description of Self-Organizing Systems might result as being faithfully conform to their “Emerging Quality”.

2. The Progressive Development of an Appropriate Formal Language

L. Boltzmann was the first who attempted at describing Self-Organizing Systems in more appropriate formal terms, by proposing the adoption of a new Thermodynamic Principle: The Principle of Maximum Exergy Inflow to the System [17] .

Some years later, A. Lotka [18] reformulated such a Principle in the form of: The Principle of Maximum Exergy Flow through the System (Lotka [19] [20] ).

Both such attempts were not perfectly successful, because still based on the concept of Exergy, which is a quantity that is strictly pertaining to Classical Thermodynamics. Consequently, it re-proposes, by itself, the concepts of efficient causality, logical necessity, functional relationships.

A really new formal language only appears with H. T. Odum, with the genial introduction of Emergy (Em), defined as Exergy (Ex) by Transformity (Tr)

$Em=Ex\cdot Tr$ (2.1).

Equation (2.1) clearly shows that Emergy is still based on “Exergy”. However:

1) Quality Factor Tr “Transforms” Ex into a new physical quantity: Emergy;

2) The latter in fact is not defined in “functional terms”, but only by “assignation Rules” [22] ;

3) This is precisely because Tr is expressed by means of a non-conservative Algebra;

4) Consequently, the output “excess” of the three Fundamental Process in Emergy Analysis (Co-Production, Inter-Action, Feed-Back) is always understood as being “irreducible” to its specific inputs in mere functional terms.

This means that Emergy is able to represent the “Emerging Quality” of Self-Organizing Processes. Consequently, the general enunciation of the Maximum Em-Power Principle (Odum [6] [7] [8] ) can equally be referred, at a phenomenological level, to the corresponding maximization tendency of the “Emerging Quality” on behalf of Self-Organizing Systems.

The Maximum Em-Power Principle, however, had not a corresponding and specific formulation under variable conditions. On the other hand, such a formulation in variable conditions could not be given in terms of the Traditional Differential Calculus, because the traditional derivatives, as a consequence of their conceptual basic presuppositions (see Table A1), are not properly apt at representing the “generative” behavior of “Self-Organizing Systems”, and consequently they tend to partially “filter” such a “generative” behavior.

This is why, in order to achieve an appropriate mathematical formulation of the Maximum Em-Power Principle, I introduced the concept of “Incipient Derivative” [9] [10] [11] [12] [13] , defined as

${\left(\frac{\tilde{d}}{\tilde{d}t}\right)}^{\tilde{q}}f\left(t\right)=\underset{\tilde{\Delta }t:0\to 0^{+}}{L\tilde{i}m}\circ {\stackrel{\sim }{\left(\frac{\tilde{\delta }-1}{\tilde{\Delta }t}\right)}}^{\tilde{q}}\circ f(\; t\; )$

for

$\tilde{q}=\tilde{m}/\tilde{n}$ (2.2)

a definition that will be illustrated in detail in the next paragraph.

However, it is already possible to anticipate that such a definition shows that the “Incipient Derivative” is not an “operator”, like the traditional derivative (d/dt), but it could be termed as a “generator”, because it describes a Process in its same act of being born [14] .

The Mathematical Formulation of the M. Em-P. Principle in terms of Incipient Derivatives was preliminarily given in [10] , and afterwards, in a more articulated form, in a specific book co-financed by the Center for Environmental Policy [11] .

During the successive eight years (2002-2010), such a mathematical formulation was adopted in several Disciplines, such as Classical Mechanics, Quantum Mechanics, General Relativity, Chemistry, Biology, Economics and the corresponding results were reunited in two books (titled: “Lightness of Quality” [23] and “Ascendency of Quality” [24] ).

At the end of this wide range of applications, I realized that it was possible to give a more general formulation of the Maximum Em-Power Principle, in the form of the “Maximum Ordinality Principle” [3] .

For the sake of clearness, the Rational of such a generalization process, articulated in a few logical steps, is recalled in the next sections.

3. The Incipient Derivative of Ordinality $\tilde{q}$

The “Incipient” Derivative of a given Ordinality $\tilde{q}$ , whose definition previously introduced is here recalled for the sake of clarity

${\left(\frac{\tilde{d}}{\tilde{d}t}\right)}^{\tilde{q}}f\left(t\right)=\underset{\tilde{\Delta }t:0\to 0^{+}}{L\tilde{i}m}\circ {\stackrel{\sim }{\left(\frac{\tilde{\delta }-1}{\tilde{\Delta }t}\right)}}^{\tilde{q}}\circ f(\; t\; )$

for

$\tilde{q}=\tilde{m}/\tilde{n}$ (3.1)

will be illustrated by considering first its general properties and, immediately after, its more specific properties.

To this purpose, it is worth preliminary pointing out that the concept of “Ordinality” refers to two “distinct” concepts, which however are considered as being one sole entity, that is as a Whole. These are: its “cardinality” and its “ordinal genetic relationships”. This means that the Ordinality $\tilde{q}$ , synthetically represented as $\tilde{q}=\tilde{m}/\tilde{n}$ (as in Equations (2.2) and (3.1)), in reality it has to be more properly understood as

$\left\{\tilde{m}/\tilde{n}\right\}=\left\{k,\left(\tilde{m}/\tilde{n}\right)\right\}$ (3.2)

in which:

- k represents its cardinality;

- While $\left(\tilde{m}/\tilde{n}\right)$ represents its Ordinal Genetic Relationships, where the round brackets expressly indicate that they represent only a part of the concept of Ordinality, understood as a Whole. In fact, the first member of Equation (3.2) is represented in curly brackets, precisely because this symbol is usually adopted to indicate the concept of a Whole.

The Ordinal Genetic Relationships $\left(\tilde{m}/\tilde{n}\right)$ can also more synthetically termed as “Ordinal Relationship”, not only because they are not “functional” Relationships, but especially because the adjective “Ordinal” also indicates that they are precisely those Relationships that give the most significant contribution to the definition of the general concept of Ordinality understood as a Whole.

3.1. General Properties of the “Incipient” Derivative of Ordinality $\tilde{q}$

Definition (2.2) clearly shows what we have synthetically anticipated, that is: the “Incipient Derivative” is not an “operator”, like the derivative (d/dt) in the Traditional Differential Calculus (TDC), but it could be termed as a “generator”, because it describes the Generativity of a given Process, in its same act of being born [9] - [14] . In fact:

1) The sequence of the symbols is now interpreted according to the direct priority of the three elements that constitute its definition (from left to right). This is the reason why they acquire a completely new different meaning with respect to the traditional one;

2) The three symbols, in fact, do not represent “three” distinct operations, but a unique and sole Generative Process;

3) The symbol $L\tilde{i}m$ , whose etymological origin comes from the Latin word “Limen” (which means a “threshold”), represents the “threshold” of that “ideal window” from which we observe and describe the considered phenomenon;

4) The symbol $\tilde{\Delta }t:0\to 0^{+}$ now indicates not only the initial time of our registration, but also the proper “origin” (in its etymological sense) of something new which we observe (and describe) in its proper act of being born, as a Generative Process;

5) It is then evident that the “operator” $\tilde{\delta }$ now registers the variation of the observed property $f\left(t\right)$ , not only in terms of quantity, but also, and especially, in terms of Quality (as the symbol “tilde” would expressly remind). So that the ratio which appears in Equation (3.1) indicates not only a quantitative variation in time, but both the variation in Quality and quantity;

6) Consequently, when we take the incipient (or “prior”) derivative of Ordinality $\tilde{q}$ of any $f\left(t\right)$ , the Exit of such a process will keep “memory” of its genetic origin. This is because, besides its quantity, it will result as being Ordinally structured (as shown at the next Paragraph 3.2.2) according to the indication of such an exponent. The latter in fact precisely expresses how each part of the output is genetically Ordered to the Whole and, at the same time, how each part is related to all the others in terms of Ordinal Harmony Relationships (illustrated at Paragraph 5.6);

7) In this way, the “incipient” derivative represents the Generativity of the considered Process, that is the output “excess” (per unit time) characterized by both its Ordinal Genetic Relationships and its related cardinality, while the sequence of the symbols in its definition can be interpreted as representing a unique inter-action process between the same;

8) The above-mentioned reasons clearly show why the “Incipient” Derivative, precisely because of such properties, is able to unify (and, at the same time, to specify) the description of the various Self-Organizing Processes of the surrounding World, when they are explicitly understood in terms of Quality.

3.2. Specific Properties of the “Incipient” Derivative of Ordinality $\tilde{q}$

Let us start from considering first its specific cardinality k.

3.2.1. The “Incipient” Derivative of cardinality k

On the basis of Definition (3.1), the exit of the incipient derivative of Ordinality k is [12]

$\frac{{\tilde{d}}^k}{\tilde{d}{t}^k}f\left(t\right)={\stackrel{\sim }{\left(f^{\prime }\left(t\right)/f\left(t\right)\right)}}^k\cdot f\left(t\right)$ (3.2.1)

In fact, through successive formal passages, we have that

$\left(\tilde{\delta }-1\right)f\left(t\right)=f\left(t+\tilde{\Delta }t\right)-f\left(t\right)=f\left(t\right)+f^{\prime }\left(t\right)\tilde{\Delta }t-f\left(t\right)=f^{\prime }\left(t\right)\cdot \tilde{\Delta }t$ (3.2.2)

and, consequently

$\left(\tilde{\delta }-1\right)/\tilde{\Delta }t=f^{\prime }\left(t\right)/f\left(t\right)$ (3.2.3)

Such an expression, when introduced in the Definition (3.1), gives

$\underset{\tilde{\Delta }t:0\to 0^{+}}{\mathrm{l}\tilde{i}m}{\stackrel{\sim }{\left(\frac{\tilde{\delta }-1}{\tilde{\Delta }t}\right)}}^k\cdot f\left(t\right)={\stackrel{\sim }{\left(f^{\prime }\left(t\right)/f\left(t\right)\right)}}^k\cdot f\left(t\right)$ (3.2.4)

Such an explicit formal process shows that the definition of the “Incipient” Derivative of cardinality k is based on a concept of limit, which however is “prior” with respect to the considered function. In fact, it is specifically referred to the considered function only after the corresponding evaluation of the latter.

It is also worth adding that in Equations (3.2.2) and (3.2.3), we have adopted the simple notation $f^{\prime }\left(t\right)$ , which in reality is more typical of TDC. It is thus now particularly important to point out that, apart from the similarity of the symbol, the traditional derivative $f^{\prime }\left(t\right)$ presents specific differences with respect to the “Incipient” Derivative $\stackrel{\sim }{f^{\prime }\left(t\right)}$ .

In fact, if we consider the “Incipient” Derivative of cardinality k of the exponential function, that is, if we assume that $f\left(t\right)=e^{\alpha \left(t\right)}$ , on the basis of Equation (3.2.4), we get

${\left(\frac{\tilde{d}}{\tilde{d}t}\right)}^k{e}^{\alpha \left(t\right)}\stackrel{*}{=}{e}^{\alpha \left(t\right)}\cdot {\left[\stackrel{\circ }{\alpha }\left(t\right)\right]}^k$ (3.2.5)

in which the specific symbology adopted $\stackrel{\circ }{\alpha }\left(t\right)$ is finalized to point out that, even if on the basis of Equation (3.2.4) the first order “Incipient” Derivative (now indicated with $\stackrel{\circ }{\alpha }\left(t\right)$ ) coincides with the traditional derivative ${\alpha }^{\prime }\left(t\right)$ , the logical processes that lead to such identical (quantitative) results are radically different. A difference which, in particular, is also pointed out by the adoption of the symbol $\stackrel{*}{=}$ , which reminds us that any “Incipient” Derivative is always the Exit of a Generative Logical Process and not of a necessary logical process.

Equation (3.2.5) can thus preferentially be adopted as the general definition of the “Incipient” Derivative of cardinality k. This is because any function $f\left(t\right)$ can always be written in the form $f\left(t\right)=e^{\ln f\left(t\right)}=e^{\alpha \left(t\right)}$ .

Such a formal representation, in fact, leads to the same result as that of Equation (3.2.5). However, such a formal representation will reveal the “Ostensive” Valence of the “Incipient” Derivative of cardinality k when, in the next paragraphs, we will introduce the general definition of Relational Space and, even more, when we will deal with the explicit solution to the Maximum Ordinality Principle.

At the same time, such a definition is also particularly apt at showing the deep differences between the cardinal values of the “Incipient” Derivatives and those pertaining to the traditional derivatives.

In fact, if we compare the traditional derivative of order n of the function $e^{\alpha \left(t\right)}$ , evaluated according to Faà di Bruno’s formula

${\left(\frac{d}{dt}\right)}^n{e}^{\alpha \left(t\right)}=e^{\alpha \left(t\right)}{\displaystyle \sum \frac{n!}{k_1!k_2!\cdots k_n!}\cdot }{\left(\frac{\stackrel{\dot{}}{\alpha }}{1!}\right)}^{k_1}{\left(\frac{\ddot{\alpha }}{2!}\right)}^{k_2}\cdots {\left(\frac{{\alpha }^{\left(n\right)}}{n!}\right)}^{k_n}$ (3.2.6)

with the “Incipient” Derivative of the corresponding cardinality n

${\left(\frac{\tilde{d}}{\tilde{d}t}\right)}^n{e}^{\alpha \left(t\right)}\stackrel{*}{=}{e}^{\alpha \left(t\right)}\cdot {\left[\stackrel{\circ }{\alpha }\left(t\right)\right]}^n$ (3.2.7),

we can easily recognize that they are deeply different. And, even if in some cases the two derivatives of the same order k coincide (for instance when $\alpha \left(t\right)$ is linear), such a coincidence has always to be seen in the light of the symbol $\stackrel{*}{=}$ in Equation (3.2.7), which reminds us that any “Incipient” Derivative is always the Exit of a Generative Logical Process and not of a necessary logical process. A concept that is contextually and specifically underlined in Equation (3.2.7) by the explicit adoption of the “notation” ${\left[\stackrel{\circ }{\alpha }\left(t\right)\right]}^n$ .

3.2.2. The Ordinal Genetic Relationships $\left(\tilde{m}/\tilde{n}\right)$ of the “Incipient” Derivative of Ordinality $\tilde{q}$

As already anticipated, beside its proper cardinality k, the “Incipient” Derivative of Ordinality $\tilde{q}$ , according to Equation (3.1), is characterized by the genesis of its corresponding Ordinal Genetic Relationships, whose specific indication is represented by $\left(\tilde{m}/\tilde{n}\right)$ .

In this respect, it is worth pointing out that the symbol $f\left(t\right)$ does not represent anymore a simple “function”, such as in the case of TDC, but it represents a Physical Entity, of Generative Nature. Consequently, a more appropriate symbol should be $\tilde{f}\left(t\right)$ , where the “tilde” notation specifically reminds us its Generative Nature.

More specifically, in the general context of Self-Organized Systems, the symbol $\tilde{f}\left(t\right)$ will be more properly understood as being representing the Relational Space of a given System, as it will be shown in the next paragraphs.

After these due premises, we can assert that the “Incipient” Derivative of Ordinality $\left\{\tilde{q}\right\}=\left\{k,\left(\tilde{m}/\tilde{n}\right)\right\}$ describes a Generative Process which, with reference to a given System, is characterized by both its cardinal and “internal genetic properties”, and it can be represented as follows

${\left(\frac{\tilde{d}}{\tilde{d}t}\right)}^{\left\{k,\left(\tilde{m}/\tilde{n}\right)\right\}}{e}^{\alpha \left(t\right)}\stackrel{*}{=}{e}^{\alpha \left(t\right)}\cdot \left\{\left(\begin{array}{c}{\left[\stackrel{\circ }{{\alpha }_{11}}\left(t\right)\right]}^k\\ {\left[\stackrel{\circ }{{\alpha }_{21}}\left(t\right)\right]}^k\\ ⋮\\ {\left[\stackrel{\circ }{{\alpha }_{m1}}\left(t\right)\right]}^k\end{array}\right),\left(\begin{array}{c}{\left[\stackrel{\circ }{{\alpha }_{12}}\left(t\right)\right]}^k\\ {\left[\stackrel{\circ }{{\alpha }_{22}}\left(t\right)\right]}^k\\ ⋮\\ {\left[\stackrel{\circ }{{\alpha }_{m2}}\left(t\right)\right]}^k\end{array}\right)\left(\begin{array}{c}\begin{array}{l}\cdots \\ \end{array}\\ \begin{array}{l}\cdots \\ \end{array}\\ \begin{array}{l}⋮\\ \cdots \end{array}\\ \end{array}\right)\left(\begin{array}{c}{\left[\stackrel{\circ }{{\alpha }_{1n}}\left(t\right)\right]}^k\\ {\left[\stackrel{\circ }{{\alpha }_{2n}}\left(t\right)\right]}^k\\ ⋮\\ {\left[\stackrel{\circ }{{\alpha }_{mn}}\left(t\right)\right]}^k\end{array}\right)\right\}$ (3.2.8)

where:

- k represents the cardinality of the “Incipient” Derivative;

- $\stackrel{\circ }{{\alpha }_{ij}}\left(t\right)$ are the genetic characteristics of the considered system, which are highlighted by the Generative Process described by the “Incipient” Derivative. For this reason, they should more properly be represented as being characterized by a “tilde” notation. However, for the sake of a simpler notation, the “tilde” notation has been omitted, and thus it is simply understood;

- Such genetic characteristics $\stackrel{\circ }{{\alpha }_{ij}}\left(t\right)$ are generally referred to the specific properties of the Relational Space $\alpha \left(t\right)$ and are evidently characterized by the initial and boundary conditions of the System;

- At the same time, the “matrix” which appears in the second member of Equation (3.2.8) is not a traditional matrix. In fact, it is an “Ordinal” Matrix, whose various elements are related between them through Ordinal Relationships, of Genetic Nature, in the form N Co-Generated genetic properties (vertical columns), further related between them in the form of N Interaction Ordinal Relationships (parallel sequence of the N column). The “Ordinal” Matrix thus represents an Ordinal Cooperation of N Co-Productions and their associated N Inter-actions.

In this way, the various elements form One Sole Entity, faithfully represented by the abovementioned Ordinal Matrix. A concept that is explicitly pointed out, also in this case, by the adoption of curly brackets.

In addition, in order to distinguish such an Ordinal Matrix from a traditional matrix, from now on, for the sake of brevity, it will be simply termed by means of the single term “Matrioska”.

The structure of the “Incipient” Derivative (3.2.8) is then able to Ostend even more clearly the concepts previously anticipated. That is:

- The symbol $\left(\tilde{m}/\tilde{n}\right)$ represents the Ordinal Genetic Relationships that characterize the “Incipient” Derivative, where the round brackets expressly indicate that they represent only a part of the concept of Ordinality, which vice versa is understood, by itself, as a Whole;

- In fact, for this reason, in Equation (3.2.8) the latter concept is represented by means of the adoption of curly brackets;

- The Ordinal Genetic Relationships can also more synthetically be termed as “Ordinal Relationship”, both because they are not, in themselves, “functional” Relationships, but especially because the adjective “Ordinal” clearly indicates that they are precisely those that give the most significant contribution to the definition of the general concept of Ordinality;

- In addition, Equation (3.2.8) allows us to point out that, when we preliminary introduced the concept of cardinal “Incipient Derivative”, this was represented as a simple and proper mathematical concept, which, in this sense, has some similarities with that of a traditional derivative. This is why it was possible to continue to adopt the term “function” and the correlative symbol $f\left(t\right)$ , even if it was well clear the profound difference between the correlative Logical Process adopted;

- Vice versa, when we consider the “Incipient” Derivative of Ordinality $\tilde{q}$ , its meaning, when considered in the descriptive context of Self-Organizing Systems, is more properly referable as the description of a Generative Process;

- Consequently, in such a case it is more appropriate to consider Equation (3.2.8) as representative of a Generative Process, which highlights the Genetic Properties of a Physical Entity that, in the case of a Self-Organizing System, it is usually represented by the proper Relational Space of the System;

- So that, to take into account the abovementioned different aspects between the two considered Derivatives, in general it is preferable to adopt the synthetic tilde notation $\tilde{f}\left(t\right)$ , in order to more specifically indicate, in addition, that the considered System is already the Exit of a previous Generative Process.

3.2.3. Specific Properties of the “Incipient” Derivative When Understood of Higher Ordinality

The Ordinality of the “Incipient” Derivative, as previously defined (see Equation (3.2)), represents the most frequent form of Ordinality of the Self-Organizing Systems usually considered.

However, in particularly cases (especially in “Living” Systems), it may be characterized by a more “articulated” structure. For example, its cardinality can directly be associated to a correlative Ordinality $\tilde{2}/\tilde{2}$ , corresponding to an “additional” Coproduction-Interaction Process.

In such a case, the Ordinality $\tilde{q}$ will be then represented as

$\tilde{q}=\left\{k\uparrow \tilde{2}/\tilde{2};\left(\tilde{m}/\tilde{n}\right)\right\}$ (3.2.9)

in order to have, in such a way, a more adherent representation of the Internal Generativity of the System under consideration.

In this respect, however, some examples of more articulated forms of “Incipient” Derivative, with reference to particularly complex “Living” System, are illustrated in [25] .

4. Mathematical Formulation of the Maximum Ordinality Principle

The Maximum Ordinality Principle (MOP), whose verbal enunciation asserts that “Every System tends to maximize its Ordinality, including that of its surrounding habitat”, is formulated by means of two fundamental equations, which are so strictly related to each other, so as to form a Whole [13] [16] [21] .

4.1. The First Fundamental Equation of the Maximum Ordinality Principle

On the basis of the previous concept of “Incipient” Derivative, the First Fundamental Equation is formulated as follows

${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{m}/\tilde{n}\right\}}\left\{\tilde{r}\right\}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (4.1)

$\left(\tilde{m}/\tilde{n}\right)\to Max\to \left\{\tilde{2}/\tilde{2}\right\}\uparrow \left\{\tilde{N}/\tilde{N}\right\}$ (4.1.1)

where $\left\{\tilde{r}\right\}$ is the Relational Space of the System under consideration (see Paragraph 5.1), while $\left\{\tilde{m}/\tilde{n}\right\}=\left\{k,\left(\tilde{m}/\tilde{n}\right)\right\}$ represents its corresponding Ordinality, while $\left(\tilde{m}/\tilde{n}\right)$ indicates the Ordinal Genetic Relationships characterized by $\tilde{m}$ Ordinal Co-productions and $\tilde{n}$ Ordinal Interactions, and the Maximum Ordinality is reached when $\left(\tilde{m}/\tilde{n}\right)$ equals $\left\{\tilde{2}/\tilde{2}\right\}\uparrow \left\{\tilde{N}/\tilde{N}\right\}$ (as indicated in Equation (4.1.1)).

In this respect, it is worth noting that:

1) The underlined symbol ${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s$ explicitly indicates that the Generative Capacity of the System (more appropriately termed as Generativity) is “internal” to the same System. This is because it is precisely that which gives origin to its Self-Organization as a Whole;

2) The symbol “ $\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ ” represents a more general version of the simple figure “zero”, as the latter systematically appears in the traditional differential equations. In fact, it now represents, at the same time:

- The specific “origin and habitat” conditions associated to the considered Ordinal Differential Equation (4.1);

- While the symbol “ $\stackrel{[\to }{=}$ ” indicates that the System, during its Generative Evolution, is persistently “adherent” to its “origin and habitat” conditions.

4.2. The Second Fundamental Equation of the Maximum Ordinality Principle

It is formulated as follows

${\left(\tilde{d}/\tilde{d}t\right)}^{\left(\tilde{2}/\tilde{2}\right)}\left\{\left\{\tilde{r}\right\}\circledR {\left(\tilde{d}/\tilde{d}t\right)}^{\left(\tilde{2}/\tilde{2}\right)}\left\{\tilde{r}\right\}\right\}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (4.2)

and it can be considered as representing a global Feed-Back Process of Ordinal Nature, which is internal to the same System. Equation (4.2), in fact, asserts that the Relational Space of the System $\left\{\tilde{r}\right\}$ , which “emerges” as a solution from the First Equation, interacts in the form of the Relational Product $\circledR$ (defined at Paragraph 5.1) with its proper Generative Capacity ${\left(\tilde{d}/\tilde{d}t\right)}^{\left(\tilde{2}/\tilde{2}\right)}\left\{\tilde{r}\right\}$ . In such a way as to originate a comprehensive Generative Capacity, which at any time, is always adherent to the origin and habitat conditions of the Second Fundamental Equation.

This is an aspect which is particular important for the Ordinal Stability of the System, especially when the latter interacts with other surrounding Systems understood as being part of its proper habitat.

The Maximum Ordinality Principle, in its two fundamental equations, always presents an explicit solution.

The latter will now be presented:

1) By preliminarily illustrating its basic elements;

2) Then, by formulating the correlative solution in explicit terms;

3) Finally, at the end of Appendix A, the general explicit solution to the MOP will also be presented and structured in a corresponding operative form, so that it may result as being more directly and easily adopted in analyzing any System under consideration.

5. Explicit Solution to the Mathematical Formulation of the Maximum Ordinality Principle

In order to show the explicit solution to the Maximum Ordinality Principle, it is worth recalling the fundamental concepts pertaining to the Relational Space of a System.

5.1. The Relational Space of a System (and the “Proper Space” of the System)

In this respect, it is fundamental to recall that the symbol $\left\{\tilde{r}\right\}$ in Equation (4.1) represents the Relational Space of the System, which obviously depends on the Nature of the System analyzed.

We can then start from the consideration of a System whose Relational Space is characterized, for example, by the following three topological coordinates $\left\{\tilde{\sigma },\tilde{\phi },\tilde{\vartheta }\right\}$ .

Such a hypothesis is surely valid in the case of a “non-Living” System. Nonetheless, it is also valid in the case of a “Living” System too. Whereas, in the case of “Conscious” Systems, the three coordinate will surely be different.

For example, in the case of the Economic Analysis of European Community, with its 27 States, the variables could be (K, L, N), that is Kapital, Labour and Natural Resources, as shown in [26] .

In all cases whatsoever, the three topological coordinates $\left\{\tilde{\sigma },\tilde{\phi },\tilde{\vartheta }\right\}$ are always considered as the Exit of a Generative Process (this is the reason for the tilde notation), and we always have that

${\left\{\tilde{r}\right\}}_s=e^{\tilde{\alpha }\left(t\right)}=e^{\left\{\tilde{\sigma }\circledR \tilde{i}\oplus \tilde{\phi }\circledR \tilde{j}\oplus \tilde{\vartheta }\circledR \tilde{k}\right\}}$ (5.1.1)

This is because, on the basis of a generalized form of De Moivre representation, it is always possible to write

$\begin{array}{c}{\left\{\tilde{r}\right\}}_s=\left\{\tilde{\rho }\circledR \tilde{i}\circledR e^{\tilde{\phi }\circledR \tilde{j}}\circledR e^{\tilde{\vartheta }\circledR \tilde{k}}\right\}=\left\{e^{\tilde{\sigma }\circledR \tilde{i}}\circledR e^{\tilde{\phi }\circledR \tilde{j}}\circledR e^{\tilde{\vartheta }\circledR \tilde{k}}\right\}\\ =e^{\left\{\tilde{\sigma }\circledR \tilde{i}\oplus \tilde{\phi }\circledR \tilde{j}\oplus \tilde{\vartheta }\circledR \tilde{k}\right\}}=e^{\tilde{\alpha }\left(t\right)}\end{array}$ (5.1.2),

where the traditional versors $\stackrel{\to }{i},\stackrel{\to }{j},\stackrel{\to }{k}$ are now replaced by three unit spinors $\tilde{i},\tilde{j},\tilde{k}$ , which are defined in such a way as to satisfy the following Relational Product Rules:

$\tilde{i}\circledR \tilde{i}=\oplus 1$ $\tilde{i}\circledR \tilde{j}=\tilde{j}$ $\tilde{i}\circledR \tilde{k}=\tilde{k}$ (5.1.3)

$\tilde{j}\circledR \tilde{i}=\tilde{j}$ $\tilde{j}\circledR \tilde{j}=\Theta 1$ $\tilde{j}\circledR \tilde{k}=\tilde{k}$ (5.1.4)

$\tilde{k}\circledR \tilde{i}=\tilde{k}$ $\tilde{k}\circledR \tilde{j}=\tilde{k}$ $\tilde{k}\circledR \tilde{k}=\Theta 1$ (5.1.5)

where the symbols $\oplus$ and $\circledR$ express more intimate relationships between the same spinors: both in terms of sum $\oplus$ and in terms of (relational) product $\circledR$ with respect to the case of traditional versors $\stackrel{\to }{i},\stackrel{\to }{j},\stackrel{\to }{k}$ .

So that representation (5.1.1) is similar (albeit not strictly equivalent) to a system of three complex numbers, characterized by one real unit ( $\tilde{i}$ ) and two imaginary units ( $\tilde{j}$ and $\tilde{k}$ ).

5.2. The Generative Capacity of the System

As already anticipated, the incipient derivative ${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{m}/\tilde{n}\right\}}$ , when it is underlined, explicitly indicates that the Generative Capacity of the System (more appropriately termed as Generativity) is “internal” to the same System.

This is precisely because under these conditions it represents the Self-Organization of the System as a Whole.

At the same time, this is also the reason why, differently from the traditional “incipient” derivative, in our case the “Incipient” Derivative is directly referred to the exponent of the Relational Space, that is

$e^{{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{m}/\tilde{n}\right\}}\left\{\tilde{\sigma }\circledR \tilde{i}\oplus \tilde{\phi }\circledR \tilde{j}\oplus \tilde{\vartheta }\circledR \tilde{k}\right\}}$ (5.2.1)

In addition, it is also important to underline that such an exponent, according to the same symbols adopted, is understood as a Whole (see the curly brackets, together with the symbols $\oplus$ and $\circledR$ ).

This means that the corresponding derivative have to be taken with reference to such a Whole. Otherwise, its corresponding value will be generally underestimated.

If now, for the sake of clarity, we synthetically indicate $\left\{\tilde{\sigma }\circledR \tilde{i}\oplus \tilde{\phi }\circledR \tilde{j}\oplus \tilde{\vartheta }\circledR \tilde{k}\right\}=\tilde{\alpha }\left(t\right)$ , the explicit solution to Equation (4.1) will result in the form (5.2.2), when it is given in terms of an External Representation. That is, when the coordinates of the various elements of the System are referred to a Reference System of coordinates whose origin is external to the System under consideration.

${\left\{\tilde{r}\right\}}_s=e^{\left\{\begin{array}{cccc}{\tilde{\alpha }}_{11}\left(t\right)& {\tilde{\alpha }}_{12}\left(t\right)& \cdots & {\tilde{\alpha }}_{1n}\left(t\right)\\ {\tilde{\alpha }}_{21}\left(t\right)& {\tilde{\alpha }}_{22}\left(t\right)& \cdots & {\tilde{\alpha }}_{2n}\left(t\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\alpha }}_{n1}\left(t\right)& {\tilde{\alpha }}_{n2}\left(t\right)& \cdots & {\tilde{\alpha }}_{nn}\left(t\right)\end{array}\right\}}$ (5.2.2)

The “Matrioska” in Equation (5.2.2), however, as consequence of the Internal Generativity of the System ${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{m}/\tilde{n}\right\}}$ and the correlative Self-Organization Process, when the System reaches its Maximum Ordinality, transforms its initial internal structure $\left(\tilde{m}/\tilde{n}\right)$ into the form $\left(\tilde{n}/\tilde{n}\right)$ . In such a case, the corresponding ${\tilde{\alpha }}_{ij}\left(t\right)$ will always depend on the initial and boundary conditions, and in the next paragraphs, we will show how it is possible to find their explicit expressions.

5.3. Explicit Expression of the Internal Generativity ${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{m}/\tilde{n}\right\}}$

Let assume that, under the conditions previously described, the explicit expression of the Ordinality $\left\{\tilde{m}/\tilde{n}\right\}$ , in Equation (3.2), equals

$\left\{\tilde{N}/\tilde{N}\right\}=\left\{k,\left(\tilde{N}/\tilde{N}\right)\right\}$ (5.3.1)

Equation (4.1) then becomes

${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}_s^{\left\{\tilde{N}/\tilde{N}\right\}}{e}^{\left\{\tilde{\alpha }\left(t\right)\right\}}=e^{\left\{\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{11}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{21}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{N1}\left(t\right)\end{array}\right),\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{12}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{22}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{N2}\left(t\right)\end{array}\right)\left(\begin{array}{c}\cdots \\ \\ \cdots \\ \\ ⋮\\ \cdots \end{array}\right)\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{1N}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{2N}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{NN}\left(t\right)\end{array}\right)\right\}}\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ (5.3.2)

where the symbol “ $\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ ”, as previously anticipated, represents, at the same time:

- The specific “origin and habitat” conditions associated to the considered Ordinal Differential Equation (4.1);

- While the symbol “ $\stackrel{[\to }{=}$ ” indicates that the System, during its Generative Evolution, is persistently “adherent” to its “origin and habitat” conditions.

5.4. The Initial and Boundary Conditions

Given the particular structure of Equation (5.3.2), it is possible to directly explicit the term $\stackrel{[\to }{=}\left\{\tilde{0}\right\}$ in exponential form, so that it can be written as follows

$e^{\left\{\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{11}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{21}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{N1}\left(t\right)\end{array}\right),\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{12}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{22}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{N2}\left(t\right)\end{array}\right)\left(\begin{array}{c}\cdots \\ \\ \cdots \\ \\ ⋮\\ \cdots \end{array}\right)\left(\begin{array}{c}{\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{1N}\left(t\right)\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{2N}\left(t\right)\\ ⋮\\ {\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{NN}\left(t\right)\end{array}\right)\right\}}\stackrel{[\to }{=}{e}^{\left\{\left(\begin{array}{c}{\tilde{\beta }}_{11}\left(t\right)\\ {\tilde{\beta }}_{21}\left(t\right)\\ ⋮\\ {\tilde{\beta }}_{N1}\left(t\right)\end{array}\right),\left(\begin{array}{c}{\tilde{\beta }}_{12}\left(t\right)\\ {\tilde{\beta }}_{22}\left(t\right)\\ ⋮\\ {\tilde{\beta }}_{N2}\left(t\right)\end{array}\right)\left(\begin{array}{c}\cdots \\ \cdots \\ ⋮\\ \cdots \end{array}\right)\left(\begin{array}{c}{\tilde{\beta }}_{1N}\left(t\right)\\ {\tilde{\beta }}_{2N}\left(t\right)\\ ⋮\\ {\tilde{\beta }}_{NN}\left(t\right)\end{array}\right)\right\}}$ (5.4.1)

which shows that, in principle, its explicit solution can be obtained by solving N × N corresponding differential equations of the form

${\left(\underset{\_}{\tilde{d}/\tilde{d}t}\right)}^k{\tilde{\alpha }}_{ij}\left(t\right)={\tilde{\beta }}_{ij}\left(t\right)$ (5.4.2)

In reality, as we will see, from an “operative point of view”, it is not strictly necessary to integrate Equations (5.4.2), because it is sufficient the integration of the sole “couple of reference”, generally indicated as ${\tilde{\alpha }}_{12}\left(t\right)$ . This is because, as shown later on, all the other couples ${\tilde{\alpha }}_{ij}\left(t\right)$ are related to the reference couple ${\tilde{\alpha }}_{12}\left(t\right)$ in the form of “Assignation” conditions, according to the Harmony Relationships (Paragraph 5.6), this means that the next paragraph, which concerns the explicit solution in terms of Equations (5.4.2), in reality is here presented only for generality of exposition.

In fact, by considering the “Emerging Property” of the Harmony Relationships, the Solution can strictly be formulated in terms of the sole integration of the condition concerning the couple ${\tilde{\alpha }}_{12}\left(t\right)$ .

5.5. Explicit Solution to Equation (5.4.1), Understood in Terms of External Description

Equation (5.4.1), considered from a general point of view, always presents an explicit solution. This is because in the majority of the most frequent Self-Organizing Systems (both “non-Living”, “Living” and “Conscious” Systems), the general structure of the initial conditions can be assumed as being equal to

${\beta }_{ij}\left(t\right)={\left(a_{ij}+b_{ij}\cdot t\right)}^p$ (5.5.1)

in which p can also be a fractional number.

Such initial conditions always lead to the explicit solution of any unknown ${\tilde{\alpha }}_{ij}\left(t\right)$ that appears in Equation (5.4.1). This is because by considering the general definition of the cardinal “Incipient Derivative” (3.2.1), we have that

$\frac{{\tilde{d}}^k}{\tilde{d}{t}^k}f\left(t\right)={\stackrel{\sim }{\left(\frac{{\tilde{f}}^{\prime }\left(t\right)}{f\left(t\right)}\right)}}^k\cdot f\left(t\right)=\beta \left(t\right)$ (5.5.2)

in which $\beta \left(t\right)$ now represents the initial condition for the generic function $f\left(t\right)$ .

Consequently, through successive formal passages, we have

$f{\left(t\right)}^{1-k}\cdot f^{\prime }{\left(t\right)}^k=\beta \left(t\right)$ (5.5.3)

from which

$f{\left(t\right)}^{\frac{1-k}{k}}\cdot f^{\prime }\left(t\right)=\beta {\left(t\right)}^{\frac{1}{k}}$ (5.5.4)

whose integral

${\displaystyle \underset{0}{\overset{t}{\int }}f{\left(t\right)}^{\frac{1-k}{k}}\cdot {\tilde{f}}^{\prime }\left(t\right)}\cdot dt={\displaystyle \underset{0}{\overset{t}{\int }}{\left(\beta \left(t\right)\right)}^{\frac{1}{k}}\cdot dt}$ (5.5.5)

leads to

$f{\left(t\right)}^{1/k}\cdot k={\displaystyle \underset{0}{\overset{t}{\int }}{\left(\beta \left(t\right)\right)}^{\frac{1}{k}}\cdot dt}$ (5.5.6)

and, consequently, we have

$f\left(t\right)=1/k\cdot {\left\{{\displaystyle \underset{0}{\overset{t}{\int }}{\left(\beta \left(t\right)\right)}^{\frac{1}{k}}\cdot dt}\right\}}^k$ (5.5.7)

where $f\left(t\right)$ now represents any ${\alpha }_{ij}\left(t\right)$ , while $\beta \left(t\right)$ represents the corresponding associated initial condition ${\beta }_{ij}\left(t\right)$ .

The explicit solution of the generic ${\alpha }_{ij}\left(t\right)$ is then given by

$\begin{array}{c}{\alpha }_{ij}\left(t\right)=\frac{1}{k}\cdot {\left\{{\displaystyle \underset{0}{\overset{t}{\int }}{\left({\beta }_{ij}\left(t\right)\right)}^{\frac{1}{k}}\cdot dt}\right\}}^k=\frac{1}{k}\cdot {\left\{{\displaystyle \underset{0}{\overset{t}{\int }}{\left({\left(a_{ij}+b_{ij}\cdot t\right)}^p\right)}^{\frac{1}{k}}\cdot dt}\right\}}^k\\ =\frac{1}{k\cdot b_{ij}}\cdot {\left(\frac{k}{p+k}\cdot {\left(a_{ij}+b_{ij}\cdot t\right)}^{\frac{p}{k}+1}\right)}^k\end{array}$ (5.5.8)

However, as already anticipated, such a formal procedure it is not specifically required to evaluate all the couples ${\alpha }_{ij}\left(t\right)$ that characterize the explicit solution to the First Fundamental Equation (4.1). This is because The Explicit Solution does not end up at this level.

In fact, the General Solution to Equation (4.1) is characterized by an additional contribution. That is, the contribution of the Harmony Relationships, which represent an “Emerging Solution” which, correspondently, shows an “Emerging Property” of the Self-Organizing Systems: that is, the Diffusive Generativity, among the various elements of the same System, which in reality represents the proper origin of the Harmony Relationships.

5.6. The Harmony Relationships, as the Exit of a Diffusive Generativity of the System

The Process of Genesis of the Harmony Relationships can be shown by adopting two different descriptive modalities, that is: by adopting an External Representation or, alternatively, an Internal Representation.

The two Representations are substantially equivalent between them. However, the adoption of an Internal Representation is able to Ostend much more clearly the abovementioned “Excess of Quality” on behalf of the System analyzed.

This is because, as already anticipated, an External Representation is the one in which each element of the System is referred to a system of coordinates characterized by an origin which is external to the System analyzed. Whereas, in the case of an Internal Representation, the various elements of the System are referred to a system of coordinates which is internal to the same System analyzed.

In the latter case, each element ${\tilde{\alpha }}_{ij}\left(t\right)$ of the System, at its Maximum Ordinality, is preferably referred to the corresponding element of the main diagonal belonging the same row i, and, this leads to the following Representation

${\left\{\tilde{r}\right\}}_s=e^{\left\{\begin{array}{cccc}0& {\tilde{\alpha }}_{12}\left(t\right)& \cdots & {\tilde{\alpha }}_{1N}\left(t\right)\\ {\tilde{\alpha }}_{21}\left(t\right)& 0& \cdots & {\tilde{\alpha }}_{2N}\left(t\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\alpha }}_{N1}\left(t\right)& {\tilde{\alpha }}_{N2}\left(t\right)& \cdots & 0\end{array}\right\}}$ (5.6.1)

in which all the elements of the main diagonal are evidently equal to zero, whereas all the other elements ${\tilde{\alpha }}_{ij}\left(t\right)$ assume a binary-duet structure, and thus satisfy the following Specularity Relationships

${\left\{{\tilde{\alpha }}_{ij}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left\{{\tilde{\alpha }}_{ji}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (5.6.2)

which represent a much more profound concept with respect to the traditional symmetry (the symbol “ $\stackrel{\ast }{=}$ ”, in fact, does not represent an equality, but a simple assignation condition).

Such a Representation then allows us to show the Generative Process that leads the System to its Maximum Ordinality and, at the same time, to its Maximum Stability conditions, because it restructures the internal relationships between the various elements in such way as these show an additional “Emerging” Property, which is initially based on the following “topological” Relationships:

${\tilde{\lambda }}_{12}\oplus {\tilde{\alpha }}_{12}\left(t\right)\stackrel{\ast }{=}{\tilde{\lambda }}_{1j}\oplus {\tilde{\alpha }}_{1j}(\; t\; )$

for

$j=3,\cdots ,N$ (5.6.3)

together with all their associated incipient derivatives, up to the order N − 1

${\left\{{\stackrel{\circ }{\tilde{\lambda }}}_{12}\oplus {\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\right\}}^{\tilde{k}}\stackrel{\ast }{=}{\left\{{\stackrel{\circ }{\tilde{\lambda }}}_{1j}\oplus {\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\right\}}^{\tilde{k}}$

for

$k=1,\cdots ,N-1$ (5.6.4)

where ${\tilde{\lambda }}_{ij}$ represent their corresponding internal reciprocal Correlating Factors, which are clearly distinct from the values of the initial conditions, because the latter are already included in the correlative expressions ${\tilde{\alpha }}_{ij}\left(t\right)$ .

Such properties represent the bases of the previously mentioned Property of Diffusive Generativity, which is faithfully represented by the following Harmony Relationships

$\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\oplus {\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}\stackrel{\ast }{=}{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j\circledR \left\{{\tilde{\alpha }}_{12}\left(t\right)\oplus {\tilde{\lambda }}_{12}\left(t\right)\right\}$

per

$j=1,2,3,\cdots ,N-1$ (5.6.5)

whose explicit Process of Genesis is illustrated in Appendix A1, while the associated Ordinal Roots of Unity ${\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j$ are illustrated in Appendix A2.

If we now take into account the Harmony Relationships (5.6.5), together with their specific structure and the correlative symbology adopted, the Solution to the First Fundamental Equation pertaining to the System analyzed can be represented as follows

$\left\{\tilde{r}\right\}=e^{\left\{\tilde{\alpha }\left(t\right)\right\}}=e^{\left\{{\tilde{\alpha }}_{12}\left(t\right)\oplus {\tilde{\lambda }}_{12}\left(t\right)\right\}\circ \left\{{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_1,{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_2,\cdots ,{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{N-1}\right\}}$ (5.6.6)

which reflects the Self-Organization of the Systems in terms of “couples”, according to an Internal Description.

At the same time, it shows that the basic “topological” structure in terms of the reference couple “12” (see Equation (5.6.3)) has been correspondently “transformed” and, at the same time, “updated”, as a consequence of the Diffusive Generative Process which leads to the Harmony Relationships that, as anticipated, result as being substantially structured in terms of the sole reference couple “12”.

6. Explicit Solution to the Two Fundamental Equations of the MOP, Understood as a Whole

The MOP, considered in its two Fundamental Equations understood as a Whole, differently from the problems formulated in TDC, always presents an explicit solution. This is especially due to IDC and, in particular, both to the solution to the First Fundamental Equation in the form of Matrioska and the associated Harmony Relationships, which allow us to represent the System in the form of “couples”, by assuming one arbitrary couple of elements as a reference.

So that, precisely because of such specific characteristics, the MOP enabled us to reconsider and explicitly solve some “particular” problems, generally dealt with in literature in terms of TDC, which are generally considered as being “unsolvable”, “intractable”, “with a drift”. The solutions of which ended up by showing that the Maximum Ordinality Principle has an extremely general validity [16] [21] .

The Explicit Solution to the Two Fundamental Equations of the MOP, understood as a Whole, can be obtained by introducing the solution to the First Fundamental Equation (4.1), previously shown,

$\left\{\tilde{r}\right\}=e^{\left\{\tilde{\alpha }\left(t\right)\right\}}=e^{\left\{{\tilde{\alpha }}_{12}\left(t\right)\oplus {\tilde{\lambda }}_{12}\left(t\right)\right\}\circ \left\{{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_1,{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_2,\cdots ,{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{N-1}\right\}}$ (5.6.6)

into the Global Feed-Back Process represented by the Second Fundamental Equation (4.2). The latter consequently transforms into a typical Riccati’s Equation of Ordinal Nature, whose explicit solution is given by

$\left\{\tilde{r}\right\}=e^{\left\{\tilde{\alpha }\left(t\right)\right\}}=e^{\left\{\tilde{B}\left(t\right)\right\}\circ \left\{{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{13},{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{14},\cdots ,{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{1N}\right\}}$ (6.1)

where

$\tilde{B}\left(t\right)=\left\{\left(\begin{array}{c}\oplus \tilde{A}\left(t\right)\\ \Theta \tilde{A}\left(t\right)\end{array}\right),\left(\begin{array}{c}\Theta \tilde{A}\left(t\right)\\ \oplus \tilde{A}\left(t\right)\end{array}\right)\right\}$ (6.2)

and

$\tilde{A}\left(t\right)=\left\{{\left\{{\tilde{\alpha }}_{12}\left(0\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(0\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\right\}\circ {\left\{{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}^{\uparrow \left\{\tilde{N}/\tilde{N}\right\}}\right\}}^{\left(\tilde{2}/\tilde{2}\right)}\oplus \ln \left({\tilde{c}}_1\oplus \left\{{\tilde{c}}_2,t\right\}\right)$ (6.3)

where the term $\ln \left({\tilde{c}}_1\oplus \left\{{\tilde{c}}_2,t\right\}\right)$ accounts for the origin and habitat conditions of the Feed-Back Equation and, at the same time, also represents an Over-Ordinality contribution specifically due to the same Feed-Back Process.

This latter contribution, as already anticipated, is particularly important for the System stability, especially when the System interacts with another System of its surrounding Habitat.

Equation (6.1), together with Equations (6.2) and (6.3), then represents the Explicit “Emerging Solution” to the Maximum Ordinality Principle, formulated in two “Incipient” Differential Equations (4.1) and (4.2), when the latter are properly considered as being a Whole.

7. General Validity of the Explicit Solution to the Maximum Ordinality Principle

Equation (6.1), considered with the associated Equations (6.2) and (6.3), has a general validity because, at the same time, it is valid not only for non-Living Systems, but also for Living Systems and Human Systems too.

What’s more, the same fact that Solution (6.1) is always an Explicit Solution represents a very general property that evidently has a huge relevance from an operative point of view.

In addition, Solution (6.1) introduces some further fundamental novelties of gnoseological nature, which enabled us to clearly assert that “The ‘Emerging Quality’ of Self-Organizing Systems, when modeled according to the Maximum Ordinality Principle (MOP), offers a Radically New Perspective to Modern Science” [21] . This is exactly what also suggested a possible of reformulation of such a Solution into a corresponding version in operative terms.

8. Explicit Solution to the MOP Reformulated in Operative Terms by Means an EQS Simulator

In order to have an explicit solution that may result much easier to be programmed on a computer and, in particular, on a simple PC, the previous Explicit Solution can be restructured in operative terms, in order to realize an “Emerging Quality Simulator” (EQS), which, however, is not “equivalent”, by itself, to a traditional computer program. This is because, even if conceived for operative finalities, EQS always keeps memory of the genetic Ordinality of the Processes analyzed. So that the various forms of Ordinality, although considered in operative terms, will always be accounted for in terms of their “correlative associated cardinalities”.

If we then suppose for example that the Relational Space of the System is represented by the following three generative coordinates $\left\{\tilde{\sigma },\tilde{\phi },\tilde{\vartheta }\right\}$ , characteristic of a “non-Living” System, the fundamental Relationships of EQS are

a) ${\tilde{\rho }}_{1j}\left(t\right)=A\cdot e^{{\tilde{S}}_l\left(t\right)}$ (7.1)

con ${\tilde{S}}_l\left(t\right)={\psi }_{1,1}\cdot E_{l,1}\cdot \left[B_l\cdot {\tilde{\Sigma }}_0\left(t\right)-C_l\cdot \left({\tilde{\Phi }}_0\left(t\right)+{\tilde{\Theta }}_0\left(t\right)\right)\right]$ (7.1.1)

b) ${\tilde{\phi }}_{1j}\left(t\right)={\psi }_{1,2}\cdot E_{l,2}\cdot \left[B_l\cdot {\tilde{\Phi }}_0\left(t\right)+C_l\cdot {\tilde{\Sigma }}_0\left(t\right)\right]$ (7.2)

c) ${\theta }_{1j}\left(t\right)={\psi }_{1,3}\cdot E_{l,3}\cdot \left[B_l\cdot {\tilde{\Theta }}_0\left(t\right)+C_l\cdot {\tilde{\Sigma }}_0\left(t\right)+C_l\left({\tilde{\Phi }}_0\left(t\right)+{\tilde{\Theta }}_0\left(t\right)\right)\right]$ (7.3)

$E_l{}_{,i}=\frac{{\epsilon }_{l,i}+4\pi \cdot l}{N-1}$ $B_l=\cos \left(\sqrt{2}\cdot {\psi }_l\right)$ $C_l=D_l=\frac{1}{\sqrt{2}}\sin \left(\sqrt{2}\cdot {\psi }_l\right)$ (7.4)

and

${\psi }_l={\psi }_2\cdot \frac{{\epsilon }_2+2\pi \cdot l}{N-1}$ (7.5)

in which:

1) ${\tilde{\Sigma }}_0,{\tilde{\Phi }}_0,{\tilde{\Theta }}_0$ synthetically represent the Ordinal coordinates of the reference couple, generally termed as “couple 12”, which, on the other hand, can be arbitrarily chosen. So that the symbols ${\Sigma }_0,{\Phi }_0,{\Theta }_0$ stand for $\left\{{\tilde{\sigma }}_{12},{\tilde{\phi }}_{12},{\tilde{\vartheta }}_{12}\right\}$ .

Such coordinates, however, considered in transient conditions, will correspond to the solution to Equation (5.6.5), with reference to the sole couple “12”.

Consequently, in adherence to the symbology previously adopted, those coordinates can be represented as ${\Sigma }_0\left(t\right),{\Phi }_0\left(t\right),{\Theta }_0\left(t\right)$ ;

2) The Ordinal factors ${\psi }_{1,i}\cdot E_{l,i}$ originate from the assumption that the Harmony Relationships, here reproduced for the sake of clearness

${\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j\circledR {\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$

for

$j=1,2,3,\cdots ,N-1$ (5.6.5)

are modulated by the correlative Ordinal terms ${\left\{{\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ , which, apart from specific cases of given Habitat conditions, can be considered “null”, because the initial topological “assignation” of the Correlative Factors is “Transfigured” by the Diffusive Generative Process.

In this respect, the terms ${\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ , after a previous reduction of the Ordinality $\left\{\tilde{2}/\tilde{2}\right\}\to 1$ , are characterized by three different periodicities $E_{l,i}=\frac{{\epsilon }_{1i}+4\pi l}{N-1}$ , each one specific for each coordinate, which originate from the explicit expression of the Ordinal Roots of Unity and, at the same, are also characterized by the specific factors ${\psi }_{1,i}$ ;

3) In fact, after having rewritten the Ordinal Relationships in the following form

$Exp\left\{{\tilde{\sigma }}_{1j}\left(t_0\right),{\tilde{\phi }}_{1j}\left(t_0\right),{\tilde{\vartheta }}_{1j}\left(t_0\right)\right\}\stackrel{\ast }{=}Exp\left[{\left(\sqrt[N-1]{\tilde{1}}\right)}_l\circledR \left\{{\tilde{\sigma }}_{12}\left(t_0\right),{\tilde{\phi }}_{12}\left(t_0\right),{\tilde{\vartheta }}_{12}\left(t_0\right)\right\}\right]$ (7.6)

4) And after having assumed the explicit expression of the Ordinal Roots of Unity, illustrated in Appendix A2 (Equations (A2.5) and (A2.6)), here explicitly recalled for the sake of clarity

${\left(\sqrt[N-1]{\tilde{1}}\right)}_l\stackrel{\ast }{=}Exp\left\{\tilde{\alpha }\circledR \tilde{i}\oplus \tilde{\beta }\circledR \tilde{j}\oplus \tilde{\gamma }\circledR \tilde{k}\right\}$ (A2.5)

where

$\alpha =\frac{{\epsilon }_1+4\pi \cdot l}{N-1}$ $\beta =\frac{{\epsilon }_2+2\pi \cdot l}{N-1}$ $\gamma =\frac{{\epsilon }_3+2\pi \cdot l}{N-1}$ (A2.6)

the expansion series of Equation (A2.5), together with the contextual adoption of the Rules of the Ordinal Product (5.1.3), (5.1.4), (5.1.5), leads to the Ordinal Relationships (7.1), (7.1.1), (7.2), (7.3), initially introduced, with the associated coefficients given by Equations (7.4), (7.5).

For the sake of completeness, it is worth adding that:

- The symbol $\left\{\tilde{1}\right\}$ represents the Unity of the System (understood as a Whole) by means the representation of the Unity of its Proper Space of Relations;

- ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ characterize the spatial orientation of the System as a Whole, with reference to its Ordinal Proper Space and, more specifically, with respect to the Reference “Couple 12”;

- The “periodicity” of the “spinor” $\tilde{i}$ is assumed equal to 4π, because it is expressed in steradians;

- While the periodicity of the spinors $\tilde{j}$ e $\tilde{k}$ are both equal to 2π radians, because these spinors are always “orthogonal”, both between them and with respect to the spinor $\tilde{i}$ . An “orthogonality” that can be seen as a form of reciprocal “irreducibility” (as also indicated by the same Relational Products);

- While the Factor “ $\tilde{A}$ ” represents an Internal Ordinal Factor according to which all the radial “Uniances” of the various Couples are appropriately referred to the radial “Uniance” of the Reference Couple “12”. This latter concept will clearly be illustrated in a next section specifically titled “Distance and Uniance”.

At the same time, by means of the Internal Ordinal Factor “ $\tilde{A}$ ”, the cardinalities “associated” to the various “Uniances” are all expressed in terms of a desired scale of measure.

9. General Considerations on the Explicit Solution Reformulated in Operative Terms by EQS

From the previous exposition, it should result as being evident that the Harmony Relationships (further illustrated in Appendix A1) represent an “Irreducible Excess”. That is an “Exceeding” manifestation of the Generativity of the System, where the latter is at the same time Self-Organizing, of Ordinal Nature, and understood as a Whole.

This means that the same Explicit Solution reformulated in operative terms, precisely because obtained through an Ordinal Deductive Process from the Harmony Relationships and the Ordinal Roots of Unity (further illustrated in Appendix A2), represents an “Emerging Solution” from the Maximum Ordinality Principle.

Consequently, even if the single Relationships refer to each single couple “1j”, and thus to the three “distinct” variables ${\tilde{\rho }}_{1j},{\tilde{\phi }}_{1j},{\tilde{\vartheta }}_{1j}$ , the latter do not represent a simple traditional “vector”, but an “Ordinal vector”. That is a unique and sole Relational Entity, which is usually represented in curly brackets, such as $\left\{{\tilde{\rho }}_{1j},{\tilde{\phi }}_{1j},{\tilde{\vartheta }}_{1j}\right\}$ , precisely because it is understood as a Whole.

This means that the three variables ${\tilde{\rho }}_{1j},{\tilde{\phi }}_{1j},{\tilde{\vartheta }}_{1j}$ , although recognizable as being “distinct”, they are not conceptually “separable” between them.

Such an assertion is also even truer (and in particular way) with reference to the various triples of variables pertaining to all the couples which compose the System, which a fortiori are not conceptually “separable” between them precisely because the System is understood as a Whole.

In other words, the Fundamental Relations pertaining to EQS previously shown do not only furnish the N − 1 single Ordinal vectors $\left\{{\tilde{\rho }}_{1j}\left(t\right),{\tilde{\phi }}_{1j}\left(t\right),{\tilde{\vartheta }}_{1j}\left(t\right)\right\}$ that characterize each single couple of the System, but they also represent, even more, a Unified Ordinal Description of the System understood as Whole.

In other terms, the coordinates furnished by the Operative Solution are not conceptually “separable” between them, neither with reference to each single couple, nor with reference to all the various couples of the System as a Whole.

This leads us to point out another important aspect always in the Light of the Maximum Ordinality Principle.

9.1. Distance and “Uniance”

A direct and correlative consequence is that, even if at a “preliminary and intuitive” interpretation, such Ordinal Relationships could be thought as giving the “distances” between the various couples of the System analyzed, in reality, in adherence to the M.O.P, such an interpretation (and the corresponding “terminology”), should be substantially modified. In particular, by adopting a more appropriate term, such as “Uniance”, instead of that of “distance”.

This is because, as already anticipated, the concept of “distance” tends more to divide, than to unify. In fact, the same etymology of the word (from Latin “dis-stant”) indicates that “one element stays here and the other one stays there” or, equivalently, “one is here and the other one is there”.

Consequently, in an Ordinal Perspective the term “distance” should preferably be replaced by a different term, possibly able to indicate the concept of “union” of two elements, more than their “distance”.

In this respect, by introducing a neologism (that “rhymes” with the term “distance”, but it exactly indicates the opposite meaning), we could say that the same value that in a “functional” perspective represents a “dis-tance”, in an Ordinal Perspective indicates a “uni-ance”. That is, it indicates that the two elements form “one sole thing” of Ordinal Nature, precisely because they are the Exit of the same Generative Process. So that the term “Uniance” expresses an Ordinal concept, and not a mere cardinal concept, such as that of “distance”. Any “Uniance”, in fact, is characterized by its own Ordinality.

As a simple example, let us think of a couple of elements ${\tilde{\alpha }}_{ij}\left(t\right)$ whose Relationship is characterized by a Binary-Duet Ordinality ${\left\{{\tilde{\alpha }}_{ij}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ . Such a specific and proper Ordinality is precisely that which represents the Ordinal “Unity” between two elements of the System. While, at the same time, its “associated cardinality” only indicates their topological distribution in the Relational Space of the System.

Consequently, when all the various “Uniances” are considered in the context of the Harmony Relationships, they reveal that the System is a Whole of Ordinal Nature, in perfect adherence to the Maximum Ordinality Principle.

In addition, such an assertion also has an even more general sense, that is: it is precisely the Generativity of the Self-Organizing System the one which, with its proper Diffusivity, characterizes all the elements of the System in terms of “Ordinal Relationships”. In that sense, such Ordinal Relationships are all of genetic nature, like in the case of “brothers”.

“Brothers”, in fact, are termed as such not because of their “direct reciprocal relationships”, but because of their direct reference to the same genetic principle: their father (or their mother or both).

Consequently, in perfect “Adherence”, the term “Uniance” synthesizes the concept of an Ordinal Unity of Genetic Nature.

9.2. Proper Space and Proper Time of a Self-Organizing System

Another important aspect that has to be underlined, always in the Light of the Maximum Ordinality Principle, is precisely that synthetically indicated in the title.

The Maximum Ordinality Principle, in fact, shows that Each Self-Organizing System, precisely because characterized by its own “Emerging Quality”, evolves in a “time” and a “space” which are exclusive and specific of each System analyzed. Consequently, the latter can be more faithfully termed as “Proper Time” and “Proper Space” of the System [25] .

This is an aspect that is radically different from the case of the Traditional Scientific Approach, in which time and space are assumed as being absolute.

Such a difference, however, does not represent a real “obstacle” with reference to the interpretation of the output of EQS Simulator. What is important, in fact, is to know that such a “difference” exists and, at the same time, to be aware of their correlative different Nature. In this case, in fact, such a “difference” can always be dealt with in perfect analogy with the “reduction” of the Uniances, when the latter have to be compared with the correlative distances.

In addition, such a “difference” is so specific and characteristic of the Self- Orgnizing Systems, that it cannot even be “reduced” to the space-time conception of General Relativity.

General Relativity, in fact, introduces the concept of “space contraction” and “time dilatation” between two reference systems in a reciprocal movement, according to the following relationships [27]

$\Delta x^{\prime }=\Delta x\cdot \sqrt{1-V^2/c^2}$ (9.2.1)

$\Delta t=\frac{\Delta t^{\prime }}{\sqrt{1-V^2/c^2}}$ (9.2.2)

It is then possible to show that Einstein’s “space-time conception” represents a particular modality at introducing the concept of the second order “incipient” derivative [25] . Such a particular modality, however, manifests itself at a simple cardinal level, corresponding to a “reduction” process of the Proper Space and Proper Time of a given System.

This means that Einstein’s space-time conception in reality corresponds to the introduction of the second order “incipient” derivative, considered, however, at its mere “cardinal level” [ib.].

10. Two “Com-Possible” Scientific Approaches, Albeit “not Equivalent” between Them

The two above mentioned Scientific Approaches, with their corresponding formal languages, TDC and IDC, respectively, when considered with reference to their corresponding “presuppositions” (that is the subjacent “way of thinking”) result as being two different descriptive modalities which are always “com-possible”. In the sense, that they do not exclude each other. They simply co-exist.

This is because, as already anticipated, the Traditional Scientific Approach, which leads to TDC, cannot exclude (in principle) the adoption of different mental categories and their corresponding formal language (e.g. IDC), because of the absence in its presuppositions (especially “necessary” logic) of any form of perfect induction.

On the other hand, the same happens in the case of the adoption of IDC, precisely because of the same reason, although IDC is based on mental categories characterized by a different form of Logic (e.g. the “Generative” Logic).

Consequently, the two formal languages, TDC and IDC, can always be adopted independently from one another. Although this “com-possibility” does not mean that they are “equi-valent” between them (as it is clearly shown by the case of the “Three-body Problem”).

Their “in-equivalence”, in fact, can easily be shown by comparing the different consequences of their respective adoption, when such consequences are obviously considered in the light of their corresponding “mental categories”.

In fact, beside the Traditional Scientific Approach, which affirms that “Every System is a mechanism” (at a phenomenological level), there is also the possibility of a different Approach, according to which “Every System is a Self-Organizing System” (always at a phenomenological level). This is the fundamental reason why they lead to the adoption of two corresponding different formal languages, with some associated important consequences.

In the first case, in fact, the adoption of TDC leads to:

1) Unsolvable Problems in explicit formal terms (as in the case of the “Three-body Problem”);

2) Intractable Problems even by adopting the most advanced computers (as in the case of Protein Folding);

3) Problems characterized by experimental “drifts”, which always represent an indication of possible “side effects”;

4) In addition, it is worth pointing out that TDC can present some “side effects” even in the case of accurate experimental confirmations. Such “side effects”, in fact, can result as being “masked” by the same fact that all the experimental confirmations are always based on the adoption of methods, instrumentation and measurements that are conceived (and designed) in a perfect conformity with the fundamental presuppositions of TDC [21] .

Vice versa, the adoption of IDC does not present such problems, whereas, in turn, it presents several advantages.

In fact, as already anticipated, the adoption of IDC is finalized to describe the “Emerging Quality” of “Self-Organizing Systems”. This leads to the formulation of the MOP, which is able to offer a radically New Perspective to Modern Science. That is: “Every System is a Self-Organizing System” (see Table A1).

This is because IDC results as being the most appropriate language able to describe the fundamental characteristics of “Self-Organizing Systems”. In fact, the “Incipient Differential Calculus” (IDC):

1) is able to represent, in appropriate formal terms, the “Emerging Quality” of Self-Organizing Systems as an “Irreducible Excess”;

2) In this way, IDC enabled us to formulate a very general Principle, the Maximum Ordinality Principle (MOP), which can be understood as “One Sole Reference” Principle [16] ;

3) The latter in fact results as being valid in any field of analysis (from non-living Systems, to living Systems and human social Systems too);

4) In addition, the adoption of IDC always leads to explicit formal solutions (such as in the case of the “Three-body Problem”);

5) At any topological scale (e.g. from atoms (Quantum Mechanics) to Galaxies (Celestial Mechanics));

6) Both under steady state and variable conditions;

7) What’s more, the corresponding Solution to any mathematical model based on the MOP (and thus formulated in terms of IDC) always results as being an “Emerging Solution”. That is, a Solution whose Ordinal Information content is always much higher than the Ordinal content corresponding to the initial formulation of the problem;

8) As a direct consequence, this leads to the fact that any “Emerging Solution” can never be reduced to mere “functional relationships” (as previously shown in the case of the “Three-body Problem”);

9) This is also means that the adoption of IDC does not require any specific reference to the traditional Physical Laws or to the well-known Thermodynamic Principles (precisely because the latter are always understood as “functional relationships”). In this respect, see also previous Paragraph 8.1, concerning the Relationship between “Forces” and “Diffusive Generativity”;

10) Finally, the adoption of IDC never leads to “side effects”. This is because, even when an “Emerging Solution” might manifest some related “Emerging Exits” [21] , the latter can always be interpreted as being corresponding “Extra Benefits”, initially not recognized as such. This leads to point out another fundamental aspect, always in the Light of the Maximum Ordinality Principle and the correlative maximization of the Ordinality of its associated Habitat.

11. More General In-Equivalence between the Two Scientific Approaches, Especially with Reference to the Relationships between Man and the Environment

Although from a general point of view the in-equivalence between the two formal languages can preliminarily be recognized at the level of “Thinking”, such an in-equivalence is even much more marked at the level of “Decision Making and Acting”. Especially when considering, as a basic reference criterion, the corresponding different concepts of “inter-relationships” between Man and the Environment [21] .

This is because the adoption of TDC always “reflects” the general idea that “every system is a mechanism”, while the “com-possible” formal language IDC is always orientated at describing any system as a “Self-Organizing System”. This is the fundamental reason for the adoption of the three new mental categories (shown in Table A1), which are radically different from the three basic presuppositions of the former.

This easily leads to recognize that the most profound “in-equivalence” between TDC and IDC situates at the level of Decision Making and Acting, in particular with respect to the Environment. In fact:

1) At the level of “Decision Making”, the two formal languages will evidently lead to make decisions (that will become consequential future actions) in a perfect conformity with their respectively different way of thinking: TDC, in conformity with its “aprioristic” presuppositions; IDC, vice versa, in conformity with the new mental categories that, on the contrary, are adopted “a posteriori”.

Consequently, in both cases the two formal languages will suggest “decisions” in perfect conformity with their corresponding concepts of “surrounding habitat”: understood as a “set of mechanisms”, in the case of TDC or, respectively, as “a unique Self-Organizing System” in the case of IDC [21] ;

2) At the level of Action, however, it is exactly where it is possible to recognize the most marked differences between the two Scientific Approaches. This is because, in such a case, the specific different origin of each formal language, together with the associated powerful expressive capacity that any formal language is able to manifest, represent the fundamental aspects that systematically “guide” (sometimes even “force”) the research for specific practical solutions to the various problems and their subsequent actual implementation, in particular with respect to the Environment.

In other terms, the profound differences between the two Scientific Approaches, characterized by their corresponding formal languages, TDC and IDC, respectively, become particularly evident at the “level of Action”, because the corresponding formal solutions become consequential facts [28] .

In this respect, the Ostensive Examples previously considered in the various Biennial Emergy Conferences (from 2010 to 2020), are sufficiently clear to show the profound differences that may result, in practice, when adopting the one or the other descriptive formal language.

In addition, an ulterior and more radical form of in-equivalence will be analyzed in the next paragraph.

12. Radical In-Equivalence between Falsification and Relaunce

Another aspect that points out even more clearly the in-equivalence between the Traditional Approach and the Ordinal Approach is the fact that the first one is characterized by “confirmation/falsification” processes whereas the second one is characterized by “Emerging Exits”.

The “confirmation” processes, in fact, are strictly necessary in the case of Traditional Theories, which are adopted “a priori”, and are specifically based on those mental categories previously recalled. In particular, necessary logic.

At the same time, the absence of experimental confirmations of the corresponding conclusion of Traditional Theories represents a valid argumentation for their “falsification” (according to Popper’s Falsification Principle).

On the contrary, the Ordinal Approach based on the “Emerging Quality” of Self-Organizing Systems, strictly speaking does not require correlative “confirmation processes” in order to be accepted as being a “valid” Approach.

This is because the Ordinal Approach is adopted “a posteriori”, that is downstream the recognition of the Manifestation of Quality as an “Irreducible Excess”, and consequential adoption of the new correlative Mental Categories.

So that, the research for the “Maximum Adherence” of the correlative Over-Deductions (in Generative Logic) to experimental results, does not represent, properly speaking, the research for a “confirmation”. But, paradoxically, it represents the “confirmation” of a “denial”. Or better, “a confirmation” that can be termed as being “not less than”.

In fact, it is exactly such circumstance the one that properly generates the concept of Relaunce.

The latter in fact consists in recognizing that the description of the “Emerging Quality”, as performed at a preliminary given stage, if characterized by “Emerging Exits”, can be recognized as being “not less than”. That is, the description can be re-proposed at a Higher Level of Ordinality with respect to the one initially supposed and assumed to describe the Process (or Phenomenon) analyzed.

At this stage, the profound “in-equivalence” previously shown between the two formal languages, which mainly and clearly manifests at the level of “facts”, may suggest, as a possible conclusion, the consideration of an extremely important question: “where are we going”, as a consequence of the adoption of one or the other descriptive formal language: TDC or IDC?

13. Conclusion: Where Are We Going?

The afore-mentioned differences between the two Scientific Approaches and their correlative formal languages, TDC and IDC, which can preliminarily be recognized at a gnoseological level and, even more, at the level of their respective practical consequences, enable us to draw some general conclusions that can be synthetically summarized as follows.

From a general point of view, in fact, it is possible to delineate three possible answers to the previous question:

1) Modern Science is so radically rooted in TDC (and in its corresponding presuppositions) that it is extremely improbable to hypothesize, in spite of the afore-mentioned intrinsic limitations of such a formal language, a rapid change of the corresponding paradigm (as, for example, the same case of the “Three-body Problem” would suggest).

In this sense, we have to expect a generalized persistence in the adoption of the traditional formal approach (TDC);

2) This fact, however, does not prevent from thinking that some Scientists, with reference to some specific problems (for instance related to the “Three- body Problem”), will decide to preferentially adopt the innovative IDC approach;

3) Even if, more probably, because of the afore-mentioned “com-possibility” between TDC and IDC, it may be expected the adoption of both formal approaches at the same time, so as to choose the optimal operative solutions on the basis of the corresponding experimental results.

By always taking into account, however, that TDC translates, in formal terms, a “self-referential” gnoseological approach, while IDC represents, always in formal terms, a “hetero-referential” gnoseological approach (as previously illustrated and synthetically summarized in Table A1).

Appendix A1. Process of Genesis of the Harmony Relationships from a Diffusive Generativity

Appendix A1 points out, in more explicit terms, what synthetically previously asserted, that is: the Harmony Relationships represent, by themselves, an “Emerging Solution” which, in addition, is also “Exceeding” with respect to the Solution to the First Fundamental Equation.

In fact, what we presented at Paragraph 5.6 of Appendix A are nothing but the basic presuppositions for the formulation of the Harmony Relationships, which, however, do not represent a “necessary consequence” of those presuppositions, because they manifest an “Extra”, or better, an “Irreducible Excess” with respect to them.

Let us thus recall the basic elements that will enable us to show that the Harmony Relationships precisely represent an “Emerging Extra” of Generative Nature.

We have seen in fact that the Emerging Solution to the First Fundamental Equation allows us to write the following topological “Assignation Relationships”

${\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left\{{\tilde{\alpha }}_{1j}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1j}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$

for

$j=3,4,\cdots ,N$ (A1.1)

and, at the same time, their corresponding topological “Assignation Relationships”, written in terms of “Incipient” Derivatives in the form

${\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{12}\right\}}^{\tilde{k}}\stackrel{\ast }{=}{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{1j}\right\}}^{\tilde{k}}$

for

$k=1,2,\cdots ,N-1$ (A1.2)

in which, for simplicity of notation, the Ordinalities $\left\{\tilde{2}/\tilde{2}\right\}$ , which appear in Equations (A1.1), are considered as being included in the symbols of the quantities to which they refer to.

More specifically, Equations (A1.2) cannot be interpreted as a “necessary consequence” of Equations (A1.1), because the latter are obtained on the basis of “Incipient” Derivatives. Consequently, they are all of Generative Nature.

In fact, if rewritten in the following form

$\frac{{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{12}\right\}}^{\tilde{k}}}{{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{1j}\right\}}^{\tilde{k}}}\stackrel{\ast }{=}\tilde{1}$

for

$k=1,2,\cdots ,N-1$ (A1.3)

they allow to assert that the considered System is already characterized by a proper and specific “Interior Unit”, of Generative Nature, formally represented by the symbol “ $\tilde{1}$ ”.

Such a “Unity”, however, is still in the form of “Not Less Than”. This is because:

- In a Generative Contest, they are certainly not the parts that, through the Relationships “between” them, give “Origin” to the “Excess of Unity”

- Because it is exactly true the opposite: in fact, it is the Generative Unit of the System that, with its proper “Excess”, Qualifies the Relationships “between” the parts.

Consequently, the most Adherent Formulation of the Self-Organizing Generative Process is that which can be obtained by re-proposing Equations (A1.3) in the form

$\frac{{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{12}\right\}}^{\tilde{k}}}{{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{1j}\right\}}^{\tilde{k}}}\stackrel{\ast }{=}\left\{\tilde{1}\right\}$ per $\forall k$ (A1.4)

or better, even more properly, as follows

$\frac{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{12}\right\}}{\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{1j}\right\}}\stackrel{\ast }{=}{\left\{\tilde{1}\right\}}^{\frac{1}{\stackrel{\sim }{\left(N-1\right)}}}$ $j=2,\cdots ,N$ (A1.5)

in which the symbol $\left\{\tilde{1}\right\}$ now formally represents the Generative Whole, which, at the same time, is Self-Organizing and of Ordinal Nature. While its unique and sole exponent $1/\stackrel{\sim }{\left(N-1\right)}$ explicitly represents the fundamental concept previously anticipated, that is: it is the “Whole”, with its proper Generative “Excess”, the one that properly “Qualifies” the Relationships “Between” the parts.

This is obviously true not certainly in the sense of Relationships understood “two by two”, but as the specific Reflex of an Ordinal Unit, which, in any case, represents an “Irreducible Excess” with respect to the simple “composition” of the single “parts”.

Consequently, Relation (A1.5), can also be written in the form

$\left\{{\stackrel{\circ }{\tilde{\alpha }}}_{1j}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{1j}\right\}\stackrel{\ast }{=}{\left\{\tilde{1}\right\}}^{\frac{1}{\stackrel{\sim }{\left\{N-1\right\}}}}\circ \left\{{\stackrel{\circ }{\tilde{\alpha }}}_{12}\left(t\right)\oplus {\stackrel{\circ }{\tilde{\lambda }}}_{12}\right\}$ for $j=2,\cdots ,N$ (A1.6)

which, reinterpreted in terms of “Progenitor Relationships”, finally leads to the formal expression of the Harmony Relationships. The latter, written in the form

$\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\oplus {\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}\stackrel{\ast }{=}{\left(\sqrt[\stackrel{\sim }{N-1}]{\left\{\tilde{1}\right\}}\right)}_j\circledR \left\{{\tilde{\alpha }}_{12}\left(t\right)\oplus {\tilde{\lambda }}_{12}\left(t\right)\right\}$

for

$j=1,2,\cdots ,N-1$ (A1.7)

clearly show that the Diffusive Generativity of the System “updates”, by Assignation, all the couples at the first member, and, contextually, the same reference couple “12”.

Equations (A1.7) then clearly show that all the elements of the Ordinal Matrix (5.6.1) can be obtained on the basis of one sole couple ${\tilde{\alpha }}_{ij}\left(t\right)$ assumed as reference and their associated Correlating Factors.

In this respect, it is also worth noting that condition (A1.2) is properly the one that represents the fundamental presupposition of what could be termed as an Intensive Whole, precisely because of the “consonance” between all the generative derivatives up to the order N − 1, that are due to the “Diffusive Generativity” of the Self-Organizing System.

This is the specific reason why, by means of the MOP, and its correlative Harmony Relationships, it was possible to reconsider some “particular” problems that, in the Traditional Scientific Literature, are generally known as being “unsolvable” (such as, for example, the “Three-body Problem”), or “intractable”, or “with a drift”. Whose solutions ended up by showing that the Maximum Ordinality Principle has an extremely general validity [16] ?

Appendix A2. The Ordinal Roots of Unity $\left\{\tilde{1}\right\}$ and Their Role in the Harmony Relationships

In this respect, it is worth observing that previous Relationships (A1.7) are written in such a form only for reasons of clarity and exposition simplicity. In such a form, however, it could seem that the various elements that characterize the System are “still” related, “between” them, according to Relationships of the type “two by two”.

In reality, if one makes explicit the term ${\left(\sqrt[\stackrel{\sim }{N-1}]{\left\{\tilde{1}\right\}}\right)}_j$ according to its more specific meaning, that is as ${\left\{\tilde{1}\right\}}^{\frac{1}{\left\{\stackrel{\sim }{N-1}\right\}}}\equiv {\left\{\tilde{1}\right\}}^{\frac{1}{\{N-1,(\stackrel{\sim }{N-1})\}}}$ , in which N − 1 refers to the cardinality, while $\stackrel{\sim }{\left(N-1\right)}$ refers to the Internal Ordinal (N − 1)-ary Relationship, it is possible to more appropriately write (by pointing out the Ordinalities $\left\{\tilde{2},\tilde{2}\right\}$ , previously underwritten)

${\left\{{\tilde{\alpha }}_{1j}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1j}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}{\left\{\tilde{1}\right\}}^{\frac{1}{\{N-1,(\stackrel{\sim }{N-1})\}}}\circ {\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (A2.1)

that is, even more explicitly, in the form

${\left\{{\tilde{\alpha }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{1,j+1}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\stackrel{\ast }{=}\left(\begin{array}{c}{\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_1\\ {\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_2\\ ⋮\\ {\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_{N-1}\end{array}\right)\circ {\left\{{\tilde{\alpha }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}\oplus {\left\{{\tilde{\lambda }}_{12}\left(t\right)\right\}}^{\left\{\tilde{2}/\tilde{2}\right\}}$ (A2.2)

from which it is possible to recognize that the single “cardinal” values that in Equation (A1.7) appear as they were “distinct”, and, in addition, as being “separated”, in reality they are the Reflex of an Ordinal Unit that transcends them, and relates them in the form of an (N − 1)-ary Relationship.

This is the aspect that (more than others) clearly manifests that the Harmony Relationships represent an “Excess” with respect the initial Assignation Relationships (5.6.3) and (5.6.4).

In addition, as far as the “explicit” meaning of the Ordinal Routs of Unity is concerned, previously synthetically indicated in the form

${\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_j$ per $j=1,2,3,\cdots ,N-1$ (A2.3)

it is worth expressly pointing out that the symbol $\left\{\tilde{1}\right\}$ represents the Unity of the System (understood as a Whole), with specific reference to the Unity of its Proper Space (as well as its Relational Space).

Such a Fundamental Unit can be then expressed by the following Relationship

$\left\{\tilde{1}\right\}=e^{\left\{\alpha \circledR \tilde{i}\oplus \beta \circledR \tilde{j}\oplus \gamma \circledR \tilde{k}\right\}}$ (A2.4)

Consequently, the Ordinal Roots ${\left(\sqrt[N-1]{\left\{\tilde{1}\right\}}\right)}_l$ will be represented in the following form

${\left\{\tilde{1}\right\}}_l=e^{\frac{\left\{\alpha \circledR \tilde{i}\oplus \beta \circledR \tilde{j}\oplus \gamma \circledR \tilde{k}\right\}}{N-1}}$ (A2.5)

where:

- $\tilde{i},\tilde{j},\tilde{k}$ are the fundamental spinors of the Relational Space, understood in their more general sense, that is, as the specific foundation of any given System

- α, β, γ are respectively equal to

$\alpha ={\epsilon }_1+\frac{4\pi \cdot l}{N-1}$ $\beta ={\epsilon }_2+\frac{2\pi \cdot l}{N-1}$ $\gamma ={\epsilon }_3+\frac{2\pi \cdot l}{N-1}$ (A2.6)

- Where the “periodicity” of the “spinor” $\tilde{i}$ , as we already know, is equal to $4\pi$ , because expressed in steradians;

- While the periodicity of the spinors $\tilde{j}$ e $\tilde{k}$ are both equal to $2\pi$ radians (each), because these spinors are always “orthogonal”, both between them, and with respect to the spinor $\tilde{i}$ (an orthogonality that can be understood, inter alia, as a form of reciprocal “irreducibility”);

- The quantities ${\epsilon }_1$ ${\epsilon }_2$ ${\epsilon }_3$ represent specific “parameters” of the Relational Space each time considered, with specific reference to the “couple 12”.

Sometimes (for example in the case of Protein Folding), for an easier “topological” representation Equations (A2.6) can also represented as

$\frac{\alpha }{N-1}=\frac{{\epsilon }_1+4\pi \cdot l}{N-1}$ $\frac{\beta }{N-1}=\frac{{\epsilon }_2+2\pi \cdot l}{N-1}$ $\frac{\gamma }{N-1}=\frac{{\epsilon }_3+2\pi \cdot l}{N-1}$ (A2.7)

which however can always re-proposed in the previous form (A2.6) through an appropriate choice of the parameters ${\epsilon }_1$ , ${\epsilon }_2$ , ${\epsilon }_3$ .

On the basis of the previous exposition, it should be clearer that The Ordinal Roots of Unity $\left\{\tilde{1}\right\}$ contribute to manifest that the associated Harmony Relationships represent an “Irreducible Excess”, that is an “Exceeding” Manifestation of a Generative System, which, at the same time, is Self-Organizing, of Ordinal Nature, and, above all, it is understood as a Whole from the very beginning, and not vice versa.

NOTES

¹These “new mental categories” can no longer be termed as “pre-suppositions”, because they are not defined “a priori” (as in the case of Traditional Approach). In fact, they are chosen only “a posteriori”, on the basis of the “Emerging Quality” previously recognized. “Generative Causality”, in fact, refers to the capacity of a Self-Organizing System to manifest an “irreducible excess”; “Adherent Logic”, correspondently, refers to the capacity of our mind to draw “emerging conclusions”. That is, “conclusions” whose information content is much higher than the information content corresponding to their logical premises, although persistently “adherent” to the latter. “Ordinal Relationships”, in turn, refer to particular relationships of genetic nature, which will be illustrated in more details later on, with reference to any Generative Process.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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