This is a translation of the final chapter of the essay “Espíritu del Cuaternario” (Quaternary Spirit)

After so many considerations on science we need to return to a more general perspective.

It was probably as a reaction against the intrinsic idealism of the trinitarian symbol that a number of thinkers of diverse orientation turned in the twentieth century, and especially after the post-war period, towards quaternary schemes as symbols of totality. Jung was maybe the first perceiving the need for this shift, followed later by such well-known authors as Heidegger with his earth-heaven-celestials-mortals quaternity, or the Schumacher of the magnificent «Guide for the Perplexed» with his fourfold field of knowledge: inner self, inner world, outer self, outer world, opposed two by two as determinants of experience, appearance, communication and science.

Another one of those thinkers was Raymond Abellio, who also presented a quaternary model of perception and thus of knowledge, in which the mere relation between the object and the sense organ is always only a part of a larger proportion —a relation of relations— , since the object presupposes the world and the sensation of the organ a whole body that organizes it and gives it a definite sense:

The most important aspect of this exacting proportion (object/world = sense/body) is the ignored but ever-present continuity between the extremes «world» and «body», where the world is not a sum of objects, nor the body of organs, parts or entities. In short, the primitive homogeneous medium of reference for any phenomenon, motion or force, since we already know beforehand that any motion or change of density is only a manifestation of the principle of dynamic equilibrium, either as a sum or as a product.

The body from within is the seat of sensation, the undifferentiated common sensorium from which the different organs have emerged, and without which there would be neither subject nor «common sense.» In harmony with this, one can speak of two modes of intelligence, one which seems in motion and pursuing its object, and an immobile one which allows us to listen to our own mentations, and without which they could not exist. Let us try to think without listening to ourselves, and we shall see that this is impossible: the very compulsion to think is nothing but the desire to listen to oneself.

Abellio proposed an «absolute structure» of space to which the «movements of consciousness» would also be referred, and which would be nothing but the axes of ordinary flat space with the traditional six directions.

The relation of the motions of the body with respect to its isometric center of gravity as origin of coordinates is similar to the movements of the object-oriented intelligence with respect to the immutable intelligence. Certainly the «space of the mind» does not seem extensive at all, but to verify its intimate connection with the physical it is enough to put into practice any of those isometric exercises in which one remains standing and hollows out simply to perceive the balance in the micro-movements necessary to maintain the posture. If the intimate is the interpenetration of the internal and the external, we have here both a physical exercise and an exercise for the intelligence, which allows us to verify the intimate, transcendental relationship between motion and immobility.

According to Abellio, «the perception of relations belongs to the mode of vision of the «empirical» consciousness, while the perception of proportions is part of the mode of vision of the «transcendental» consciousness»: this would apply to the four parts or moments of the quaternity already expressed. As an heir of Husserl, Abellio makes a great effort to go beyond conceptualist schemes but still remains within the bondage of knowledge. Global awareness or «consciousness of consciousness» is not the object of an infinite regression only by virtue of a sort of ultimate «passage to the limit» that keeps reminding us of such things as Peirce’s «ultimate interpreter».

The ultimate dilemma of understanding, as Siddharameshwar puts it, is that without detachment there is no knowledge, and without knowledge there is no detachment. Now, this detachment is not the mere separation that we take with respect to the object, but something more deeply connected with our will. We suppose that we want to know, but know what? We do not even know that, and perhaps we do not even want to know that. Scientists strive to find the solution to inherited problems, but it is way more important to know how to put aside the questions that one has not asked oneself.

The empirical self cannot be the operator of global consciousness —consciousness without object-, and the «transcendental self» is only a name for that which never says «I» and doesn’t need it: for that body-world continuum within which sense objects appear. That continuum sometimes gives us some knowledge, if we aspire to deserve it, perhaps with the sole object of bringing about the detachment of the intelligence and the self that are normally conscious of adhesion and separation and live by their alternation, but lose ground in the middle.

So it could be said that all global knowledge is simply a grace of which the empirical self is the object in order to facilitate its detachment and give it some aptitude; and it is perfectly within logic that it can only arise beyond the desire for particular knowledge, as grace is identical with being, the non-particular par excellence. But there is also around this neutral word, «being,» a transubstantiation of intellect and will.

Abellio’s fourfold proportion points to a succession of thresholds, of ever-widening perspectives, but which we should not see only as angles of knowledge, but as degrees of participation in being, arising from a double movement of assumption and incarnation. Of course, this double movement also occurs in scientific knowledge, but as long as Nature is just an object we are talking about two types of knowledge that are not even comparable.

In this context, the metanoia or metacognition can in no way give rise to an infinite regress, because what it actually implies is a repeated transformation of the mutable or apparent with respect to the immutable that by definition cannot be seen.

On the other hand, the «turn towards the body» of the most recent philosophy is too partial not to be easy prey to instrumentalization —much has been said about sex and «desiring machines» but the truth is that desire, which is a feminine agency, acts more in the soul or even in the spirit than in the body; while the will, which is much more literally confined in bodies and is a masculine agent, is systematically ignored. Basically, the body continues to be seen as an object, even if on the other hand science prevents us from considering an external desiring, naturing Nature.

Considering the bodies it is obligatory to mention the fundamental ambiguity of the relational mechanics inaugurated by Weber with respect to the three energies —kinetic, potential, internal— which, although a mere consequence of the equations, seems nonetheless natural, and which should be taken into account with respect to certain balances and proportions.

Noskov’s longitudinal vibrations internal to the moving bodies, postulated precisely to justify the conservation of energy which in Weber was merely formal —no more and no less than in the other mechanics— are at the core of the issue, and it is easy to see how they should «fit» within the data of classical mechanics, quantum mechanics and relativity —and in the parallel transport of the so-called geometric phase. In general, in any field when distinguishing between particle and field we have a self-interaction problem under accelerations that quantitatively coincides with Noskov’s oscillations. This view in terms of resonance is in harmony with the ancient, and much more timeless, conceptions of Nature.

As we have already seen in different places, the retarded potentials and the corresponding oscillations would not only be present at the micro level, but also in complex organic systems such as respiration and blood circulation.

The Solar Verb, the arouser, Savitr, is totally incomprehensible in our objective representation of Nature without the twin notions of internal vibration and external resonance; and these notions are naturally connected when we put force and potential on the same basis. To get another view of the physical Continuum, we would have to deal yet with other questions of relevance, such as scales transitions in length and time, fractional dimensions and fractional operators, but this would lead us back to complex matters; in any case we have already suggested some relationships.

«Help me and I will help you», today as always, this is what Nature tells us. Modern science, as imposing as it is, ignores this connection, which passes immediately through one’s own body but extends to the limits of the world.


Greek science was a science of observation, while to arrive at experimental science, in which man unites physical and mathematical manipulation to take possession of Nature, we must wait for Arab science with authors such as Alhazen, whose great work is now a thousand years old and who clearly prefigures Galileo six centuries in advance —though it is clear that it was in the West where this experimental stage reached its peak.

Although today the vast majority of scientists remain confined to this stage, the only one of interest to the social apparatus and the powers that govern it, there are some for whom mastery over Nature expressed in Laws that allow prediction represents only the outermost possible limit of knowledge.

Today we are in a position to arrive at a third stage of scientific reason founded on self-observation, which reintegrates mathematics and the variable observation of phenomena in the universality of the Self, the undifferentiated space that is the basis of any intellection. The self-interaction present in that field clearly points to the reflexive aspect in the principle of equilibrium from which all mechanics can be derived.

This third stage of reason will be able to manifest itself to the extent that we understand that there is no law of nature that does not depend on self-interaction, just as there is no consistency or tautology or circularity in current physical theories that is not expressly elaborated. The concept of self-interaction should help us to see instead how and to what extent a physical system is effectively open or closed.

On the reflective path of this self-observation, mathematics, as «pure form», can still find another way of endowing itself with content when mathematics and physics seek a balance between description and prediction. And it is to be hoped that this will tend to happen spontaneously, to the extent that the obstructions created by the unilateral justification of a given purpose are removed.

Calculus is the best example of this, and since it is analysis itself that makes quantity take command to its present extreme, combining the idea of infinite division with the construction based on indivisible elements, we have counterposed two alternative ways for calculus based on the other two extreme forms of the indivisible: the unit interval and division by zero. Both would return us to a sort of continuum somewhere between the current conceptions of the physical and mathematical ones.
On the other hand, it could not be more significant that, now that the cybernetic control model takes command over the scientific and social rationality, we find the principle of feedback at the very basis of natural «systems» governed by «fundamental forces». This brings us face to face with the other essential aspect of differential equations, the boundary conditions, which are also interface conditions of the system.

All modern calculus since Kepler’s problem is a reverse engineering on the global contour of a system, which also involves an inversion of the relation between what is conceived as the «core» and the «periphery» of it —the internal domain of the function and its boundary conditions. But, from a strictly descriptive point of view, it is the boundary that has shaped the dynamics, and the integral, what may well be called a pseudo-differential; and only the predictive power and the magic of variable assignement have led us to forget this fact.

The same logic of usurpation operates in the now indiscriminate application of the «cybernetic reason» to which we are all subjected. However, applying relational logic to a system such as the blood circulation and the heart according to the Weber-Noskov retarded potentials, the difference between margin and center vanishes —in a case where such a distinction could not be more relevant.

It is then of particular value to try to bring the problems of dynamics back into the reflexive domain of an evolution detached from a homogeneous medium but still dependent on it, since any feedback can only work by the contrast of something heterogeneous with its homogeneous background. Although the crucial question would be to check how the mathematical function is connected to the the regulatory principle in its direct functional aspect, for example, through biofeedback.

The case of the pulse wave could serve as a perfect illustration of something much more general. The essential thing about biofeedback here is not its capacity for modification, which is always so limited; in fact, ideally this capacity would have to be nil, since what is involved is the interpolation of the subject himself in an objective functional reality. If the modification of organic functions can be induced, even then that serves above all to see that its principle of action or regulation can be qualified neither as conscious nor as unconscious, neither as voluntary nor as involuntary, since any of these characterizations contradict itself.

We also speak of a system traversed by a retarded potential that is open but tends as much as possible to stabilize as a closed system, and should therefore exhibit asymptotic features.

It is usually assumed that asymptotic behavior is the simplification of a more complicated particular case; but we can also see it from the other side. For every asymptotic aspect, as an exponent of a discontinuity, defines more limited boundary conditions with respect to a more general case, from which it follows. In this sense, asymptotic analysis unwittingly lies in the trace of the universality of the Self.

Applied mathematicians and physicists as competent as Kurt Friedrichs or Martin Kruskal have argued that asymptotic description is not only an appropriate tool for the analysis of Nature, but that it points to something more fundamental, and we cannot agree more with them. Undoubtedly, interest in this field is growing steadily in the age of computation, since asymptotics is the natural mediator between numerical methods and those of classical analysis; numerical analysis being more quantitative and asymptotic analysis more qualitative. But even if we are apparently talking about a method of applied mathematics, we cannot fail to notice in it a symbolic component deeper and more evocative than in the so-called pure mathematics.

The object of asymptotics, as a heuristic, are the limit cases; but the whole analysis is permeated by heuristics insofar as it is application. It would then be of great interest to see what new developments the asymptotic analysis can acquire between the two poles of the indivisible we have commented. The constant differential calculus, as a method of finite differences based on a unitary rate of change, seems much closer to numerical analysis, but allows, together with the relational criterion of homogeneity, to scope out way better the dimensional analysis and boundary of problems. It also presents a much more basic consistency than standard calculus.

Then «asymptotology,» as Kruskal called it, still has much to gain in consistency, depth and unity, and not only by virtue of the new perspectives that alternative forms of calculus may open up. There is good reason to believe this, beyond the fact that the consistency of classical analysis is one of results rather than of procedure: because it points as directly as possible to simplicity from the point of view of the limit, and the simplest use to hide the most folds and unnoticed dimensions; because it lies at the basis of field and potential theories besides thermodynamics, because it tends to connect different theories and to define the transition between different scales, because it connects with the more general projective arguments of geometry and is at the very heart of pure arithmetic as well as the most elementary calculus, and because it is a signature of equilibrium, stability and self-interaction.

«Asymptotic analysis» is practically everything the applied mathematician does when he is not doing numerical analysis. For historical reasons asymptotics is associated with perturbation theory —first in celestial mechanics and then also in quantum mechanics- as well as the study of singularities. However, constant differential calculus makes it possible to treat celestial mechanics without the perturbative approach, and the velocity-dependent forces in relational gravitodynamics make black holes unfeasible. Like many other branches, the emergence of asymptotic analysis in perturbation theory has much to do with contingency, reflecting the punctual fact that it was the first important question in mechanics outside the direct scope of the conventional methods of calculus.

Throughout the eighteenth century, in the wake of Newton, absolute space was seen as the element of contact between the conditioned and the unconditioned, between the things of the world and their Creator. In contrast Leibniz, the other founder of calculus, expounded the idea of space as a set of relations, and both notions had a decisive influence on the Kantian transcendental doctrine of the ideal character of space and time, which attempted to propose a synthesis. However, Pinheiro has shown that neither of these two positions explains satisfactorily something as basic as the whirlpool of water in Newton’s bucket experiment.

The asymptotic features would be essential, and not merely incidental, insofar as every observable system is a separation from the continuum or homogeneous medium in which space, time, matter and motion cease to be discernible. In the absence of a complete characterization, they shows us some of the most notorious aspects of its relative differentiation, some of its «boundary layers», to use a very general concept from fluid mechanics.

Thus, certain aspects of asymptotic evolution would be both an index of individuation of systems and the most hyperbolic symbol of the absolute that is in the midst of everything and has no contact with anything. Gaudapada spoke in his Mandukya Karika of «contactless» union, asparsha, although it seems clear that what does not admit of contact does not require union either. And yet Nature seeks in its own way and in every way the unattainable that resists nothing —surely to compensate for the fact that it has separated itself from it. Venis’ morphological sequence also echoes this evolution as an individuation process, and it is no small matter that the asymptotic enters the projective hypercontinuum of appearances.

Let us return to the circuit of the blood pulse seen as a delayed potential with an internal wave a la Noskov. Let us recall that for Noskov these oscillations permeate everything, very much in the style of the stoic pneuma that conveys the logos: «It is the basis of the structure and stability of nuclei, atoms, and planetary as well as stellar systems. It is the main reason of the occurrence of sound (in particular, the voices of people, animals, and birds; the sound of wind instruments etc.), electromagnetic oscillations and light, whirlwinds, pulsations of flowing water, and gusts of wind. It, at last, explains elliptical orbital motion…»

Certainly these are great claims for something that has not even been recognized —except for, alas, those de Broglie’s matter waves so faithfully verified at any scale. The much more massive predicament of relativity, much more than the irreducible ambiguity of the three forms of energy in these equations, suffice to explain this overt non-admission. However, they seem to fit perfectly into the known process of the arterial pressure wave, which is also amenable to subjective interpolation via biofeedback.

As we have already said, the geometric phase is an index of the “geometry of the environment», of how the system is not reducible to the conservative idealization of the theory; it is not by chance that Berry is a specialist in asymptotic methods and semiclassical approximations, which he has also tried to apply to the central problem of number theory. The idea emerges that the ambient geometry of the retarded potential traverses as an oscillation precisely the effective aperture that the system has with respect to a closed conservative system, and that it is due to this aperture that the system tends to appear conservative or closed. At least in a manifestly open system such as our organism, with the dependence of the heartbeat on the effect that respiration has on the circulation, the idea of self-induction acquires its full meaning.

Since it goes without saying that in a closed system self-induction is null and needless: this is the tautological element presupposed in a concept such as «mechanics». And yet Maxwell’s equations, the proverbial exponent of a tautological symmetry, give rise to self-induction. It is only with Noskov that the “closure” of the system take place inside the bodies themselves, that what has been ignored outside, and relegated to the convenient nebulosity of the field, is literally embodied.

But we have already said that electromagnetic induction can legitimately be considered as a mere particular case of mechanical induction —even if the quite reproducible experimental evidence has not been properly addressed. The heart then is a «pump», but not a mechanical pump: it is not a «vacuum pump” either, but a pump effect in the ordinary vacuum, which is nothing but the ambient environment. We are talking about the fundamental physical vacuum —there is no other one-, which obviously has nothing to do with the wild guesses of energy levels of theoretical physics.

And since we will never measure this vacuum except in bodies, all mechanics must, by definition, be the average of what happens between bodies and the vacuum. There is no possible escape from this, and yet we are incapable of assuming it. The word «mechanics» has acquired undesirable tones because of the restricted domain we have imposed on it, just like the word «automatic,» which Aristotle still used as a synonym for spontaneous, for that which moves «of its own accord.» It is by leveling force and potential that mechanics and dynamics become equivalent terms, as the study of the relationship between bodies and the vacuum that can never be reduced to mere motion. This is the main caveat to be made to a poorly understood «relational mechanics».

The pulsations of the Sun and the other stars, an ever-expanding area of study, also raise the same question as the heart: that its entire body is traversed by the non-uniform resonances or oscillations of the Noskov potential. Here we can very clearly identify the main sources of variation of the potential on the planets —even if the assessment of interplanetary density raises other kind of questions. In any case, and including a number of factors such as the relation of the solar sphere to the barycenter of the system, these resonances would be associated with the back-coupling of the central body with the global field; some will be surprised that an interpretation that cannot be more straightforward is not even considered.

The modern idea of mechanics, in tune with the overall trend, consists in reducing «the other» to «the same»; here we want to indicate at least how «the same» that is not contemplated is the genuine principle of differentiation, of directly appreciable and measurable forms and profiles. A sameness, an aseity that, far from reducing anything, neither to analytical conditions nor of any other kind, would be the condition of openness by which any entity breathes.

Thus, the «interpolation» of the subject in a self-induced mechanical system is only a detour to embrace something already fully present and operative, since the entire mechanics, and not only that of biological organisms, is a reflexive balance that includes the environment at any given moment. Conservation symmetries refer in the penultimate instance to the Third Principle, but ultimately to the First Principle of inertia, the same principle that proposes «an isolated object that is not isolated». By simply dispensing with the idea of inertia, and replacing it with that of dynamic equilibrium, all this would cease to sound strange.

Truly, that the order we observe, from light to atoms, to complex biological molecules, cells, organisms, planetary systems and galaxies, can subsist even for a single instant without this reflexive principle of equilibrium that is present in everything, still seems to me a chimera. The idea of inertia, or that of accumulation over time of biological events synthesized in inheritance, are mere ghosts without an immediate principle of actualization, which obviously cannot consist of «blind forces». This spontaneous balance would already include statistical mechanics and entropy, as implied by Pinheiro’s thermomechanical reformulation of dynamics.

Studied as a biomechanical subject, the dynamics of the circulatory system allows us to short-circuit between our mechanical notions trained by habit and the consciousness prior to thought. This would already be quite an achievement. In addition, there are many open questions to keep thought busy.

For example, the seemingly anecdotal fact that both the ratio between diastole and systole time intervals in humans and other mammals closely approximates the ratio between maximum systolic and minimum diastolic pressure, the continuous ratio 0.618/0.382, allows us to directly connect asymptotic and numerical analysis, the physics of discrete events and continuous values, a feedback loop for the numerical and continuous methods themselves, or an optimization with a particular recursive ratio for algorithmic measurement theory. It further invites us to connect all these aspects with a thermomechanical description including entropy like Pinheiro’s.

Since the heart functions here as a regulator rather than as a pump, and its action is the effect of global motion rather than its cause, it can be clearly seen in what sense it is the «internal clock» that rather than marking indicates the system’s own time: such is the sense of the proper time of dynamic systems that would have to replace the «global synchronizer» of the old mechanics. This return of the local measure to its effective configuration is something generalizable and far-reaching, because to perceive the creative power expressed in Nature one needs to see beyond the global synchronizer.

Such a description shows that the two usual proposals to explain the observable order, or the metaphorical «watchmaker» of complex biological systems, be it the random mechanics cut over time by natural selection, or the «intelligent design» attributed to an ultra-mundane creator, lead us away from the essential, the spontaneous principle of organization which is also the principle of immediate actualization. It is always from the immediate openness with the environment that islands of organization and stability are reached.

This same biomechanical subject allows us to define as far as possible the relationship between the arterial pulse wave, a palpable and concrete biological illustration of Noskov’s hypothesis, the effective openness of the system with respect to the conservative case and the corresponding asymptotic approximation. It should also make it possible to define the internal tone of the system, the essential element of the nonexistent definition of health, and to investigate the quantities that allow its characterization.

We have also seen that the three gunas of samkhya, that peculiar coordinate system, seem to correspond also to the three principles of mechanics extrapolated to open systems with conservation of momentum. Their most tangible aspect, even if reactive or derivative, the doshas of pulsology, can serve to investigate the relationship with the threefold manifestation of energy and what we call its «relational ambiguity». What is the logic presiding over the recursive scaling of the three qualities of material nature within a precise but open quantitative domain? What interface conditions are they defining?

Of course samkhya, insofar as it is a «dualistic» system, has a genuinely asymptotic character: the gunas or modalities of Nature mark a tendency, a path of ascent and descent without in any way coming into contact with Purusha, «the pure spirit» or consciousness, the same described in the Rig Veda as a giant from whose dismemberment the parts of the cosmos emerged. Only the total, non-dynamic equilibrium of the three modalities would produce their fusion with the absolute. Under an entirely similar logic, also the joint relationship of the three feet or letters of the sacred syllable points asymptotically to the fourth foot.

Asymptotics, like so many other forms of analysis, has proposed its own «uncertainty principle» between simplicity and exactness via localization, pure approximation itself. On the other hand, classical mechanics is the first asymptotic approximation of quantum mechanics, but the latter cannot be defined without reference to the former: a fine example that there is something fundamental in asymptotics. It would be opportune also to investigate in depth the connection between the indistinguishability of the three energies in relational mechanics and the various uncertainty relations —for there is not one, but many- of quantum mechanics, developing Noskov’s arguments.


The relativistic continuum, in either of its two versions, is notorious for its asymptotic profiles, to which it owes much of its fascination and speculative impulse. We only need to contrast how the Weber-Noskov mechanics (not to mention Pinheiro’s here) deviates from such aspects, to begin to see the other side of the question: the asymptotic not as an approximation to infinity but as a relative, partial detachment from unity.

The same can be said of the so-called «holographic principle» originating from entropy considerations in the boundary conditions of a singularity or black hole, which would reduce the entire universe to its projection on a mere two-dimensional surface. This principle cannot but be true insofar as we know nothing except through light, the universal, unavoidable mediator. But, obviating the fact that such a principle is due to the very pertinence and universality of the geometric phase, it is clear that the question does not beg any singularity: as Mazilu puts it, even the surface of any extended particle such as the electron, the day one tries to describe its ephemeral configuration, will have to reflect these limits, as intrinsic as the very spin of the particle.

In this sense it is idle to think of what may occur inside bodies, inside any body. And yet light faithfully reflects the interaction of each body with the void in which the other bodies are placed —its innermost pulsation, if that is how we want to call it, if the void and the bodies cannot exist separately.

Let us enjoy for a moment this supreme irony of the history of physics. Physicists who have insisted that the geometric phase is only a small appendage of quantum mechanics without the slightest fundamental entity, now appeal to this same «phenomenon» in its last frontier to wrap the whole universe in a thin film and set it up as a great principle with which otherwise one does not even know what to do. So much has been discussed about the limits of information density, about quantum fluctuations of position, about quantum gravity and holographic noise; so many attempts to verify it, including tabletop experiments, have been proposed.

And one may ask, why look for “quantum black holes» to verify a phenomenon that by definition has to encompass absolutely everything, from the beating of my heart to my perception of colors? «Because a global argument is of little use if we cannot predict new things», any expert so well trained to make predictions would immediately tell us. Of course, all field theories, and not only the electromagnetic field and light, have been built from the outside to the inside —from the boundary conditions to the «corpus of the theory», the equations that allow us to calculate. To then separate the corpus of the theory from its conditions, to turn it into Natural Law, leads us to forget their mutual dependence.

Predictions are blinding if this circumstance is ignored. Why is there so much discussion about the range of energies to experimentally verify this holographic principle? Because there is not the least certainty about the background that determines its scale, the Planck scale itself being a gigantic extrapolation. Even the uncertainty of the energy of a photon has nothing to do with it, as the most elementary analysis shows, since it does not change at all if we change the value of the constant; which does not prevent from using it to make dimensional analysis of the entire universe. If the estimate of the energy of the vacuum has turned out to be such a tremendous nonsense, no one will dare to say it is written on stone. This reminds us another infamous global principle, deserved object of so many jokes: the so-called «anthropic principle» proposed to answer the enigma of the fine-tuning of the magnitude of the great constants. Neither estimates nor predictions serve to shed light on the context. The homogeneity principle tells us that constants with dimensions are not universal, but the by-product of a conventional cutout from the background, an «emancipation» from ambient conditions —which is precisely what the geometric phase, mother of the holographic principle, reflects. There is no other “fundamental physical vacuum» than the ambient environment.

The geometric phase and the very structure of gauge fields suggest a sliding knot in energy constants and scales, and several mutually ignoring theories seem to point in this direction. Not to mention that the black hole hypothesis pretends to give us at once «ultimate boundaries», singularities, and processes beyond the singularity, which is like wanting to give a cake, have it and eat it: no longer two, but three incompatible things at the same time.

«There is plenty of room at the bottom», but surely not the way it is expected, stretching the Planck scale to the limit. On the other hand, if today we know from the gauge theory of gravity that we can dispense with curved space-time to describe its field, all the more reason we can dispense with the elementary Minkowski continuum. This should make us think more about the theories following Weber, which become field descriptions just by integrating over the volume and do not require neither a continuum with two flavors nor additional dimensions.

It should be clear that current prediction-centered theories throw into the sea the key to describe scale transitions, which are the very «background» that the word «fundamental» would like to appropriate. Nottale, and later Mazilu and Agop, have proposed a meritorious theory of «scale relativity», but one can also dispense with relativity itself with much more straightforward arguments. Mazilu and Agop’s development is expressly neoclassical, and yet goes into the jungle of fractal geometry and the non-differentiable continuum: a good example that one can have simple ideas that nevertheless allow to envision the superabundant complexity of Nature.

«Nature cares naught for analytical difficulties»: few words more certain than these of Fresnel’s immortal phrase. If the non-differentiable continuum is possible, one can be sure that Nature uses it with real profusion; and the same path integrals of light would be the best exponent of it. Mazilu employs it directly to try to create an operational model of the brain based on the inexhaustible geometry of light and de Broglie’s wave mechanics, which vividly recalls Bandyopadhyay’s «fractal mechanics» devised for the same purpose. In any case, Nottale’s or Mazilu’s use of the Planck scale deserves some examination, for it should be clear that the common use of it as if it were a rule is nothing but the extension of the global synchronizer to all the more amorphous domains of physics. Nature simply evaporates where this chronometer rules, which entrusts us with the task of trying to imagine it free of this equally imaginary constraints. A belt suits her much better than a rigid ruler.

The sad thing is that the holographic principle has even been seen as a «confirmation» of the idea that the universe is a gigantic computer, to bring the empire of the global synchronizer to its apotheosis. And yet, one can be quite sure that were the world a computer it would not even have lasted long enough to explode into pieces, let alone overheat. If this thing we are involved in and participate in «works» to any extent, it will have to be to the extent that it is not a computer and is not based on our idea of computation. And yet «quantum computation» itself, understood as the most exquisitely minimalist modulation of that which they now call «individual quantum states”, even if there is nothing individual in that domain and precisely because of it, is at the very crux of the matter at hand, doing an extraordinary job to ignoring it. It is enough to carefully follow the thread that leads from local prediction to global description for the geometric phase to go from being a control parameter to begin to impact and resonate in the entire sphere of knowledge.

If there is something «automatic» in the universe, it must be precisely in the sense of spontaneity, of that which moves by itself in the open: the principle of dynamic equilibrium already guarantees it just by dispensing with inertia. And yet this trifle, apparently a mere game of definitions, takes us as far as we can walk, for even if we can dispense with inertia in absolute terms, nothing prevents us from still counting on it in relative terms. The «three and a half principles» can walk the walk.

Scale relativity physics moves between two asymptotic scales invariant under dilation —the Planck length and a maximum cosmological length associated with the Lorentz transformation- so that resolution requires explicit variables. Reworked by Mazilu and Agop, it also becomes a theory of the infrafinite, the finite and the transfinite without leaving the domain of light. It is at least an idea of great interest with strong reminiscences of Leibniz’s monad, though it is clear that it has not been sufficiently developed. And if it is undoubtedly speculative, it is still much less so than the black hole sagas, which enjoy the greatest hold even if they scandalously violate the fundamental convention of the genre, claiming among other impossibilities that physical reality can go beyond mathematical singularities. The Riemann zeta function has often been used to regularize the vacuum energy levels and divergent series at the event horizon of these holes, including the light path integrals, by an asymptotic expansion of the temperature evolution under scaling transformations of the background metric. This may sound like pure scale relativity but it is actually its antithesis from the perspective of global synchronization.

The daunting scale differences between particles and the size of the universe, or even between it and the Planck length, are insignificant compared to the difference between any number that can ever be computed by a machine and the numerical infinity that constitutes the integrity of the Riemann zeta function. One could even say that they are absolutely negligible, but that would be to disdain the evidence accumulated with so much work by mathematicians. In any case, if the minuscule evidence of the computed zeros of the function has any meaning, it would be precisely because the entire function and its underlying «dynamics» involve some kind of scale relativity within it, although to approach it and find its resolution would require taking into account a number of factors that we cannot even list here.

It is a theorem that, if the Riemann hypothesis is true, the zeta function allows to approximate any analytical function of the infinite possible ones with any degree of resolution. In fact, the function would be a concrete representation of any text and accumulation of knowledge that the human being or any intelligent being can achieve, which would also be repeated in it an infinite number of times. Since the non-differentiable continuum also contains «all that», but the zeta function itself is infinitely differentiable and incomparably more structured, it can be assumed that this function and its large family of associated functions would constitute the widest and narrowest bridge between the differentiable and the non-differentiable —although what is differentiable and what is not, also depends crucially on the criterion of the calculus we use. Since the major problem of this function is to relate the local information to the global condition, if there is any way to gradually bound its dynamics, even if it is an indefinite process, it would have to be by reversing the relationship that physics and calculus have always posed between the derivative and the boundary conditions. At least the turn in orientation should not pose so many problems, if one admits that modern calculus as it is used is already the product of a thorough inversion.

The so-called scale relativity is a very general principle using present-known physics as a guide, but it is incomparably simpler to begin with the contrast between relativity and the Noskov equations when we take them to the supposed limits, starting always from the Lagrangian in Kepler’s problem, the real keystone of modern physics; it is not very advisable to try to swallow the Whole when we do not reasonably understand any of the infinitely more modest totalities appearing everywhere.

There is already something singular enough in the evolution of any entity in relation to the background from which it emerges, in which it is maintained, and to which it returns: if we are able to see this, even with just the imagination, we will have already achieved a great deal. In fact, it is for not understanding the unobstructedness of this ephemeral and self-sustained «singularity» that we plunge headlong in search of crossing singularities that are non-holes by definition. One speaks of «the wave function of the universe” without blinking an aye, and the same holographic principle makes one think of a wave front of inconceivable complexity; but the simplest wave in three dimensions is already a challenge to the imagination.

Think again of the awesome ingenuousness of Venis’ wave-vortex, who has not made the slightest use of physics or mathematics to unveil the most “symplectic» of morphologies. Although his transformation sequence also seems to show a wave front as a characteristic surface phenomenon, its tremulous boundaries fluctuate between full and empty with more delicacy than the best brushstroke. But how is it that a wave that is supposed to exist in an infinite number of dimensions, including fractional ones, still exhibits such a recognizable profile in six dimensions, and in three, and even in two? The only conceivable answer is that the totality that escapes our perception is still reflected in a point of dynamic equilibrium, which is a line or a plane of equilibrium, and so on.

Venis’s sequence shows without profanation in what sense Nature is always equal to itself: fleeting like a whirlwind and immutable like the sphinx. Between one aspect and the other there are infinite layers. What is that amorphous bulb that insinuates itself everywhere, shows in the contour and hides in its center? These surfaces and profiles are beyond measurement —they are purely projective in nature- and yet they are reflected in all kinds of phenomena that can obviously be measured. Following the curves of the sequence one can guess when this imaginary flow speeds up or slows down, gets slower, and slower still, until the wave front roars and breaks and bursts in the midst of silence.

Light between space and matter; between the full and the empty, the light. We have already indicated why the categorical separation between the vibration of sound and light is due first of all to a misunderstanding —although it is unquestionable that light is vibration emancipated from matter and sound is not. But, can light be heard? It is a good question, although from the above it would be easy to judge rather the contrary: light would not only be a principle of expression or actualization, now it looks like a global organ of perception and an all-encompassing tympanum in which matter itself resonates. But, as vibration, who or what could listen to it? Nothing that can be exposed, nothing that is manifest; which may concern the bodies, the void, both, or neither. There is a point from which all that recedes to mean nothing, and yet there is still a long slipknot not only related to the scale of energy but also to the ambiguity inherent in its threefold manifestation.

This «relative detachment from unity» is an omnipresent question and by no means confined to physics or mathematics, which can nevertheless reflect it. For the formal logic that seeks self-consistency, unity only reaffirms us in the tautological; but from the point of view of applied mathematics, to which all calculus belongs, unity is never a given formal question, and in pursuit of it much old knowledge can be transformed into something new. From the very moment it is assumed that calculus or analysis necessarily proceeds from the whole to the part and from top to bottom instead of the reverse, the general disposition of the sciences will have changed.


Heir to a large extent to de Broglie, David Bohm made an interpretation of physics openly challenging the prevailing reductionism and spoke, between marked extremes of eloquence and vagueness, of “wholeness and the implicate order». Bohm cannot be reproached for being theoretically conservative, for he was already heterodox enough among his contemporaries; and yet such a discourse as his finds much less echo today even if in all this past time we all have gained an invaluable perspective.

Bohm was not really aware enough of the universality of the geometric phase and its relevance to classical mechanics —though we have already seen that present-day theorists are no better off. Nor did he conceive of the effect of potentials as a vibration or resonance that runs thoroughly traversing matter, but, following in this the spirit of the age, as a non-dynamical information field. The latter aspect is important for interpretation; the former has innumerable consequences yet to be explored. On the other hand, Bohm saw the importance of the measurement problem but he could not question its general relations with calculus or analysis as can be done today with utmost clarity. Finally, his work lacks a real discussion of the fundamental principles of physics as such.

In proportional terms, the universe is almost entirely empty; not only in the intergalactic immensities but even in our own body. Of course, this physical vacuum is still far from being mere empty space, revealing itself to the closest inspection as a tenuous sea of radiations. For general relativity, space tells matter where to go and matter tells space how to curve; but the space-time continuum is not empty, unconditioned space, but a totally different dynamic animal subject to curvature within a specific metric discipline.

We have already mentioned that Poincaré, the first clear proponent of the principle of relativity, preferred to think of the bending of light rather than of space, and at least it is now accepted that field theories, including gravity, can be formulated in flat space. After all, light was always seen to bend in water, but space never, and there was no need to artificially condition what had always been perceived as unconditioned.

The density of the physical vacuum with radiation is almost zero with respect to matter, but the density of really flat space, without the slightest trace of energy in it, is strictly zero with respect to the radiating physical vacuum. Waves attenuate indefinitely, and on the other hand, even a particle of matter such as an electron tends to dilate without limit when there are no other bodies in its proximity, as has been proven time and again. We would thus have three asymptotic echelons, from matter to radiation and from radiation to space without curvature and even the least metric restriction.

For the Vedanta, pure space has no qualities or even dimensions; in modern geometry the closest we have would be the primary projective space. Pure space is the perfect symbol of spirit in many traditions, but it is clear that space is not a sentient being that perceives itself. So the sense of aseity that is inherent to me must come from something other than that space prior to concepts of which I nevertheless still have some notion.

Today it is customary to call field excitations «particles» —in accelerators, for example- but the relaxation of an electron that can extend meters or kilometers, or to infinity if the environment does not prevent it, is not usually called a «wave». There is thus a very human prejudice in favor of high energies that leads us to overlook a powerful fundamental tendency. Moreover, this tendency of matter to relax as a function of its surroundings beats in unison with the most elementary interpretation of entropy, since molecules also tend to spread out as much as they can, going from the hottest to the coldest regions and regularly occupying the available empty space.

In this sense, so different from that of relativity, space is telling matter where it has to go, but matter tells space nothing, because space, both in the limit and in absolute terms, has no curvature. Just as I know my body but my body does not know me, there is no possible reciprocity here. There is however an unquestionable reciprocity between matter and radiation, which is something completely different. In reality, relativistic space-time encompasses the dynamical relations of the physical vacuum based on interactions, and its potential theory remains on undefined ground: that is why there are alternatives in flat space such as the modern gauge theory of gravity or those derived from Weber line of thought.

In other words, relativity dispenses with absolute space but maintains the absolute time of the global synchronizer —when with the theory of the “retarded potential» the opposite can be done: maintaining absolute space without further qualifications and allowing everything to be governed by its own internal time, which implies the environmental variation of the forces and constants that now are postulated as fixed.

Absolute space can still «speak to matter» with two different languages: that of the asymptotic tendency of matter and radiation to expand as a function of the environment, which implies a mediated and dynamic mode that also includes thermodynamics, and the immediate language of the potential, which is simply derived from position. And this last language supposes the intimate vibration of matter.

Let us recall that almost all the production of radiation is dispatched as «spontaneous emission», which is the most opposite that can be proposed to a mechanical explanation. And yet this spontaneous emission would be nothing but the manifest form of the internal vibration of the particle’s body. In this sense, one could also speak of «spontaneous absorption», even if we say simply absorption.

We have then a circularity between the immediate or instantaneous, completely independent of motion, manifested by position or potential, and the dynamic or mediated aspects of the interaction between matter and radiation. This resolves the aporia presented by both Vedanta and samkhya: how the absolute and the conditioned can coexist without the slightest problem. The dualism of the modern age can also be seen in a different light.

Absolute space without qualities or dimensions corresponds to Prajña, the state of deep sleep in which all determination disappears. However, one knows that the aseity is even closer than that, that there is something even more general than the most undifferentiated, which logically implies that it is also not separated from the rest, from that which manifests itself. That is why we have speak of a primitive homogeneous medium as something somewhat different from undivided space. In the spontaneous response of matter and radiation to something that by definition does not move, we have a very simple way of grasping in what sense the cosmos is not creation but manifestation, the manifestation being nevertheless something truly spontaneous and creative.

If physics is an inexhaustible matter, it is because it can never be reduced exclusively to motion or extension. To want to explain everything by motion or extension is, besides being the ultimate expression of nihilism, completely trivial. The Cartesian res extensa was conceived expressly as a theater for motion, at the same time that for Galileo rest ceased to have proper entity. We are speaking, precisely, of the two fathers of the principle of inertia. And yet, when we return to the original perception of space as the immobile by definition, independent of any dynamics, the internal and the external once again interpenetrate.

It is only a matter of perceiving rest in motion, in order to see the activity in the rest. To achieve such feats, which many will believe to be miracles, it is enough to duly probe the theory of the potential, which will take us into as deep waters as we wish. Most may protest and argue that a potential is never independent of dynamics, but physicists called the geometric phase by such a name precisely to separate it from the realm of forces or interactions, and there was a closed consensus on this. As we have already pointed out, even if we wanted to insert it into the principles of dynamics, it would have to be through the internal articulation of «the three states of rest» with the principle of dynamic equilibrium at its center, which, depending on the case, can be the same or something very different from the multiple and complicated criterion of the relativistic principle of equivalence.

Indeed, from the very Kepler problem on down, there are all sorts of important cases in macroscopic physics that are not well covered by current field theories; but even ignoring that now, we have the fact that quantum entanglement exhibits instantaneous correlations, and the geometric phase, which appears at all scales, is only a lower resolution expression of the same order of correlations. Gravity itself, if it is not really a force, could perfectly well be found in this case. Then, even general relativity, which also does not consider gravity as a force, would be describing only interactions from the point of view of the holographic principle —which we already know where it comes from. It seems that physicists do not know what to do with this reasoning either.

Relativity would be a pure theory of surfaces, but so is the picture of interactions in quantum mechanics. Instantaneous correlation has to be the true and genuine global synchronizer —a synchronizer independent of interaction, but echoed by any interaction. This reveals the extent to which comparisons of the universe to a large computer are irrelevant. The true global synchronizer is as ungraspable as predynamic space; its local translation in wave mechanics is the de Broglie hypothesis of the particle «internal clock”, an oscillation no doubt, which does admit of indirect experimental confirmation but attracts little interest. It may well be said that the only global synchronizer is the one that does not need to synchronize or take measurements of anything, but in which everything resonates.

I have shown, in the most general way, in what sense «that which is above» can guide «that which is below» without the need for contact; but that which we imagine above is also always below and at the bottom of what appears. I have also indicated what kind of changes make it easier for the scientific mentality to feel comfortable with this idea, which has its roots in something prior to thought. The problem of finality, which physics always eluded as much as it could, and which it misinterpreted by projecting its own shadows from above, has a very simple solution that nevertheless has two sides.


According to what has been said, time is a more superficial phenomenon than the space without qualities, and if it seems subjective to us, and even if it could seem to Kant to be pure form, it is precisely because it flows continuously through that domain independent of motion which is at the basis of subjectivity. It is only that almost all of our attention is pending on the innumerable variations of the contents of experience in time, while the other, though invariable, determines the immediate vibration of novelty.

For the persistent dream that is wakefulness, subjectivity must reside in the variation and novelty of its elements; but deep sleep not only does not dream, it does not even sleep. It is already fully present in the midst of this busy wakefulness —except, of course, for my lack of attention. Attention is then enough for that third stage which is always in the background to become the fourth which is nowhere in the sequence.

In physics the demarcation of these superimposed planes of experience and time would have a similar pattern; although the incompatibility between the two great dominant theories, in addition to their own internal contradictions and paradoxes, makes everything much more difficult to arbitrate. The universality of the geometric phase makes it a natural bridge between the micro and macroscopic, but the global synchronizer, which with the principle of inertia shapes the whole account of physics, prevents the local translation of what the conversions of scale imply.

To get the slightest idea about this, it is necessary to leave the prevailing theories. We have already pointed out several by way of contrast and now we will bring up another one, which is not even based on physical or mathematical, but purely morphological considerations. The wave-vortex sequence in Venis’ hypercontinuum is based on projective arguments but cannot avoid being reflected in planes as varied as living beings or the great cosmic formations.

Following the compelling logic of his vortices, Venis concludes that the cosmological redshift does not indicate a general expansion of the universe but only a local contraction by a gradual shift in the dimension of our local group: if we were to shrink we would not notice it in ourselves, but would only see the size of the environment increase. This displacement would have a speed that can be estimated, but it would involve an internal time that could only be verified by leaving the group in question, be it the solar system or the galaxy, and passing to another time branch to return again to the place of origin.

This is simply speculation —as is most of cosmology- but it serves perfectly well to illustrate the gulf between the exigences of the global synchronizer and a physics that takes the circumstances of the environment seriously. Surely the stellar immensities do not exist for everything to be as it has been measured in the corner of a laboratory; but, even so, for a science as mathematical as physics, overcoming its own yardsticks represents an extraordinary challenge.

But we don’t need to leave the galaxy to see how the environment modifies the ruler instead of the opposite: the «retarded potentials» already tell us this at any scale. The spiral traced by the planets of the solar system only deviates from an exact profile by the same order of differences as the value of the Lagrangian of each orbit, which is already eloquent, and this spiral approximates the plane of a vortex. And so we can continue down to the atoms and the orbits of the electrons, vortex-waves themselves. At all scales we find entanglement of the potentials, be they classical or quantum.

The anthropocentrism or immodesty involved in speaking of universal constants might well be called comical were it not for the difficulty of getting out of both the logic of global synchronization and the legitimization of appearance in the name of the irreducibility of the observer’s position; and in this case both are mutually justified. Even those who are most aware of the unfoundedness of this claim have the greatest difficulty in imagining what the implications would be if there were no synchronization in such an obviously anthropocentric sense and what might lie on the other side.

Today the famous redshift of galaxy light is directly associated with the microwave background radiation to reinforce the big bang narrative. What the story does not usually tell is that Gamow’s 1952 prediction was neither the first nor the most accurate, and that a good number of physicists had already estimated its temperature with greater accuracy, many years in advance and with much less data, from Guillaume in 1896, to Regener, to Nernst, to Herzberg or Max Born, among others. All of them had in mind a universe in dynamic equilibrium.

Venis also presupposes a dynamic balance between local expansion and contraction, but invokes the third principle of action-reaction in justifying the balance. However, the third principle alone does not amend the principle of inertia, which is the background that dominates this whole narrative. Indeed, if what is observed everywhere is motion, and with Newton’s three laws nothing moves without having been moved by something else, we are forced to believe that everything proceeds from an original impulse, no matter that it is the most grandiose violation of the principle of conservation of energy.

To become aware of the extent to which principles determine the plane of contact between physics and metaphysics, it suffices to recall the contrast that Roger Boscovich, the great precursor of field theories, proposed for the postulate of absolute magnitudes: “A motion which is common to us and the world cannot be recognized by us —not even if the world as a whole were increased or decreased in size by an arbitrary factor”. Even if the whole world were to shrink or expand in a matter of days, Boscovich continues, with an identical variation in forces, there would be no change in our impressions or in our mind’s perception. Joint change has no rank of experience, only differences count.

Of course, this would have nothing to do with the big bang if in it there is no joint evolution of the constants. But if the constants modified their mutual relationship in the past, why don’t they do so right now? In fact, we have been assured for many years now that the «expansion of the universe» is accelerating.

Venis obviously distinguishes velocity of motion and velocity of flow in time, but he does not know how the correlation could be established, nor what other factors would be involved. Even after more than a century of relativity, that there can be different temporal velocities determined from the outside —which has nothing to do with «time travel»- will seem impossible to many. Now, what has been said here is that the background of all experience, including the experience of time, has nothing to do with motion and its infinite possible relations. A velocity in time does, whether it is a measurable physical time or the subjective time that in dreams can contract or dilate without the slightest apparent rate.

In short, the background of experience is completely formless and common to all beings, be they gods, humans, galaxies, planets or atoms; there is nothing individual about it, but it is what makes the subject subject. That is the great difference between non-duality and the variants of modern idealism. On the other hand, time itself is part of the individuation of beings, indeed, it is incorporated in their form and that of their body and its «resident matter”, as Venis calls it.

It can be seen then that Venis, almost unwittingly, is proposing a far-reaching strategy for seeing time beyond the fiction of the global synchronizer and its universal flattening. And it does not matter that the connection of its morphology to physics and mathematics is entirely unexplored, since its foundation is in projective space itself prior to metrics and determinations. Even as pure phenomenology, it provides a connection between understanding, imagination and vision that the present sciences would like for themselves.

Moreover, Venis sequence of transformations poses fine and profound problems for calculus and mathematical analysis, since it exhibits a smooth and differentiable evolution in the hypercontinuum, which can take any real number among the integers of the dimensions. Today they use to work with «fractional» dimensions in countless areas of applied analysis, and they are commonly known as fractals; however these fractals are not differentiable. On the other hand, only very recently a bridge is beginning to emerge between the fractal geometry so common in Nature and the so-called «fractional calculus» which is not restricted to operators with integers. Fractals belong to space, but fractional operators affect the time domain.

In short, fractional calculus covers intermediate time domains in processes of all kinds, including wave processes. Fractals are nonlinear, while fractional dynamics governs linear processes, —and yet it poses a disturbing anomaly for calculus: it exhibits nonlocal dependence on history and long-range spatial interactions. And so, fractional calculus itself raises a huge problem of interpretation that neither physicists nor mathematicians have been able to solve. Igor Podlubny proposed to distinguish between an inhomogeneous cosmic time and a homogeneous individual time. Podlubny admits that the geometrization of time and its homogenization are due first of all to calculus itself, and warns that intervals of space can be compared simultaneously, but not those of time, since we can only measure them in sequence. What may be surprising is that this author attributes non-homogeneity to cosmic time, rather than to individual time, since in reality mechanics and calculus evolved in unison under the principle of global synchronization. In his reading, individual time would be an idealization of the time created by mechanics, which is to turn everything upside down: in any case it would be the time of mechanics that is an idealization.

It is clear that mathematicians also fail to rise above these aporias since calculus is not just an accomplice of synchronization but its principal agent. Fractals are a geometrical expression of equally ubiquitous power laws of scale, but the operators of fractional dynamics would govern their evolution and what is called their temporal «memory». Now, Venis vortices evolve smoothly and linearly through fractional dimensions while describing a temporal evolution that drives logic, intuition and imagination, showing a common thread amidst an impenetrable jungle of figures. The enveloping boundary layers of their profiles also invite a competition between the various modes of calculus to see which one best fits their contours.

Given that Venis’ concept of vortex encompasses not only whirlwinds, but things as amorphous as a spherical bulb or drop, the formless is emerging as logic and process within something much more formless still. This is why it is possible to think of a universal morphological scope, even assuming that the forms do not cease to be fleeting ghosts, apparitions in the domain of the impermanent. And if Venis has been undoubtedly inspired by Far Eastern philosophy, his phenomenology is, as he himself calls it, an “infinity theory», heir to that unmistakable Western impulse to see beyond any limit. Here, however, there is a tendency to perceive the infinite in the very limits of the concrete forms.

Most real processes, for example the stresses and deformations of a material or its fracture, are highly non-polar, as they cannot be characterized like fields in terms of vectors, displacements and forces, nor do they show any detectable axis in their evolution; fractals, on the other hand, are suitable for describing them. In contrast, fractional dynamics may retain the mathematical polarity of fields, so that the connection of the two is a way, still very abstract today, of exploring the transition from chaos to order. Vortices, instead, are present in all phases of matter and light, and define the most visible transition between chaos and order, turbulence and form.

The zeta function, with its pole at unity and its abysmal critical line, is associated with the so-called «quantum chaos» which would be softer, but as in any other circumstance, it has never been possible to specify the boundary of its transition. It is extraordinary that such a universal «mathematical object» as this function maintains such isolation from any visible geometry, not to mention logic; the multiple analogies with advanced physics or the innumerable analytical graphs of the function cannot make us forget its rigorous isolation in arithmetic, the pure domain of time. This «pure time» is nothing but time absorbed and without any external relation, the most opposite we can conceive of to the time we experience. Since Venis wave-vortices are also of extraordinary universality, and have a deep and countless-layered link to time and entropy, it is impossible for them not to have a connection to this function, and a close connection at that, too, even if no one has yet explored the subject.

The intellect that perceives objects never perceives itself; that is why it is not the self, precisely, otherwise both would inevitably tend to merge. This irreducible fact is also reproduced, of course, in the simple act of counting, at the base of the whole arithmetic. True geometry is always synthetic and elementary; arithmetic, when it is not merely trivial, always has analytical depth. In this sense one may well speak of an explicate and an implicate order, in quite a different sense than that given by Bohm to these words; Poincaré preferred simply to say that geometry is a posteriori and arithmetic a priori. The geometricians show a good hair, and purebred analysts were bald, such was the infallible rule in the old days. But those who turn to number theory have their hair again, and today even algebraists and logicians can wear a long mane; the interconnection between all branches of mathematics has reached such a point that one no longer knows what will happen with the scalp of its practitioners.

Venis vortices are inevitably nested within one another at very different scales, making it no simple task to identify the «substance» of the temporal flow. In this sense, they would be as evanescent as the forms themselves, even if they are not confused with them in any trivial sense. If we think a little about this circumstance, we realize that a time is emerging from here that is both material and formal, in a sense different from those that physics and philosophy have so far handled. Even if we understand that the subjectivity of its flow does not ultimately depend on motion at all.

It is quite possible that the Riemann hypothesis will never be proved, as it is quite possible that many other incomparably simpler conjectures of arithmetic, such as the Collatz conjecture, will never be proved. Nevertheless, the zeta function has already intrigued mathematicians enough, and will intrigue them much more, and in that sense it will have fulfilled its mission; but for the moment it has only summoned algebra and analysis, which have hardly offered any real contrast to ti. Instead of continuing to grope about in the darkness of the mouth of the cave, they could confidently advance backwards beholding the light at the entrance, geometry at any rate. Even elementary geometry is infinite, when it knows how to find its problems. Modern calculus, on the contrary, is a grinding machine that gives predictions, but hardly knows anything about the physical geometry of the problems.

Something much more probable but much more disturbing than the demonstration of this hypothesis can then be foreseen: its contact with geometry and its entry into the human imagination, and not only into the mathematical imagination. Some analysts have already detected spiral waves in the domain of the function and exercise themselves trying to imagine their morphogenesis, but such abstract patterns still require many degrees of decompression to make contact with the genuinely geometric. This requires a radical change in the orientation of calculus, along some of the lines mentioned above.

It is just reasonable to think that the Riemann hypothesis is only the consequence of the asymptotic behavior of the prime numbers tending to infinity, yet this has very little to do with its demonstration; however this is not the main interest of this function. There is, in any case, an absolutely elementary geometry of the primes within the line, described by the periodic curves from the origin intersecting each number and its multiples, and the primes being cut only by their own curve and that of unity. Even from this very simple representation surprising patterns gradually emerge, and the complexity in the superposition of the curves increases as we plunge into the depths of the number line. The most basic patterns can only be seen by extracting them from the line and displaying them in at least two or three dimensions. It does not take long for the first spiral motifs to emerge. It would be necessary to continue in 4, 5, or 6 as in Venis sequence, and in the intermediate domains, modulating the possible dynamic components. After all, there are also infinite dimensions that fold and compress in the number line and the zeta function.

The time-form that accompanies time-matter is not only in the vortices that describe particles or galaxies, but a little everywhere and nowhere. When a wave is broken a spiral wave appears that persists and excludes all concentric rings; this was precisely Arthur Winfree’s simple observation as he began to model the geometry and resonances of biological time. Spiral waves can be found even in the motion of the heart.

Venis transformation sequence is a game of balance between contraction and expansion; these are the «three feet» of his vision. The fourth foot must necessarily be beyond the forms but in the middle of them. In any case, equilibrium seen only in terms of action-reaction does not take us out of the limitations of the principle of inertia: it gives us the explained order of a closed system, not the order that an open system implies. For, indeed, dynamic equilibrium is implicate order by its very definition, «holomovement» in Bohm’s parlance, even if its implications can never be extracted. The geometric phase, on the other hand, can be described as a torsion or shift in dimension, which thus acquires not only a geometrical but also a morphological connotation; although it is understood that it is the dynamic component that is deformed, not space or potential.

Astrophysicist Eric Chaisson has observed that the density of the energy rate is a much more decisive and unambiguous for a complexity metric and its evolution at all scales than the various uses of the concept of entropy, and his arguments are very simple and convincing. Contrary to what one might think, this rate, measured in ergs per gram per second, averages 0.5 in the Milky Way, 2 in the Sun, 900 in plants, 40,000 in animals and 500,000 in human society; the relative value in an atom or molecule is missing here. This density of energy flux, as we say, indicates not only the complexity, but also the individual evolution of an entity, its transformation throughout the process of birth, maturity, aging and death. It must therefore be linked to what is commonly understood as «subjective flow of time» although it is still just a transfer, an exchange with the environment within something alien to motion. No doubt this flow rate can be applied to Venis’ vortices if physical values were assigned to them, since they too reflect the evolution of individual entities and their appearance. As in the whirlwind of life, the increasing restriction in entities, the essence of aging, is itself a matter of great subtlety and multiple concentric layers that only gradually unveil themselves. In-depth study of the subject would make it possible to see to what extent we are talking about an incarnated time, a time in matter and form. Returning to the physical world, another question would be how this rate relates to the principle of maximum entropy or to the thermomechanical equilibrium that can account for the various Lagrangians of atoms, planetary orbits and galaxies.

Increasing restriction is the touchstone of the individual evolution of an entity and its aging, which so little attention has yet deserved despite the fact that its profiles are always before our eyes. Even in the almost insurmountable abstraction of number theory can be found multiple domains and orders of increasing restriction, counterpoint of the various measures of numerical entropy, much more irreducible and singular than what in calculus is meant by «approximation» and «boundaries»; for calculus only uses arithmetic as an instrument, while number theory seeks in it its own physiognomy. Entropy is an extensive property proportional to the logarithm of states; the quantity of prime numbers is inverse to the logarithm of the integers, and the same non-trivial zeros of the zeta function on the critical line with exact real value of 1/2, would reflect the principal domain of an increasing constraint that wraps as a product the total sum of the prime numbers up to infinity. Fechner’s law tells us that the intensity of sensation is proportional to the logarithm of the stimulus, and the time of life and memory unwinds with logarithmic spiral rhythm. As Tolstoy wrote, there is only one step from the five-year-old child to the one I am now, but from the five-year-old child to the newborn there is an immense distance, an abyss from the newborn to the embryo, and the inconceivable between the embryo and the non-being.

It may seem counterintuitive that the energy flux density is much higher in a blade of grass than in the incandescent atmosphere of the Sun; but this is closely related to the aforementioned fact that entropy tends to the maximum and order produces more entropy. Both facts give away that there are very fundamental things that we are looking at from the wrong side, not to mention the more than possible contribution of entropy to the very order of dynamics. Also in all this, one would have to stand with one’s back to the mine to appreciate the light that floods the cave. In any case, these considerations allow us to see that even aspects of time that we believe to be highly subjective can have physical and mathematical expressions with their own entity; and it goes without saying that in Venis’ morphology there are many more motifs than spirals.

Considering special relativity, it has been said since Pearson that an observer traveling at the speed of light would not perceive any movement and would live in an «eternal present»; but it is clear that light is transmitted and moves, and even pulsates with more precision than the best clock, so this is simply false. We also know from Huygens’ principle that its propagation implies a continuous deformation at each point, the very opposite of timelessness expressed in the most graphic way. Precisely for relativity, there is no other way of conceiving time than motion, so this claim contradicts its own assumptions, although it expresses superbly how the principle of global synchronization would like to be above all contingency and even above the very time it imposes over everything else. Presumably, physicists have given up on such illustrations, which only expose the theory. If anything physical is to be outside of motion and time, it certainly cannot be this.

The multiple inertial frames of reference of general relativity only succeed in creating a permanent confusion, even about the very meaning of the word inertia, which already speaks for itself. To see and consider, with eyes wide open, that there is no inertia in this world: there can be no better suspension of the mechanically learned, no deeper meditation. The most intense, too, that the permanent agitation of thought does not bear for a moment. Is that hard to assume that there is no inertia in this world, to understand that the dancing principle of dynamic equilibrium is telling us that everything is interpreting its position at every instant spontaneously and with all its being, beyond freedom and necessity?

Venis readily admits that we still know almost nothing about how, if there is only a hypercontinuous infinite-dimensional space, our perception is so severely limited to three-dimensional space. Also the Hilbert space of quantum mechanics is initially infinite-dimensional, but are we talking in this case about abstractions? Venis’ inferences can be very risky, and yet they have the peculiarity, today more than ever exceptional, of being guided almost exclusively by the specific spirit of form. The infinite dimensions of an abstract space leave us indifferent, but what is implied here is that the dimensions of concrete forms in physical space are only a minimal section of entirely formless domains, much closer to us than we think; of course, physical reality has never allowed itself to be reduced to the visible either. In any case, any supraformal or infraformal domain must be connected to our familiar world of form by the same thread and principle.

Is there a physical process in the reduction of dimensions, or will it never cease to be for us something purely imaginary? Both things are not that incompatible, but they depend on the starting point; physicists need measurements, imagination is an individual exercise that does not require instruments. Finally, we can always count on that which does not even need imagination.

Our hyperkinetic culture cannot conceive anything outside of motion; thus, the geometric phase is given a highly abstract and derivative status when it reflects in the most immediate way the place from which we look, which is certainly not an inertial frame. However, it is the description of the dynamics that has become rarefied to the point of representing practically nothing. Since for the physicist the immediate is not worth anything either, there is no choice but to bring in another type of argument.

The dynamics of a system is supposed to control the geometric phase, which plays an entirely passive role; and this is the part now routinely assigned to it in the laboratory. The unavoidable question is, how can something that merely follows motion be independent of it? But, on the other hand, an instantaneous correlation cannot be associated with motion at all. How to decide the question? There are multiple ways of approaching this new abyss; all that is required is to change the idea of what is to be measured.

Nor there is any need to smash particles in gigantic accelerators, but only to get in tune with the exquisite sensitivity of potentials in field theories and beyond. In principle, the difference between an ordinary potential and an «anomalous» potential such as that of the geometric phase seems very clear; in practice, as always when we talk about energy, the question is much more delicate and subtle. Heisenberg’s misnamed «uncertainty principle» has been disproved countless times and its various relations are subject to successive corrections depending on the experimental framework. Not only high-precision experiments, even phase-sensitive frequency meters of color TVs show an accuracy many times higher. But, following Binder’s suggestion, and beyond this ailing principle so in need of the simplest dimensional analysis, all kinds of phase-locked loop devices can be designed to create a feedback or back-coupling between the dynamic and geometric phase, and study where is the equilibrium and what is the boundary condition. Binder interprets that it is the geometric phase that controls the strength and sign of the coupling constants of the fundamental forces, and even conceives of scenarios to shift the Planck scale to the nuclear range. According to Binder, what would vary and «curve» would be the dynamical space-time; but the instantaneous velocity of the geometric phase cannot curve at all. Velocity is the fundamental quantity of physics; physical space and time are notions derived from its measurement.

We know very well how little attention is paid to proposals such as Binder’s, and we know also why. But, quite simply, that which is instantaneous cannot be secondary with respect to that which takes time to operate; that which is pure act cannot be passive with respect to that which takes time to actualize. This simple reflection makes one think that physics has been inverting all this time the notions of act and potency, but it will take a long time to recalibrate their relation, and perhaps even longer to assimilate their consequences. All matter is interpreting, spontaneously recreating that pure act. Light is vibration emancipated from matter, but to transcend the global synchronizer amounts to step out of any temporal order linked to motion, be it that of the solar system, our galaxy or the entire cosmos, observed or unobserved. Physicists, scientists in general, can still choose between control and wisdom; between closing themselves over man and Nature, or opening for them the vastest of horizons.

Everything that moves is but a reflection of that which does not move, a ripple upon its waters. It is not the potentials that are retarded, it is the non-fundamental forces that are varying. We do not know whether gravity has a speed limit or not, nor do we know how other completely different forces that may exist at other scales, in galaxies or beyond, would behave; but we do know that what is instantaneous is not retarded and is the reference for anything showing speed limitations. As some like to say, «you can’t beat that”, though you can ignore it. And, after letting the question sink in, one might add: «you are that».


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