If mathematics, physics, complexity theories and computer science continued to expand and cross-fertilize at the same rate as today for a thousand years, they still would not find the key to morphology; not even in two thousand years either. Perhaps this may give some idea of its value, although we all know that nothing can be found without actively searching for it. That is the point: the mentioned sciences already have their own momentum and inertia that nothing can change, only a creation of a new science from scratch could overcome without obstruction the deficiencies of its predecessors.
Introduction
The morphology of organisms has its origin in Goethe’s 1790 study on the metamorphosis of plants, in which he describes in detail the continuity between different forms that we have called homology since Owen. The term «morphology» was coined by Goethe himself, but the notion of homology, sporadically noticed, would go back at least as far as Aristotle. The concept of homology was later subsumed within the theory of evolution, but as Ronald Brady points out, in it it becomes a static similarity of explanatory order, not a dynamic transformation in time self-contained in its description. Morphology is not comparative anatomy.
The situation has not changed to this day. Basically, we have «hard» predictive sciences, such as physics, and explanatory sciences such as cosmology or the theory of evolution. And mediating between them there are other disciplines based on probability, such as thermodynamics or statistical mechanics, used more than anything else as mathematical tools that in no way serve to describe phenomena on the same level they appear. That is to say, between the predictive sciences and the explanatory sciences there is only a connection at an abstract level, but not at the proper descriptive level, which is what we need to make the world intelligible, to give it more and more accessible layers of intelligibility.
Goethe, who was at the antipodes of mathematical talent, did not fail to express some hope that someday his ideas would find their Lagrange; that is, that mathematics would finally be able to stick to the phenomena themselves, besides assisting in the foundation of physical hypotheses and theories motivated by prediction as it has always done. But there has been plenty of great mathematicians since then, and yet the distance between the phenomenal world and the queen of sciences has not ceased to grow. And the alibi has been precisely physics, with which it is assumed that the real world is sufficiently accounted for.
In biology there have been some notable exceptions to this general mismatch, such as the work of D’Arcy Thompson, Waddington or René Thom on the morphogenesis of organisms, who prolonged a certain Aristotelian line of thought using increasingly modern mathematical tools, but their efforts have had no continuity and today are seen as little more than occasional deviations from the mainstream of this science. Even before them, in 1892, Otto Snell had formulated the classical allometric equation, but allometry, which studies the relationships between size, shape, anatomy and physiology, although neatly developed, is still just another analytical tool within the much broader trend of applying statistics and data analysis to biology. Current ontogeny and developmental biology connect directly to the basic questions of morphogenesis but are abducted by the causal and explanatory models of embryology, molecular biology or cell differentiation, without even considering the possibility of more general principles that might mediate between the organic and the physical.
And, of course, there is today a thriving area known as mathematical morphology, but since it is basically concerned with digital image processing, not only is it not a general study of shapes, but it is not even a branch of mathematics, being just another application of computation. Waiting for who knows what that might give it its long-awaited birth certificate, morphology as a science on its own remains in the limbo of its pure possibility. More than two and a half thousand years after the birth of geometry, morphology remains in no man’s land.
What does morphology need to be born? Obviously, first of all, the desire for it to exist. Then, to have a clear awareness of the kind of gap that exists among the present sciences and that none of them cares to fill, for this gap is bound to be its natural place. Beyond this, there is no need to speculate too much because we already have an accomplished anticipation of what morphology can be in the exploratory work of Peter Alexander Venis.
1. Vortex morphology and the Venis transformation sequence
Image by Peter Alexander Venis
The simplest change of shape or transformation that we can observe in a continuous medium is a wave. What we see above is a series of sections in fractional dimensions of the evolution of a wave-vortex according to Venis’ reconstruction. Venis describes this evolution in six spatial dimensions, but it should be kept in mind that he does not speak simply of whole dimensions such as length, width or depth but of the continuum of all the «fractional» dimensions, in fact dimensions with any real number. It should also be noted that within this sequence the term «vortex» includes both rotational and non-rotational flow movements.
We cannot even intuit well the evolution of a spherical wave in 3 dimensions, so why try it in 6? Because there is something in six dimensions, or at least in a certain way of understanding them, that is better intuited than in three. But for a physicist or a mathematician the most unusual thing about the new meaning of the word «dimension» introduced here is not the possibility of fractional orders, something that is already often used today, but the fact that they are also linked to material density: dimension 0 and dimension 6 of the sequence would correspond to maximum and minimum extremes of density as well as to changes of size or scale.
Even physicists continue to separate space and matter in spite of the sophisticated field theories they now hold. For example, general relativity assumes, in the terms of differential geometry, that «space tells matter how to move, and matter tells space how to curve»; and this theory is after all very similar to fluid dynamics, with an extra dimension for space-time. It is an attempt to get out of a dualism that has nothing to do with Aristotelian hylemorphism, but with the inheritance of the atomistic vision, which was also Newton’s, of corpuscles of matter moving in the void. Quantum mechanics, too, which has so little to do with ancient atomism, has adopted this view in spite of itself and in spite of admitting that the particle is meaningless without its field. Modern physicalism tries to get rid of the dualism of Plato and Descartes by importing it into its conception of physical reality; which is even more paradoxical if one considers that both philosophers, like Aristotle, were monists with respect to physical reality and explicitly denied the separation between space and matter.
In this unfolding in six dimensions of the Venis sequence, the three dimensions that we consider our ordinary space would be only the midpoint of equilibrium between the extremes of the sequence, which in terms of vorticity, would correspond to the most formless point, that in which any polarity or axis of rotation is neutralized. From this morphological perspective, our three familiar dimensions are not just one possible dimension between the point and infinity, but the center between the extremes of our perception. It is very probable that the number of dimensions of the “One Field» is infinite, but the fact that its projection seems to be reduced to 6 would be due to our own perception, which, as we know, always involves an intellectual component in its organization.
One can look for many concomitant reasons for this hexadimensional arrangement, but what is really important is the relocation of spatial experience with respect to oneself. In our geometrical conception of space, which is something so different from our experience, one represents oneself as a point within coordinates ruling for an infinity of different and interchangeable points. Even if one stands at the origin of those coordinates, it is only as a pure abstraction, not as a center in the middle of pairs of opposites, between full and empty, contraction and expansion, dense and subtle, stress and strain, birth and death; and it is these attributes what colors all the time our experience of reality.
We will never appreciate sufficiently the scope of this reinterpretation of space, which once again places man at the center in the midst of conditions by virtue of his own nature and of the larger Nature that envelops him. After all, the famous Copernican turn into which we are still thrown off is just an intelectual outburst that cannot last long, while the alignment of forms with perception and of this one with the formless that gives it its being is inscribed within a timeless return.
It is thus understandable why Goethe did not find «his Lagrange» before, and why he will never find him if we insist on seeing things from the angle of established specialties. Venis’ is the inference of a naturalist, not that of a physicist nor that of a mathematician in possession of a great arsenal of analytical methods; and certainly Lagrange himself, the father of analytical mechanics who plugged the great holes in Newton’s celestial mechanics, contributed like few others to the divorce between the mathematical and the natural. Goethe made his remote appeal to mathematics within his discourse on color, but all his observations on Nature are morphology as they consciously deal with the boundaries of phenomena.
Image by Peter Alexander Venis
Although Venis does not seek a demonstration, his sequence of transformations is more eloquent than a theorem. It is enough to look at it carefully for a while for its evolution to become evident. It is a general key to morphology, regardless of the physical interpretation we want to give it. Venis does not rely on formalisms at all, but he has detected for the first time something that has an irresistible attraction for mathematicians and physicists: a new type of dynamic symmetry. However, this symmetry does not fit in any obvious way into the dynamical variables that physics use to handle.
For Venis the appearance of a vortex on the physical plane is a phenomenon of projection of a wave from a single field where dimensions exist as a compact whole without parts: the “One Field» as he puts it is just another way of speaking of the primitive homogeneous medium as a reference for dynamic equilibrium. It is clear that a completely homogeneous medium can be characterized neither as full nor as empty, and it is equally valid to say that it has an infinite number of dimensions as to say that it has none.
Here, however, the homogeneous and undifferentiated is as much in the background that surrounds the contour of the forms as in their central point of equilibrium. For us, three-dimensional space is synonymous with depth, but here it seems that everything moves on the surface, and it is precisely in three dimensions that the amorphous reaches its maximum. «A vortex is the only part of a wave within the One Field that intersects our physical world»: strange definition for anyone who ignores that here projection is everything. But what is projecting it?
We said that Analysis became too abstract and that biological ontogenesis has never been able to go beyond the particular. Well, the morphological sequence of Venis is at exquisite equidistance between the most basic mathematical physics and biology, between the concreteness of the latter and the abstraction of the former; and this is already something, when one admits the striking inadequacy of mathematics for living processes. Venis’ wave-vortex is form and process indissolubly linked, but its natural place is in the projective space free of measurement, which could only descend to the quantitative through affine geometry and metric geometries. The forms we observe in Nature, from mushrooms and spirals to rings, bulbs, horns, chameleons, clouds, galaxies and whirlpools, typically involve a definite portion of their evolution, but we will never find the complete matrix exemplified in a phenomenon before our eyes.
Image by Peter Alexander
One has the impression of contemplating an interdimensional ouroboros, a hysteron proteron, a womb that turns itself inside out as it forms, a still distilling itself, that which gives its form to the matrix of all forms. It is true that there is something inevitably subjective in any appreciation of forms, but this matrix seems to be expressing an irreducible reciprocity between the subjective and the objective.
Venis calls his explorations an “Infinity Theory», not so much for the pretension of «having a theory» as to apologize for not finding more justification for it. But the truth is that if we stick to his procedure there is no more need of justification beyond its correspondence with phenomena, for as it emerges it is undoubtedly a phenomenological morphology inseparable from our perception of appearances. The whole subject relies entirely on the primary plane of the aesthetic, and this cannot be seen as a limitation but as proof that we are dealing with something broader than the always limited domain of measurement.
«The human being himself, to the extent he makes sound use of his senses, is the most exact physical apparatus that can exist»; this statement by Goethe can only be understood as another way of saying that the most important thing for the human being is already summed up in an unsurpassable way in what he can perceive by himself. The flood of ultra-sensory data that all kinds of devices provide us with today cannot change this, but they do help us to forget it. When the German poet speaks of the experiment as a mediator between the object and the subject, he is not so much looking for a bridge between the two as for the needle of the balance that helps us to remember their unity. There is in the fidelity of description a Middle Way along which science has barely walked.
In our writings we have spoken of a certain retroprogressive method of return to the Principle and of an infinity as simplicity that would be at the antipode of the present infinity as complexity. The construction of any number of dimensions with orthogonal axes and points seems infinitely simple, but the determination of their possible contents and evolutions becomes infinitely complex. On the contrary, the variety of forms of Nature seems arbitrary to infinity, but starting from its global conception, a simplicity irreducible to the quantitative can be attained.
If one attends to Goethe’s experimental method, two stages can be observed: the analytical one that goes from the unlimited complexity of phenomena to simpler ones that in turn seek to converge in their principle, and the synthetic one that proceeds in reverse order seeking the relationship between the principle and the complex phenomena. Venis also proceeds in a similar fashion; but since Leibniz and Newton the reverse engineering of calculus, instead of determining the geometry from the physical considerations, in order to derive from them the differential equation, what does is to first establish the differential equation in order to seek therein the physical answers. These two procedures need not even be equivalent, which authorizes us to think that there must be a physical geometry very different from the one considered today by Analysis, both in the interpretation and in the relation to the Principle; and this is what really matters to us.
2. Morphology and Causality
Physicists and mathematicians would never have detected the symmetry of the Venis sequence since they move in more defined fields and with other historical lines of motivation, but as soon as they become aware of it they will not cease to make questions trying to circumscribe it. It is something desirable, and on the other hand to be feared insofar as there is the the risk of throwing the baby out with the bathwater. In any case, Venis’ great achievement is that since the appearance of his sequence, morphology can now pose questions to physics and mathematics standing on its own ground. Venis himself tells us that before starting his journey of discoveries he was a game programmer working with graphics and algorithms, and no doubt that has left an imprint on his style of exposition, which is as clear and logical as something that has infinity in its background can be. And the truth is that programmers determined to follow this Ariadne’s thread of visual logic could often be more helpful in developing and completing this theory than those who want to subdue it with the usual arsenal of quantitative analysis methods.
Image by Peter Alexander Venis
In the transformation sequence there are 5 basic types or modes of vortices mirrored, but the interplay of their variants, combinations and metamorphoses has a richness that aspires to be up to the challenge of the inexhaustibility of phenomena. In any case this is not a mere classificatory typology like those so abundant in the life sciences, but it is imbued with a logic that connects the modes causally, and this causality is in a certain sense stronger and more univocal than that found in the fundamental laws of physics, not to mention complex systems in general, as here causality coincides with the direction of the flow.
The notion of physical causality has long been in crisis, and it can even be said that this crisis is one with the relentless expansion of physics since Newton’s time. As is well known, the fundamental physical laws, insofar as they depend on variational principles such as the Lagrangian or the Hamiltonian, admit an infinite number of causes to explain them; which is the main reason why physicists have ceased to concern themselves with them at all. Already in the 21st century it was claimed that with the date deluge “correlation supersedes causation”. This is plainly false and does not even need a refutation, since the more the mass of data increases the more necessary are interpretations and models; but this does not matter because it is the very inflation of causal models that causes the devaluation of any idea of causation. If physics, from gravity to electromagnetism to entropy to quantum mechanics to cosmology, was the first to dispel faith in causes, lately it has been the excess of information and complexity.
In physics we use to associate the word «cause» with the well-defined notion of force, but the truth is that in relativity force ceases to be primary and in quantum mechanics as well although for different reasons. And in spite of this, the order of conception of physics still prevails and we cannot help thinking in terms of forces, and specifically of controllable forces, since merely measurable forces are not within the competence of this science. Physics continues to interrogate Nature with and through controllable forces, which thus, if only because of our unavoidable anthropocentric procedure, get in our mind the status of causes.
However, in the new morphology emerging with Venis, other meanings of causality and generation appear. On the one hand there is the apparently univocal causality of flow, but on the other hand there is the generation of the wave in the One Field and its projection as a phenomenon in the form of a vortex; being the appearance a dimensional flowing intersection of the wave. One of the most basic distinctions Venis finds is between central wave/vortex and peripheral wave/vortex; central waves are transverse, like those of electromagnetism, and peripheral waves are longitudinal like sound waves. This distinction is very important from the descriptive point of view although we will see that it still hides much more than it suggests.
But before contemplating the possible connections of this general morphology with physics or mathematics, a difficult and slippery subject that presupposes an adequate knowledge of the tenets of the former and of the descriptive shortcomings of the latter, we may ask how and to what extent the transformations in the Venis sequence are representable —directly or indirectly representable. Representation and description are, for what matters here, practically equivalent; but rather than taking it to the always pertinent levels of philosophical discussion, we prefer to keep these notions in their original context: computer graphic design and our limits in the representation and perception of different dimensions.
Venis, who is well aware that his approach is just the beginning of a long journey, attempts to arrange the basic motions in a series of charts involving the number of dimensions, the apparent shape of the vortex motion, and the motion that might correspond to the wave in the One Field prior to the projection. For motion the most basic distinction would be that of translational and rotational movements; and it is already remarkable, to begin with, that physics has treated such motions almost always separately and has avoided a concrete view of their connection. Let us not forget, for example, that, at the statistical level, the virial theorem eliminates the spin or rotation in its balance of kinetic and potential energy, so that the derivation of a supposed moment of inertia is just one more of the many accounting tricks of physics.
What would we see in a graphical animation of the transformation sequence in a virtual reality application? It would obviously depend on the program and the concepts we introduce, since Venis does not intend to settle the matter and is more concerned with appearance than with its possible recreation; while, on the other hand, the requirements of physics at the descriptive level have always been secondary. Classical mechanics, for example, routinely ignores Heinrich Hertz’s distinction between a material particle, which is an irreducible physical point in space at a given moment, and a material point as it appears to our senses and which as a volume can contain any number of material particles —a planet, a star or a galaxy. Nicolae Mazilu reminds us that this purely classical distinction also partially coincides with the wave-particle duality of quantum mechanics.
Reducing a body or a volume to a point, either in quantum mechanics or in classical mechanics, is a way of omitting three dimensions, not in the ideal empty space but in the ordinary space-matter continuum. If in quantum mechanics the description of extended particles is not possible, it is, first of all, because special relativity does not admit them, in spite of being a macroscopic theory. Maxwell’s equations are only useful for portions of the field with extension; special relativity is only valid for point events, and in the field equations of general relativity point particles are again meaningless. However, in a Weber-type relational mechanics, which can also be extended to fields by integrating over the volume, one can work with both point and extended particles, which allows it to cross the gulf between the micro and macroscopic without the need of curvatures or pretending to convert time into another dimension, among many other contortions.
The electromagnetic transverse waves of Maxwell or Hertz do not indicate that the electric and magnetic components are perpendicular in the literal geometrical sense but in the sense of a statistical average between space and matter, and this is why such waves have resisted all attempts of sound geometrical representation. However, it is enough to understand that electricity implies charged matter separated by space, and magnetism space separated by matter, for the irreducibly statistical character of the geometrical attributes of the wave to become self-evident.
Weber’s relational mechanics, an inexcusable reference for the latter Maxwell’s electrodynamics, raises since 1848 the problem of the retarded potential and its justification of the conservation of energy. Following Weber’s logic, Nikolay Noskov postulated a longitudinal vibration internal to bodies in motion to cover the difference between the kinetic and potential energy of the retarded potential. Since Weber’s mechanics can contain terms proportional to the square of velocity, one cannot distinguish always between these energies and the internal energy of the body —and one would have to dwell on the link between this indistinction and the quantum indeterminacy relations. This internal wave coincides with de Broglie’s famous «matter waves»; the Schrödinger wave equation itself is a mixture of different equations describing waves in a medium and waves inside the moving body, but as we know this ceases to make sense in the relativistic Dirac equation.
However much the differential calculus has accustomed us to believe it, a point has no physical reality, but is a purely mathematical concept, or as Leibniz put it, a simple modality. Going from a point to a sphere with volume adds three more dimensions to the physical characterization of a particle, just as allowing freedom of rotation of an oriented point with inertia we have six physical dimensions instead of three. The rotation of a point in all three dimensions allows us to generate vortices and volumes along its translational motion. If we wanted to visualize the internal wave of matter in an extended particle, it would not be something as simplistic as a corpuscle traversed by a longitudinal vibration, but a configurational process in six dimensions that would include the spin in its wave-particle field.
Returning to the graphic representation of the transformation sequence, if I surmise here that, starting from a poin vortex, there must be a continuous movement of folding and unfolding, of mutual immersion and eversion between the three dimensions of translation and the three of rotation, it is not for any sound evidence of it or because Venis puts it in those terms, but simply because I suppose that an ideal primordial wave that emerges from the undifferentiated whole to return to it must exhaustively cover all the combinations, all the possibilities of differentiation. But of course the multiple intersections of a single wave allow many other variants. At any rate, the simplest way to describe that combination of translation and rotation, immersion and eversion is with the concept of torsion. A morphology of vortices is by necessity a morphology of torsion.
We propose the possibility of virtual representation of the sequence not as a mere computer application or an interactive exploration but trying to find its own place within our imagination. The starting point is the idea that each organism recreates the whole as much as it can; this double movement outside in and inside out is reminiscent of the afferent and deferent processes of the nervous system, and therefore they should be deeply related with the specialization of the cerebral hemispheres. If, in addition, the two aspects of torsion can be conveniently connected with our own interacting cycles of action and perception, our ability to conceive morphogenesis and morphodynamics would be greatly enhanced.
Venis himself assumes that the wave-vortex must have a close relationship with the Hopf fibration, a well known mathematical object relevant for the two-dimensional harmonic oscillator, fluid dynamics, magnetohydrodynamics, quantum entanglement, the Dirac monopole, the geometric phase of a potential and many other aspects of physics. Its analysis requires complex variable, but before approaching the connection with physical processes, it would be necessary a careful study of the mathematical correspondences starting with the calculus itself. It goes without saying that here we must content ourselves with just some remarks.
For example, passing from projective or synthetic geometry to a synthetic differential geometry that includes motions, the descriptive as well as analytical deficit of ordinary calculus cannot be ignored. Without good analysis there is no good synthesis, and standard analysis does not even correctly analyze the physical geometry of problems, as evidenced by the fact that a mobile accelerating in a straight line is represented by a curve. Miles Mathis has shown conclusively that classical differential calculus, no matter whether it is based on infinitesimals or limits, always has at least one dimension less than the physical problem, and this cannot be overlooked when we seek a reasonably complete representation. Calculus had a humble origin in the study of curves but with the algebraization of analysis the primary relationship between calculus and geometry was buried forever. Mathis proposes a constant differential calculus based on a unit interval to put an end once and for all to instantaneous velocities and fictitious points. Otherwise, a synthetic differential geometry starting from points may be discretionary but in any case unspecific.
Can a virtual graphical representation help us to visualize higher dimensions or complex hypersurfaces? This question has a catch and the catch is the essential point. Since any number of dimensions is «just» representation in motion, but on the other hand what we want is to rearrange that representation and bring it closer to our consciousness. On the one hand, it would be perfectly illusory to think that morphology can illuminate that secret chamber of Nature of which Kant spoke and which would always elude our senses, since nothing that is based on concepts and representations can ever do so. But the search for an accurate description brings us closer to the balance between explanation and prediction, arithmetic and geometry, participation and distance, analysis and imagination; since, moreover, imagination already springs directly from that same chamber which is also in each one of us. The balance zone between these two extremes is often crossed by a great deal of data traffic, but it is also potentially a zone of silence.
As for hypersurfaces in the complex plane, although this one does not need any justification in mathematics, in physics it should, and in a morphodynamics of a descriptive nature they are completely necessary; and it is obvious that the most direct generator of complex numbers and non-commutative properties is the rotation of vectors. This would bring us back to the dynamics of the oriented point in 6 dimensions. The use of complex numbers in physics is nothing but a way to have more degrees of freedom and extra dimensions without the need to justify them. What we don’t use to ask is wether we are able to conceive balance in terms of dimensions, and yet this is at the heart of the transformation sequence.
If we think of morphology in terms of projection and intersection, two opposing but almost equivalent ideas emerge: the absence of causality, and vertical causality. The term «vertical causality» has been proposed by Wolfgang Smith in his perennialist, neo-Aristotelian interpretation of quantum mechanics. Although it may surprise many, reading quantum perplexities in an Aristotelian key is nothing new, and already Heisenberg wrote that the quantum domain, «between being and non-being», corresponds quite well with what has been understood since Aristotle under the concept of potentiality.
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Smith simplify things and affirms that the world that physics and quantum mechanics in particular are concerned with is not the world of bodies, but that of matter under the sign of quantity —physics is concerned with the measurable, which would be a completely different ontological domain than the one we perceive with our senses. Returning to the traditional body-soul-spirit triad, Smith concludes that the corporeal is limited in time and space, the animic only by time, and the spirit by neither of them; the three would be like the boundary of a sphere, its interior and its center.
Smith’s argument may be acceptable for quantum mechanics, since no physicist claims it is on the same plane of reality than the world of our perceptions, but for classical mechanics and cosmology does not hold water. One cannot simply appeal to «substantial forms» while ignoring the formative capacity of forces, motions and potentials. Few physicists doubt the real character of hydrodynamics or astronomical processes, even if their causal status becomes more problematic as we move away further in space and time.
In any case standard quantum mechanics, that of the Copenhagen interpretation and point particles without extension, is clearly incompatible with all our notions of macroscopic reality, and therein lies the whole force of Smith’s argument, which does not need to create alternative realistic or causal descriptions, such as that of de Broglie-Bohm. He does not need it because there is simply no such thing as a «quantum reality,» but only a potentiality that we actualize from our macroscopic corporeal world composed of matter and form. After all, and with a very elementary argument, Smith would like to bring us back to realism.
But there is another totally different way of contemplating vertical causality in Nature, which does not involve separating matter and form beforehand, as Smith does, in order to reunite them later through a mere concept. In addition, physicists have never said that hylemorphism is wrong, but rather that it is of no use for modern physics. Here we have repeatedly observed that, not only quantum potentials, but physical potentials in general, being independent of any velocity of transmission, can only be pure act, while on the contrary the interactions with which dynamics is concerned, being limited by velocity, always require time. Physics, understood as dynamics, subordinates potential to force, but it cannot be that what is instantaneous must be subordinated to what needs time to react.
It is legitimate to consider the existence of a vertical causality over and above the efficient or horizontal causality of dynamic interactions, since potentials are not reducible to forces and involve an implicate order of a different kind. Bohm’s famous implicate order is the resurrection of Leibniz’s pre-established harmony in the wake of de Broglie; the great difference is that the theory of the retarded potential allows us to see that this also holds at the macroscopic scale. But, since a potential cannot be retarded because it has no transmission velocity, what is not simultaneous is the action and reaction between forces, blowing up the global synchronizer on which all modern physics is tacitly built. The implications of this are hard to conceive.
We have already seen in other places that Weber’s relational mechanics allows a reformulation of the three principles of classical mechanics that do not pass through the law of inertia, nor through the constancy of the central forces, nor through the simultaneity of action and reaction or global synchronization. In this sense, relational mechanics, which has its precedent in the ideas of Leibniz, would be simply an acausal phenomenalism. Thus it is not necessary to understand vertical causality literally since the idea that there is a horizontal causality that extends virtually without limits would be simply another illusion. There is neither horizontal nor vertical with respect to a homogeneous and undifferentiated primitive medium, the only true infinite; the contrast only arises when we assume that there can be a global synchronization in dynamics, which besides being metaphysical involves a contradiction in terms. Vertical causality is an implicate order with its own temporal lines and layers.
The idea that the phase shift of the “retarded potential” cannot be such and actually implies a proper time would be indirectly supported by the cosmological implications of Venis’ own morphology. Venis speaks of time branches within a hierarchy of scales in which only larger scale vortices can affect the time branch. Even anomalies such as the deceleration of deep space probes would be due to a transition of scales. For Venis the reality of these temporal branches could only be verified by traveling great distances, leaving the sphere of influence of a star or a galaxy; but following the thread of relational mechanics it can be interpreted that each phase shift of the potential entails a certain deviation. The difference between the phase shift of the «retarded potential» and that of the so-called geometric phase is that the former is already included in our equations, while the latter is not and has to be considered as an additional curvature.
But, of course, the geometric phase is not unique to quantum mechanics, as physicists themselves often claim, and exists at all scales, and can be clearly verified in animal locomotion or in simple experiments with vorticity on the surface of water. Interference affects the flow and the simplest way to conceive the geometric phase is as an interference phenomenon. But, being a universal phenomenon, depending on the case and description we can interpret the geometric phase in countless ways: as a parallel transport, as a self-induction, as a curvature or flow of the symplectic form, as a conical intersection between potential energy surfaces, as a transition between dimensions, as a transition of scales, as a torsion or a change in density, as a phase transition, as a degeneracy point, as a retarded potential, as the difference in the Lagrangian, as a resonance, as a holographic interference, as a loop, as a principle of enslavement, as a hole or singularity of the topology of motion, as a conversion between spin and orbital angular momentum, as a proper time or timeline, as a memory, as an interface, or even in other ways that need not be exclusive.
Only the degree of fidelity in the description will allow us to decide which of these interpretations best fit the case; but several of them fit perfectly in the logic of Venis’ morphology. It may well be said that, if variational principles allow for an infinite number of causes, the geometric phase gives us infinite ways of seeing the absence of causality. And this absence is the only light we can shed on the so-called vertical causality; but in any case it is something that is open to both mathematical description and experimental verification.
In the mechanics of contact and controllable forces we associate force with deformation; but in the so-called «fundamental forces», as in gravity, there is no deformation when motion occurs, as in free fall, but only in the potential situation, as in solid bodies on the surface of the globe. If we attend to the form, it is as if there were a third state of rest; but mechanics based on inertia can draw no lesson from this striking circumstance. In relational mechanics even the shapes of ellipses, whether in planetary or atomic orbits, depend crucially on the so-called retarded potential. Potentials have a definite effect on shape; we can imagine them as the landscapes and valleys through which matter travel.
Continuing with the idea of projection, we have seen before the supreme irony that the holographic principle, arising from the most extreme conditions that theoretical physics can conceive, tells us, after having raised the number of dimensions for a theory of gravity to four and for string theories to ten or even twenty-six, that ultimately any physical evolution with all its information is reducible, no longer to a volume, but to a surface; when it was assumed that processes such as gravity or Huygens’ principle of light propagation cannot operate in two dimensions. However, the greatest irony is that the holographic principle is based on the geometric phase, a supplement to quantum mechanics that does not even properly belong to it: a loop or curvature added to the unitary, closed evolution of the Hamiltonian. The geometric phase does not belong to Hilbert’s projective space, but reflects the physical geometry of the environment that is not included in the definition of a closed system. And this openness of a closed system to its environment is what is now supposed to define the limits of our experience of the world.
If the geometric phase is equivalent to a torsion, it is an entirely trivial truth that the torsion of a surface is sufficient to describe any conceivable form. Since the holographic principle is universal, to speak of the universality of the morphology of torsion would be equally justified; the question is how many degrees of simplification or synthesis this general principle admits, which ultimately only tells us that we know nothing except with light and in light. There has been a great deal of speculation as to how could be possible to check the holographic principle experimentally, but if it really is universal and we have it everywhere, all that is needed is to pull the slipknot that should exist between the «retarded potential» and the geometric phase; for what this lag tells us is that there is a feedback loop that depends on the environment. In fact, if the retarded potential is applied to gravity, the black holes that brought the holographic principle to the foreground become impossible to begin with because as velocity increases the force decreases proportionally.
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It is almost an assumption of Venis’ theory that all waves would have their origin in a single process, changing only the circumstances of projection or intersection. Here we share this view, the truth of which would have wider consequences than any treatise can summarize. Even today there is a general consensus among physicists that the longitudinal waves of sound and the transverse waves of electromagnetism and light are completely different processes with no possible connection. However, in October 2021 Shubo Wang‘s team in Hong Kong demonstrated the existence of transverse sound waves created with a metamaterial that couples the spin of the wave and its momentum.
This discovery probably marks the beginning of the end of the preconceived ideas we still have about waves, and soon will find its due counterpart with the verification of longitudinal electromagnetic waves. In fact, a few months earlier a St. Petersburg team had arranged another metamaterial to easily produce longitudinal electromagnetic waves. Longitudinal electromagnetic waves are not entirely unknown, as was the case with transverse acoustic waves, but their theoretical status is still debated. It should always have been obvious their link to Maxwell’s equations since these are only a special case of Euler’s fluid equations as well as Weber’s —even if Weber never predicted electromagnetic waves of any kind. There are of course Noskov’s longitudinal waves associated with the retarded potential, which the Russian author always linked directly to sound and the wave mechanics of gases and fluids; Shubo’s team considers the scattering field of the waves as a spherical projection of the incident field in which it induces non-commutative rotations and «leads to geometric phases that account for the spin to orbital angular momentum conversion”.
We have mentioned these recent discoveries because they open a new way to contemplate the unity of Nature, a way that does not go through the «unification» of the known fundamental forces but through the exploration and reconstruction of the unity of form when trying to represent the physical geometry of processes. Of course, this physical geometry can and must include statistical aspects. This search of unity has nothing to do with the umbrella project of unification of the fundamental forces, for to approach the characterization of this primordial wave is to approach an acausal perspective and the vertical causality implied by the projection. And since acoustic holography offers a complete analogy with optical holography, and the geometric phase allows controlled wavefront shaping, it should be much easier to study the critical connection between “normal” retarded potentials and “anomalous” geometric phases, that in addition provides us with the key link between wave mechanics, dynamics and morphology.
From the relational point of view there are no causes in the mechanical sense but relations, and the co-implication of these relations has a depth still entirely to be explored. A new synthetic perspective opens up, but the physical geometry of which we speak is obviously more than kinematics; this would be the major limitation of a relational mechanics such as that of André Assis, who on the other hand raises an issue of great depth by replacing the principle of inertia by the zero-sum dynamic equilibrium of forces.
With an approach related to the Duhem’s attempt to join analytical mechanics and thermodynamics, although based on an argument suggested by Landau & Lifshitz, Mario Pinheiro has proposed an ergontropic reformulation of mechanics, defined by a balance between minimum energy variation and maximum entropy production. Since these are the two most fundamental tendencies we can observe in Nature, it is just reasonable to treat them together integrating consistently rotational vortical motion and translational motion.
Entropy, conceived by Clausius as a tendency to a maximum, already entailed a spontaneous finality; what physicists have never answered is what is the reason behind the blatantly teleological component of action principles governing physics. With a formulation of equilibrium between both, this second part seems more understandable. Boltzmann created a formidable confusion by equating entropy with the subjective concept of disorder, but, as R. Swenson already said more than thirty years ago, «the world is in the order production business, including the production of living beings and their capacity for perception and action, because order produces entropy faster than disorder.»
If entropy is not a by-product of mechanics, but is within the same equilibrium conditions of the latter, the meaning of physical processes changes completely. An internal equilibrium formulation such as Pinheiro’s assumes that there is always free thermodynamic energy in the environment, an «external» disequilibrium; by the same token, mechanics itself would emerge from an open and irreversible background. In the ergontropic framework, systems are subject to external forces but react to external constraints with vortical rotational motion and dissipation, with a direct conversion of angular motion into linear motion via topological torsion. This is therefore of interest both at causal and morphological level.
Pinheiro revisit Newton’s rotating bucket experiment and gives us a rendition that is neither the absolute space interpretation of the author of the Principia nor the relational one along the lines of Leibniz or Mach, who attributes the formation of the vortex to the frame of distant stars. Since we believe that the principle of inertia is entirely dispensable, we have always leaned toward the relational interpretation; however Pinheiro’s interpretation adds a necessary element, since in a case like this we cannot simply dispense with local causality. «What does matter is the transport of angular momentum (imposing a balance between the centrifugal force, pushing the element of fluid to outside) counterbalanced by the fluid pressure.”
The transfer of matter to regions of higher pressure is already an indication of the second law of thermodynamics, and the fact that it takes time for water to acquire its concavity speaks eloquently of the presence of friction. But, on the other hand, the vorticity within the water cube also reminds the hypothetical case of an extended particle, with intrinsic rotation while being traversed by a wave according with the “retarded potential”. What we now call local causality can never be understood more geometrico, but, like the same electromagnetic waves and so many other physical processes, somewhere halfway between geometry and statistics; local causality would be the non-deterministic reaction to global causality, which is another name for vertical causality or unity. Even today, an experiment as simple as that of Newton’s cube allows us to make a different reading of the connection between central forces and non-central or peripheral potentials.
We have frequently spoken of three fundamental equilibria in physics and in electrodynamics in particular: the zero-sum dynamical equilibrium, the ergontropic equilibrium between minimum energy variation and maximum entropy, and the equilibrium of densities with a unit product. Such equilibria are rarely contemplated in current frameworks and when one of them is eventually considered it is not properly related to the other two. These three such basic equilibria should be aligned with the morphodynamic balance in the midst of the Venis sequence, which in a sense is reminiscent of the so-called «potential flow» of fluid dynamics. Naturally, this central equilibrium has ramifications in a wide variety of different circumstances.
We said that Venis mentions as a possible mathematical approach to his morphology the Hopf fibration with use of complex variable. In a morphology of flow and torsion that keeps the emphasis on description the relation between the use of complex numbers and rotation, of these two elements with time and the retarded potential, and of all of them with timeless projection and vertical causality should be investigated. Of course, any idea of causality is part of a complexion of concepts, but once one begins to delve into morphology as knowledge of figure and configuration, this will affect our ideas about space, time and causality; matter, form and motion.
3. Individuation and Singularity
In Venis’ hexadimensional wave-vortex the most concrete and the most abstract merges. Concrete, because it is possible to follow the evolution of its form in a way that is both intuitive and logical, and because it is based entirely on the simplest notion of flow. Abstract, one should perhaps say in quotation marks, not because it would involve a hypersurface in six non-integer dimensions, nor because there always seems to be something in its movement that eludes us, but because as a whole it does not correspond to any static «thing» or object, but only to processes. Indeed, nothing could better illustrate something usually so unrepresentable as the philosophy of process and becoming, from Heraclitus to Whitehead or Simondon.
It is surely no coincidence that it was Whitehead the first to formulate a point-free geometry, which was later followed, for different motivations, by point-free topologies. Moving away from set theory, the mathematical theory of categories finally fulfilled Aristotle’s program of making the categories of concepts explicit. Synthetic differential geometry was developed, and, in general, the modern mathematical concept of topos coincides surprisingly well with that defined by the Greek philosopher: the boundary of the enveloping body according to where it touches what it envelops. René Thom also moved closer and closer to Aristotelian concepts in his «sketch of Semiophysics»; these are just examples of how the attempt to think forms and the organic, our most immediate tangible reality, has generated a movement of mathematical abstraction at the most fundamental level —of new concepts that nevertheless connect in depth with the first thinker of the organism. The now ubiquitous language of homology and cohomology has great affinity with morphology as it is natural to treat a vortex as a topological defect, but the extreme algebraic abstraction of the specialty has succeeded in separating it from its intrinsic connection with directly observable formative processes.
Our notion of projection ideally starts from a point, but we have already seen that the idea of point in differential calculus is dissociated from physical geometry and that such dissociation has a clear solution. Synthetic differential geometry emerged as an attempt by Lawvere to give an axiomatic basis to continuum mechanics but it leaves intact the dimensional analysis of classical calculus and cannot access the physical geometry that morphology demands.
The notion of the point particle, which as we have already seen is prescribed by the constraints of special relativity, is one example among many of physical pseudoindividuation of an entity. The Copenhagen interpretation of quantum mechanics believes in the identity of individual particles, but the very characterization of its aspects inevitably passes through statistical criteria. Thus we see that an irreducible mathematical concept, such as the point, has a deficient physical and differential application that is also allied with the congenital nominalism of all modern science. From a configurational point of view, focused on the physical geometry, there should be no contradiction between geometric and statistical characterization, as rather both imply each other.
To put it more directly, for us to be an individual is to be singular, but one is not irreducibly singular without the contribution of the idea of point, which implies a very long and abstract evolution of concepts. The same is true in our cosmology, with the idea of initial and final singularities, not to mention black holes. These singularities only seem possible to us not only because of our notions of point in space and time but also due to taking the principle of inertia and central forces to their ultimate consequences, and we have already seen that in a relational mechanics there are no such evolutions. Physicists who speculate about black holes speak of irretrievable information, but what is not realized is that the same principle of inertia makes irretrievable aspects of physical geometry that contain information of a morphological order.
The last great theorist of individuation was probably Gilbert Simondon. Simondon made an exhaustive critique of the Aristotelian principle of individuation and in general terms his contribution is of great value, although one may sometimes doubt if the Aristotle he criticizes is really the Greek philosopher or rather his scholastic derivations. In any case we will not discuss this since what matters to us now is the alternative he offers to this principle with his idea of the process of individuation and the ultimate limit of its contour in this our age of extreme nominalism.
Simondon, not by chance either, is one of the thinkers who most clearly aspires to situate himself in a middle ground between physics and biology; at least that is where he wants to situate physical individuation, since he clearly sees psychic and social individuation as processes with great similarities but of a different order. For the Frenchman, the individual cannot be thought of without equally contemplating the pre-individual and the transindividual. The individual would emerge as a change of phase in the potential; this formula is a way of passing from physical causality to the cause of individuation, that is, to its very process. Simondon does not directly relate this gap to the geometric phase shift, not least because in 1958 even physicists did not use the term. Some have questioned whether his application of physical concepts is legitimate, but what is certain is that the philosopher proves to be more aware of the importance of this phase shift, not only than the physicists of his time, but also than those of present times.
It is necessary to insist, given the deep-rootedness of our preconceived ideas: that the geometric phase closes the contour of any form is, in principle, as trivial a fact as saying that every form must be circumscribed by its environment. But this would only be trivial if the physics with which the system has been defined had not been constructed from the start obviating the influences of the background environment. That is why phenomena such as the so-called Aharonov-Bohm effect, which is by no means exclusive to quantum mechanics, created such perplexity among physicists. In a relational mechanics like that articulated by Assis reformulating the three principles, neither the forces are constant because they always depend on the environment, synchronization depends on the phase of the potential, and the principle of inertia is replaced by dynamic equilibrium. The principle of inertia is contradictory in itself, since it asks us to consider «a closed system that is not closed». And yet this contradiction is at the heart of the dynamism of modern physics, of its «creativity» for better or for worse. But within the inertial context of modern physics, in which also quantum mechanics remains embedded, it is impossible to appreciate the temporal differentiation of each entity. The principle of inertia demands that everything is out there: it is the basis of the construction of the Law as pure exteriority. But in itself there is nothing necessary about it, it only transcribes the phenomena to the most external interpretation possible. According to this principle, nothing can have a proper time, and therefore its form must be at any rate accidental.
If this is well understood, it is much easier to understand Simondon’s various approaches to the concept of individuation, both physically and morphologically. For example: «The individual is the partial and relative resolution that manifests itself in a system containing potentials and involving certain incompatibility in relation to itself, an incompatibility composed of forces in tension as much as by the impossibility of an interaction between extreme terms of the dimensions». Or: «Individuality is a resolution of an initial incompatibility rich in potentials». Or: «one could say that the only principle by which one can be guided is that of the conservation of being through becoming». Or his vision of the individual as communication between different orders of magnitude, macro and microscopic; all these verbal formulas are perfectly in tune with the descriptions of the transformation process in Venis.
Venis, always cautious in his judgments, is the first to provide indications that the transformation sequence may not be complete. He also rightly points out that if the vortical morphology does not translate easily to the domain of organic matter, it is due to the presence of several simultaneous equilibrium states. The coexistence of these equilibria and their compromise could be, perhaps, the most important key to seemingly irreducible biological complexity —and also to a general theory of development and aging.
Today biomathematics is developing exponentially but, like so many areas, not because of theoretical advances but because of the brute force of computation and massive data mining. Massive biological data analysis can in no way be confused with theoretical biology, for the latter must aspire to understand the problem already posed by Aristotle, namely, how life is possible as the existence of autonomous agents. The theory of evolution does not deal with this, but rather takes life for granted. Neither does molecular evolution, which revolves around self-replication of DNA or RNA; for not only short peptides, but even vortices of inorganic matter can replicate under certain conditions of stratified flow. These vortices are so robust that they can survive indefinitely even under subsequent turbulent conditions.
Even helical amino acids have a degree of vorticity, can be subjected to torsion and exhibit defined geometric phases: that is, they have a critical sensitivity to their environment which is the only thing that can explain that the same enzymes can create different proteins depending on the environment. But we have already seen that Pinheiro’s ergontropic vortex or systems with retarded potential have a capacity for self-adjustment or feedback that is not contemplated in either classical or quantum mechanics. This is prior and much more basic than self-replication. The idea that matter is blind is untenable: matter sees through form, so to speak. And in a sense, it sees much more than we do.
Matter itself would be only a configuration. If today it is a cliché to contrast physicalist reductionism, which conceives observable properties as resulting from the interaction of particles of matter, with emergentism, which, descending in a direct line from Aristotle, affirms the appearance of irreducible configurations and that the whole is always more than the sum of its parts, it is clear that morphology will always be on the latter side. And if we distinguish between a «weak emergence», in which the higher level cannot modify the lower one, and a «strong emergence», in which descending causality is possible, it is not hard to support this second thesis from the point of view of a vortex morphology.
The so-called downward causality, still in the midst of controversy, does not necessarily have to be synonymous with vertical causality, although the two can easily overlap. Perhaps the main difference is that vertical causality is at the moment a rather negative and too open-ended term, whereas when we speak of downward causality we want to pay more attention to the details and positive aspects of mediation between different physical strata. If what we understand by vertical causality is rather the implicate order and the intrinsic synchronicity of the potential at all scales, one and the other can coincide without the slightest difficulty since in any case the downward causation also takes place through the mediation of the environment, which it is not to say that there is only one means of causation, but rather the opposite. As we said, if the variational principles of dynamics admit an infinite number of causes, the geometric phases admit infinite ways of interpreting the absence of causation, that is, of interaction; everything depends on whether we attend to the controllable force or to the potential. That quantum mechanics is not complete is conclusively demonstrated by the geometric phase; what we do not know is how far the loop of its feedback can be extended.
Complexity researchers such as Stuart Kauffman or Danko Nikolić stress that downward causation cannot be consistently defended without a deep respect for the details and circumstances of the «system» or organism in question. Recourse to downward causation might allow substantial, if not decisive, advances in specific areas and problems, whether in chemistry, biology, psychology, neuroscience or sociology, and indeed the concept was first proposed by Donald T. Campbell, a social scientist, in the context of hierarchically organized biological systems —although already Kant postulated that organisms are cause and effect of themselves. But there are also well-known and robust physical processes, such as Bénard convection or the self-replicating vortices mentioned above, that would exemplify it in the most obvious way. That a vortex can survive turbulence is no less worth of respect, a respect that also demands attention to the details of how this is possible.
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In dealing with the principle of individuation, Leibniz makes another great concession to nominalism and settles the problem by affirming that in reality only individuals exist. Long before postulating the existence of monads or simple substances, Leibniz also contemplates the physical and mathematical aspects related to the surfaces of individuation of bodies in the context of a fluid plenum to reach the conclusion that forms lack real entity and always have an imaginary component inextricably connected with our perception. The position underlying Venis’s morphology essentially coincides with this, even though he tries to avoid metaphysical statements. However, Leibniz’s position does not deny the objective component either; as we have seen, in relational mechanics there is always feedback with the environment, so that one can rightfully interpret action-reaction cycles as action-perception cycles. Each physical potential is not merely a position, but an authentic perspective: the circumstance of the environment from the agent’s perspective.
The individual implies the pre-individual in the same way that a form presupposes the environment or the phase shift a previous potential. This surplus on course in the individual seeks to transcend itself both on the horizontal, or transindividual, plane and on the vertical plane, which has been so often called transcendent; and yet both would be projections on the self, the homogeneous and undifferentiated medium. Aristotle appealed to an entelechy to explain the autonomy of the organism; Leibniz and Goethe stick to the same idea. Simondon prefers to shift the question to the process itself, but in Venis’s morphology the very symmetry of flow suggests a reflexive principle or self even in the midst of the formless. This would be the most immanent way of grasping something that is both wholly immediate and entirely metaphysical, or, if one prefers, transcendental.
At the heart of the individual, as at the heart of consciousness, is the simple and undivided, rather than the singular; any singularity or form can only exist in contrast with this background without qualities. If in a homogeneous medium with a unit density we imagine the appearance of two separate volumes with a reciprocal change of density, this does not seem possible without the emergence of a torsion connecting them. The fact that in this background or homogeneous medium it is impossible to distinguish the full from the empty, nor consciousness from matter, makes it easy to speak of a «neutral monism», but of course this is only a philosophical position, which, like any other, becomes complicated in the way of a more precise definition. For that it is better to think that emptiness is form and form is emptiness, and stick to it.
Every form is mind communicating with a much more minute and vast one, but from the point of view imposed by inertia, it is not possible to «recover the information». All mind is spirit because spirit is intellect and will, but for intelligence it is not possible to recover the will that has been invested in it.
A vortex is a self-contained process, which seems to have its own entity and mass while remaining connected to the whole —for it cannot exist without permanent contact with the background from which it emerges. It opens and stretches, dilates and contracts always with its own rhythm, much like the heart, which in reality is a spiral muscular band acting like a spring regulating the flow of blood also in helical trajectories. But all animal and plant organs can be considered as different types of frozen vortices in different states of evolution and equilibrium.
Undoubtedly, the transformation sequence can be seen as an archetype of the individuation process and its stages, an ephemeral configuration within the homogeneous environment with which it never loses contact. However, the correspondence between the evolution of the sequence and the process of growth, maturity and aging remains to be interpreted. It becomes necessary to distinguish the abstract from the concrete in the sequence of the vortex before reuniting it, and the same should be done for the development of the organism in terms of flow and obstruction, in order to be able to appreciate their common ground and differences.
This may seem an extremely complex problem and yet it contains a very simple principle. Modern medicine, which has accumulated such mountains of information on diseases, has not even bothered to find a functional definition of health, such as Ehret’s efficiency principle, which says that vitality is equal to power minus obstruction (V = P — O); power P being synonymous with pressure and energy, and obstruction O with tension and matter, having at a constitutive biomechanical level, the boundary of deformation between both. Since these variables fall squarely within the most elementary logic of flow, they can also be associated with the transformation sequence without much difficulty.
There is a fundamental principle of development, including economic and social development, which, for some good reason, is permanently ignored: the principle of increasing restrictions for complex systems, which can also be related with the efficiency principle and the transformation sequence. And other basic aspects such as the principle of minimum action understood as movement along paths with least curvature or restriction, the principle of maximum action in size or scale, the principle of maximum entropy production, or the energy density flow per mass, which as Eric Chaisson points out is a much simpler and more reliable indicator of the degree of complexity than entropy production. Although these are very different aspects, all of them and some others converge in the evolution of an individual entity as a flow process.
Surely the most essential aspect of this process is something very simple and perfectly intelligible to anyone. Surely it is also a qualitative, inner law that can be understood without the need for measurements, physical quantities and mathematical apparatus. If we mention such late and elaborate physical concepts, it is because, at best, they also can benefit from a convergence in a much simpler complexion. The internal law of individuation is reflexively simple but allows us to connect very complex aspects of organisms as well as abstract mathematical structures; that is why the Venis sequence suggests a true symplectic morphology, taking this last word not in the mathematical sense Weyl gave it, but attending to the complexion of the causal in the phenomenon of form. It was Goethe himself who said that causality had not only length but also width; now we give it height, and it would be this unfolding of the implications of the causal dimensions that makes it possible to simplify the complex.
But what is an «internal law»? To date, still under the influence of the law of inertia, all physical laws are considered external. However inertia is not a law, but a mere principle that can be replaced by others, such as the principle of dynamic equilibrium: this demands a zero sum between all the forces acting at any point in any state of motion. It is not the same to see any process, even organic aging, as being governed by inertia, as it is to see it like an equilibrium between factors.
External law is and always will be odious to the living. This is the critical aspect of religion that is still perpetuated in its entirety in science and in the social. But the «inner law» of the living of which we speak is so inner that it is not even possible to internalize it, as has always been done with religions, laws or instruction. In fact, it may well be said that it is not even internal, but that it is intimate, which is another way of saying that in it the interior and the exterior cease to be different. It is this law alone that interests us, because it is the only one that connects us directly with the formless of the Principle. The external law binds us and appears to us as such only to the extent that we do not know our own law, that which the Sanskrit language called Swadharma.
4. A Middle Way
Venis’ morphology, a sort of nondual hylemorphism, sees the flow in terms of yin and yang, but the dialectic of pairs of opposites exists in any culture and is only one method of approximation among others; Goethe himself speaks of systole and diastole, syncrisis and diacrisis. What is really important is to seek a balance between explanation and prediction by means of adequate description. We insist that the great deficit of science is still at the level of description of phenomena, as our mathematical models have accustomed us to replace phenomenal infinity by a mathematical infinity with a very poor correspondence.
The dialectical method proceeds by assimilation and in this sense easily allows us to bring a problem to the level of experience; it also has the advantage of allowing us to notice early on the fundamental ambiguity of any question posed between extremes. There is no important question that cannot be simplified to the extreme while maintaining a patent nucleus of significance that usually coincides with its fundamental ambiguity; an ambiguity that nevertheless soon escapes the threshold of attention of consciousness.
Morphology lies in a no-man’s land between mathematics, physics and natural philosophy, and thus also in their junction, something from which different knowledges can benefit. Richard McKeon defined with great clarity the four basic methods of philosophy —dialectical, problematic, logistic and operational- and noted that there is no question arising in any one of these positions that cannot be translated into the terms of the other three. So far philosophy has scarcely been able to draw lessons from this internal architecture, but for a general overview mathematics may find it more useful than the conventional division of its domains into arithmetic, geometry, algebra, and analysis.
There is certainly nothing imaginable that mathematics cannot translate into its language of equivalences, equalities and identities; the problem is rather the opposite, that mathematics has left far behind the world we are able to perceive and represent. If it would only devote a small part of its forces to faithfully account for appearances, science would be a very different issue. Even postulates that seem as unscientific as that of Goethe’s polarity of colors, with its sound perceptual basis nevertheless, would have found their corresponding «analytical continuation»; for color is between light and darkness as light and darkness are between space and matter. Man can only perceive his own world, but in his world everything else is already implicated; any attempt to transcend his own limits by ignoring them or trying to overthrow them only leads to the loss of meaning and the dissolution of his own sphere. And so, almost everything in current science is actively and passively a force of dissolution.
If mathematics, physics, complexity theories and computer science continued to expand and cross-fertilize at the same rate as today for a thousand years, they still would not find the key to morphology; not even in two thousand years either. Perhaps that may give some idea of its value, although we all know that nothing can be found without actively searching for it. That is the point: the mentioned sciences already have their own momentum and inertia that nothing can change, only a creation of a new science from scratch could overcome without obstruction the deficiencies of its predecessors.
Something that has such strategic potential will not go unnoticed for long by the powers seeking to capture the symbolic initiative in any field. And since today it is impossible to have the slightest illusion about science and what it serves, it is obligatory to keep in mind the worst and the best that can happen to morphology in this last journey, since morphology, as a theory of individuation, is really the most finished product of nominalism as well as its turning point.
Spengler, the first proponent of a morphology of cultures, noticed a century ago that we are basically indifferent to the idea of causes and predicted the advent of a general «physiognomic science» which, beyond the millennium, would finally envelop any particular knowledge, transcending the petty ideas of causality; but his words were soon forgotten and no one knew what to do with such a prophecy. And yet, as we have already seen, the gradual erosion of the idea of cause in the sciences themselves has continued its unstoppable process, a process that began to roll down as early as the theory of universal gravitation and its subordination of description to prediction.
But Spengler was pointing to something both anterior and posterior to the idea of causality. The reciprocal arrangement of space, time and causality is undoubtedly a late achievement, but the sense of inner direction and irreversibility has always existed and will always exist. Spengler opposes History to Natural Science and blood to calculus, but in the morphology of flow and torsion causality in the most elementary sense and internal direction coincide by necessity.
Spengler’s physiognomic science, that last Faustian unfolding now freed from the idea of causes, would be a comparative historical morphology in which physics plays no part. Perhaps what Turchin baptized with the name «Cliodynamics» has something to do with this; although chronologies, statistics, sociometry, ecology, evolutionary biology, etc., would constitute, at best, only a preliminary phase in trying to access the interior of the historical heartbeat, that which the German philosopher called «the soul of cultures». But, given that these transdisciplinary attempts always have a precarious basis —like any other theory of complexity- if they also aspire to predict the future, they can hardly be situated within a process, since prediction requires stepping out of the current and abstracting from the given context.
Although Spengler opposed physiognomics to the natural sciences, the rise of correlation and its paracausal character can easily be extrapolated to what is now happening with computing, data mining, expert systems, artificial intelligence, surveillance or personalized marketing and medicine. What machine learning networks do today, with individual databases and their exhaustive and systematic survey, from medical data and images to fingerprint or facial recognition —often with the assistance of mathematical morphology- would fall squarely into the preliminary phase that we could call «analytical physiognomics,» «statistical physiognomics,» or «soulless physiognomics.» But it is clear that, in this relentless process, no soul will be born by spontaneous generation of statistics.
Thus, although no one speaks of «physiognomics» today, it is evident that within the great traffic of data there is an overwhelming tendency towards algorithmic and biometric control that wants to descend from the instances of power to the innermost recesses of the individual. The same computation feeds back and reshapes itself in trying to reproduce the various semantic fields of the individual, which, understood in Simondon’s way, can go far beyond human biological individuals. Naturally, if all this tendency could already be foreseen by Spengler himself, it must be part of the fatum, and therefore has nothing to do with the symplectic morphology we are dealing with. This one is not a convergence, but an aside from all that came before, and even a conscious distillation of what has been rejected by the other sciences.
There is also something in China that gravitates intensely towards both morphology and physiognomics, with a still very wide margin of indecision. Chinese culture has a marked interest in the process of individuation that largely follows from its extreme nominalism and from the fact that, geometry and morphology being largely antithetical, the historical lack of development of geometry in China will tend to be compensated spontaneously by a great development of morphology. On the other hand, it is just one of those mirrorific counterpoints that the origin of the binary code can be traced back to Leibniz’s logistic interpretation of the Chinese hexagrams of the Book of Changes.
We have said recently that the limits of a science such as mathematics are given by its criterion of application and its degree of receptivity to phenomena; but our idea of causality also depends on both. The basic criterion of application is given by our idea of calculus or analysis; we have also seen some of the reasons why it is conceivable that we have not even reached the end of the first stage of analysis and general application of mathematics, which would be the realization that differential calculus descends from the global to the local without any real foundation. If we succeed in getting down to the bottom, the end of this phase would be followed by another one of ascent in abstraction from the physical geometry of phenomena, followed by a final one of descent according to its knowledge of vertical and downward causality.
If these phases are intended to be conceived as a historical development, then it is impossible to know what spans of time separate them, whether such as that between Newton and us, that from Archimedes to Leibniz, or even way longer periods; but they can also be viewed in a perfectly timeless way, as movements of the spirit that are often potentially present even within the proof of a simple theorem.
By way of contrast, we can compare these 3 phases with the three levels of civilization according to the Kardashev scale. This is a scale of development based on the amount of energy a civilization can extract from the environment: a type I civilization would use all the energy available on a planet —something we would still be far from-, a type II civilization would use all the energy radiated by its own star, and a type III civilization would use all the energy available in its own galaxy. It is a shame that the original scale does not include the energetic harnessing of black holes, which is where this kind of logic invariably ends up, although I understand that later cosmologists have corrected this shortcoming. And as nothing is too much, others have added a type IV and a type V, which would control universes and “collections of universes».
In any case it is clear that a scale based simply on extraction of resources is the ultimate scale of the poor devils. It is unlikely that if we had a deep knowledge of Nature and of ourselves we would think of interstellar or intergalactic travel, and in fact the latter can be taken as good evidence to the contrary. Higher knowledge, including technical knowledge, learns to dispense with detours and intermediate steps —with mediations – and even mathematics attests to this, although on the other hand mathematics such as the present ones are incapable of containing their own expansion in all directions.
It is doubtful that the archons of each solar system are willing to see how the interplanetary space junkyard grows, but neither will they need to take violent measures: the increasing restrictions of scale of a civilization as an individual phenomenon are extremely effective to contain it by themselves, and these restrictions reach their limits of contradiction and incompatibility on the intellectual plane earlier than on the material plane. Already now it can be seen that the internal logic of these limits imposes an alarming inefficiency in the rational use of our limited intellectual resources.
Other lovers of speculation have proposed different metrics for such scales. Based, for example, on information, although it is doubtful that this is a less barbaric category than energy. Or Barrow’s inverse metric, based on the mastery of decreasing, microscopic scales rather than increasing or macroscopic scales. All this is supposed to be of interest in order to ponder our own anthropological case, but I don’t think that is the point: all it reveals is the limitations of the current scientific knowledge. Because not even our idea of limit as a basis for analysis is sound to begin with.
A symplectic morphology like that of Venis, even if it is in its earlier stages, gives us at least other different ideas about the relationship between macro, micro and mesocyclic scales: ideas that remain faithfully linked to the phenomena we perceive. The very idea that going beyond phenomena, with the mediation of machines and high mathematical technology, leads us to transcend our limitations is the basic fantasy that most distances us from knowing our place in the Cosmos. Venis, for example, does not believe that gravity dominates the universe on a large scale, but considers it a mere local phenomenon of what he calls orbs or spherical regions within a flow regime, and even within them its action diminishes perceptibly; the motion of stars in galaxies escapes the domain of those regions and is due to a vortical motion like that of the hurricane although on a larger scale. But the best example of transmutation of meaning without changing anything of what we know is the interpretation of the cosmological redshift: within Venis’ dimensional logic this cannot imply the expansion of the universe, as our comical egocentrism surmises, but simply the contraction of our local group.
Naturally, Venis is not in a position to quantitatively justify his judgments, but he is much more likely to be right than wrong. The whole of Western science is built on an inordinate and unwarranted extension of predictions that in reality have been a reverse-engineered copy of known results that no one knows how to justify. This generalized reverse engineering without guarantee or justification, and leaving in the shadows most of what that does not fit with it, is what creates the persuasive illusion of «the unreasonable effectiveness of mathematics». And that absolutist extension without guarantees, that speculative inflation to infinity is what creates the price, not the value, of its symbolic capital.
We see for example how today the geometric phase is routinely used in control theory or in attempts of quantum computation when it places before our noses the umbilical cord that connects an individual being with its environment, the very thing that denies that we can consider a system as inert and closed. Between the living and the dead, we choose the dead without hesitation. We want to control Nature but not to know how it regulates itself, and if we exclude this, we cannot recognize our own law either.
We have already seen before that calculus has oscillated between the idealization of infinitesimals and the rationalization of limit theory; the unwary might think that by reaching both extremes nothing could be missing, but the most important thing that remains in the middle still is being ignored. We have also spoken of the rights of the description of phenomena to find the balance between prediction and explanation, which is but another angle of the same question. Without the proper assessment of phenomena, we will never find our own vertical. Finally, others, like Kant himself, have also believed that there must be an internal law as surely as there is an external law, but the truth is that there is neither, and that both thinker and thought are only the activity of thinking, the Logos we have always been looking for. Thus there are some good reasons to think that there is a Middle Way in the sciences and scientific knowledge, and that we need not to look very far, but only cease to adhere to a willfully unbalanced conception.
As we said, a vortex is both a very concrete and abstract process, in a different sense to the one we usually give to these words. And this has many consequences if studied in the proper perspective. The Indian samkya, for example, calls vrittis, vortices, to any mental modification, that is, to the ephemeral par excellence, thoughts. It may seem a figurative way of speaking, but if we start from the idea of a still medium without modifications, the comparison is inevitable and implies much more than what we associate with such vague terms as «mental waves» or «brain waves». The point is not that an interference of waves can be seen as a vortex, but that the degree of abstraction in the way a vortex is considered can echo the degrees of abstraction of thought itself. Moreover, mental representations require discrete states for there to be cognition: to have clear and distinct cognition the first thing is not to have it, and this also has a correspondence in terms of vortices and even their possible self-replication.
We have already seen that in the third quarter of the last century there was a one more revolution in the fundamental degrees of mathematical abstraction with category theory, algebraic topology and other developments; latter there were even pioneers, such as Robert Rosen, who tried to understand the autonomy of the organism by applying these new mathematical categories. It is true, as Kauffman has said, that the world is not a theorem and that the mathematics of set theory falls short to describe life, but it is also true that there is life in mathematics beyond set theory. However, here again a middle ground between the concrete and the abstract has been lacking so that neither goes too far and admit the rectification of an agile, “gentle empiricism», so different from the positivism on the industrial scale of refutation and specially confirmation of hypotheses.
Mathematics is the science of pure forms, in the purest intellectual sense; morphology can only be a science of the forms of pure phenomena. This defines the extreme affinity and contrast between both ways of knowledge. But the most universal of phenomena is change, and the branch of mathematics that studies change is calculus, its main application, through which mathematics ceases to be pure form at all. Yet it is clear that the calculus only deals with change in a very definite sense that eludes descriptive requirements. «What moves does not change, and what changes does not move»: taken at its symbolic extremes, and with due license, one could understand by «motion» translation, and by «change» rotation. Since physics has been modeled by an idea of differential calculus based on a baseless difference, we should see what happens when we use calculus as a function of non-difference, as constant differential calculus does, and apply it to the symplectic morphology of torsion.
We have alluded to two extreme forms of nominalism: a Western nominalism that is intellectually overdetermined, and an Eastern nominalism that is underdetermined but more receptive to phenomena. The former greatly overestimates the value of theories by trying to exploit them to their maximum conceivable extent, while for the latter everything that is theoretical pretension already seems to be an excess of systematization. But the truth is that morphology opens a free path to the knowledge of forms, ideas or essences, inasmuch we abandon the Platonic pretension of a separate, immaterial and immutable reality. Given the one-sidedness of its analysis, physics has not been able to get rid of its own misplaced Platonism either, just substituting metaphysics for mathephysics. There is another analysis and another object of analysis that can change our idea of zero, infinity and unity; of number, measure and equilibrium.
Seeking reintegration into unity has nothing to do with the elucidation of intellectual aspects unless we start from an intellectual overdetermination; in the same way that simplicity would not be interesting if we did not start from a circumstance marked by complexity and differentiation. Knowledge in the first person by itself is not the same as that same knowledge examined from the third person, but no analysis of the latter can recover the immediacy of the former. However, we have seen that the constant differential calculus is in close correspondence with the execution of immediate actions whose procedure cannot even be explained by those who perform them, as for example catching a flyball in the outfielder problem. In this respect we have spoken of the possibility of a fourth-person knowledge or an immediate non-intuitive knowledge which is at the basis of all our assumptions; the aim is to connect third-order knowledge with this other knowledge in the most direct way possible, avoiding with the greatest care distractions and detours. The only certainty that natural philosophy has is that the Principle is not only the starting point, but also the end of all our inquiries.
There are two ways open to the scientist today: to continue on the descending path as a microserf under the protection of the great technocratic powers, or to take the ascending path towards new forms of intelligible simplicity. The first way, in which young dwarfs carry old giants on their shoulders struggling to advance one centimeter towards nowhere, though socially rewarded has increasingly insignificant yields; the second does not promise technical applications and when it does, one must know how to elude them, but it allows a free advance towards what is most significant for man, provided one knows how to stick to the most basic and does not expect a particular return. No matter how many advantages it grants me, knowledge as mastery over external things always distances me from the mastery of myself.
The social individual is a type of socially induced contraction and reaction; the biological individual has a more extensive physical background but is also a reaction to the environment. The individual personality is a mixture of biological and social individuation but is still reactive. The person perceives the individual and his personality, but they do not perceive the person, just as I perceive my body and my thoughts but neither my body nor my thoughts perceive me. The center of the person is not a singular consciousness, but an impersonal simple consciousness that is nevertheless aware of the singular in the person and the individual; just as the formless envelops the form and at the same time is hidden in the balance of its manifestation process. From the degree of alignment between our perception of phenomena, the individual, the person and the impersonal consciousness depend the degrees of abstraction and concreteness that our consciousness of being may attain.
The Middle Way in science is like the axial mountain of ancient cosmographies, which despite being at the center of the world also surrounds it and defines its contours. The way of extremes in science as we know it today seems on the contrary to be all-encompassing and yet blind to the essential. Science can be a mountain of opacity or a mountain of transparency depending on the point of view we adopt, but for most the historical perspective we have reached is of no use. Those seeking the mountain at the center have two ways of approaching it, either by asking for its location, or by wondering how it became possible that they cannot see it.
References
Peter Alexander Venis, Infinity Theory
Ronald H. Brady, Form and cause in Goethe’s Morphology (1987)
Miles Mathis, A redefinition of the derivative —Why the calculus works, and why it doesn’t (2003)
O. M. Dix and R. J. Zieve, Vortex simulations on a 3-sphere (2019)
Nikolay Noskov, The phenomenon of retarded potentials
Mario J. Pinheiro, A reformulation of mechanics and electrodynamics (2017)
K. T. Assis, Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force ( 2014)
Gilbert Simondon, La individuación a la luz de las nociones de forma y de información (1958)
Philip S. Marcus, Suyang Pei, Chung-Hsiang Jiang, and Pedram Hassanzadeh, Self-Replicating Three-Dimensional Vortices in Neutrally-Stable Stratified Rotating Shear Flows (2013)
Shubo Wang et al., Spin-orbit interactions of transverse sound (2021)