A well-known mathematician said once that it would take a million years to understand prime numbers well, if ever. One should have asked him what he aspired to understand about prime numbers that could take that long. As far as I know, no mathematician has even asked the question why the prime numbers, being at the core of arithmetic, do not seem to have any importance in Nature —when yet the Riemann zeta function, so intimately linked to their structure, seems to be reflected in many different kinds of physical systems.
Learning to count does not require counting to a million. Prime numbers only appear in retrospect as a reflection on the order involved in their succession. This is in itself eloquent, if we agree with Poincaré that, unlike geometry or logic, arithmetic follows from a synthetic a priori principle: the recursive definition of the sum or product is irreducible to the logical definition. And what this means, if one thinks well, is that the retroprogressive method has in mathematics as much course as its forward advance by formal generation.
Poincaré comes to say that one demonstrates by logic whereas by intuition one invents; although for demonstration, as opposed to mere verification, which is purely analytic, intuition would also be necessary. I quite agree with his position; but there is still potentially another double retrogressive movement that points beyond logic and intuition. We have called it fourth-person knowledge and also, following the terminology of Jakob Fries, immediate non-intuitive knowledge. Yes, intuition invents because it is synthetic and creative; but all this is under the sign of the impulse of formal generation that is connatural to mathematics, and here we would like to point to something beyond it. For it has long been the case, according with the general tone, that mathematical invention is too speculative, and formal logic too exploitative.
In his reflections on mathematical discovery and invention, Poincaré speaks of a conscious ego that does the arduous work of preparation and of an unconscious or subliminal ego that would make possible combinations on the basis of the previous work. We can verify this not only in mathematical work, but in all kinds of problems with which we sufficiently occupy our mind. But these two egos only indicate the activity of the outermost and intermediate layer of our consciousness; the most basic level does not even deal with forms. This pre-formal level is responsible for our sense of identity and identification, it is just the Self without any other quality than its capacity for identification.
This coincides with the three avasthas of nondual Vedanta: the waking state, the dreaming state and deep sleep state, which are respectively concerned with the external world and the body, the internal world and the mind, and that intimate which is neither internal nor external and which we usually call «I» or “self”, although this very I tends to identify itself with the body and the external experience of our conscious life. Anyway this ternary scheme still remains useful to overcome the dualistic oppositions such as objective/subjective, body/mind, conscious/unconscious, and in any case provides a route to delve into our present theme.
This «third I» at the bottom of consciousness is indeed the most primary, and if we don’t acknowledge it as such is because this same I without qualities is the very principle of identification that commonly adheres to qualities. Nevertheless this self is beyond the domain of form, including mathematical form, and this is the main reason why it has not attracted more attention from this science or any other, such as, for example, psychology. If for Poincaré it was already a very delicate matter to discern the part of the subliminal ego in knowledge, and his ideas, such as the central place of convention in physics and mathematics, have never been popular, something more subtle is still needed to detect the role of this naked self in knowledge beyond the objective and subjective.
Elsewhere we discussed the basic concordance of these three moments of the self with the three categories of Peirce’s ternary logic, as well as with the three principles of classical mechanics, and we also advanced, without much justification, that there is a certain correspondence between fourth-person knowledge and the so-called «fourth principle of mechanics,» which comes to play with a reformulation of the three classical principles in terms of a relational mechanics without inertia valid for open systems with a feedback loop including the geometric phase. Of course, Poincaré already showed that the principles of mechanics are a matter of convention —which, of course, does not mean that they are simply arbitrary; though the same could be said of other branches of applied mathematics, not only geometry, but the very same calculus.
If the conventional character of mechanics has hardly been admitted, much less has been done with the principles of analysis, despite the fact that various forms of non-standard analysis are well known and used. The discovery of non-Euclidean geometries is considered a critical moment in the history of mathematics, yet non-standard forms of analysis have had nowhere near the same impact. And yet, if instead of contemplating only the new forms of infinitesimal calculus, which are still new rationalizations of the original idealization, one were to take into account an elementary redefinition such as the constant differential calculus based on the unit interval discovered by Miles Mathis, this would affect much more closely the immediate object of mathematical thought which is the natural number. For, as we will point out later, and although Mathis himself does not consider it, his redefinition potentially implies the rebirth of the theory of proportion at the heart of analysis.
Let us return to the role of the «third self», which is actually the first one, in knowledge in general and in mathematical knowledge in particular. Mathis rightly speaks of how the problem that his calculus tries to cover has not only defied any solution, but “has defied detection”. And this alone suffices to give us a correct idea of the role of the most basic self, even if Mathis here does not concern himself in the least with questions of discovery. Poincaré’s subliminal ego performs a work of selection. The Frenchman invokes the random collisions of the kinetic theory of gases, but, as he admits himself, this is just a crude simile; it would be more appropriate to speak of the overlapping or superposition of the innumerable possible options. The fundamental I, the same one that operates identifications, is also the only one capable of inhibiting itself from realizing them, and can thus detect both new questions and new truths, which are completely different things. And also, since this I is simple in the sense that it would be absurd to attribute a composition to it, it is responsible for detecting the most common, basic and recurrent facts: the simplest facts, which exist in innumerable ways but which are also the most difficult to recognize since they are veiled by the multiple structures to which we adhere and with which we identify ourselves.
If the primary I is the source of identification, it is also the source of the objectification of thought, and by the same token, it alone makes possible an objectivity without objectification that permits the detection of new questions within the most familiar. Of course Mathis does not reach this point by equanimity alone, but by sheer rectitude in reasoning and by the prior conviction that there must be a much simpler way to justify calculus; but all this facilitates that equanimity which is the condition of recognition “by lifting a single curtain”. Classical calculus is full of recipes and artifices, but no artifice can ever replace rectitude. As for simplicity, which in itself is a sign of the degree of truth and its quality, the challenge is there for anyone who wants to accept it: to show a way for calculus simpler than his.
Of course, if the mathematical community ignores such an amendment in full, it is not because it lacks relevance but just the opposite; at this point, it cannot afford a revision of the fundamentals at such a basic level, which should affect all the main branches of this science. But this tells us enough about the economy of knowledge. Today’s mathematicians use to work at the tactical level, and even when they propose bold connections in their most ambitious programs, for example of geometry with number theory, their evolutions do not go beyond the operational level. The real strategic level is forbidden to them because, as it is easy to understand, going back to the fundamentals one is very careful not to touch the foundations.
The constant differential calculus also has a surprising correspondence in its procedure with the way we act when we try to catch an object describing a parabolic trajectory. Although someone running after a fly ball cannot explain how he catches it, it has been shown more than convincingly that he simply moves in such a way as to maintain a constant visual relationship, naturally without making any complicated time estimates of acceleration or anything of the sort. Incidentally this expose the computational paradigm, many preconceived ideas about cognition and other associated issues. However, here too, Artificial Intelligence experts have made heroic efforts to ignore and minimize these simple facts.
If the purely formal procedure, of simple numerical analysis, of constant differential calculus corresponds to immediate actions involving clearly informal knowledge, we are reconnecting third order knowledge with first-person knowledge, and the mere fact that this is possible justifies the claim of the existence of fourth-person knowledge and thus of non-intuitive immediate knowledge.
Pythagoras, Plato or Euclid still seriously considered the contemplative value of number and proportion. However, contemplation has never disappeared, and it is still easily perceived in the fact that even today mathematician can isolate themselves from the world without a sense of sacrifice and with the best fruits for their activity. Regardless of the era, the mathematician is a spontaneous recluse and a natural ascetic.
Poincaré distinguished three purposes of mathematical work: mathematics by and for itself, mathematics that studies Nature, and mathematics as a philosophy that tries to delve into basic concepts such as number, space, time or logic. He also rightly insisted that mathematics should not be navel-gazing since much of its development is due to what its application raises, and anyone can see to what extent physics, hand in hand with analysis, has favored its growth. But in this classification of goals it can be seen that the primary plane of mathematics is an aesthetic plane —just like the logical category of firstness in Peirce, only the other way around: not with a minimum, but a maximum intellectual component. This is where the autarchy of mathematics, this self-sufficiency that invites contemplation, comes from. Mathematics applied to Nature would be secondary in Peirce’s logical sense, and the philosophy of mathematics tertiary in that same sense. Also for Peirce, however far he was from intuitionism, mathematics had an irreducibly aesthetic component in this relatively immediate sense.
In this very fact that mathematics, an eminently intellectual third order activity, always returns to the first plane as its own, lies not only the autarchy of mathematics but also its indefinite capacity for synthesis. Yet mathematics has expanded too much for anyone, least of all its practitioners, to be seen as a self-contained whole. It is this uncontainable expansion that prevents us, above all, from contemplating mathematics in the same spirit as Plato. Now, our intention here is not to recover the Platonic spirit for this activity, since in fact it has already preserved it more than one would think, and not only in mathematics but also in physics. Rather, we would like the Platonism in which we still move to close the full circle of its transformations and misunderstandings in order to open a new cycle after more than two thousand five hundred years.
The unstoppable expansion, both cumulative and accelerated, of mathematics begins with modern calculus. As we use to remind, if knowledge doubled regularly every fifteen years, today the body of mathematics would be four million times larger than in Newton’s time. There are two thousand years from Euclid to Newton, but although minds were anything but idle during all that long period, one cannot speak of a remotely similar growth. The great difference lies in the emergence of calculus.
With calculus there is a radical inversion of priorities with respect to the external world that at the same time disrupts the primary plane of mathematics. In this first attempt to reverse engineer the change in Nature from known results, instead of determining the geometry from the physical considerations, in order to deduce from them the differential equation, the differential equation is first established and then the physical answers are sought in it. This, which seems the definitive overcoming of Greek mathematics, is in reality only the misplaced maintenance of some of its illusions. In fact Mathis rightly attributes the basic misunderstanding of calculus no longer to Newton or Leibniz but to Archimedes himself.
And this way we have passed from Platonic metaphysics to present day mathephysics with its mirages and rationalizations. Analysis, which soon forgets the humble origins of calculus in the study of curves, replaces physical geometry by another parallel reality. And finally, as a consequence of the attempt to found analysis by arithmetic, the very idea of number dissolves and is replaced by set theory, a new branch of logic. The new mathematical continuum, irreducible to the perceptual continuum that had motivated it, also puts an end to the primacy of number.
Modern mathematics has swallowed too much to be able to digest it properly; not only in the total mass of accumulated knowledge, but in each of its parts, which contains too many operations that are far from being obvious when they are not openly illegal, as Mathis shows even with the most elementary functions of calculus. And just as knowledge in every field multiplies, manipulations of this kind increase in the same proportion. How is it possible that in such conditions, so far removed from evidence, the present-day mathematician is still capable of losing the sense of time for long hours while absorbed in a problem? Poincaré gives us again the answer when he observes that it is not only the order that matters, but the unexpected order —the renewal of identity through different things.
The excess of expansion and the fading of mathematical evidence is one with the primacy given to prediction over description; an imbalance that has been compensated by a justification or foundation that has never gone beyond rationalization. If calculus had been adjusted to the physical geometry of the problems, our idea of Nature today would be way too different, who knows to what extent. Mathis applies his finite method to the most direct solution of a great number of crucial problems of physics, but this is only one side of the question, and in fact the most difficult thing in science is not to want to solve problems that one has not asked in the first place.
In its relation to Nature, the extremes that define the range of action of mathematics are its criterion of application to external problems, which affects in particular the calculus, and its degree of receptivity to natural phenomena. The quality of the connection between both depends on the fidelity of description, which in modern physics is entirely subordinate to prediction. The characterization of phenomena in physics is always much more tentative and approximate than in mathematics, more qualitative as it depends on the domain of measurement —if it is true, as Hegel wanted, that measurement is the synthesis of quality and quantity. But at the same time the intimacy of the physicist with the methods of calculus has created a pre-selection of the phenomena most susceptible of controllable measurement.
Peter Venis’s projective morphology, which is a phenomenology of Nature, gives us a modern contrast of what a truly qualitative approach to the domain of Form can be without the quantitative pre-selection from which physics starts. Unlike Mathis, Venis does not try to solve famous problems, he discovers new questions in need of a more rigorous formulation. But in what sense rigorous? In the same sense that we have been applying calculus to physical problems while ignoring their physical geometry? Let us hope that Mathis’ warning will do some good. Mathis’ critique of calculus goes far beyond the well-known objections of finitists and constructivists, and this is why it is ignored. Mathis and Venis perfectly illustrate the two extremes of the criterion of application and responsiveness to phenomena, and the quality of the connection established between both will also define the quality of their development.
As we have seen before, the physical meaning of dimensions in Venis’ morphology has yet to be clarified. But Mathis makes it clear that analysis has never been able to correctly analyze even the number of physical dimensions of the most basic problem, as shown by the mere fact of representing a straight-line acceleration by a curve. The question is not only the representation since in any case the analysis of change still depends on the given point and instant. According to Venis, however, all geometric manifestations, whether points, lines or planes, have reality only as projections and sections of a homogeneous field or medium that can have zero or an infinite number of dimensions.
Venis conjectures that the study of the wave-vortex will require the use of complex numbers and techniques such as those derived from the Hopf fibration. Mathis on the other hand sees no need for wave mechanics in quantum mechanics, nor for the use of complex numbers in physics, not to mention that he also shows that even the application of the most elementary trigonometric functions is not understood. The perspective of the two authors is diametrically opposed and yet should also be complementary. Venis’ morphology puts forward another idea of the continuum that reconnects with the perceptual continuum, but this perceptual continuum need not coincide with the treatment of continuous media in physics, nor can it refer to the classical use of calculus in them.
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 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.
If we go back to the role of the three egos in mathematics, the conscious ego is on the level of form, the subliminal ego on a pre-formal level, the primary self on the formless level. In India the manifestation of any impermanent entity is seen in terms of nama-rupa, the concurrence of name and form. In the West we do not usually contemplate this pair, which has no dualistic connotations, but we do speak of mind and matter, which for us does; a more akin translation would be that of essence and accident. Mathematical form is detached from matter, but the form contemplated by Venis’ morphology is one with it. This view of morphology cannot be dualistic because form depends on a continuum of non-integer dimensions and these dimensions depend on perception. However this does not exclude an objective description because each physical potential is not merely a position, but a perspective: the circumstance of the environment from the perspective of an agent.
Put another way, number as a mathematical form is detached from the Word, from language, but the most qualitative aspect of forms, which in physics has always been considered contingent, is a direct expression of the language of change, that same change that has always escaped like water from the web of artifices woven by the calculus. The question is how it could not escape. This can only be explained by the continuous change of dimensions and by the shift of the potential. For the former there is already the fractional calculus, widely used but lacking credible physical interpretation; for the latter we have in physics the geometric phase and the theory of the retarded potential, aspects undoubtedly linked by their dependence on the environment but which physics has not yet been able to connect.
Morphology opens an indirect way to the knowledge of forms, ideas or essences, precisely insofar as we abandon the Platonic pretension of a separate, immaterial and immutable reality. In this wave-vortex morphology, the substantive is the wave emerging from the homogeneous medium, while the adjective is the cut or dimensional section that produces the appearance of the vortex. That which gives name can only be that which identifies, and only that which is formless and lacks identity can identify itself with something. Physics has been unable to grasp the process of individuation of an entity, but that is precisely the question of morphology; answering that question, it overcomes the nominalist reduction of everything possible to the individual and undertakes the way of return.
Economy of thought is essential in any science and even more so for mathematics, but the degree of economy of thought depends crucially on the right choice of terms, and precision presupposes prior indeterminacy or ambiguity. Accurate naming allows one to assume things that would otherwise be much less tractable, and Poincaré sentenced that mathematics was the art of giving the same name to different things. The importance of this maxim should not be underestimated, even though the relationship of mathematics with names and with the art of naming is anything but fortunate: one only needs to realize the absurd naming of such important and honed concepts as the «real numbers», «imaginary numbers”, «complex numbers», or “topology» —which actually deals with that which is independent of position— to be convinced of this. One more symptom is the unstoppable proliferation of proper names that strip the new concepts of their very conceptual contribution.
The economy of terms, the search for synthesis through the name, remains an almost entirely untrodden path for mathematics. Analysis and synthesis are supposed to be reciprocal, but in practice analysis has overwhelmingly predominated in modern mathematics. This, however, holds a sort of promise, since this overwhelming predominance has nothing to do with the quality of analysis, on the contrary, if we duly appreciate the fact that analysis does not know how to analyze even the dimensions of the simplest changes. Since without good analysis there is no good synthesis, we should be concerned with the quality of the steps, instead of the quantity as a technified mathematics, blinded by the achievement of results, has done.
Today emphasis is put on the unity of mathematics above the multiplicity of its manifestations, applications and branches. In this sense, we would then speak of arithmetic and geometry, of algebra or analysis more as a tribute to the history of the development of the subjects, the contingency of their appearance and the present state of knowledge, than out of deep necessity. But on the other hand, mathematics without its form is pure indeterminacy and potentiality. Mathematics in act is pure form, but without form there is no objectivity. Science must be objective if it wants to have a social function and not fall into mysticism, but the subject cannot stop reading continuously between lines and planes, displacing the value of signs.
To follow any mathematical argument one needs to know the underlying structures, which partly depend on the contingencies of historical development and partly are subject to progressive rearrangements. Mathematics is thus eminently historical not less than eminently timeless, but the relation between the synchronic and the diachronic within is subject to constant redefinition. In Philosophic Semantics and Philosophic Inquiry, Richard McKeon characterized with great clarity the four basic methods of philosophy —dialectical, problematic, logistic, and operational— and suggested that there is no question raised 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 from a material rather than a formal perspective mathematics may find it more useful than the conventional division of its domains into arithmetic, geometry, algebra, and analysis.
McKeon’s modes of inquiry —assimilation, discrimination, construction, and resolution— are themselves a recapitulation in a historical key of Aristotle’s famous four questions at the beginning of the second book of the Posterior Analytics: whether it is, what it is, what sort it is, and why it is —or questions of experience, of existence, of that which is, and of being. As it is well known, these four questions became the four constitutiones of Roman rhetoric, thus entering into legal and political philosophy; however, this foundational work of the Aristotelian canon is the first explicit attempt to formalize the method of scientific knowledge at a time when it was not formalized at all, and no doubt the geometrical analysis of contemporaries such as Eudoxus is at the basis of his reasoning. Aristotle does not do geometry but he expounds what geometricians do.
The four cardinal questions of the Posterior Analytics —to which is added the knowledge of what the name means- do not seem to have an internal symmetry, but the modes that McKeon derives from them can be conceived themselves as the four angles of the activity of knowing: the knowledge, the knower, the known, and the knowable. The modes of inquiry thus “serve to unravel the tangled history of the methods of induction and deduction, analysis and synthesis, discovery and proof which first emerged from distinct modes and were variously merged with each other and inverted.»
The four questions and their associated modes are on the intermediate, pre-formal plane of semantics and heuristics, between the formal and the formless, between logic and metaphysics, between what is known by definition and what the name hides. But if Aristotle, reasoning within an unformalized science in which almost everything quantitative was still expressed verbally, tries to descend towards the undeveloped formal plane, today we can use the intermediate plane to ascend the scale of generality of the mathematical concept, from the unnameable in actual mathematics to the unknowable in the name.
Everything in our modern age was born out of a pronounced imbalance, lives out of imbalance, and will end with imbalance. Much has been said about the separation between «the two cultures» of the sciences and the humanities, but little notice has been taken of the fact, and McKeon barely does it, that the modern sciences have given almost all their preference to the logistic and operational methods, vehemently rejecting the dialectical method exemplified by Plato and the problematic method inaugurated by Aristotle; the latter two have been relegated to the verbal realm of politics and the humanities. Even the life sciences, from biology to medicine, have been reduced in their theory to the combination of the first two, leaving the other two just for «the discussion of their social and human aspects” —for what is now understood as «rhetoric».
Accordingly, it could be thought that the first two methods are more suitable for quantitative analysis, which is so predominant today, while the second two would be better adapted to the subtleties and nuances of the qualitative. However, mathematics is a complete whole that includes the quantitative and the qualitative. The question is similar and is linked to the purely conventional distinction between «pure mathematics» and «applied mathematics»; mathematics always tends to go full circle between theory and application, their conflation being only a matter of time.
Poincaré said that there are not solved and unsolved problems, but more or less solved problems, and history proves him right since the very idea of what is an acceptable proof or solution changes insensibly with the times, just as other key notions of mathematics change. On the other hand, any important mathematical concept points in multiple directions at the same time, and the four questions and the four modes would be a synthesis of the possible fronts from the point of view of the contents.
It has been said that every great culture has had its own concept of number, but ultimately every mathematician and every person has his own concept, which results from his perception of the whole of mathematics and its unity, or rather his lack of it. This perception, and the will to mobilize it in one direction, is what determines the spirit of a research program —if we understand by spirit the unity of intellect and will. The four questions and the four modes can be used to choose a formal course of investigation, or else to find a more neutral and equanimous perspective, an «objectivity without the will to objectify». The first is a descending path directed towards terms and solutions, the second is an ascending path that tries to distill the ideas enclosed in the conventions liberating the will locked in the contents.
The four modes admit as many variants in their principles, methods and interpretations, as well as selections, so it is not difficult to elaborate a notation of their 4 x 4 x 4 combinations, which in any case should be seen as ideography or mnemonics rather than as a formalism. Applying these schematism to modern mathematics may seem an eccentricity at this point, but, if so, it would rather be due to the fact that it is this mathematics that has had an eccentric and unbalanced development. Justified by specialization, there is an interest, essentially unintentional but no less jealous of its forms, in keeping the methods of the sciences and the humanities isolated in such a way that they can never communicate or translate meaningfully.
The still recent mathematical theory of categories, with its successive elaborations and extensions, has become a natural language of mathematics at both the most basic and the highest levels of abstraction, and yet it is still the fulfillment of Aristotle’s old project of making explicit the content of the categories of concepts. In fact his book on the categories or predicates is much more metaphysical and remote from mathematics than the Posterior Analytics, but the idea that persists from this foundational text basically boils down to the trite scheme facts→ induction → principles → deduction. McKeon himself applied the modes to the historical approaches in modern physics but avoided to deal with mathematics, and yet there would be nothing «out of place” in it but for the places where modern developments have led. Apart from the fact that theoretical and applied mathematics always go full circle, we argue here, moreover, that it is only by delving into its application to the physical world that mathematics yields its ultimate philosophical value, including in it that of the philosophy of mathematics. And it is precisely the failure to take this seriously enough that most limits the present mathematical theory of categories.
The mathematical theory of categories, according to its initiators a metamorphosis of Klein’s Erlangen program, is a deepening of the common basis of logic and geometry as spaces of transformations. Of all the advocates of this program, it has surely been Lawvere who has endeavored most to bring the viewpoints of mathematics and philosophy closer together, having tried to model even dialectical logic and the identity of opposites in calculus and physics. Lawvere has always tried to connect categories and physics, especially continuum physics, but if we duly ponder the «selectively unbalanced» development of modern sciences, physics included, it is easy to see why these attempts have not gone far despite the permanent growth in the applications of the language of categories.
To what extent do the four questions and the four modes of thought define an axis or common ground for the selected topics, beyond the nature of the topic itself taken in its first generality? This would have to be investigated with a sufficiently large series of examples to come close to an answer. The questions have no apparent symmetry, but the modes are not entirely exclusive and admit a common ground. It would be desirable for mathematics to explore this in a heuristic fashion, not so differently from what it has done in its theory of categories, as this last one had a pronounced heuristic profile right from the start. If the mathematical theory of categories acquired relevance in its day because many of the concepts that mathematicians handle were far from being explicit, the semantic and research schemes are particularly relevant because since the scientific revolution two basic modes of investigation have taken precedence at the expense of the other two, and it would be necessary to bring to light what has been left in the shadows in order to have a broader and more neutral perspective.
Any subject or problem should be able to admit four cardinal formulations, but the most important thing is what their superposition may illuminate. Today, for example, the analysis of dualities is not just a method but almost a general principle of research in physics and mathematics that has allowed great advances in territories that would otherwise have remained obscure. The contrast between the four modes may bear some superficial resemblance to the analysis of dualities, morphisms, homologies, maps, correspondences and transformations in general but leads to a different domain of possibilities.
There are not a few who, like Dingler, consider Poincaré’s conventionalism «naive»; one readily concedes that principles such as those of mechanics are not directly dictated by experience even though they are drawn from it, but it is not admitted that there can be any other possible result. We have however good reasons to think that it is the latter assumption that is naive: we have repeatedly seen a relational alternative to the three principles of mechanics that for the most part allows the same predictions as Newton’s principles, even if it must inevitably yield discrepancies as well, which are otherwise not difficult to justify. The overlay of modes of investigation should show both the discrepancies and the common ground; but in contrasting them we are also contrasting developments that tend to exclude each other in time.
Despite its understandable limitations, Aristotle’s method in the Posterior Analytics is essentially retrogressive, since it tends to go from conclusions to premises and principles; and yet it remains analytic as its title promises. The main interest of explicitly contrasting modes of investigation is heuristic and intuitive, but as we note, this can be understood in a double sense and not only in the currently predominant sense of «solving problems». In fact also here it is possible to go back from our intuitions, always conditioned by habits and acquired knowledge, to a «non-intuitive» base that is momentarily excluded but which is actually broader.
The pertinence of this synoptic approach can only be verified through the study of concrete cases, cases in which the material implications of a problem leave ample room for ambiguity. Almost all applied mathematics falls into this category, but also concepts and problems of pure mathematics. Just as in the sciences, beginning with physics, the constructive and the operational approaches have taken hold, so in mathematics the presence of analysis and algebra has grown at the expense of arithmetic and geometry which were at their origin; one need only see that in such vast areas as current algebraic geometry or non-commutative geometry any figurative notion of geometry was lost long ago.
Both well-defined solved theoretical problems and unsolved ones can benefit from this unfolding and overlapping of the four modes of investigation. Although contrasting the modes requires a conscientious work of preparation and composition, what emerges again can be seen as a distilled form of paralogy if not analogy. Viewed from the outside, Aristotle’s four cardinal questions are a way of systematically present the results of scientific investigation; but the constructive interference of the four modes of thought contains something that is too internal even to the very activity of investigation and tends to be reduced or rejected. This «something» is something cognitively relevant and points beyond systematic organization: it points to the inarticulate arcana at the origin of language and name.
At the other extreme, this breakdown can be applied to problems which resist univocal definition and where application and theory, the quantitative and the qualitative, are totally mixed and muddled: a glaring example of this is money and the monetary system which regulates it. No field can be less pure than this, and yet the directed technological evolution, the politics behind the technology, and the juncture of global geopolitical confrontation place this instrument of domination at the most acute of crossroads. However, the digitalization of money, which is only one more episode in the digitalization of consciousness, albeit a key episode, also runs, as a presumed technological solution, within a publicized but secretive meroscopic or logistical-operational axis that blocks the holoscopic, dialectical-problematic axis. Naturally, those who wish to increase the control of the ruled have to argue that up to now a fair monetary system has not been possible due to technical difficulties which are now definitely overcome, while trying to obviate the long history of collusion between money and debt which is still reinforced by the new tools. However, the simultaneous contrast of these four approaches should reveal something quite different from a mere historical analysis.
Technological solutions to the money problem always tend to optimize control of the population by a tiny minority. But much of what is conceived as the best alternative solution is also a hostage to technological solutionism in one way or another. Surely the best options are those that depend less on the degree of technological development, since those that defend the logic of the irreversible only want to consolidate a capture operation. We believe that the contrast of the four modes suggests at all times a certain timeless axis, a possibility and a way of evading the mechanisms of coercion based on the supposed irreversibility of progress. Needless to say, forcing an irreversible framework only bring us closer to the catastrophe.
According to the myth, human language has undergone successive stages of degradation, and propaganda is the best proof that language can not sink any lower. In addition, the quantitative and statistical has always been one of the most basic tools of such endeavor. Economic theory and its analysis is the supreme exponent of this radical perversion of means by ends using quantity as an alibi, of an ideological elaboration deliberately constructed to divert and conceal, but it would be very naive to think that the rest of the sciences that rely on quantity have not exploited its enormous potential for duplicity; when economics began to make intensive use of mathematics physics had already been doing so for more than two centuries.
Logic and geometry can always be reconciled, but physics always takes us beyond extension and motion, and it is this intensive aspect that gives it all its interest. The intensive is no less subject to quantity than the extensive, but its nature is not reducible and for us it is like an adjunct to our representation. Yet even natural language seeks to reflect in its words such intensive qualities as color, density, or tension. There is something profoundly physical in the name, and it is precisely that which escapes the domain of form and extension.
In science and in physics in particular, besides deduction and induction we have abduction or hypothesis, but this is only a veiled form of analogy to which an attempt is made to give a mathematical expression. Analogy plays on the other hand a genuinely functional role in modern physics since the Lagrangian and Hamiltonian variational principles, which are at the basis of the fundamental laws, are nothing but exact analogies.
The so-called fundamental laws and the hypotheses that have given rise to them ignore almost everything about the simple fundamentals of mathematical proportion. Their elegant-looking equations not only ignore the principle of homogeneity of physical proportions, but over time they pile up more and more heterogeneous quantities, as many knots to be untied but that are never untied, remaining at the basis of the inexplicable numbers and constants that must be entered by hand for calculations to work. An analogy without proportion is just a black box.
Mathis’ constant differential calculus, being the natural form of the method of finite differences, reintroduces the possibility of a theory of proportion into the analysis of change. The theory of proportion was basic to Greek mathematics, and without it number and quantity are disconnected from the Cosmos and the idea of the good. On the other hand, it has been reproached to Eudoxus that his theory of proportion, which passed to Euclid and Archimedes, only succeeded in separating geometry from arithmetic, blocking the development of algebra and calculus. But today we can interpret all this in a new light.
When Aristotle says that a problem of geometry must seek its principles in geometry and not in arithmetic, on the one hand he is only vindicating his own philosophical mode which demands that each problem have its specific reflective principle, unlike, for example, the global principles of Plato’s dialectical system, or the simple or actional principles of the other two prototypical modes; but on the other hand he is also echoing the dominant idea of the time, endorsed by Eudoxus, that the two are categorically different sciences.
Today there is an emphasis on the unity of mathematics but only on a large scale, since at the most basic level analysis has become completely dissociated, much more so than in Greek times, from geometry. If anyone ever succeed in bringing them together again, it is Mathis, and yet he does not consider the implications of his work for completely recasting the theory of proportion in Analysis —when in fact he has found so many elementary transparencies and correspondences for the «mysterious numbers» of physics. Certainly no one is obliged to take these explanations for granted, but it must be recognized that they are much simpler than most speculations, and that means something.
Mathis also takes pleasure in showing multiple examples of how physics is incapable of identifying the causes of even the simplest questions, things like why a sail allows one to sail, and it would not take long to see that his basic criticisms fully affect not just one, but all the four cardinal questions Aristotle poses within the context of geometry and astronomy. Since, unlike the Greeks, here we are not only convinced of the unity of mathematics, but believe that the degree of unity of its concept directs reasoning at higher levels of elaboration, it would be valuable, rather than embarking on large programs connecting vast areas, to attempt to connect the four major branches of mathematics and the four modes at the most basic level, where decisive divergences and complementarities already arise.
What can be seen in the current “grand unified programs” of mathematics, such as Langlands’, is a radical separation between extremely elementary questions and extremely sophisticated theories; and such a lack of middle ground, so necessary for science, forces one to doubt not only the theoretical subtleties but also the choice of basic elements. If applied mathematics exhibits holes such as those revealed by Mathis, one cannot expect theoretical speculations of the nth degree to have more consistency, but rather the opposite. One must search for the minimal matrices of meaning that are representative of a totality attending the foundation, not ignoring it.
The Aristotle of the Posterior Analytics was already doing metascience and metamathematics, as the modern category theorists would do again twenty-three centuries later. And in both cases we have a struggle between the search for the organic and the attempt to organize vast theoretical panoramas. For also the mathematical theory of categories emphasized from the beginning the idea of natural transformations; but within the context of the great mathematical industry, it is difficult to see what can be «natural» and what «machinery». All present-day “grand unified theories,» whether in physics or mathematics, have a clearly conservative function, exhibit common ills, and are expressly designed not to rethink the fundamentals except where it is harmless: they are the escape route for the manifold contradictions of modern research. Mathematicians have been enormously perspicacious in exploring the possibilities of their science, but only in a definite direction —the one that preserves the achievements and limitations of a model that, although it seems to transform continuously, has not changed in essence since Newton.
So the «natural transformations» of mathematics have at least as much of history as of nature, a history that is not only made by its «impressive successes” but must necessarily have a counterpart. We have seen one way to unravel, at least partially, these compromises. Another way is to apply from the outset the natural transformations of concepts to a framework that deals as directly as possible with the very forms of phenomena, such as Venis’ projective morphology —a morphology of the vortex that has been formulated in terms of pairs of opposites. This morphology is still awaiting proper formalization without the historical connotations that analytic and algebraic tools have acquired, but on the other hand it has a far superior analogical potential, a potential that could still be much better realized if calculus and differential geometry can naturally incorporate the theory of proportions. Vortices, as topological defects, are a natural object for the modern language of homology and cohomology, but in projective morphology they acquire an intuitive value they never had before.
Venis’ morphology naturally incorporates the idea of morphodynamic equilibrium, and a proper idea of equilibrium is necessary to connect and give physical content to the theory of proportions. Of course, equilibrium by itself is such a general concept that it cannot but be present in a thousand ways in physics and mathematics; but again we meet with truncated modalities of equilibrium from the very beginning —from the same three principles of Newton’s mechanics onward. Equilibrium is as principal in open systems as are conservation laws in closed ones; but, contrary to what is supposed, we have reasons to think that the latter emerge from the former. So we have insisted on the importance of replacing the intrinsically contradictory principle of inertia by the principle of dynamic equilibrium, with the consequent modification of the other two; just as we have emphasized the importance of ergontropic equilibrium, and the equilibrium of densities with a unit product. The connection of these four equilibria is already a whole research program, and quite different from the present ones; the golden scale of Zeus has nothing to weigh in closed systems.
These other forms of equilibrium play a fundamental role in the morphological process of individuation, and thus it is possible to connect equilibrium, proportion, form, proper name and transformation. This would in turn create a new constellation for the relationship between quantity and quality, language, logic and number. The untruncated study of balance in Nature and in human activity, including individuation processes, points to an understanding of purpose and time that transcends teleology and dysteleology, our prejudices about the perfection and imperfection of individual phenomena of any order, be it a wave, a sun or a civilization.
Whereas the great “unified programs” point toward an indefinite unity, here we consider that what directs intuition is always behind judgement, in the minimal matrices capable of condensing a balanced totality. The balance of the four questions and the four modes are some of these matrices, but even the three principles of mechanics are the extreme contraction of the three basic modes of the principles as starting points, basic distinctions and foundations of unity, which determine common limits of global closure.
Mathematics self-exam is revealed in a particular way in its diagrammatic dimension, something that Peirce rightly emphasized. As a visual icon, the diagram reverts reasonings of higher level towards the most primary dimension of thought, and this retrograde movement is essential not only for analysis but also for new syntheses; Peirce wanted to use it above all for logical analysis and had to assist with resignation to the most involutive effects of this process. The last decades of the twentieth century have seen a resurrection of diagrams with the study of Riemann surfaces and their topological and group correspondences, but this too must have a diminishing and limited return, because, as Alfonso De Miguel Bueno claims, the first thing that should be translated to the basic plane of the diagram are the most elementary aspects of numbers, groups and models of physical systems.
We start from the obvious but little recognized fact that current mathematics is full of leaps into the void, sleights of hands and the most varied subterfuges. In these conditions of rarefied abstraction, diagrams must be, to use Peirce’s terminology, degenerate icons. For diagrammatic reasoning to reach an optimum of iconicity —of relation between the sign and the object— it should try to meet those conditions that are now so much despised: descriptions with homogeneous quantities, correct physical dimensions, analysis with finite quantities, and so on. When diagrams are sufficiently immediate, as are some of Bueno’s, there is no need to attend such demands.
In any case, since the icon is the most primary thing in mathematics, and is opposed to the arbitrariness of both the tertiary symbol and the secondary index, the search for an optimum of iconicity supposes a maximum of suchness, of participation in knowledge, and an approach to that which we call fourth-person knowledge and non-intuitive immediate knowledge, with the possibility of connecting elaborated knowledge and immediate participation. The intelligibility of the real depends on continuity. The homogeneous medium cannot be the ultimate interpreter insofar as it rests in itself; but to the extent that demands the maximum degree of continuity between its most distant moments of intelligibility such as Peirce’s three categories.
It should be emphasized that diagrams with high iconic value are much more than heuristic aids and even reverse the sense of the current research. If it is true that each culture has treasured its own intimate sense of number, and only in the final phase of civilization does an interest in the «mathematical results» of other developments arise, the conversion of its great motifs into icons of high purity is equivalent to the moment when the seeds of a tree come back into contact with the primordial soil. For example, number theory has been shaped by the contributions of three very different cultures, represented by the names of Euclid, Diophantus and Gauss or Riemann; what would it mean to find critical diagrams within these motifs, as Bueno seeks to do? It would mean going beyond the search for results, beyond the will oriented in a certain direction; it would allow the sense of the language of Nature and the meanings of our languages to coincide in ways hitherto forbidden by the very direction of development. In a retroprogressive sense, diagramatology can be both science and technology, although opposed to the present trend towards complexification.
Surely the only open problem in mathematics that seems to have by itself a potential to change our basic idea of numbers is the Riemann hypothesis. And this for several more or less obvious reasons: because it is the most basic relationship between addition and multiplication; because it directly affects the integers, the relationship between natural and prime, rational and real numbers, and by extension the other number fields; because it has multiple physical and mathematical connections; because it invites us to interpret the relationship between real and complex numbers in the physical plane; because it establishes the most irreducible link between chaos and order, chance and necessity, simplicity and complexity; and because it is the only possible unifying motif, together with its large family of associated functions, for a theory, that of prime numbers, which would otherwise be the least unitary area of mathematics. The only kind of unity that it is reasonable to expect from the prime numbers are asymptotic laws, otherwise of the most diverse orders, for which there can be no more general common motif than the Riemann zeta function.
Faced with a question like this one, some strive to achieve a proof and others seek a better understanding and intuition of the problem. Arithmetic and geometry continue to be basically dissociated, even though the efforts to connect them are stressed today. Thus, for example, there have been attempts to build bridges between the theory of the zeta function and the most sophisticated algebraic geometry, which has so little geometry in it after all. Since it is a commonplace that «all geometry is projective geometry», that the latter is simpler than affine or Euclidean geometry, that it shows a persistent tendency to reappear in unconnected fields, and that it provides an undoubtedly intuitive basis of the widest range for asymptotic motifs, one may wonder why this function has not been successfully linked to the immense possibility of the projective in general.
The way to establish the link would have to come from physics, and electromagnetic theory in particular. It is well known that multiple correlates of the Riemann zeta and the zeros of its critical line have been found in electrodynamics, from the potential of electrostatic fields to far radiation patterns; as it is also known that electrodynamics has resisted every attempt of geometrization and that there is an irreducible statistical component in it. Delphenich observes however that the metric theory of electromagnetism may well conceal a more general geometrical question of complex projective geometry commensurate with the transition from point mechanics to wave mechanics, and his arguments are worth pondering. On the other hand, it is assumed that the zeros of the Riemann zeta emerge from the difference between a sum and an integral. The Lorentz transformation in electrodynamics shows a one-to-one correspondence with the projective Möbius transformation, but the same can be said of Weber’s electrodynamics with retarded potential, which in addition gives us a different interpretation of time.
In a word, if there are electrodynamic correlates of the zeta function, there should also be non-trivial correlates associated with the projective geometry of waves, and this, which in principle has nothing to do with any kind of proof, does have the potential to alter our qualitative conception of numbers and their use in the physical world. Obviating harmonic analysis, this longitudinal cut would make it possible to connect the function with other questions more or less unrelated: the fundamental irreversibility of radiation, equilibrium in open systems, Weber electrodynamics and its relational mechanics, the relation between retarded potentials and the geometric phase, the shaping of a wave front, the projective morphology of the Venis wave-vortex and its transition between dimensions, the theory of proportion in differential projective geometry, and so on.
For identical reasons to the holographic principle, the torsion morphology can trivially produce any surface or pseudo-surface, while according to Voronin’s conditional universality theorem, the Riemann zeta function can reproduce any analytic curve with any degree of approximation an infinite number of times. The zeta function is already an extreme numerical synthesis that is looking for another synthesis in its extreme counterpart. And there would be a «projective arithmetic» quite different from those that have been developed since the time of von Staudt, with its theory of proportions and correspondences, its algebra and its own “science of the balance”.
Riemann’s express motivation was prime numbers counting but the origin of his intricate elaboration is in the theory of functions; as for the prime numbers themselves, which can already be counted in the classical manner without much difficulty, the German mathematician could only expect them to behave fairly, just as in the probabilities of flipping a coin an infinite number of times. But the great mass of implications of the function cannot come merely from this ideal balance, but from its fusion with complex analysis, and the converse question is whether by recomposing the idea of the function and the differential one can draw lessons for the very idea of number. Projective geometry is purely qualitative, and evidently, the interest of finding a projective, as well as a physical, analogy for the function, consists in distilling the qualitative aspects of a purely analytic numerical expression. The concept of function should naturally correspond to that of process, and yet, due to the rupture of analysis with description, there is a huge divide between them. If truly the Riemann hypothesis can tell us something qualitatively new about number, this would be the best way to investigate it.
The simplest interpretation of the hypothesis as infinite balance must maintain an essential continuity with the countless less obvious, deeper interpretations; but since according to the present standards it does not serve to prove anything, it cannot be taken into account. Here, on the contrary, we believe that the proof is secondary to our capacity of understanding and assimilation, and this increases by maintaining the same principle at different levels and observing the transformations of its application.
We have seen that it is difficult to justify the critical line and the calculation of the zeros from the very criteria of the Cauchy-Riemann equations, however much the mathematical community has considered them unproblematic; we have also conjectured that the same identification of a critical line and its zeros could arise from a projective analogy in the context of the densities of an electromagnetic theory with retarded potential such as Riemann’s. And it is inevitable to think that if mathematicians have accepted Riemann’s calculations it was not so much because of their analytical rigor as because without them there would be no axial motif in number theory.
But even if these connections on a large scale were achieved, and took place at a much more intuitive level than in the present macro-programs, their qualitative impact on such things as our idea of number would be limited to what we are able to assimilate in the minimal matrices of meaning and in the maximum contraction of principles —they depend on the degree of harmony and solidarity with them, just as in mechanics the basic disposition have much more weight than the results of the nth level. So today it is perfectly illusory to expect «revolutions» from higher research, which is nothing but an institutional luxury and a last precipitate of civilization. Also in the teaching of mathematics this is the most important thing, no matter how much the institutions try to preserve the inherited forms. But the scandal of modern mathematics is not the institutional luxury of high-level research, but the way in which the quantitative has become an instrument of control and of closure of the system on itself.
The modern Technopolis wants to conceive itself as Cosmopolis but it exists to the extent that it reduces and denies the autonomy of the powers of the Cosmos. And so, for example, it insists on linking climate with human behavior as if more important factors could not exist.
Nothing encodes the fate of our technical civilization better than the myth of the city of Tripura. The triple city has a date with itself but evades it as best it can because the alignment of its three levels also implies its destruction. Technology demands the perfection of its own sphere, however alien it may be to human nature; but perfection cannot be expected without the concurrence of that cosmic element which it has tried to evade during the symbolic thousand years that close the cycle of its development.
This open cosmic element, can be either outside or inside the great social animal; it hardly matters as is bound to remain ignored till the end. At the entrance of the temple of modern science one can read the two great precepts: «Ignore yourself», and «Everything in excess», and its countless priests and dependents try to keep them with the most abnegated jealousy. Since the worst of this science so little our own is its spurious sophistication and its exacerbated disconnection with almost anything that matters, we have always tried to re-establish some kind of link between the idea of knowledge that has been held in other ages and modern scientific knowledge, even being aware that trying to marry prudence with folly need not be of any benefit for neither sides. There is, however, an organic link between all generations, and reinforcing it when many seek to destroy it is more important than any isolated result.
A crucial feature of modern science is that it generally sustains an entitative interpretation, a combination of operational and logistic methods, and comprehensive or global principles that descend directly from the principles of Newtonian mechanics. Since these in turn have as their point of departure the closure with respect to open systems and as their tacit point of arrival the global synchronization implicit in the action-reaction principle, this circuit draws the horizon of their whole teleonomy. We have seen, however, how these same principles can be reformulated in terms of equilibrium for open systems without the need for imposed closures or metaphysical instances.
In the XVII century, at the start of the scientific revolution, people like Stevin or Newton still believed that Europe was just beginning to recover a lost original wisdom; but the progressive formalization and specialization, together with the amount of accumulated knowledge, made that idea definitely implausible. Today the most that can be admitted is that in other times man relied much more on the power of analogy, which strictly speaking is always an inverse relation; but we already see that scientific knowledge cannot do without its ascendancy either, neither in hypotheses, nor in mathematical procedures nor in interpretations, and in fact it exploits it systematically, although only in a definite direction. Newton himself came to his greatest discoveries in optics, calculus, gravity or the principles of mechanics reflecting about inverse problems.
Thus, inverse problems were at the fulcrum of his natural philosophy, and it is the direction given in this turn, more than anything else, what created the drift of modern science. That is why it is necessary to reverse the order of the sequence knowing full well that a mere return is neither possible nor desirable. Analogy has always been and will always be the most powerful tool of theory; the important thing is to know how to consistently give it another direction when it has been sufficiently seen where it leads us with its present orientation. There are still those who believe that science can be partial in its application but never in its principles; but this is to ignore completely the symbolic range of the framework of knowledge itself. And domination consists precisely in the framework that one manages to impose. Much more than the results, what matters is to control the direction in which the spirit of others works.
To give another orientation to science does not pass through the latest theoretical by-products but through the conscious restitution of what is fundamental. Today «theory» is synonymous with the speculative operation of knowledge, but we know that for the Greeks it basically meant contemplation. The very idea of number can only be linked to the good, beauty and contemplation by recovering its nature and value as proportion, which cannot be limited to the sphere of geometry. Today, restoring the meaning of proportion to number means appreciating its natural presence in calculus, which is no longer impossible after more than three centuries of endless analysis.
This unexpected emergence of the theory of proportion in analysis should be closely connected with a general theory of equilibrium in open systems, which will return us by the straightest way to the ancient Science of the Balance. Arabic algebra was literally born as «the Science of Restoring and Balancing”, but separated already in al-Khwārizmī from natural philosophy, it soon rolled down the slope towards the science of substitutions and the indiscriminate logic of universal equivalence. Duly linked to the theory of equilibrium in open irreversible systems, algebra can leave the realm of universal exchange to become again the science of the irreplaceable.
The geometric phase shows that quantum mechanics is incomplete as is any conservative mechanics, but most prefer to believe that this is a secondary detail. Nor has it ever been seen that the same ray of light returns to the flame or to the light bulb, and yet everyone speaks calmly of the temporal reversibility of the fundamental laws. Thus physics would like to maintain for its object the same timeless status of mathematical truth, but not without an irreparable loss in our perception of reality. This inverted Platonism is at the origin of modern technologic optimism and its selective disconnection with many aspects that are not only essential but also vital.
The geometric phase allows to shape a wavefront modulating its potential instead of using controllable forces as is prescribed in physics. This consolidates the existence of a third domain of experience in the history of the discipline, since, alongside perturbative physics, such as that of cannons, machines or accelerators, and non-perturbative physics, such as that of astronomy, there is now physics that operates by tuning or modulation. However, everything is still interpreted in the old framework, even if the force is no longer fundamental to both quantum mechanics and relativity. But the connection between the retarded potential and the geometric phase, or between the potential in the dynamical equations and the one beyond them, directly affects the theory of equilibrium as well as the morphology of observable processes; it even has to find an explicit correlate in biological feedback processes connected to our consciousness.
Consider the relationship between respiration, its bilateral cycle and the phase shift of the retarded potential. Since, as we have seen, the retarded potential is a ubiquitous phenomenon and is equally present in biological functions, the study of its equilibrium in a case like this, which is also linked with consciousness, and its analysis in terms of finite intervals offers us a limited but precious example of what the internal side of the mathematical law entails.
It is in open systems that equilibrium can acquire something of its true dimension, and it is in a finite calculus such as constant differential calculus that questions of proportion reappear in modern mathematics and physics. The ancient Science of the Balance deals in addition with the proportion between the manifest and the unmanifest, which in physics finds some partial correspondence with extensive and intensive properties. There is, of course, a whole gulf between the intention and quantification of the one and the other, but nevertheless a certain continuity still persists which is based on the generality and multiple planes of the very concept of equilibrium. By contrast, with the tacit principle of global synchronization in conservative mechanics it would seem that everything is on the same plane of causality, which incidentally cannot be specified because it is already established beforehand.
The Science of the Balance would also be naturally related to projective geometry and morphology, and, assisted by these, would allow us to grasp a whole scale of equilibria. Since the scope of equilibrium theory is completely different when we deal with open and closed systems, like any other issue the very idea of balance can be analyzed by contrasting the four questions and four modes of investigation, as they consider open and closed systems, in order to have a broader perspective of the matter, and to better grasp how access to certain approaches has been systematically blocked. The four modes have an archetypal value because they not only trace the watersheds of thought but of will also.
Everything has an equilibrium in its specific interval of change, an equilibrium over time and an equilibrium beyond time. A theory of equilibrium in the sense indicated here must necessarily establish other connections between physics, mathematics and natural philosophy. On the other hand, modern analysis and the theory of point particles, perfectly dispensable in other frameworks, prevent the specific study of the processes of formation or individuation of an entity. Mathematically, there is more incommensurability between a point and the radius of a particle than between it and the dimensions of the known universe. We have spoken of a «symplectic morphology» not in the sense now given to the word “symplectic” in differential geometry, but in terms of the symploké of the individual, its complexion, which implies both a connection with the medium and a differentiation and independence with respect to it.
It is not that the individual has an experience, but that the experience has made the individual, who in turn is usually engaged in a second-order experience. This uncontrolled experience, which Nishida called «pure experience,» is the original consciousness prior to thought. There is a natural analogy between this original consciousness and a homogeneous medium prior to the conventions that determine metric, affine or projective spaces, although thoughts themselves can be conceived as projections, not on an extended space but on a line of duration that need not be an external timeline.
Nor is the analogy between consciousness and a physical field entirely unfounded, but the word «field» only indicates a portion of space with certain associated quantities, and mechanics is after all more fundamental; especially if we recognize that there are basic mechanical facts that are not captured by field theories. Nor has it been possible to demonstrate the stability of matter, but like so many other things it has been rationalized by relying on phenomenological arguments such as the exclusion principle. The idea of equilibrium is more general and has more scope than all modern theories, as long as we know how to remove the obstructions and blockages they have created.
The idea of balance connects as directly as possible the conditioned consciousness with the unconditioned, our direct physical experience, our mediated intuition and our immediate non-intuitive knowledge. In symplectic morphology, balance regulates the whole process of transformation of matter and form in the beginning, in the end and in the middle. For Yabir the highest part of the Science of the Balance was the Balance of Letters, synthesized by three of them, corresponding to these three moments. Impossible not to think of the sacred monosyllable of the dharmic traditions proposed as the name of the absolute, where being and consciousness are always in perfect equilibrium without affecting in the least the unknowable infinitude in the background.
The infinitude of language and its transformations, as well as the infinitude of everything manifested, is only a minute part of the infinitude named by the Name, and yet it is more inherent to us than any object of knowledge. The interesting thing for knowledge is to what extent it can come to terms with this metacognition; a metacognition that is not knowledge of knowledge but intimate transmutation of intellect and will.