Categorial Logic – Paracompleteness versus Paraconsistency. Thought of the Day 46.2

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The fact that logic is content-dependent opens a new horizon concerning the relationship of logic to ontology (or objectology). Although the classical concepts of a priori and a posteriori propositions (or judgments) has lately become rather blurred, there is an undeniable fact: it is certain that the far origin of mathematics is based on empirical practical knowledge, but nobody can claim that higher mathematics is empirical.

Thanks to category theory, it is an established fact that some sort of very important logical systems: the classical and the intuitionistic (with all its axiomatically enriched subsystems), can be interpreted through topoi. And these possibility permits to consider topoi, be it in a Noneist or in a Platonist way, as universes, that is, as ontologies or as objectologies. Now, the association of a topos with its correspondent ontology (or objectology) is quite different from the association of theoretical terms with empirical concepts. Within the frame of a physical theory, if a new fact is discovered in the laboratory, it must be explained through logical deduction (with the due initial conditions and some other details). If a logical conclusion is inferred from the fundamental hypotheses, it must be corroborated through empirical observation. And if the corroboration fails, the theory must be readjusted or even rejected.

In the case of categorial logic, the situation has some similarity with the former case; but we must be careful not to be influenced by apparent coincidences. If we add, as an axiom, the tertium non datur to the formalized intuitionistic logic we obtain classical logic. That is, we can formally pass from the one to the other, just by adding or suppressing the tertium. This fact could induce us to think that, just as in physics, if a logical theory, let’s say, intuitionistic logic, cannot include a true proposition, then its axioms must be readjusted, to make it possible to include it among its theorems. But there is a radical difference: in the semantics of intuitionistic logic, and of any logic, the point of departure is not a set of hypothetical propositions that must be corroborated through experiment; it is a set of propositions that are true under some interpretation. This set can be axiomatic or it can consist in rules of inference, but the theorems of the system are not submitted to verification. The derived propositions are just true, and nothing more. The logician surely tries to find new true propositions but, when they are found (through some effective method, that can be intuitive, as it is in Gödel’s theorem) there are only three possible cases: they can be formally derivable, they can be formally underivable, they can be formally neither derivable nor underivable, that is, undecidable. But undecidability does not induce the logician to readjust or to reject the theory. Nobody tries to add axioms or to diminish them. In physics, when we are handling a theory T, and a new describable phenomenon is found that cannot be deduced from the axioms (plus initial or some other conditions), T must be readjusted or even rejected. A classical logician will never think of changing the axioms or rules of inference of classical logic because it is undecidable. And an intuitionist logician would not care at all to add the tertium to the axioms of Heyting’s system because it cannot be derived within it.

The foregoing considerations sufficiently show that in logic and mathematics there is something that, with full right, can be called “a priori“. And although, as we have said, we must acknowledge that the concepts of a priori and a posteriori are not clear-cut, in some cases, we can rightly speak of synthetical a priori knowledge. For instance, the Gödel’s proposition that affirms its own underivabilty is synthetical and a priori. But there are other propositions, for instance, mathematical induction, that can also be considered as synthetical and a priori. And a great deal of mathematical definitions, that are not abbreviations, are synthetical. For instance, the definition of a monoid action is synthetical (and, of course, a priori) because the concept of a monoid does not have among its characterizing traits the concept of an action, and vice versa.

Categorial logic is, the deepest knowledge of logic that has ever been achieved. But its scope does not encompass the whole field of logic. There are other kinds of logic that are also important and, if we intend to know, as much as possible, what logic is and how it is related to mathematics and ontology (or objectology), we must pay attention to them. From a mathematical and a philosophical point of view, the most important logical non-paracomplete systems are the paraconsistent ones. These systems are something like a dual to paracomplete logics. They are employed in inconsistent theories without producing triviality (in this sense also relevant logics are paraconsistent). In intuitionist logic there are interpretations that, with respect to some topoi, include two false contradictory propositions; whereas in paraconsistent systems we can find interpretations in which there are two contradictory true propositions.

There is, though, a difference between paracompleteness and paraconsistency. Insofar as mathematics is concerned, paracomplete systems had to be coined to cope with very deep problems. The paraconsistent ones, on the other hand, although they have been applied with success to mathematical theories, were conceived for purely philosophical and, in some cases, even for political and ideological motivations. The common point of them all was the need to construe a logical system able to cope with contradictions. That means: to have at one’s disposal a deductive method which offered the possibility of deducing consistent conclusions from inconsistent premisses. Of course, the inconsistency of the premisses had to comply with some (although very wide) conditions to avoid triviality. But these conditions made it possible to cope with paradoxes or antinomies with precision and mathematical sense.

But, philosophically, paraconsistent logic has another very important property: it is used in a spontaneous way to formalize the naive set theory, that is, the kind of theory that pre-Zermelian mathematicians had always employed. And it is, no doubt, important to try to develop mathematics within the frame of naive, spontaneous, mathematical thought, without falling into the artificiality of modern set theory. The formalization of the naive way of mathematical thinking, although every formalization is unavoidably artificial, has opened the possibility of coping with dialectical thought.

The Locus of Renormalization. Note Quote.

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Since symmetries and the related conservation properties have a major role in physics, it is interesting to consider the paradigmatic case where symmetry changes are at the core of the analysis: critical transitions. In these state transitions, “something is not preserved”. In general, this is expressed by the fact that some symmetries are broken or new ones are obtained after the transition (symmetry changes, corresponding to state changes). At the transition, typically, there is the passage to a new “coherence structure” (a non-trivial scale symmetry); mathematically, this is described by the non-analyticity of the pertinent formal development. Consider the classical para-ferromagnetic transition: the system goes from a disordered state to sudden common orientation of spins, up to the complete ordered state of a unique orientation. Or percolation, often based on the formation of fractal structures, that is the iteration of a statistically invariant motif. Similarly for the formation of a snow flake . . . . In all these circumstances, a “new physical object of observation” is formed. Most of the current analyses deal with transitions at equilibrium; the less studied and more challenging case of far form equilibrium critical transitions may require new mathematical tools, or variants of the powerful renormalization methods. These methods change the pertinent object, yet they are based on symmetries and conservation properties such as energy or other invariants. That is, one obtains a new object, yet not necessarily new observables for the theoretical analysis. Another key mathematical aspect of renormalization is that it analyzes point-wise transitions, that is, mathematically, the physical transition is seen as happening in an isolated mathematical point (isolated with respect to the interval topology, or the topology induced by the usual measurement and the associated metrics).

One can say in full generality that a mathematical frame completely handles the determination of the object it describes as long as no strong enough singularity (i.e. relevant infinity or divergences) shows up to break this very mathematical determination. In classical statistical fields (at criticality) and in quantum field theories this leads to the necessity of using renormalization methods. The point of these methods is that when it is impossible to handle mathematically all the interaction of the system in a direct manner (because they lead to infinite quantities and therefore to no relevant account of the situation), one can still analyze parts of the interactions in a systematic manner, typically within arbitrary scale intervals. This allows us to exhibit a symmetry between partial sets of “interactions”, when the arbitrary scales are taken as a parameter.

In this situation, the intelligibility still has an “upward” flavor since renormalization is based on the stability of the equational determination when one considers a part of the interactions occurring in the system. Now, the “locus of the objectivity” is not in the description of the parts but in the stability of the equational determination when taking more and more interactions into account. This is true for critical phenomena, where the parts, atoms for example, can be objectivized outside the system and have a characteristic scale. In general, though, only scale invariance matters and the contingent choice of a fundamental (atomic) scale is irrelevant. Even worse, in quantum fields theories, the parts are not really separable from the whole (this would mean to separate an electron from the field it generates) and there is no relevant elementary scale which would allow ONE to get rid of the infinities (and again this would be quite arbitrary, since the objectivity needs the inter-scale relationship).

In short, even in physics there are situations where the whole is not the sum of the parts because the parts cannot be summed on (this is not specific to quantum fields and is also relevant for classical fields, in principle). In these situations, the intelligibility is obtained by the scale symmetry which is why fundamental scale choices are arbitrary with respect to this phenomena. This choice of the object of quantitative and objective analysis is at the core of the scientific enterprise: looking only at molecules as the only pertinent observable of life is worse than reductionist, it is against the history of physics and its audacious unifications and invention of new observables, scale invariances and even conceptual frames.

As for criticality in biology, there exists substantial empirical evidence that living organisms undergo critical transitions. These are mostly analyzed as limit situations, either never really reached by an organism or as occasional point-wise transitions. Or also, as researchers nicely claim in specific analysis: a biological system, a cell genetic regulatory networks, brain and brain slices …are “poised at criticality”. In other words, critical state transitions happen continually.

Thus, as for the pertinent observables, the phenotypes, we propose to understand evolutionary trajectories as cascades of critical transitions, thus of symmetry changes. In this perspective, one cannot pre-give, nor formally pre-define, the phase space for the biological dynamics, in contrast to what has been done for the profound mathematical frame for physics. This does not forbid a scientific analysis of life. This may just be given in different terms.

As for evolution, there is no possible equational entailment nor a causal structure of determination derived from such entailment, as in physics. The point is that these are better understood and correlated, since the work of Noether and Weyl in the last century, as symmetries in the intended equations, where they express the underlying invariants and invariant preserving transformations. No theoretical symmetries, no equations, thus no laws and no entailed causes allow the mathematical deduction of biological trajectories in pre-given phase spaces – at least not in the deep and strong sense established by the physico-mathematical theories. Observe that the robust, clear, powerful physico-mathematical sense of entailing law has been permeating all sciences, including societal ones, economics among others. If we are correct, this permeating physico-mathematical sense of entailing law must be given up for unentailed diachronic evolution in biology, in economic evolution, and cultural evolution.

As a fundamental example of symmetry change, observe that mitosis yields different proteome distributions, differences in DNA or DNA expressions, in membranes or organelles: the symmetries are not preserved. In a multi-cellular organism, each mitosis asymmetrically reconstructs a new coherent “Kantian whole”, in the sense of the physics of critical transitions: a new tissue matrix, new collagen structure, new cell-to-cell connections . . . . And we undergo millions of mitosis each minute. More, this is not “noise”: this is variability, which yields diversity, which is at the core of evolution and even of stability of an organism or an ecosystem. Organisms and ecosystems are structurally stable, also because they are Kantian wholes that permanently and non-identically reconstruct themselves: they do it in an always different, thus adaptive, way. They change the coherence structure, thus its symmetries. This reconstruction is thus random, but also not random, as it heavily depends on constraints, such as the proteins types imposed by the DNA, the relative geometric distribution of cells in embryogenesis, interactions in an organism, in a niche, but also on the opposite of constraints, the autonomy of Kantian wholes.

In the interaction with the ecosystem, the evolutionary trajectory of an organism is characterized by the co-constitution of new interfaces, i.e. new functions and organs that are the proper observables for the Darwinian analysis. And the change of a (major) function induces a change in the global Kantian whole as a coherence structure, that is it changes the internal symmetries: the fish with the new bladder will swim differently, its heart-vascular system will relevantly change ….

Organisms transform the ecosystem while transforming themselves and they can stand/do it because they have an internal preserved universe. Its stability is maintained also by slightly, yet constantly changing internal symmetries. The notion of extended criticality in biology focuses on the dynamics of symmetry changes and provides an insight into the permanent, ontogenetic and evolutionary adaptability, as long as these changes are compatible with the co-constituted Kantian whole and the ecosystem. As we said, autonomy is integrated in and regulated by constraints, with an organism itself and of an organism within an ecosystem. Autonomy makes no sense without constraints and constraints apply to an autonomous Kantian whole. So constraints shape autonomy, which in turn modifies constraints, within the margin of viability, i.e. within the limits of the interval of extended criticality. The extended critical transition proper to the biological dynamics does not allow one to prestate the symmetries and the correlated phase space.

Consider, say, a microbial ecosystem in a human. It has some 150 different microbial species in the intestinal tract. Each person’s ecosystem is unique, and tends largely to be restored following antibiotic treatment. Each of these microbes is a Kantian whole, and in ways we do not understand yet, the “community” in the intestines co-creates their worlds together, co-creating the niches by which each and all achieve, with the surrounding human tissue, a task closure that is “always” sustained even if it may change by immigration of new microbial species into the community and extinction of old species in the community. With such community membership turnover, or community assembly, the phase space of the system is undergoing continual and open ended changes. Moreover, given the rate of mutation in microbial populations, it is very likely that these microbial communities are also co-evolving with one another on a rapid time scale. Again, the phase space is continually changing as are the symmetries.

Can one have a complete description of actual and potential biological niches? If so, the description seems to be incompressible, in the sense that any linguistic description may require new names and meanings for the new unprestable functions, where functions and their names make only sense in the newly co-constructed biological and historical (linguistic) environment. Even for existing niches, short descriptions are given from a specific perspective (they are very epistemic), looking at a purpose, say. One finds out a feature in a niche, because you observe that if it goes away the intended organisms dies. In other terms, niches are compared by differences: one may not be able to prove that two niches are identical or equivalent (in supporting life), but one may show that two niches are different. Once more, there are no symmetries organizing over time these spaces and their internal relations. Mathematically, no symmetry (groups) nor (partial-) order (semigroups) organize the phase spaces of phenotypes, in contrast to physical phase spaces.

Finally, here is one of the many logical challenges posed by evolution: the circularity of the definition of niches is more than the circularity in the definitions. The “in the definitions” circularity concerns the quantities (or quantitative distributions) of given observables. Typically, a numerical function defined by recursion or by impredicative tools yields a circularity in the definition and poses no mathematical nor logical problems, in contemporary logic (this is so also for recursive definitions of metabolic cycles in biology). Similarly, a river flow, which shapes its own border, presents technical difficulties for a careful equational description of its dynamics, but no mathematical nor logical impossibility: one has to optimize a highly non linear and large action/reaction system, yielding a dynamically constructed geodetic, the river path, in perfectly known phase spaces (momentum and space or energy and time, say, as pertinent observables and variables).

The circularity “of the definitions” applies, instead, when it is impossible to prestate the phase space, so the very novel interaction (including the “boundary conditions” in the niche and the biological dynamics) co-defines new observables. The circularity then radically differs from the one in the definition, since it is at the meta-theoretical (meta-linguistic) level: which are the observables and variables to put in the equations? It is not just within prestatable yet circular equations within the theory (ordinary recursion and extended non – linear dynamics), but in the ever changing observables, the phenotypes and the biological functions in a circularly co-specified niche. Mathematically and logically, no law entails the evolution of the biosphere.

The Sibyl’s Prophecy/Nordic Creation. Note Quote.

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The Prophecy of the Tenth Sibyl, a medieval best-seller, surviving in over 100 manuscripts from the 11th to the 16th century, predicts, among other things, the reign of evil despots, the return of the Antichrist and the sun turning to blood.

The Tenth or Tiburtine Sibyl was a pagan prophetess perhaps of Etruscan origin. To quote Lactantus in his general account of the ten sibyls in the introduction, ‘The Tiburtine Sibyl, by name Albunea, is worshiped at Tibur as a goddess, near the banks of the Anio in which stream her image is said to have been found, holding a book in her hand’.

The work interprets the Sibyl’s dream in which she foresees the downfall and apocalyptic end of the world; 9 suns appear in the sky, each one more ugly and bloodstained than the last, representing the 9 generations of mankind and ending with Judgment Day. The original Greek version dates from the end of the 4th century and the earliest surviving manuscript in Latin is dated 1047. The Tiburtine Sibyl is often depicted with Emperor Augustus, who asks her if he should be worshipped as a god.

The foremost lay of the Elder Edda is called Voluspa (The Sibyl’s Prophecy). The volva, or sibyl, represents the indelible imprint of the past, wherein lie the seeds of the future. Odin, the Allfather, consults this record to learn of the beginning, life, and end of the world. In her response, she addresses Odin as a plurality of “holy beings,” indicating the omnipresence of the divine principle in all forms of life. This also hints at the growth of awareness gained by all living, learning entities during their evolutionary pilgrimage through spheres of existence.

Hear me, all ye holy beings, greater as lesser sons of Heimdal! You wish me to tell of Allfather’s works, tales of the origin, the oldest I know. Giants I remember, born in the foretime, they who long ago nurtured me. Nine worlds I remember, nine trees of life, before this world tree grew from the ground.

Paraphrased, this could be rendered as:

Learn, all ye living entities, imbued with the divine essence of Odin, ye more and less evolved sons of the solar divinity (Heimdal) who stands as guardian between the manifest worlds of the solar system and the realm of divine consciousness. You wish to learn of what has gone before. I am the record of long ages past (giants), that imprinted their experience on me. I remember nine periods of manifestation that preceded the present system of worlds.

Time being inextricably a phenomenon of manifestation, the giant ages refer to the matter-side of creation. Giants represent ages of such vast duration that, although their extent in space and time is limited, it is of a scope that can only be illustrated as gigantic. Smaller cycles within the greater are referred to in the Norse myths as daughters of their father-giant. Heimdal is the solar deity in the sign of Aries – of beginnings for our system – whose “sons” inhabit, in fact compose, his domain.

Before a new manifestation of a world, whether a cosmos or a lesser system, all its matter is frozen in a state of immobility, referred to in the Edda as a frost giant. The gods – consciousnesses – are withdrawn into their supernal, unimaginable abstraction of Nonbeing, called in Sanskrit “paranirvana.” Without a divine activating principle, space itself – the great container – is a purely theoretical abstraction where, for lack of any organizing energic impulse of consciousness, matter cannot exist.

This was the origin of ages when Ymer built. No soil was there, no sea, no cool waves. Earth was not, nor heaven above; Gaping Void alone, no growth. Until the sons of Bur raised the tables; they who created beautiful Midgard. The sun shone southerly on the stones of the court; then grew green herbs in fertile soil.

To paraphrase again:

Before time began, the frost giant (Ymer) prevailed. No elements existed for there were ‘no waves,’ no motion, hence no organized form nor any temporal events, until the creative divine forces emanated from Space (Bur — a principle, not a locality) and organized latent protosubstance into the celestial bodies (tables at which the gods feast on the mead of life-experience). Among these tables is Middle Court (Midgard), our own beautiful planet. The life-giving sun sheds its radiant energies to activate into life all the kingdoms of nature which compose it.

The Gaping Void (Ginnungagap) holds “no cool waves” throughout illimitable depths during the age of the frost giant. Substance has yet to be created. Utter wavelessness negates it, for all matter is the effect of organized, undulating motion. As the cosmic hour strikes for a new manifestation, the ice of Home of Nebulosity (Niflhem) is melted by the heat from Home of Fire (Muspellshem), resulting in vapor in the void. This is Ymer, protosubstance as yet unformed, the nebulae from which will evolve the matter components of a new universe, as the vital heat of the gods melts and vivifies the formless immobile “ice.”

When the great age of Ymer has run its course, the cow Audhumla, symbol of fertility, “licking the salt from the ice blocks,” uncovers the head of Buri, first divine principle. From this infinite, primal source emanates Bur, whose “sons” are the creative trinity: Divine Allfather, Will, and Sanctity (Odin, Vile, and Vi). This triune power “kills” the frost giant by transforming it into First Sound (Orgalmer), or keynote, whose overtones vibrate through the planes of sleeping space and organize latent protosubstance into the multifarious forms which will be used by all “holy beings” as vehicles for gaining experience in worlds of matter.

Beautiful Midgard, our physical globe earth, is but one of the “tables” raised by the creative trinity, whereat the gods shall feast. The name Middle Court is suggestive, for the ancient traditions place our globe in a central position in the series of spheres that comprise the terrestrial being’s totality. All living entities, man included, comprise besides the visible body a number of principles and characteristics not cognized by the gross physical senses. In the Lay of Grimner (Grimnismal), wherein Odin in the guise of a tormented prisoner on earth instructs a human disciple, he enumerates twelve spheres or worlds, all but one of which are unseen by our organs of sight. As to the formation of Midgard, he relates:

Of Ymer’s flesh was the earth formed, the billows of his blood, the mountains of his bones, bushes of his hair, and of his brainpan heaven. With his eyebrows beneficent powers enclosed Midgard for man; but of his brain were surely all dark skies created.

The trinity of immanent powers organize Ymer into the forms wherein they dwell, shaping the chaos or frost giant into living globes on many planes of being. The “eyebrows” that gird the earth and protect it suggest the Van Allen belts that shield the planet from inimical radiation. The brain of Ymer – material thinking – is surely all too evident in the thought atmosphere wherein man participates.

The formation of the physical globe is described as the creation of “dwarfs” – elemental forces which shape the body of the earth-being and which include the mineral. vegetable, and animal kingdoms.

The mighty drew to their judgment seats, all holy gods to hold counsel: who should create a host of dwarfs from the blood of Brimer and the limbs of the dead. Modsogne there was, mightiest of all the dwarfs, Durin the next; there were created many humanoid dwarfs from the earth, as Durin said.

Brimer is the slain Ymer, a kenning for the waters of space. Modsogne is the Force-sucker, Durin the Sleeper, and later comes Dvalin the Entranced. They are “dwarf”-consciousnesses, beings that are miðr than human – the Icelandic miðr meaning both “smaller” and “less.” By selecting the former meaning, popular concepts have come to regard them as undersized mannikins, rather than as less evolved natural species that have not yet reached the human condition of intelligence and self-consciousness.

During the life period or manifestation of a universe, the governing giant or age is named Sound of Thor (Trudgalmer), the vital force which sustains activity throughout the cycle of existence. At the end of this age the worlds become Sound of Fruition (Bargalmer). This giant is “placed on a boat-keel and saved,” or “ground on the mill.” Either version suggests the karmic end product as the seed of future manifestation, which remains dormant throughout the ensuing frost giant of universal dissolution, when cosmic matter is ground into a formless condition of wavelessness, dissolved in the waters of space.

There is an inescapable duality of gods-giants in all phases of manifestation: gods seek experience in worlds of substance and feast on the mead at stellar and planetary tables; giants, formed into vehicles inspired with the divine impetus, rise through cycles of this association on the ladder of conscious awareness. All states being relative and bipolar, there is in endless evolution an inescapable link between the subjective and objective progress of beings. Odin as the “Opener” is paired with Orgalmer, the keynote on which a cosmos is constructed; Odin as the “Closer” is equally linked with Bargalmer, the fruitage of a life cycle. During the manifesting universe, Odin-Allfather corresponds to Trudgalmer, the sustainer of life.

A creative trinity plays an analogical part in the appearance of humanity. Odin remains the all-permeant divine essence, while on this level his brother-creators are named Honer and Lodur, divine counterparts of water or liquidity, and fire or vital heat and motion. They “find by the shore, of little power” the Ash and the Elm and infuse into these earth-beings their respective characteristics, making a human image or reflection of themselves. These protohumans, miniatures of the world tree, the cosmic Ash, Yggdrasil, in addition to their earth-born qualities of growth force and substance, receive the divine attributes of the gods. By Odin man is endowed with spirit, from Honer comes his mind, while Lodur gives him will and godlike form. The essentially human qualities are thus potentially divine. Man is capable of blending with the earth, whose substances form his body, yet is able to encompass in his consciousness the vision native to his divine source. He is in fact a minor world tree, part of the universal tree of life, Yggdrasil.

Ygg in conjunction with other words has been variously translated as Eternal, Awesome or Terrible, and Old. Sometimes Odin is named Yggjung, meaning the Ever-Young, or Old-Young. Like the biblical “Ancient of Days” it is a concept that mind can grasp only in the wake of intuition. Yggdrasil is the “steed” or the “gallows” of Ygg, whereon Odin is mounted or crucified during any period of manifested life. The world tree is rooted in Nonbeing and ramifies through the planes of space, its branches adorned with globes wherein the gods imbody. The sibyl spoke of ours as the tenth in a series of such world trees, and Odin confirms this in The Song of the High One (Den Hoges Sang):

I know that I hung in the windtorn tree nine whole nights, spear-pierced, given to Odin, my self to my Self above me in the tree, whose root none knows whence it sprang. None brought me bread, none served me drink; I searched the depths, spied runes of wisdom, raised them with song, and fell once more from the tree. Nine powerful songs I learned from the wise son of Boltorn, Bestla’s father; a draught I drank of precious mead ladled from Odrorer. I began to grow, to grow wise, to grow greater and enjoy; for me words from words led to new words, for me deeds from deeds led to new deeds.

Numerous ancient tales relate the divine sacrifice and crucifixion of the Silent Watcher whose realm or protectorate is a world in manifestation. Each tree of life, of whatever scope, constitutes the cross whereon the compassionate deity inherent in that hierarchy remains transfixed for the duration of the cycle of life in matter. The pattern of repeated imbodiments for the purpose of gaining the precious mead is clear, as also the karmic law of cause and effect as words and deeds bring their results in new words and deeds.

Yggdrasil is said to have three roots. One extends into the land of the frost giants, whence flow twelve rivers of lives or twelve classes of beings; another springs from and is watered by the well of Origin (Urd), where the three Norns, or fates, spin the threads of destiny for all lives. “One is named Origin, the second Becoming. These two fashion the third, named Debt.” They represent the inescapable law of cause and effect. Though they have usually been roughly translated as Past, Present, and Future, the dynamic concept in the Edda is more complete and philosophically exact. The third root of the world tree reaches to the well of the “wise giant Mimer,” owner of the well of wisdom. Mimer represents material existence and supplies the wisdom gained from experience of life. Odin forfeited one eye for the privilege of partaking of these waters of life, hence he is represented in manifestation as one-eyed and named Half-Blind. Mimer, the matter-counterpart, at the same time receives partial access to divine vision.

The lays make it very clear that the purpose of existence is for the consciousness-aspect of all beings to gain wisdom through life, while inspiring the substantial side of itself to growth in inward awareness and spirituality. At the human level, self-consciousness and will are aroused, making it possible for man to progress willingly and purposefully toward his divine potential, aided by the gods who have passed that way before him, rather than to drift by slow degrees and many detours along the road of inevitable evolution. Odin’s instructions to a disciple, Loddfafner, the dwarf-nature in man, conclude with:

Now is sung the High One’s song in the High One’s hall. Useful to sons of men, useless to sons of giants. Hail Him who sang! Hail him who kens! Rejoice they who understand! Happy they who heed!

Duality’s Anti-Realism or Poisoning Ontological Realism: The Case of Vanishing Ontology. Note Quote.

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If the intuitive quality of the external ontological object is diminished piece by piece during the evolutionary progress of physical theory (which must be acknowledged also in a hidden parameter framework), is there any core of the notion of an ontological object at all that can be trusted to be immune against scientific decomposition?

Quantum mechanics cannot answer this question. Contemporary physics is in a quite different position. The full dissolution of ontology is a characteristic process of particle physics whose unfolding starts with quantum mechanics and gains momentum in gauge field theory until, in string theory, the ontological object has simply vanished.

The concept to be considered is string duality, with the remarkable phenomenon of T-duality according to which a string wrapped around a small compact dimension can as well be understood as a string that is not wrapped but moves freely along a large compact dimension. The phenomenon is rooted in the quantum principles but clearly transcends what one is used to in the quantum world. It is not a mere case of quantum indeterminacy concerning two states of the system. We rather face two theoretical formulations which are undistinguishable in principle so that they cannot be interpreted as referring to two different states at all. Nevertheless the two formulations differ in characteristics which lie at the core of any meaningful ontology of an external world. They differ in the shape of space-time and they differ in form and topological position of the elementary objects. The fact that those characteristics are reduced to technical parameters whose values depend on the choice of the theoretical formulation contradicts ontological scientific realism in the most straightforward way. If a situation can be described by two different sets of elementary objects depending on the choice of the theoretical framework, how can it make sense to assert that these ontological objects actually exist in an external world?

The question gets even more virulent as T-duality by no means remains the only duality relation that surfaces in string theory. It turns out that the existence of dualities is one of string theory’s most characteristic features. They seem to pop up wherever one looks for them. Probably the most important role played by duality relations today is to connect all different superstring theories. Before 1995 physicists knew 5 different types of superstring theory. Then it turned out that these 5 theories and a 6th by then unknown theory named ‘M-theory’ are interconnected by duality relations. Two types of duality are involved. Some theories can be transformed into each other through inversion of a compactification radius, which is the phenomenon we know already under the name of T-duality. Others can be transformed into each other by inversion of the string coupling constant. This duality is called S-duality. Then there is M-theory, where the string coupling constant is transformed into an additional 11th dimension whose size is proportional to the coupling strength of the dual theory. The described web of dualities connects theories whose elementary objects have different symmetry structure and different dimensionality. M-theory even has a different number of spatial dimensions than its co-theories. Duality nevertheless implies that M-theory and the 5 possible superstring theories only represent different formulations of one single actual theory. This statement constitutes the basis for string theory’s uniqueness claims and shows the pivotal role played by the duality principle.

An evaluation of the philosophical implications of duality in modern string theory must first acknowledge that the problems to identify uniquely the ontological basis of a scientific theory are as old as the concept of invisible scientific objects itself. Complex theories tend to allow the insertion of ontology at more than one level of their structure. It is not a priori clear in classical electromagnetism whether the field or the potential should be understood as the fundamental physical object and one may wonder similarly in quantum field theory whether that concept’s basic object is the particle or the field. Questions of this type clearly pose a serious philosophical problem. Some philosophers like Quine have drawn the conclusion to deny any objective basis for the imputation of ontologies. Philosophers with a stronger affinity for realism however often stress that there do exist arguments which are able to select a preferable ontological set after all. It might also be suggested that ontological alternatives at different levels of the theoretical structure do not pose a threat to realism but should be interpreted merely as different parameterisations of ontological reality. The problem is created at a philosophical level by imputing an ontology to a physical theory whose structure neither depends on nor predetermines uniquely that imputation. The physicist puts one compact theoretical structure into space-time and the philosopher struggles with the question at which level ontological claims should be inserted.

The implications of string-duality have an entirely different quality. String duality really posits different ‘parallel’ empirically indistinguishable versions of structure in spacetime which are based on different sets of elementary objects. This statement is placed at the physical level independently of any philosophical interpretation. Thus it transfers the problem of the lack of ontological uniqueness from a philosophical to a physical level and makes it much more difficult to cure. If theories with different sets of elementary objects give the same physical world (i. e. show the same pattern of observables), the elementary object cannot be seen as the unique foundation of the physical world any more. There seems to be no way to avoid this conclusion. There exists an additional aspect of duality that underlines its anti-ontological character. Duality does not just spell destruction for the notion of the ontological scientific object but in a sense offers a replacement as well.

Do there remain any loop-holes in duality’s anti-realist implications which could be used by the die-hard realist? A natural objection to the asserted crucial philosophical importance of duality can be based on the fact, that duality was not invented in the context of string theory. It is known since the times of P. M. Dirac that quantum electrodynamics with magnetic monopoles would be dual to a theory with inverted coupling constant and exchanged electric and magnetic charges. The question arises, if duality is poison to ontological realism, why didn’t it have its effect already at the level of quantum electrodynamics. The answer gives a nice survey of possible measures to save ontological realism. As it will turn out, they all fail in string theory.

In the case of quantum-electrodynamics the realist has several arguments to counter the duality threat. First, duality looks more like an accidental oddity that appears in an unrealistic scenario than like a characteristic feature of the world. No one has observed magnetic monopoles, which renders the problem hypothetical. And even if there were magnetic monopoles, an embedding of electromagnetism into a fuller description of the natural forces would destroy the dual structure anyway.

In string theory the situation is very different. Duality is no ‘lucky strike’ any more, which just by chance arises in a certain scenario that is not the real one anyway. As we have seen, it rather represents a core feature of the emerging theoretical structure and cannot be ignored. A second option open to the realist at the level of quantum electrodynamics is to shift the ontological posit. Some philosophers of quantum physics argue that the natural elementary object of quantum field theory is the quantum field, which represents something like the potentiality to produce elementary particles. One quantum field covers the full sum over all variations of particle exchange which have to be accounted for in a quantum process. The philosopher who posits the quantum field to be the fundamental real object discovered by quantum field theory understands the single elementary particles as mere mathematical entities introduced to calculate the behaviour of the quantum field. Dual theories from his perspective can be taken as different technical procedures to calculate the behaviour of the univocal ontological object, the electromagnetic quantum field. The phenomenon of duality then does not appear as a threat to the ontological concept per se but merely as an indication in favour of an ontologisation of the field instead of the particle.

The field theoretical approach to interpret the quantum field as the ontological object does not have any pendent in string theory. String theory only exists as a perturbative theory. There seems to be no way to introduce anything like a quantum field that would cover the full expansion of string exchanges. In the light of duality this lack of a unique ontological object arguably appears rather natural. The reason is related to another point that makes string dualities more dramatic than its field theoretical predecessor. String theory includes gravitation. Therefore object (the string geometry) and space-time are not independent. Actually it turns out that the string geometry in a way carries all information about space-time as well. This dependence of space-time on string-geometry makes it difficult already to imagine how it should be possible to put into this very spacetime some kind of overall field whose coverage of all string realisations actually implies coverage of variations of spacetime itself. The duality context makes the paradoxical quality of such an attempt more transparent. If two dual theories with different radii of a compactified dimension shall be covered by the same ontological object in analogy to the quantum field in field theory, this object obviously cannot live in space and time. If it would, it had to choose one of the two spacetime versions endorsed by the dual theories, thereby discriminating the other one. This theory however should not be expected to be a theory of objects in spacetime and therefore does not rise any hopes to redeem the external ontological perspective.

A third strategy to save ontological realism is based on the following argument: In quantum electrodynamics the difference between the dual theories boils down to a mere replacement of a weak coupling constant which allows perturbative calculation by a strong one which does not. Therefore the choice is open between a natural formulation and a clumsy untreatable one which maybe should just be discarded as an artificial construction.

Today string theory cannot tell whether its final solution will put its parameters comfortably into the low-coupling-constant-and-large-compact-dimension-regime of one of the 5 superstring theories or M-theory. This might be the case but it might as well happen, that the solution lies in a region of parameter space where no theory clearly stands out in this sense. However, even if there was one preferred theory, the simple discarding of the others could not save realism as in the case of field theory. First, the argument of natural choice is not really applicable to T-duality. A small compactification radius does not render a theory intractable like a large coupling constant. The choice of the dual version with a large radius thus looks more like a convention than anything else. Second, the choice of both compactification radii and string coupling constants in string theory is the consequence of a dynamical process that has to be calculated itself. Calculation thus stands before the selection of a certain point in parameter space and consequently also before a possible selection of the ontological objects. The ontological objects therefore, even if one wanted to hang on to their meaningfulness in the final scenario, would appear as a mere product of prior dynamics and not as a priori actors in the game.

Summing up, the phenomenon of duality is admittedly a bit irritating for the ontological realist in field theory but he can live with it. In string theory however, the field theoretical strategies to save realism all fail. The position assumed by the duality principle in string theory clearly renders obsolete the traditional realist understanding of scientific objects as smaller cousins of visible ones. The theoretical posits of string theory get their meaning only relative to their theoretical framework and must be understood as mathematical concepts without any claim to ‘corporal’ existence in an external world. The world of string theory has cut all ties with classical theories about physical bodies. To stick to ontological realism in this altered context, would be inadequate to the elementary changes which characterize the new situation. The demise of ontology in string theory opens new perspectives on the positions where the stress is on the discontinuity of ontological claims throughout the history of scientific theories.

Conjuncted: Gadamer’s Dasein

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There is a temporal continuity in Dasein. This is required for the revelation of a work of art through interpretation, both as understanding which already was, and as the way in which understanding was. Understanding is possible only in the temporal revision of one’s standpoint through the mutual relations of author and interpreter which allow the subject-matter to emerge. Here, the prejudices held by the interpreter play an important part in opening an horizon of possible questions.

Subsequent understanding that is superior to the original production, does depend on the conscious realization, historical or not, that places the interpreter on the same level as the author (as Schleiermacher pointed out). But even more, it denotes and depends upon an inseparable difference between the interpreter and the text and this precisely in the temporal field provided by historical distance.

It may be argued that the historian tries to curb this historical distance by getting beyond the temporal text in order to force it to yield information that it does not intend and of itself is unable to give. With regard to the particular text in application, this would seem to be the case. For example, what makes the true historian is an understanding of the significance of what he finds. Thus, the testimony of history is like that given before a court. In the German language, and based on this reason, the same word is used for both in general, Zeugnis (testimony; witness).

Referring to Gadamer’s position, we can see that it is in view of the historical distance that understanding must reconcile itself with itself and that one recognize oneself in the other being. The body of this argument becomes completely firm through the idea of historical Bildung, since, for example, to have a theoretical stance is, as such, already alienation; namely, dealing with something that is not immediate, but is other, belonging to memory and to thought. Moreover, theoretical Bildung leads beyond what man knows and experiences immediately. It consists in learning to affirm what is different from oneself and to find universal viewpoints from which one can grasp the thing as “the objective thing in its freedom,” without selfish interest. This indicates that an aesthetic discovery of a thing is conditioned primarily on assuming the thing where it is no longer, i.e., from a distance.

In this connection, we can extend critically Gadamer’s concept of the dynamism of distanciation from the object of understanding which is bounded by the frame of effective consciousness. This is based on the fact that in spite of the general contrast between belonging and alienating distance, the consciousness of effective history itself contains an element of distance. The history of effects, for Ricoeur, contains what occurs under the condition of historical distance. Whether this is either the nearness of the remote or efficacy at a distance, there is a paradox in otherness, a tension between proximity and distance which is essential to historical consciousness.

The possibility of effective historical consciousness is grounded in the possibility of any specific present understanding of being futural; in contrast, the first principle of hermeneutics is the Being of Dasein, which is historicity (Geschichtlichkeit) itself. In Gadamer’s view, Dasein’s temporality, which is the basis for its historicity, grounds the tradition. The last sections of Being and Time claimed to indicate that the embodiment of temporality can be found in Dasein’s historicality. As a result of this, the tradition is circularly grounded in Dasein’s temporality, while also surpassing its borders in order to be provided by a hermeneutical reference in distance.

We must study the root of this dilemma in so far as it is related to the sense of time. This is presupposed by historical consciousness, which in turn is preceded essentially by temporality. This inherent enigma in the hermeneutics of Dasein’s time led Heidegger to distinguish between authenticity and inauthenticity in our relation to time. The current concept of time can never totally fulfill the hermeneutical requirements. Ricoeur considered that time can be understood only if grasped within its limit, namely, eternity, but because eternity escapes the totalization and closure of any particular time, it remains inscrutable.

On the other hand, a text can be seen as temporal with regard to historical consciousness since it speaks only in the present. The text cannot be made present totally within an historical moment fully present-to-itself. It is in its a venir that the presence of the text transpires, which can be thematized as revenir (or) return.

Based on this aspect, each word is absolutely complete in itself, yet, because of its temporality, its meaning is realized only in its historical application. Nevertheless, historical interpretation can serve as a means to understand a given and present text even when, from another perspective, it sees the text simply as a source which is part of the totality of an historical tradition.

For Heidegger, the past character of time, i.e., the ‘pastness’ (passétité) belongs to a world which no longer exists, while a world is always world for a Dasein. It is clear that the past would remain closed off from any present were present Dasein not itself to be historical. Dasein, however, is in itself historical insofar as it is a possibility of interpreting. In being futural Dasein is its past, which comes back to it in the ‘how’. This is the ontological question of a thing in contrast to the question of the ‘what.’ The manner of its coming back is, among other processes, conscience. This makes clear why only the ‘how’ can be repeated. According to Ricoeur, history presents a past that has been as if it were present, as a function of poetic imagination. On the other hand, fictive narration imitates history in that it presents events as if they had happened, i.e., as if they occurred in the past. This intersection between history and fiction constitutes human time (le temps humain) whence an historical consciousness develops, where time can be understood as a singular totality.

Since the text can be viewed temporally, interpretation, as the work of art, is temporal and the best model for hermeneutical understanding is the one most adequate to the experience of time. Nevertheless, against Ricoeur, Gadamer found the identity of understanding not to be fixed in eternity. Instead, it is the continuity of our becoming-other in every response and in every application of pre-understanding that we have of ourselves in new and unpredictable situations. On this issue, it can be asked whether there is a way to reconcile Gadamer and Ricoeur on the issue of hermeneutical temporality.

The authentic source in the eternal return to Being can be discovered in Heidegger’s position: the eternal repetition of that which is known as that which is unknown, the familiar as the unfamiliar. The eternal return introduces difference which is disruptive to our conceptions of temporal movement. However, identity and difference must be destabilised in favor of the performance of a new concept of hermeneutics. In this a temporal event requires that one cross over to another hermeneutics of time that cannot be thought restricted only in temporalization since it is beyond when one begins. This concept is called by Heidegger the nearness of what lies after.

In addition, understanding is to be taken not as reconstruction, but as mediation in so far as it conveys the past into the present. Even when we grasp the past “in itself,” understanding remains essentially a mediation or translation of past meaning into the present situation. As Gadamer states, understanding itself is not to be thought of so much as an action of subjectivity, but rather as the entering into an event of transmission in which past and present are constantly mediated. This requires not detaching temporality from the ontological preconception of the present-at-hand, but trying to distinguish that from the simple horizon phenomenon of temporal consciousness. The event of hermeneutics never takes place if understanding is considered to be defined in the arena of the temporalization of time in the past in itself. 

Gadamer sees one of the most fundamental experiences of time as that of discontinuity or becoming-other. This stands in contrast to the “flowing” nature of time. According to Gadamer, there are at least three “epochal” experiences that introduce temporal discontinuity into our self-understanding: first, the experience of old age; second, the transition from one generation to another; and finally, the “absolute epoch” or the new age occasioned by the advent of Christianity, where history is understood in a new sense. 

The Greek understanding of history as deviation from the order of things was changed in medieval philosophy to accept that there is no recognizable order within history except temporality itself. (Nonetheless, the absolute epoch is not to be taken merely as similar to a Christian understanding of time, which would result in a technological conception of time in terms of which the future is unable to be planned or controlled.) The new in temporality comes to be as the old is recalled in dissolution. In recollection, the dissolution of the old becomes provocative, i.e., an opening of possibilities for the new. The dissolution of the old is not a non-temporal characteristic of temporalization.

Physical Congruences of Nāgārjuna’s Mūlamadhyamakakārikā, Yukti-sastikâ, śūnyatā and Pratītyasamutpāda. Note Quote

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The Middle Way of Mādhyamaka refers to the teachings of Nāgārjuna, very interesting are the implications between quantum physics and Mādhyamaka. The basic concept of reality in the philosophy of Nāgārjuna is that the fundamental reality has no firm core but consists of systems of interacting objects. According to the middle way perspective, based on the notion of emptiness, phenomena exist in a relative way, that is, they are empty of any kind of inherent and independent existence. Phenomena are regarded as dependent events existing relationally rather than permanent things, which have their own entity. Nāgārjuna middle way perspective emerges as a relational approach, based on the insight of emptiness.  śūnyatā (emptiness) is the foundation of all things, and it is the basic principle of all phenomena. The emptiness implies the negation of unchanged, fixed substance and thereby the possibility for relational existence and change. This suggests that both the ontological constitution of things and our epistemological schemes are just as relational as everything else. We are fundamentally relational internally and externally. In other words, Nāgārjuna, do not fix any ontological nature of the things:

  1. they do not arise
  2. they do not exist
  3. they are not to be found
  4. they are not
  5. and they are unreal

In short, an invitation do not decide on either existence or non-existence (nondualism). According the theory of  śūnyatā, phenomena exist in a relative state only, a kind of ’ontological relativity’. Phenomena are regarded as dependent (only in relation to something else) events rather than things which have their own inherent nature; thus the extreme of permanence is avoided.

In the Mūlamadhyamakakārikā, a tetralemma is pointed out: “Neither from itself nor from another, nor from both, nor without a cause, does anything whatever anywhere arise”. In the Yukti-sastikâ, Nāgārjuna says, “That which has arisen dependently on this and that that has not arisen substantially (svabhavatah, स्वभावतः). What has not arisen substantially, how can it literally (nama) be called ‘arisen’? […] That which originates due to a cause and does not abide without (certain) conditions but disappears when the conditions are absent, how can it be understood as ‘to exist’?”

By the notions of ‘to arise’ and ‘to exist’, Nāgārjuna does not mean the empirical existence but the substantial existence. When in many passages of Mūlamadhyamakakārikā Nāgārjuna states that things do not arise (7.29), that they do not exist (3.7, 5.8, 14.6), that they are not to be found (2.25, 9.11), that they are not (15.10), that they are unreal (13.1), then clearly this has the meaning: things do not arise substantially. They do not exist out of themselves; their independence cannot be found. They are dependent and in this sense they are substantially unreal. Nāgārjuna only rejects the idea of a substantial arising of things which bear an absolute and independent existence. He does not refute the empirical existence of things as explained in the following: “It exists implies grasping after eternity. It does not exist implies the philosophy of annihilation. Therefore, a discerning person should not decide on either existence or non-existence”. (15.10)

For Nāgārjuna, the expression ‘to exist’ has the meaning of ‘to exist substantially’. His issue is not the empirical existence of things but the conception of a permanent thing i.e. the idea of an own being, without dependence on something else. Nāgārjuna refutes the concept of independent existence which is unchangeable, eternal and existing by itself. Things do not arise out of themselves, they do not exist absolutely and are dependent. Their permanent being or existence cannot be found. The many interpretations of Nāgārjuna which claim that he is also refuting the empirical existence of objects, are making an inadmissible generalization which moves Nāgārjuna near to subjectivism, nihilism and instrumentalism. Such interpretations originate in metaphysical approaches which themselves have a difficulty in recognizing the empirical existence of the data presented. This is not at all the case with Nāgārjuna. Nāgārjuna presents the dependence of phenomena mainly in images.

Pratītyasamutpāda (Sanskrit: प्रतीत्यसमुत्पाद; Pali: पटिच्चसमुप्पाद paṭiccasamuppāda) is an indication of dependence. Dependent bodies are in an intermediate state, they are not properly separated and they are not one entity. Secondly, they rely on each other and are influenced or determined by something else. Thirdly, their behaviour is influenced by something in-between, for example a mover is attracted by gravitational force, a viewer is dependent on rays of light between his eyes and the object, a piano player’s action is determined by the fine motor skills of his fingers, an agent is dependent on his act. Pratītyasamutpāda is an indication of dependence and of something that happens between the objects. One object is bound to the other without being identical to it. The implicit interpretations of Pratītyasamutpāda, are in terms of time, structure and space.

The following citations and references illustrate the term Pratītyasamutpāda. Pratītyasamutpāda is used:

1. as Dependence in Nāgārjuna’s Hymn to the Buddha: “Dialecticians maintain that suffering is created by itself, created by (someone) else, created by both (or) without a cause, but You have stated that it is dependently born”.

2. as an intermediate state by Nāgārjuna: Objects are neither together nor separated

3. as bondage in the Hevajra Tantra: “Men are bound by the bondage of existence and are liberated by understanding the nature of existence”.

4. as an intermediate state by Roger Penrose: “Quantum entanglement is a very strange type of thing. It is somewhere between objects being separate and being in communication with each other”.

5. as something between bodies by Albert Einstein: “A courageous scientific imagination was needed to realize fully that not the behaviour of bodies, but the behaviour of something between them, that is, the field, may be essential for ordering and understanding events”.

6. as the mean between things in modern mathematics: to quote Gioberti: “The mean between two or more things, their juncture, union, transit, passage, crossing, interval, distance, bond and contact – all these are mysterious, for they are rooted in the continuum, in the infinite. The interval that runs between one idea and another, one thing and another, is infinite, and can only be surpassed by the creative act. This is why the dynamic moment and dialectic concept of the mean are no less mysterious than those of the beginning and the end. The mean is a union of two diverse and opposite things in a unity. It is an essentially dialectic concept, and involves an apparent contradiction, namely, the identity of the one and the many, of the same and the diverse. This unity is simple and composite; it is unity and synthesis and harmony. It shares in two extremes without being one or the other. It is the continuum, and therefore the infinite. Now, the infinite identically uniting contraries, clarifies the nature of the interval. In motion, in time, in space, in concepts, the discrete is easy to grasp, because it is finite. The continuum and the interval are mysterious, because they are infinite.”