An extremely difficult problem: formalization of Hegel's Concept and Idea

Our paper Hegel and Modern Topology fails to address the third great part of Hegel's Logic, the one dedicated to Concept (Begriff) and its unfolding into Idea. Here we give just a few vague hints on how the theory of Concept might be mirrored in formal logic, mathematics and science. Besides the direct interest to the interpretation of Hegel we believe that such investigations may be of interest to formal approaches to phenomenology and dialectics and shed light on the problem of formal models of consciousness.

Of course Hegel states explicitly that all such formalization will pertain to the strictly finite exterior domain of the understanding and as thus be inadequate. We reply that modern developments present us with formalisms which embody multidimensional, dynamic, modal, meta-theoretical and self-referential aspects which were not evident in mainstream 19th-century logic and science.  So we will at least give it a try.

Our letimotiv is that a central role will be played by paraconsistent logics and with the phenomena of self-reference and the reflection of metatheoretical properties of a system within the system itself.  Other key concepts will be self-similarity (Hegel in the Encyclopedia Logic writes of Object that it is a division into distinct beings each one of which is the totality), process philosophy and teleology.

Let us start with the subjective logic, with Hegel's initial discussion of the concept. The following analogy occurs to us:  the universal is like a group $G$,  the particular is like a particular representation of $G$ on some vector space $V$ and  individuality is a a concrete totality (not, Hegel points out, an immediate concrete perceived individual) - and this can only be the  monoidal category of all representatiins of $G$. For certain kinds of monoidal categories the group can be extracted from the category just as the category is generated from the group by considering its representations. This expresses the flowing forth between the universal, particular and individual. 

Consider a category $C$. Then the consideration of the objects as objects corresponds to the Logic of Being wherein morphisms connecting objects appear a something both necessary and external. The Logic of Essence corresponds to considering morphisms as morphisms wherein their source and target objects are implicit.  The identity morphism $id_A$ corresponds to the principle of identity, etc.  The transitions might be understood in a 2-categorical context. In more advanced stages one considers diagrams into $C$. Finally the Logic of Concept corresponds to considering the whole category $C$.  The logical aspect is the internal logic of the category $C$. In more advanced stages we can consider the presheaf category or sheaves over a site defined with $C$ and finally come to the category of categories or other considerations from homotopy theory. Maybe the particular can be seen to correspond to slice categories.

Hegel's critical treatment of judgment and inference is curious and in particular his distinction between judgment and proposition (which might be given some justification in modern type theory) and his introduction of ontological and subjective considerations into formal logic (cf. the apple is red, the apple is edible, the rose is a plant, the rose is beautiful). This might suggest a geometric interpretation in which we distinguish for instance between accidental properties of a space (relating to fiber bundles and connections), relational properties such a cobordism and more intrinsic properties such a homotopy and cohomology.  Hegel's critique of formal logic recalls that of René Thom.  Also we could examine topological accounts of teleology given by the concept of attractor (or even organizing center) in dynamical systems theory. There is also a passage of subjective Concept in the Enclyclopedia Logic which discusses a dualism in the formula "the individual is the universal" which suggests an alteration between the discrete or chaotic topologies.  Either we trivially assert the individual is the individual or else were must assert and and indefinite number of assertions "the individual is not A, not B, not C, ...". 

From the point of view of our Analyticity, Computability and the A Priori,  there does not seem to be any clear reason to reject paraconsistent logic anymore than for rejecting a game because a player's piece can end up both on a white and a black square. Paraconsistent logic would seem to be a step towards formalizing Hegelian Reason (and, be it mentioned, furnishing a simple solution to the sorites paradox). Maybe an analysis of Fichte's Wissenschaftslehre might furnish a few clues: the ego posits the non-ego and we have $A \wedge \sim A$. From a Hegelian and dialetheistic perspective it seems reasonable to opt for the presentation of paraconsistent logic which discards the negative syllogism $A\vee B, \neg A \vdash B$ as disjunction introduction (or the disjunctive syllogism) seems to be given a more favorable view.

But reading the last sections of the Encyclopedia Logic we are lead to propose that the ultimate expression of the Hegelian Concept, the Idea,  can only be a formal system which can reflect its own metatheoretical properties  as well as of weaker systems and which can reflect its own proof theory, it can internalize the very process by which the external subject acquires metatheoretical knowledge in the system: the great example being due to Gödel or in general the classical semantic paradoxes of self-reference and truth. "This sentence is false" can be given a Hegelian interpretation. On the immediate level this expression is a subjective, spontaneous, free, immediate positing "I am freely positing that what I am saying is false" and thus in the aspect of its immediacy and positedness it is true, because it only expresses a free choice and finite determination as such (cf. omnis determinatio negatio). However by reflection and return-to-self this determination dissolves itself, its determination, its truth-claim is the seed of its own overcoming.

Can type theory and category theory encompass this kind of reflection? Are there categories or generalizations of categories whose internal logic is paraconsistent?  Or do we need to break free from type restrictions and alter some fundamental logical assumptions to be able to encompass things like Girard's Paradox (or the Burali-Forti paradox)?  Saying that $\vdash U: U$ is an expression of the Hegelian Idea.  $U$ remains the same in all its particularizations. 

The Hegelian Idea is a kind of dynamic synthesis between Sextus' Pyrrhonian dialectics and Kantian criticism - but perhaps even closer to neoplatonic dialectics. Dialectics as the life of the nous which unfolds, divides itself, circulates through itself then reunites and integrates itself within itself.

The topological dynamics of plant physiology

We have mentioned Goethe's method in the context of the history of phenomenology as well as its connection to the modern topological and geometric theory of differential equations and dynamical systems. Anyone who has studied biology, or plant biology in particular, learnt about the systems of nutrition and transport in vascular plants, the two main types of vascular tissues being xylem and phloem. The first system is easy to understand enough (and its analogy with animal systems obvious) and not incompatible with a general Goethean conception of a plant with its vertically rising gradient. But phloem is a different matter.  Topologically phloem tubes - transporting sugars and amino acids synthesized in the leaves - are spread across the entire plant in a structure analogous to animal circulatory systems - a perplexing situation which begs for deep topological and dynamical questions and explanations. For if we consider the the system of phloem tubes to be topologically equivalent to a closed ball $B^3$, then the circulation of nutrients will define a vector field on $B^3$ which is tangent to its boundary $\partial B^3 = S^2$.  So there must be sources and sinks. We can ask: where are the sources and sinks in the plant? Is there an analogy with animal digestive systems - or even neural pathways? And even more importantly: does this vector field change in time (undergo a continuous bifurcation)? Modern plant biology of course allows us to answer these questions. The point here is the great philosophical and methodological importance of asking precisely these kinds of question and the role of topological-dynamical thinking.

A point about subjective idealism

Consider the opening sentence of Schopenhauer's famous book: "Die Welt ist meine Vorstellung/The world is my representation".  How can subjective idealism seem at once so utterly certain and so dubious? The answer is quite simple and lies in the definite article "Die/the" in Schopenhauer's dictum. The world in the sense of the world in which each one of us is acquainted with through first-hand experience - my world that I take immediately as the world - this cannot be in reality any more than a modification of the subject, of consciousness, a self-projection, self-occultation of consciousness inside itself. Even from neuro-reductionist physicalist assumptions there is no way of avoiding this conclusion without dogmatism (the ruse of invoking the objectivity or public nature of logic or language) or non-physicalist explanations and we can turn Benaceraff's overrated objection against Platonism against the physicalist who wishes to refute subjective idealism:  basic neuro-physiology and gestalt psychology show that a direct causal connection between nerve impulses and the "causality" of the phenomenal subjectively constituted world - is sheer nonsense. There are physical impacts on receptors which do not produce effects on consciousness and there are effects on consciousness which are not the direct result of impacts on receptors.

The world in the sense of that world with which I am acquainted with through first-person experience, is entirely phenomenal and exists in consciousness only, despite its objectifying externalizing intentions and pretenses.  However from this it does not follow in the least that a world, any world, is my representation. It does not follow that outside my world there is not another world. Nor can we rule out that there could be some sort of cognizable connection between our world and such a world, the world which is not our world.  In Kantian terms there does not follow that there is no thing-in-itself and it does not follow that we cannot know anything about this thing-in-itself (we already know that we know that it does not necessary not exist).  Despite Kant's own arguments, we may hold that such a world-in-itself is perhaps a grey, timeless, purely mathematically structured world but in which other world-creating consciousness subsist. We can even discern such a conception in Hume's Treatise. None of our causal scientific theories or explanation depend on the phenomenal intuition or concept of time, only on a vicarious geometric abstraction, a flattened timeless time.

Consider the stereographic projection of a sphere into a plane. The sphere is the subject,  the tangent bundle of the sphere is the modifications of consciousness,  the plane is the postulated world outside consciousness.  

But is this not logical dogmatism where we endow logic, a product (or condition?) of consciousness, with the right to judge that which is beyond consciousness? If logic corresponds to the ability to understand, carry out and check rules and to meta-interpret rule-systems,  where resource constraints can be taken into account,  then perhaps continuous processes - as infinitary rule systems - transcend logic. But logic functions through necessary principles of meta-logical agreement - thus it is reasonable to endow logic with transcendental epistemic scope. If a derivation in one formal system has to be considered sufficient for the cognition of a non-finitary property of non-derivability in another formal system, then surely the belief in the extensive epistemic range of formal systems - of logic - is reasonable.

If the above considerations do not seem convincing, we can point out that the validity and transcendental scope of logic (the highest a priori condition) is an essential assumption in Hume (as we discussed in detail before), Kant, Schopenhauer and Husserl.  We can always ask Kant: what gives reason the right to judge itself and its own limits? Does not transcendental logic have to follow the rules of general logic as well (even if it includes important differences)?  And another important approach involves the formal verification principle: challenge the meaning-as-use ordinary language philosophers and neoscholastic dogmatists  to set up their theory of knowledge and mind in a formal system and then proceed to critique those systems using their own rules. The weak conclusion remains the same: either physicalism is false or (qualified) subjective idealism is true. The stronger conclusion is that if consciousness is essentially non-physical and non-local (or even trans-temporal) then this fact tends to strengthen rather than weaken the case for qualified subjective idealism. And if the dogmatists invoke a direct intuitive "given" or direct intuitive acquaintance with allegedly real exterior things then they have no right to discard the direct intuitions of consciousness being the fundamental reality that can be experienced during spiritual development and altered states of consciousness.

If we are certain that there are universal moral laws known a priori then it would seem to follow that we should treat phenomena which appear to us as being sentient beings as if they really were sentient beings (treat them according to the moral law),  regardless of the non-reality of their first-person phenomenal presentations. This is because we do not have nor could ever have certain knowledge that they do not exist (either entirely or in part) and secondarily because regardless of the situation the practice of moral deeds can only strengthen character and morality itself.  If a child is nice to its toys it will probably be nice to people and animals as well. The similarity of this argument to Pascal's wager is noted.

But subjective idealism is a misleading term because it suggest than an individual ego-subject, finite personality or self-in-the-world is at the same time some kind of substrate of the world. But in reality cosmo-ontology and egology are illusions to be dissolved together. This is the task of transcendental dissolving and liberating insight,  true philosophical phenomenism/positivism/empiricism and skeptical dialectics.

Also considerations on temporality and mereology and  intrinsic intentionality, synthesis, constitution and constructivity - applied to both the self and experience  - are quite sufficient to establish the above liberating insight with regard to the egological and cosmo-ontological illusion.

Very short note on formal concepts in science

From whence do we get the impression that some mathematical models are closer to physical and spatio-temporal intuition,  more down to earth and intimately tied to concrete applications, while others hover close to the heights of allegedly less useful  'abstract-nonsense' ? The real numbers and differential equations, these are seen as tied to dynamic-geometric intuition and of vast applicability and interest in engineering and science.  But abstract algebra and category theory appear to have no direct relevance to applicable mathematical models or to their kind of concrete geometric-dynamic intuition. We present here a few short speculations.

Perhaps the abstract models do capture fundamental levels of reality (maybe even more fundamental than the so-called concrete spatio-temporal ones) and what is required is first of all the methodic development of a special kind of intuition or cognition to grasp the planes of reality which they model (maybe something like Goethe's method is called for?). And then further models are required which can effect a mediation or transfer between these two levels, or at least allow a continuity or gradual deformation between concrete spatio-temporal reality and higher levels of reality.

A beautiful illustration of such a meditation and construction is furnished by the double fibration in Penrose's twistor theory which allows a mediation and transfer between physically significant objects on Minkowski space and abstract cohomological objects on the complex algebraic variety $\mathbb{P}^3$.This operation allows solutions to conformally invariant differential equations on spacetime (e.g., Maxwell's equations, Yang-Mills, or linearized gravity) to be identified exactly with Cech or Dolbeault cohomology classes on specific regions of twistor space. This appears to have been Penrose's attempt to give a geometric (topological) interpretation of quantum non-locality. 

Another illustration is the theory of $\mathcal{D}$-modules which presents a mediation between derived categories (and monoidal categories) and concrete models of systems of partial differential equations.

The idea of Kant's schematism of the pure concepts of the understanding can be interpreted as finding the mediation between logic and geometry (including mathematical physics). This is of immense contemporary significance as we find versions of Kantian schematism in topos theory, homotopy type theory and in areas involving monoidal categories in algebra, geometry and physics and in linear logic and computer science. 

There is also apparently a mediation and transition between the theory of bifurcations of smooth vector fields, fundamental  and ubiquitous in concrete applied mathematics,  and monoidal category and operad theory. See also Physics, Topology, Logic and Computation: A Rosetta Stone by John C. Baez and Mike Stay.

An objection can be raised that our concrete allegedly intuitive mathematical models are themselves quite abstract but are only perceived as immediate and intuitive in our present cultural context by a deliberate forgetting of the complexity of their cognitive-historical past and genesis. For instance real numbers are constructed from the rationals by quite abstract, cardinality increasing, procedures involving equivalence classes and identification. And the same goes for the concepts of continuity, differentiability and so forth. Surely the ancient Greeks had other forms of intuitive perception and concreteness in their geometry, physics and engineering. We answer this objection by our arguments that the fundamental dynamic and geometric-topological intuitions of modern science were in fact identical to those of ancient Greece. It is the modern foundations in terms of Dedekind cuts and $\epsilon$s and $\delta$s that can be critiqued from a philosophical and logical point of view and alternative foundations (locales, realizability topoi, synthetic differential geometry, homotopy type theory) can be defended which are also more aligned to the theory of space and change found for instance in Aristotle's Physics.

Meditation, phenomenology and obfuscation

There is an important missing link in the history of philosophy which has significant consequences for the history of phenomenology. This is the work of the young Rudolf Steiner. It is not fair to judge this work by his later abhorrent and manifestly false views as an occultist and cult leader any more than it would be fair to judge Heidegger's Sein und Zeit by his later similar political and cultural-anthropological views. The most important work of the young Steiner (who attended lectures by Brentano) is a work which usually goes under the title of Philosophy of Freedom (1894) but was published and revised numerous times under different titles.  Along with this work there is also much value in Steiner's earlier work on Goethe's scientific method and worldview. 

There can be no doubt that The Philosophy of Freedom, like Sein und Zeit, is a very important work and one of the masterpieces of phenomenological philosophy. In this work (and in portions of his later books such as the one on "Initiation") we find something similar to an exposition of the approach and methodology involving the transcendent self-transparency and self-illumination of thought and consciousness which we have repeatedly attempted to expound, the correct combination of "epokhê" and "satipatthâna".  

Equally important is Steiner's Goethean method of variation and "imagination". This seems to us to be the original and correct form of what later was called the method of "eidetic variation" (and the affinity for Ruskin's approach in Modern Painters seems to have gone unnoticed).  But the consequences are enormous. The combination between the philosophical-spiritual insight offered by Steiner's exposition of the Goethean method on one hand and on the other the development of modern geometric and topological theory of differential equations and dynamical systems would have been a "marriage made in heaven", two complementary aspects of the same essence,  whose union would have furnished the key to a powerful new science.

But the harmful influence of 20th-century materialist cults and their pseudoscientific dogmas (Freudian psychoanalysis, evolutionary psychology) hindered the emergence of such a science. This is the tragedy of René Thom whose work flawed by Freudian and materialist-evolutionary dogma did indeed point to such a qualitative topological science with a method based on developing higher dimensional dynamic and geometric intuition. The fact that Thom never mentioned Theodor Schwenk's Sensitive Chaos: The creation of flowing forms in water and air (preface by Jacques Cousteau) which predates Structural Stability and Morphogenesis can only be seen as an example of intellectual dishonesty. As far as I know, Thom never referred to the wonderful book by A.T. Winfree, The Geometry of Biological Time (1980) which shows how advanced topological concepts in applied science can be presented in an accessible, rigorous and clear way (cf. the explanation of phase singularities on p.26).

A major fault of René Thom was also ironically to be found on the mathematical plane. Thom, like V.I. Arnold, engaged in obfuscation and sloppy presentation of mathematical ideas.

An expert in a field has all the right to develop a conceptual shorthand for their own use or that of fellow experts. In particular, a shorthand for writing down ideas to be developed and unfolded in lectures. Such texts can also employ beautiful drawings and diagrams which may impress even the non-expert. However all this has nothing to do with writing a book whose aims is to open the gates of a knowledge to those who, while not having such a knowledge, seek earnestly for it. A book that is a genuine introduction (i.e. initiation) in the full sense of the word, and which does not only "bring in" the neophyte but presents the beauty, clarity and order of the temple in all its details and adornments.  This is the kind of book, the kind of spirit, style and approach, which every student deserves, and certainly the kind of spirit. style and approach which should be employed when presenting important philosophical, scientific and mathematical ideas to the world. To present the young student with any other kind of book can only be considered a pernicious and immoral act of deliberate obfuscation and perversion. On another occasion we hope to develop the topic of cryptography, deliberate obfuscation and "secrets of the trade". Let it just be said that this has traditionally been a not infrequent vice of academia, the deliberately ponderous, opaque, pedantic jargon which not only hinders understanding of comparatively simple concepts and arguments but also masks the author's debt (or even plagiarism) to their contemporaries or predecessors. The aim is also to appear outwardly as outstandingly intelligent, original and sophisticated and to close the doors of science to anyone outside a select few.

Besides in phenomenological philosophy and Goethean science, a tragedy occurred regarding "meditation".  There is no such thing as "meditation" in the popular sense, there is no secret powerful technique of self-help where one sits down, closes their eyes, breathes in and out, focuses their mind or attempts to detach themselves from troubling thoughts, and thus finds a cure for mental anguish or a way to gain paranormal faculties or blissful states. This goes in particular for everything connected to the term "mindfulness".  And there is evidently no easy quick philosophical-phenomenological method for the self-observation and self-transparency of consciousness.

This is not what "bhavana" or "katharsis" as in neoplatonism and the ancient mysteries was about. Rather bhavana can be likened to the building of a beautiful high-towered palace or temple, including the preparation and purification of all the materials needed as well as the ground and foundations. The "mindfulness" at work here cannot be separated from action. The popular concept of "meditation" actually only corresponds to the final royal stage in which one enters the newly built palace and climbs up many flights of stairs to survey the beauty of all the sparkling rooms and luminous windows until reaching the most wonderful of all views at the top. Before that it is the long toil and active work of the artisan and soldier - which can be be interwoven with periods of peaceful surveillance and contemplation of yet unfinished work. We do not meditate as some kind of easy, all-sufficient, way to overcome our desires, rather we first diligently and actively uproot desire-images-complexes in order to attain the peace and illumination of "meditation".  Using a software engineering metaphor, we must start with cultivation at the abstract object-oriented level before proceeding to reverse engineering at the assembly level. Thus taught both the Pali suttas and Plotinus.

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From phenomenology to a philosophy of consciousness to spiritual science

Alas there is no clear definition of what phenomenology is (which should not to be confused with the specific philosophical work of Edmund Husserl) or even what phenomenism, empiricism, positivism and the theory of knowledge exactly are.  There is no clear definition of what the philosophy of consciousness is or how it could differ from the philosophy of mind. And there is no clear idea of what a spiritual science could be or a science of consciousness. We are in possession however of the bodies of knowledge encompassed by the various branches of psychology and cognitive science - and there is surely much of value there.

In our writings we have emphasized a few important points relevant to the entire interwoven enterprise of all these subjects. The centrality of first-person experience, its descriptive insight and analysis and most importantly:  there can be no genuine progress without a radical and thorough form of personal transformation aimed at at self-transparency and self-transcendence of consciousness. We attach importance to the first-person experience of the body and the various feedback mechanisms involved, specially those related to sensory systems and the peripheral nervous system.

Is there any relevance or legitimacy in the idea of mathematical (or formal) models of consciousness? 

This question, which we will not attempt to give any definite answer for now, seems related to the circumstance of there being two distinct paths (but not necessarily non-complementary or non-convergent)  passing through all the subjects mentioned above, whose ultimate aim is not only spiritual science but the attaining of ultimate spiritual knowledge. We might characterize these two paths as pertaining to the Buddhist and the (neo)Platonic philosophical and spiritual sciences - it being understood that these terms should be read in a very universal sense encompassing in particular a huge range of philosophical literature from antiquity to the present, both from the east and the west. Both their differences and their common core (we have written on the connection between the Platonic-Socratic dialogues, Pyrrhonism, Damaskios and the Pseudo-Dionysian literature) are important.

The differences manifest in the kind of mathematical or formal model of consciousness we could associated with each one of them. The (neo)Platonic path would seem to point to geometric and topological models of consciousness (due to its synthetic, global, anagogic methods) while the Buddhist path (analytic and actively illuminating through insight) would prefer event and process-oriented computational models. Its western embodiment has great luminaries such as Sextus Empiricus, Hume, Kant and Brentano. Dialectics is found in both paths but with different modes of employment and emphasis. For both thorough psycho-somatic peace, calm and relaxation is a precondition for any progress.The dichotomy between the paths is not meant to be universal and disjoint as it is easy to find traditions that sit somewhere in the middle or incorporate core elements of both approaches (Yoga, Samkhya, various schools in the Upanishads and Bhagavad-Gita).

About the geometric-topological model of consciousness. This resolves around the ancient notion of topos (cf. Aristotle's Physics Book IV), the modern concepts of stratified space, and the granting legitimacy to traditional cartographical, architectural, horticultural and other metaphorical language regarding consciousness and the diverse faculties and regions of the soul (cf. the ars memoriae).  Its foundation is  the work of Plotinus (which integrates the best elements in the Platonic and Aristotelian texts). There are also interesting elements from neuroscience and psychology that can be integrated into such an account.

We can cautiously suggest that Plotinus' process of spiritual simplification, illumination and reversion (we would say rather 'involution' and 'inversion')  involves a radical shift in the self-perception of consciousness and at the same time a radical shift of perspective regarding what is perceived and conceived as spacetime and the laws of physics. We propose that this transformation, this shift of perspective, involves the Penrose transform of complexified (and compactified) Minkowski spacetime. The new reality is twistor space $\mathbb{P}^3$ and the previous Minkowski spacetime is something secondary and generated (its points correspond to lines in $\mathbb{P}^3$). As regard to the hyperboloid hypersurface which appears naturally in twistor theory (it corresponds to the image of real Minkowski space and stratifies twistor space), we can recall this verse by Alan Turing:

Hyperboloids of wondrous Light
Rolling for aye through Space and Time
Harbour those Waves which somehow might
Play out God's wondrous pantomime. 

 

The initial process of complexification and compactification can be likened to unveiling a more fundamental ontological domain, the intentions, meanings or processes of the world-soul at work in physics - which corrresponds to a parallel process in consciousness. Fiber bundles are perhaps a natural mode for the human mind to grasp high-dimensional spaces. The isomorphism of the fibers at different points represent their ubiquity and unity in the Plotinean sense.

 

Spiritual science will involve a geometry and cartography of  consciousness employing a stratification into spaces with their own dynamics and deformative processes which become aligned within a global anagogic process.  However we must begin with the most humble processes related to the psychosomatic domain and their geometry and dynamics. The geometry of biophysics furnishes the clues and can be transposed - through a process of variation, complexification and compactification -  into the domain of a geometry of consciousness. The branches of mathematics which offer a unification, fusion, of logic and geometry, are surely of great relevance here. 

 

There are several interrelated geometries of the mind. The geometry of the architecture of our cognitive-semantic spaces. The geometry of our personal narrative (our biography) and of narratives in general, specially the initiatic journey. The geometric physiology of the soul's powers and functions. An important aspect of such geometries is not only their stratification, multidimensionality, dynamic nature, relation to projective space,  but also their foliated nature, all forms manifest as a simultaneous united continuum of many different levels. An opposing view would see all this as a cultural inheritance of forms of illusion (prolongations of a standard naive pre-biochemical understanding of biology), of aggregation and identification and put in its stead Hume's sublime concurrent process vision of consciousness. We distinguish between objects and beliefs constituted by function (mistaken for essence) and true essence revealed in the reality of the underlying concurrent processes which do not reveal themselves necessarily as they are when they construct belief-objects subordinate to function.

 

But perhaps indeed the vision between the two paths is not so significant in comparison to what they have in common - the science of the subtle and spiritual bodies. Also recall also how Hegel and Husserl would hold that, as strange as it may seem, Plato, Sextus Empiricus and Hume  present complementary aspects of the one truth and are capable of unification at a higher level. But also postulating unity can be harmful at the level of spiritual paths, contrary to a common trend. Rather one must be lucky to find a path that is 'beautiful in the beginning, beautiful in the middle and beautiful in the end'.

 

Before the question "how can we gain higher knowledge of consciousness itself?" we should ask "what is the best knowledge for consciousness?"  It is the knowledge that lets it see through itself and liberate itself from itself. A central object of the best knowledge is the difference and unity between perception, feeling, volition, thought and the sense of self and awareness. Also of temporality and temporal intentions and the construction of objects and concepts. And is not logic the science of consciousness par excellence? And even more importantly: the best knowledge is not just knowledge of the highest object but the best knowledge is the knowledge of the best knowledge itself in its illuminating and transcending function. Consciousness must be unbounded light and its knowledge be knowledge of its own knowledge and of its own knowledge's illumination. Recursion and reflection-into-self permeate the Pali Canon and its path of spiritual development. Consciousness lives usually passively immersed in and hypnotized by its own self, forgetting its own power not only of freeing insight but its ability to act directly upon itself, dissolving the mistaken belief in the necessity of certain mental content.

 

And we can give a symbolic-anagogic formalization of consciousness properly understood, or at least partially mirror its self-clarifying, self-transcending, self-subsuming process. We have sketched some of this in our "Hegel and Modern Topology".  We could add thereto some considerations on monoidal categories (which furnish a common ground for linear logic,  concurrent processes, algebraic geometry, PDEs and quantum theory). What is important here is the internalization of their own "boxes" into wires (input and output) allowing the modelling of dynamic rewiring (closed monoidal categories).

 

About the coherence conditions in monoidal categories. The coherence condition can be viewed as the necessary amnesia which forgets the history of a process to look only at the final state and  conditions the connection isomorphism only on the final result (cf. Kant's example of surveying a house in the CPR). This is what allows objectivity to emerge: objectivity is the postulated forgetfulness of the plurality of different presentations and the postulate of their essential equivalence. As Aristotle wrote: the road from Athens to Thebes is the same road as that from Thebes to Athens. Coherence expresses the emergence of objectivity (hence order) by a voluntary forgetfulness of a system's memory via focusing only on the final state. The condition of (shallow) objectivity is that something is what it is independently of its process of coming to be.  If you start with a certain composition of $\otimes$-products X and switch them around using tensorings of ids and associators to obtain a new composition of $\otimes$-products Y then the resulting composed isomorphism linking X and Y will always be the same, no matter the intermediate route. Without a partial repulsion of memory one will be constantly becoming something different - this act is also that of the Understanding in Hegel's Logic that forgets, attempts to cancel,  its own process of coming to be.

Short note on philosophical methodology

Cultivating what we should call not transcendental consciousness, but transcended consciousness, the true epokhê in the school of Pyrrho and Hume, we realize the following.

That the formal mathematics and formal philosophy projects and the philosophy developed in our paper Analyticity, Computability and the A Priori are themselves perfectly legitimate forms of transcendental philosophy and neutral superpositivism.

That the purpose of life is to die well, finis vitae bene mori. 

Transcended consciousness is not only the power of being beyond time and its manifold constructions, but to have the liberty and freedom to concretely imagine and to virtually live consequences and alternative possibilities and to change perspective regarding a given contingent time.

Thus desire is dissolved by the simple transcendental consideration of: what if I had that which I am now desiring? Thus the taking-for-granted, the masking, of the good and blessings we have in the present and the twisted mirage of having a better past are dissolved by the recognizing that we will almost always live in our past, in the golden age, relative to the total temporality of our life. 

Thus we may long to live again in the world before the advent of LLMs.  But if we had the total perspective of how the world will change in our lifetime, we would have greater appreciation of living in the precise present that is present to us.

Yet more short considerations on AI

The philosophical significance of LLMs and a potentially powerful philosophically based critique of LLMs have not yet been developed and perhaps their importance has not even been recognized yet.

Could LLMs be tied to a certain philosophical view regarding the mind, language and the world? And if such a view is manifestly erroneous could not this furnish a sound basis for acknowledging - alongside numerous other reasons - the social and cultural harm of LLMs?

For instance, could we not explore the relationship between LLMs and Quinean extensionalism and meaning-as-use theories? Are not LLMs based on the rejection of the irreducible intensionality of human symbolic activity? Behind every symbol there is an intension. And a formal theory of intensionality must itself acknowledge the intensions of its meta-symbols.  But there is no intensionality in LLMs beyond that of the humans involved in their creation. Searle's Chinese Room ignores the deeper philosophical meaning of computation - or a potential associated geometric theory of meaning - but may be of interest for a critique of LLMs.

Do LLMs compute? Computation is a primordial intensional human activity (see our paper Analyticity, Computability and the A Priori). 

Perhaps there are other machine learning models of language of a more geometric or even combinatorial-algebraic nature which would have interested Riemann who wanted to relate meaning to a kind of cognitive geometry.   Perhaps statistical regularities should be traced back to geometry.

What are LLMs really, formally? Can we formalize the critical values wherein they become 'adequate' for their proposed task? What exactly in 'large'? How can this be formalized rigorously? Can LLM techniques be used for formal axiomatic-deductive systems and automated theorem proving? Or can we prove certain fundamental limiting theorems about the powers of LLMs akin to the unsolvability of the halting problem and Gödel's incompleteness theorems?

LLMs only exist because of the Internet. The Internet and LLMs are part of the same historical-cultural-technical process.  This ontological process might be described as the datafication of humanity. Language ceases to be a tool of human thought, communication and culture-building but rather a tool for the reduction, degrading, emptying, perversion and commodification of humanity itself. The Internet and LLMs are the anti-Gutenberg. Man has become text, sound, image, data, statistics. LLMs are a counterfeit reality, a monstrosity, the world becomes one big corporate controlled screen.

 If we compare the training data and the resulting LLMs is there or not a loss of information? Would it not be more worthwhile to develop sophisticated search algorithms and querying language to access the training data directly?

 LLM culture is the culture superficiality, atomization, banality, cosmetics and deception (a LLM is almost a trained deceiver in the biological sense - it detects and mimics human patterns). There is a loss of the multiple layers of meaning behind every symbol which cannot be reduced to statistical correlations with other symbols. Meaning is replaced with arbitrary social-statistical emergent correlative patterns.

The realm of pure mathematics - and that of pure logic, combinatorics and computability - is a pure realm which LLMs cannot touch or corrupt.  So the formal mathematics and formal philosophy projects, contrary to popular misconception likely resulting from deliberate propaganda and deception - are the antithesis of and antidote for LLM culture. There is a pure universal computational-mathematics-akin language (far beyond the natural language or the audio-visual data  that can be perverted and imitated by LLMs) and a pure logic and a pure thought and mankind may indeed hope to attain them. 

LLMs are not intelligent and statistics cannot solve formal computational problems nor can they encompass the pure a priori synthetic principles of formal computational systems (i.e. the cognitive certainly of the foundational principles for metatheoretic knowledge) as detailed in our paper mentioned above.

No statistical pattern analysis of the shadows are sufficient to lead to knowledge of the object. Pixel injection in image recognizers demonstrates this fact, and similarly for formal reasoning.  Besides lacking intensionality, LLMs lack reference and context (despite the misleading terminology of context windows). 

And most important of all since the true intellect is inseparable from morality, empathy and compassion, completely beyond the reach of LLMs. LLMs do not have the bondage to an illusion of a self.

A major task of philosophy is to destroy the evil empire of LLMs and a good starting point is deconstructing and refuting the worldviews (extensionalism, meaning-as-use) which LLMs embody. 

Is a lawyer someone trained in a specific system of laws or someone who has developed the skill to study, interpret and apply any given system of laws or perhaps be able to cope with significance changes in present laws?  Such a metalawyer is the analogue of a universal Turing Machine. A LLM could never be a judge or a metalawyer - it could not grasp the spirit of legal institutions or the deeper meaning of a given legal context.

Billions use light-bulbs without understanding the underlying physics. Billions could use LLMs thinking they are conscious.

LLMs will become more interesting in the measure in which the multilayer perceptrons models are replaced by geometrically and mathematically more sophisticated models (KANs are a step in the right direction).

Short philosophical considerations on AI

Hegel and Heidegger were thinkers about their own time, thinkers about historical events and happenings. Few have the insight and courage to fathom the full depth of the meaning of an historical event, the coming to be (coming of age?) of an historical process. Tragically,  it is only some time after the event (the time of monsters?) has hit humanity with full force that Hegel's famous owl can spread her wings. Is it not true that some of what the prophetic author of Sein und Zeit wrote about technology only makes full sense at the present?

This seems to us particularly true of the emergence of the age of the Internet and the age of generative AI which is its logical development.

And yet nothing could be further from our own philosophizing than any form of historicism or historical philosophy. As such both Hegel and Heidegger, for all their interest and insight, must be considered as having crafted systems based on an incorrigible error. 

Social progress is not a law of nature but a legitimate hope - even if at present it seems a distant one - and it is our moral duty to work towards it in the midst of uncertain outcomes.

The advent of the internet was the advent of connection between people. This connection carried rhetorically moral undertones and echoed enlightenment ideals about the desirability of sharing and making knowledge available.  In the present age of the generative AI based Internet powered and controlled by corporations and governments aligned to anti-enlightenment ideals, it may be that it is morally called upon us to practice instead the process of disconnection and the purification and preservation of knowledge(not obviously in the sense of the 'great simplification' of the Canticle of Leibowitz).   

The most basic step is ensuring locality of core information. That is, to be in possession of machines onto which have been downloaded significant portions of Internet Encyclopedias (despite their serious shortcomings) as well as some decently performing LLM. To this we add, it needs not be said, massive of digital preservation of human cultural artifacts, notably libraries.

One can use Kiwix and download for offline viewing the most recent English Wikipedia (50GB text-only 150GB with pictures).  With a AMD Ryzen 7 5825U processor with 16GB RAM and 2GB Radeon Graphics one can use Ollama and download and use some decently performing LLMs (gemma4 comes in E2B, E4B, 31B and 26B A4B). 

Most living beings alternative between states of being awake and of sleep. Can would we design a dynamic LLM which similarly alternates between states of user interaction and re-training based on this interaction? The most important being the correction and/or updating of knowledge or perhaps the removal of harmful and biased content and "thought patterns". If LLMs can improve then it is not only  a question of having the number of parameters equal to the number of neurons of the human brain.

Computers can enhance and aid human cognition as well as hinder and destroy it (there is a growing body of evidence concerning the disastrous effect of excessive or inappropriate generative AI use for individual mental health and cognitive development, not to mention for society as a whole). 

But the harms of generative AI have little to do with lesser-known extremely powerful and beneficial aspects of the computer for human cognition. We cannot go into this in detail here. Let it just be said that it involves using adequate software for the rigorous formalization of human scientific theories and concepts (specially logic and mathematics and formal methods in the sciences) and the vital feedback-loop between human thought and the software interface (IDE) which results in the simultaneous enhancement of human understanding and production and the quality of the software-based formalization and implementation. 

The software in question includes not only Rocq (formerly Coq), Agda and functional programming language but such languages as Python, Javascript and C/C++. Python is a multi-paradigm and highly versatile language with an elegant syntax. Python comes close to achieving the ideal of a universal language in the Leibnizian sense and is a wonderful tool for formalization, implementation,  verification and exploration in a variety of areas in mathematical logic and finite mathematics. 

We note also the importance of minimalism (using as few dependencies as possible) and building things from the ground up - this goes for scientific and philosophical projects, not of course for commercial and industrial ones.  We will address in the future the question of the possible role of machine learning in this process.

The geometry of cognitive biases and fallacies

 

Categorizations of objects are certainly a useful, even necessary, part of human cognitive development. We cannot underestimate the interest of so-called Venn diagrams which while going back to Leibniz arguably are strongly yet somewhat implicitly present in Aristotle's Organon. Venn diagrams can describe human categorical-cognitive architecture but with the caveat that we should be aware of the extreme geometric complexity of the sets involved as well as how a vast array of psychological tendencies and schemata are at work in shaping our perception and reasoning about such sets. 

Here is one example of cognitive fallacy (linked to geometric inadequacy) which doubtlessly occurs in political propaganda and statistically flavored pseudoscience. We have two categories of human beings A and B (the universe U can represent a given human population). A can be a desirable or desirable trait while B can represent a self-identified human group. The situation in the diagram above represents a situation in which for a standard notion of probability we can say both that 'most As are Bs' and yet  'most B's are not A's. A typical fallacy (perhaps linked to a tendency for the mind to generate a symmetrical geometric scenario) is to assume that if  'most As are Bs' then most 'Bs are As'.   We can however say that B is over-represented in category A and seek a sociological explanation. In some instance a category may appear to be defined precisely by its intersection geometry with other categories.

The above considerations show that while categories are useful cognitive constructs one must be aware of  fallacies and biases - some linked to a poor grasp of geometry - and never lose sight of what is cognitively of far greater importance: the individual. 

On generative AI

Is generative AI corrupting human knowledge and language and by extension human thinking and human culture themselves?

A wikipedia dump is around 100 GB. Wikipedia could be improved and be semantically formatted to be computer readable and advanced query systems could be developed. Would not this be better for the acquisition of knowledge and the advancement of science? Are AI generated summaries of books or papers valid replacements for human ones? What justifies our trust in generative AI as compared to a search engine?

Generative AI is corrupting the internet. Maybe it is a zombie or Frankenstein of human language and knowledge. Or a bland blend of stolen and adulterated intellectual property. By adulterating human language and knowledge it adulterates thought and culture. In the old internet one could generally become aware of the source and context of bad material. But in generative AI the poison is injected and dissolved into the whole body in an often subtle, not immediately detectable way. The 'neutral' sounding language and fake 'objectivity' are misleading. The term 'subjective' is used ad nauseam. Due to the nature of the training data, in generative AI the truth of a belief-system is a function of the power of the people upholding or promoting it.

The real danger of AI has to do with the advent of systems which no single person can fully understand or control. This is the case for standard operating systems which due to their size and hardware and firmware-linked complexities, have passed beyond being able to be understood by a single person. And generative AI is a black box.

And yet there is no reason why a slim, efficient OS with readable kernel code could not be running on most devices. Would such a kernel, understandable by a single person, be more secure than current bloated constantly updated ones? And is there a reason to abandon the semantic web project? Would the semantic web be better than both the ordinary internet and LLMs?

But we must acknowledge that philosophically the advent of LLMs is something profoundly uncanny and thought-provoking. We hold that 90% of valid criticism consists in just criticism of the poor quality, the fatal presence of previous AI-generated 'slop' and biased nature of the training data, while only 10% is criticism of LLMs as AI.

Are LLMs an emergent phenomenon caused by the size of linguistic data and hardware power capable of processing it? An emergent phenomenon for massive linguistic data in which it becomes possible to talk to data? An uncanny situation wherein a uniquely human trait (linguistic communication) is convincingly mimicked by a machine as it spontaneously emerges, in a way still little understood, statistically from massive linguistic data. As if the unique prerogative of the logos had been stolen from humanity. Maybe a human super-logos needs to be developed to prevail against the AI-logos which offers the illusion of a divine oracle, of having a god as a friend.

Relations in the Early Works of Meinong and Husserl (C. Ierna)

Both Alexius Meinong and Edmund Husserl wrote about relations in their early works, in periods in which they were still influenced by Franz Brentano. However, besides the split between Brentano and Meinong, the latter also accused Husserl of plagiarism with respect to the theory of relations. Examining Meinong’s and Husserl’s early works and the Brentanist framework they were written in, we will try to assess their similarities and differences. As they shared other sources besides Brentano, we will consider very carefully whether we should speak at all of influence or plagiarism. Despite Meinong’s accusations it seems that both he and Husserl took over some elements from Brentano and, partially through him, from John Stuart Mill, who appears to be the most probable source on relations. 

https://philpapers.org/rec/IERRIT

The following philosophers are of central importance to the student of genuine philosophy: Hume, Kant, J. S. Mill, Brentano, William James, Meinong, Marty, Mally, Husserl and Carnap.  For logic we should add Krause and Bolzano.

Dennis Schulting on Robert Hanna’s “Cognition, Content, and the A Priori”

https://3ac1ccf5-0a0f-4d25-bd7d-c2bc30197d56.filesusr.com/ugd/aa4405_673ee990a9c343d8b0d100714edf5054.pdf

Principia Logico-Mathematica

https://mally.stanford.edu/principia.pdf

Zalta uses second-order logic extended with what might be called 'third-order' predicates which express the encoding relation between $n$ first-order objects and $n$-ary predicates. These he calls the 'encoding' relation (inspired by Meinong and Mally). To us the great interest of the work of Zalta and his collaborators is that it offers a solid example of 'formal philosophy', philosophy carried out in entirely in a formal language standing in for natural language. But the real significance of Zalta's work is not so much a setting up a particular metaphysical-philosophical system or giving plausible charitable (re)interpretations of Plato, Leibniz and Frege but rather furnishing a framework for purely formal dialectics, argumentation and debate. The challenge is to formalize and express in this context the process of debate (we could take for instance the formal game outlined in Aristotle's Topics). This will involve a certain theory $T$ which both players must accept (which will include the core axiomatic-deductive system), a sequence of choices of assumptions which one player or both players must be forced either to accept or reject.  Perhaps the resulting logic is like temporal logic? We have given an example of the formalization of philosophical dialectic (debate) in our preprint "A formalization of a fragment of Plato's Lysis".

Another interesting example of formal philosophy is given by R. M. Martin' first-order formalization of Whitehead's philosophy. 

The heart of madhyamaka - kicking away the ladder

Philosophical discourse may be logically, linguistically, psychologically, sociologically and ideologically analyzed. One can question whether philosophical discourse has any claim to veracity or value. One can be amazed by the hubris and confidence of philosophers and question the epistemic or intrinsic public value of the discourse they produce.  One can inquire whether philosophers fail to question the assumptions they tacitly assume, or whether some positions be never stately clearly but only insinuated (cf. Ernst Gellner's Words and Things). One can question whether self-proclaimed "rigorous" philosophy is actually as "rigorous" as it claims to be and question whether such a "rigor" be not entirely illusory and only based on stylistic mimicry and convoluted jargon masking the lack of any possibility of formalization in an axiomatic-deductive system. One can question whether philosophical discourse be not a mere language game (like the crossword) based on vagueness, ambiguity and unconscious association, tolerated due to the loftiness of the object of discourse, the stylistically "scientific" semblance of the discourse itself, specially when focused on technical topics in logic and language. Surely we must reject a game based on the dynamics and structure of human consciousness which yet denies outright the underlying role of human consciousness in the spirit of: pay no attention to the man behind the curtain! Philosophy can well seem scientific yet be not so and produce arguments that on the surface seem to be like a logical mathematical proof but fall utterly short of it. And, most importantly, be completely lacking criteria for veracity and value. And what are we to make of philosophers who downplay  or trivialize introspective psychology, mathematical linguistics and mathematical linguistic analysis (the only way to go beyond circularity in linguistics) or the universal ethical principles of human and animal rights? All human activities with value have  public, objective criteria. Philosophy does not have such criteria. Contrary to most academic critiques of academic philosophy the central flaw is not ignoring fundamental domains of human experience and the human condition (which is a valid enough point) but the claim to be analytical and rigorous while being not so and lacking any objective public criteria of veracity or value. A mathematical proof can be formalized and checked in a proof assistant such as Agda. Introspective psychological descriptions can be confirmed, verified or corrected by comparison and discussion among researchers.  A theory in natural science can be confronted with experimental data and evidence. A medication or medical procedure has the objective criterion of the data of clinical trials.  A work of art is received by the public.. But philosophy has no criteria beyond "deemed acceptable to be published in journal X by the editor and reviewers A,B,C,...". A tacit assumption and error is that the mathematical formalization of (a relevant fragment of) natural language is neither possible nor desirable for philosophy. Another one is (often tied to physicalist dogmatism): introspective psychology (as in Hume, William James and Brentano)  is of little value and not a source of philosophical insight and progress. And if  to this we add to this the rejection of the marvelous metaphoric, poetic and analogical capabilities of natural language, we come to the conclusion that such philosophy is mere verbiage, a linguistic production without veracity or value which merits careful linguistic, psychological and cultural diagnosis and analysis.

A note on the formalization of natural language. When two people sit down to play a game they have reached an agreement. The pieces on the board and the rules will represent the game but are not themselves the game.  The agreement is that the board, pieces and rules will stand for the game. A formalization of a fragment of natural language will take the form of a formal system incorporating a series of formal definitions which a group of people will agree represents a certain fragment of natural language. This group will agree (based on a perceived contextual adequacy) to abide by the rules of the game of such a formal system for the purposes of certain arguments or debates - they agree that the formal system and its moves stand  for the linguistic activity of the debate. The formalization of natural language is in a way a return to the original essence of natural language. Indeed as mentioned in the conclusion of our Analyticity, Computability and the A Priori, it is through the reliving and showing forth of the process of our psycho-cognitive development that we may hope to ultimately see the light regarding the problems of philosophy - without naively confusing the developmental path of a cognitive structure with its intrinsic constitution(*). Just as Kant distinguished between a legitimate and illegitimate use of reason (the transcendental illusion once the basic foothold of logic and introspective intuition are abandoned) so too we must distinguish between a perfectly legitimate use and application of language (in which the adequacy of language is unquestionable, independently of any formalization) and an illegitimate use and domain as exemplified by the naive linguistic productions of philosophy.  The formalization of natural language is a necessary critique, a judgment, a challenge and a sifting.

(*) A complementary method will involve what we described as a "linguistic phenomenology". We are given an object and a restricted vocabulary (which may be or not be a set of proposed semantic primes) and the task is to define, describe or tell a story about that object using only such a vocabulary. 

The three times in human aging

In human aging we can distinguish three 'times'. Physical time, biological time and intrinsic, lived time (ontochronology).  The biological time can be written as a smooth function $b(t)$ of physical time. Ontochronology - which is in a sense the real age for a human being - is measured by the length of the graph $b(t)$ starting from the moment of birth $(t_0, b(t_0))$. Thus

$o(t) = \int_{t_0}^t\sqrt{1 + b'(t)^2} dt$

If we consider the total chronological time of a person's life as fixed, then corresponding to this constraint there can be great variability in the corresponding smooth graph for $b(t)$ (subject to further biological constraints). For instance $b(t)$ may for a significant portion of physical time lie beneath the diagonal $\Delta(t) = t$ and thus that person will have in total a longer youth determined by periods in which their biological age is less than their chronological age. There is a certain similarity between these considerations and the measurement of time in special relativity.

The method of meditation and the philosophical method

There is something to be said about books presenting a "basic method of meditation" such Ajahn Brahm's "Mindfulness, Bliss and Beyond". Such books are part of marketing ploy to sell retreats organized by religious institutions (which serve ultimately as recruiting grounds for new monks and nuns). As such the methods described - which may have unquestionable value - are only suited to the specific conditions of the retreat. Such conditions are inaccessible to the vast majority of us. This includes those who have the luck of having access to the seclusion and peace of nature for some limited amount of time, but certainly not for weeks - the time needed for such methods to be effective. These methods count on philosophical insight about consciousness manifesting on its own during the long periods of seclusion and silence, or they count on the guidance of a teacher. These methods in themselves fall under the category of  "samatha" rather than "vipassana", even if there are also "vipassana" -themed retreats. 

We note that Ajahn Brahm himself probably became aware of the Herculean task of detaching oneself from the world (abandoning past and future to abide in the eternal present) and so introduced an alternative method which amounts to calming the body and mind through body-scanning, a sweeping focus built on psycho-somatic feedback (this method can also be traced to the earliest suttas). Rather than trying to tear the stubborn agitated child from his toys or tearing the toys from the child one engages in calming and lulling the child to sleep.  What is involved here is a doctrine concerning the fundamental importance of the first-person experience of the body and concerning the fundamental unity and cyclic dependency between this first-person experience of the body and all aspects of consciousness. The first-person experience of the body involves localized 'feeling'  and the cultivation of consciousness of such feeling can open up vast domains of very subtle and significant experience. The doctrine is that such domains of subtle feeling in first-person body experience are the key to total mastery of all domains of consciousness (cf. how the flux of consciousness is woven with sense impressions and sense impressions are intimately psycho-somatically related to sense organs, receptors and their neural pathways). The doctrine appears to be present in a more-or-less veiled way for instance in the sutta MN 119. We can think of the psycho-somatic feedback as creating a kind of field, container (cf. the simile of the soap and bath in MN 119) in which the rest of consciousness is then immersed to become pacified and purified (cf. again the similes in MN 119). One can say that the initial process of both local and global psycho-somatic feedback prepares a thoroughly purified and pacified 'container',  'field' or  'body' (there are many meanings of khaya in the suttas).  Once this is thoroughly achieved then piti and sukha will 'dawn' and 'well up'.  We then interpret the similes in MN 119 as involving the further purification, 'drenching',  'irrigation' and total 'covering' of this 'container-field-body' with piti and sukha. There are interesting connections that could be made with Plotinus as well.

Hence the necessity of a "method of meditation"  which is still powerful and effective in the context of ordinary life in the world.  Such a method must be based - alongside the considerations above -  on introspective-philosophical psychology founded on phenomenological insight. A method of philosophical psychological analysis, introspection and insight which aims at liberation from both the "world" and the "self" whose starting point is the positivist neutral consideration of consciousness, the facts of consciousness, as they are in themselves, as manifest in their manifestation, without the screen of metaphysical reification or the habitual oblivion of their fundamental co-conditioning factors such as temporality or the underlying mental directedness in recollection and anticipation. A powerful tool for this method is hearing or reading the original suttas of the Pali Canon or foundational philosophical texts of the Mahâyâna tradition. It is in this spirit that much of what we have written should be read.

Another point concerns the difference between Western and Eastern art. It does seem to us indeed that in some schools and works of Eastern Art (specially traditional Japanese art inspired by Zen) have a very strong connection to the pure neutral phenomenological-philosophical insight discussed above. In some sense, there are schools of Eastern Art which manifest anatta and the holistic flux of a neutral phenomenological consciousness (for instance the traditional Shakuhachi)  which is rarely found in classical Western art (but we do find such for instance in more contemporary compositions such as Brian Eno or the Polish ambient band How to Disappear Completely). 

Should Type Theory Replace Set Theory as the Foundation of Mathematics?

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative (Thorsten Altenkirch).

https://www.researchgate.net publication/368474630_Should_Type_Theory_Replace_Set_Theory_as_the_Foundation_of_Mathematics 

We refer the reader also to Bealer's interesting critique of sets in his book Quality and Concept. Our own finitist and computationalist position concerning foundations, as we have discussed before, is very critical of ZFC and highly favors dependent type theory (but not necessarily homotopy type theory) but also attaches value to Girard's system F (or its extensions) and second-order logic (cf. Simpson's reverse mathematics project) and to systems such as $HA^\omega$ (Heyting Arithmetic in all finite types) built over Gödel's system T.  We consider the concept of list (finite sequence) to be more fundamental than that of finite set - this is reflected in combinatorics. It would also be interesting to investigate a 'dialogical' foundations of mathematics (inspired by linear logic and functional interpretations as well as automatic theorem provers) in which proofs are replaced by dialogues in the style of Plato's Meno.

Category theory is not a foundations of mathematics, but it is close. Some suitable extension of dependent type theory can serve as a good foundations which we can liken to machine code (or assembly language) together with the system calls of an operating system kernel.  Category theory is then like the C programming language and its compiler.  

It is interesting to note that it does not seem that second-order logic or even certain extensions of second-order logic are natural or suitable as a foundation for even basic areas of modern mathematics, even from a finitist (enumerable) perspective. Even if we consider only countable rings of cardinality $N$ - or consider elementary number theory - we are lead to quickly to the cardinality $PPN$ if we wish to consider something as simple as a quotient ring $R/ I$ which is a set of subsets of $R$. Also in modular arithmetic we are in fact dealing with relations between congruence classes, thus going beyond second-order logic. The same phenomenon is likely true for category theory, even if quotient constructions can be formalized without the leap into $PPN$.  It would be interesting to check if Euclid has natural third-order constructions. The lesson is that we should abandon extensionalism and use a functional interpretation of such higher-order constructions. This is what Coq and Agda do and we should investigate in detail anticipations of such an approach in ancient philosophy.

Is a mathematical model of consciousness possible?

No, a mathematical model of consciousness is not possible. First we must distinguish between the natural consciousness of Dasein studied according to Heidegger's analytic (in which there are indeed certain proto-topological notions present) and the awakened insight-imbued ego-absolving consciousness. In both cases the idea of a mathematical model of consciousness is a serious mistake because all mathematics assumes that the fundamental problems regarding circularity in the a priori and analytic we discussed have been solved (i.e. by modelling consciousness via mathematics we are already making assumptions about consciousness necessary to justify the use of mathematics). This is analogous to the error of trying to give a philosophical account of language via meaning-as-use theories. As we wrote in the conclusion of our paper on Analyticity, Computability and the A Priori:

How can we shed greater light on the questions and problems discussed in this note? What methods should be employed to go beyond the seemingly unbreakable circularity between logic, computation, arithmetic and combinatorial intuition? A possible answer lies in adopting a rigorous phenomenological approach or the combination of such an approach with something along the lines of Piaget's genetic epistemology. We need to investigate thoroughly what it means for us to learn something (like counting, calculating), to know how to do something, to understand rules, to know how to play a game and to understand a game. All of this is at present unclear and difficult. It is naive and dogmatic to take these topics that demand extensive investigation and elucidation as some kind of philosophical explanation as done for instance in meaning-as-use theories. The elucidation in question needs to proceed not by formal abstract reification (which leads to the same circularity) but by investigation of the matter itself in its concrete lived experience, both personal and interpersonal.

 

We of course must be careful about what we mean by 'model' or in what sense we are discussing the 'modelling' of something. The specific sense we had in mind here is that of theoretical physics: the model, a theoretical system consisting predominantly of mathematical structures, is posited as representing or reflecting the actual fundamental structure and process of the physical world, so much so that physicists in possession of complex mathematical theories (which do not involve any trace of ordinary physical intuition) usually speak of attaining a deeper 'understanding of nature' . This is, of course, very different from the 'models' used in engineering simulations or in applied mathematics.  All talk about consciousness, from the theories used in behaviorist or Gestalt psychology, theories of personality, Heidegger's analytic of Dasein, Piaget's genetic epistemology, the Buddhist abhidhamma - are in a sense 'models' of consciousness and as such perfectly legitimate expressions of the human desire to understand. There is nothing wrong with some of the most technical or sophisticated of these models employing mathematics as an extension of their language, mode of description and theoretical articulation (*). However this all very different from the idea of a mathematical model of consciousness conceived in a fundamental, philosophical, all-embracing sense analogous to the way mathematical models of theoretical physics are posited as models of nature. This is, as we saw above, a fundamental error and category mistake (we cannot relegate consciousness to a self-contained isolated regional ontology as done in the domains of the sciences) which ignores the fundamental question of the foundations of mathematics itself (in the circle of the analytic a priori). Mathematics is a fundamental problem of consciousness, not itself a key to solving the philosophical problems of consciousness (in Kantian language: the question of how a priori synthetic judgments are possible is essentially a problem of consciousness). There can be consciousness, conscious experience, without mathematics or mathematical activity, but there can be no mathematics or mathematical activity without consciousness. Thus consciousness, not mathematics, is the ontological primitive. Again, consciousness is Turing complete and has inherent meta-postulates regarding meta-interpretation, reflection and hypercomputation: in particular it represents a term-rewriting system which can produce a $n$-length derivation of any term-rewriting system for a given initial word.  But any mathematical model represents only a particular term-rewriting system (recursive axiomatic-deductive system) which cannot express the universal computational capacity of consciousness or the general inherent meta-postulates of consciousness. Thus no mathematical model can be an adequate model of consciousness. Or, in more concrete terms, consciousness is the power of playing all games, while a mathematical model can represent only a particular game. Thus a mathematical model cannot adequately capture consciousness.

 

Positions which would deny that mathematics is a fundamental problem of consciousness are among the following. Positions like those of Carnap, Quine and meaning-as-use we have critiqued in our Analyticity, Computability and the A Priori. Our critique also applies to versions of Fregean logicism. But the most obvious examples are  forms of physicalism: eliminationism, theories of emergence, functionalism and its (computationalist) neuro-reductionist variants. In this context one needs but to be in possession of a sophisticated enough physical-mathematical-computational model (which may be stochastic or quantum) of the brain in order to have an adequate mathematical model of consciousness, for consciousness is considered to be an emergent phenomenon and its fundamental structures and processes are considered to be theoretically deducible from such a model. Some theories introduce a twist in which the ontological primitives are considered to be 'mind-like' (for instance so-called panpsychism) but which in their articulation and theories of actual consciousness are indistinguishable from physicalism.

 

(*)  In the domains of consciousness involved in psycho-somatic feedback the theory of partial differential equations (the study of oscillatory, radiative, diffusive, equilibrium, deformation and transport phenomena) may play an important role in the study of certain aspects of consciousness - thus lending a scientific basis to many of the linguistic metaphors employed for describing the mind (and the process of meditation). The design of certain complex software such as the Linux Kernel or a RPG game may reflect the structure of consciousness once we abstract away the underlying architecture of the hardware and consider other forms of hardware (truly concurrent hardware, neural nets or celullar automata - something like contemporary GPUs). This being the case, it is very plausible that formal systems accounting for dynamic concurrent interacting processes are important for studying consciousness. While some aspects of LLMs may be interesting for the study of cognitive-semantic architecture (and these aspects very likely could be much improved) there is also much which clearly is meant to have only a practical design and significance. 

The path to peace

On second thought the considerations put forward previously could be quite mistaken both on an existential and historical-anthropological level. The golden chain in the West must indeed proceed from the East. But not through neoplatonism and certainly not through Origen, Eckhart and Heidegger. Rather it is in such figures as Sextus and Hume and Kant. There are true gems in the West but part of Western hubris is the great difficulty in coming to terms that its mainstream traditions are flawed and rotten to the core. They cannot be redeemed or purified or idealized or abstracted. The western religious spirit is so rotten than even in its own physicalist negation it propagates the same essence. Dasein is sakkayaditthi (cf. Merleau-Ponty). A detailed phenomenological analysis of an illusory limiting experience is no less illusory for that. But more brilliant is the Buddhist philosophical insight on how to overcome Dasein. Thus much of our work can be described as attempting to present some aspects of such a super-phenomenology. Anatta and Atheism in its genuine sense (the absence of introducing false constructions, separations and selections in the purity of thusness), are at the heart of Buddhism and is found in many priceless original sources in the West, and are humanity's fundamental step to individual and collective authenticity, progress and liberation.  The direct liberating knowledge and insight of anatta is inconceivable in the context of Heidegger's analytic of Dasein. Anatta is the joyous dissolving of Dasein.

The currencies with which we pay taxes to the world: attention and erotic interest. Do not pay. The task of Hercules is the first stage described in Ajahn Brahm's book Mindfulness, Bliss and Beyond. The rest is easy. Overcoming the world, not engaging in the world, is authenticity - which renders the ego transparent and flimsy and ready to be transcended. Care is the fundamental ontological-existential power. Do not care beyond what is necessary for your daily bread or helping others according to your means. The first ontological-existential step, detachment from the world, relinquishing the world, abandoning the world, returning unto the authentic being-there which is a being-beyond, a being-far-away, a being which is reconciled with itself. An authentic being which does not pay the tax and tribute to the world (which is attention, care, concern, as well as erotic interest). The authentic being which does not sully its care and concern with the world but has returned into the freedom and purity of its own self. The inauthentic being-in-the-world is being thrown in time, drowning in the ocean of time, dragged by time and its storm of past and future. In authentic being the world-time and time-in-the-world are dissolved into the present moment illuminated by the pure manifestation of the universal flux of consciousness.  In this state of transcendent withdrawal, the fabric of temporality becomes transparent and the eternal present shines through.This authentic being-there is the pure manifestation of the world as the universal flux of consciousness, manifest as such. The inauthentic being is thrown into the world and in world-time and also throws itself and loses itself in this same world and its stormy ocean of time. In authentic being, the transcendent being-there, in the universal flux of consciousness shining in the eternal present, the self itself becomes flimsy and tenuous and ripe to be overcome. There is the beacon of freedom. Although a given religion or spiritual practice can be the occasional cause for the launch into authentic being-there can be no doubt that freedom and authentic being in itself are radically incompatible and antagonistic to any religion. Religions are essentially part of the world and dependent on worldly care and intention and their talk is only intelligible for human inauthentic being-in-the-world.  

Philosophical updates

Heidegger's Being and Time must be considered a brilliant and profound work and if not a vast improvement over Husserl then at least a realization of what in Husserl was just reified meta-talk and illusive promise. Being and Time actually shows in a clear, concrete and beautiful way how phenomenology is possible and indeed phenomenology itself is rectified in several essential ways (by recovering the indisputable central importance of being and "existence") and set on solid and clear foundations which in fact are deeply rooted in Hellenic thought.

Being and Time also offers a section which might be described as the proto-topology of the Dasein's Umwelt which certainly echoes some of our previous work.

This work retains great value even if we consider the Dasein considered to be deluded and unenlightened Dasein (the essentially worldly Dasein). Also we can argue that Heidegger misread Hegel and that the Hegel-Heidegger connection might be a rich field of investigation.

Heidegger's approach to language, logic, thought and intellect is deep and illuminating and suggests (besides offering overwhelmingly powerful tools for the strangely non-embarked upon task of  destroying analytic philosophy once and for all)  some radical developments for our paper on Analyticity, Computability and the A Priori. What does it mean for us to learn something, to learn to do a thing, for instance, to learn to add, to learn to calculate, to learn the rules of game vs. to learn how to play the game? This is the principal question of our paper and Heidegger's method (combined perhaps with the work of Piaget) seems well worth pursuing. When we learn how to do something then when we do a thing, the thing itself becomes far from us having been close when we struggled to learn it. We have at present no clear understanding what it means for us to calculate or to play a game.  

Heidegger also offers some interesting insight to our spiritual and anthropological investigations: a shift of focus from early eastern traditions (like Pali Buddhism) to the overwhelming deep luminous source of Hellenic traditions - in which genuine questioning and freedom is found without the shadow of despotism and projected absolutism as in the typology of a 'lord of the world' (which was overcome somewhat in the purely philosophical Mahayana schools and more so in the sublime transformation in Chinese Buddhism grounded in Daoism: if you meet a Buddha on the road, kill him - could not have been easily uttered by an Indian Sage, only a Chinese or a Greek one).  A study of the mysteries, Orphism, Greek Drama, the pre-Socratics, the Platonic texts reveals more and more and shameful plagiarism and perversion which Hellenic thought and traditions (and their sources and influences) were subject to by certain later currents of Alexandrian and Mediterranean Hellenistic cultures (specially the so-called "gnostics").

We point out that Heidegger also has some very disappointing and serious anthropological blunders wherein he reveals himself to be a child of his own time blinded by typical German pseudoscientific racism and nationalism. In the Black Notebooks we find anti-universalist  anti-humanist   Blut und Boden type utterances which are unworthy of a philosopher. But at the same time the correction of these errors is of immense philosophical importance - and it seems it is up to us to engage in this.

There can be no authentic inquiry into being and into man unless the selfsame dignity of every single human being (and indeed animal) is acknowledged without exception (to reject the Dasein of another is to reject one's own Dasein).  All human beings are equal before Being though evidently not all human beings and cultures at a given time have the same purity and openness to Being. 

There can be no valid onto-theology without the fundamental doctrine of Apokatastasis. The eventual universal salvation of all beings can be taken as the fundamental ground of the very essence of divinity without which we have nothing but monstruosities and human calculation and construction.

Heidegger's reading of Eckhart suggests that Eckhart is deeply Hellenic and - if we allow some key clarification and rectification so as to leave no doubt about the unconditional adoption of Origen's doctrine of Apokatastasis - then Eckhart can indeed by considered an important spiritual guide providing a concrete practical complement to Plotinus. Happiness consists in dwelling peacefully in the clearing of Being. We must show in detail how Eckhart's thought is admirably compatible and suggestive of Origen's doctrine of Apokatastasis. Indeed in Wagner's Parsifal the redemption of all of nature is celebrated, even of the redeemer Himself. Spiritual love means liberation from the limitations of place, time and finite fixed embodiment and yields a measure of intemporality, omnipresence and multidimensional integration, reflection, flow and radiation.

And of course with regards to the philosophical understanding of technology it almost seems that Heidegger was a philosopher that was prophetically born not so much for his own time but for our present time. Heidegger is vastly relevant and important to understand the horrors of the contemporary technocratic dystopia and what is designated by the misnomer artificial intelligence. In fact generative AI, large language models, those electrically animated Frankenstein corpses of adulterated human language and instrumentality are the most perfect embodiment of everything Heidegger wrote about technology.

Another interesting anthropological-spiritual source concerns the Persian and Hermetic traditions understood through Henry Corbin's interesting Heideggerian hermeneutics.

When was modern computer architecture invented and by whom?

John Mauchly, 1979, ‘Amending the ENIAC Story’, Datamation, Vol. 25, No. 11

Jean Jeanings Bartik, 2013, ‘Pioneer Programmer: Jean Jennings Bartik and the Computer That Changed the World’, 1st Edition, Truman State University Press: Kirksville.

Gardner Hendrie, 2008, ‘Oral History of Jean Bartik’, Computer History Museum.

Gödel showed that proofs are strategies in a game

 We write $A_D( x; y)$ with $x,y$ lists of typed variables interpreted as follows. The variables in $y$ represent the type of a challenge to formula $A$ and those in $x$ represent the type of a defense of $A$. A proof of $A$ amounts to a term $t(y)$(functional) which expresses that every challenge can be answered.

Now suppose we have $A_D(x;y)$ and $B_D(v;w)$ and we consider $A \rightarrow B$. Then a challenge to the implication must consist in defending $A$ and at the same time challenging $B$. Thus it has template the list $x,w$. The defense, given such a $x;w$ must consist in a defense of $B$ given the defense $x$ of $A$ (a functional $g(x)$) and in a challenge to $A$ given its defense $x$ and the challenge $w$ to $B$ (a functional $f(x,w)$). Thus $(A\rightarrow B)_D(f,g; x,w) \equiv A_D(x; fxw) \rightarrow B_D(gx, w)$.  Note that $f$ depends on $x$ and $w$ because we are only interested in challenging $A$ if the opponent is at once defending $A$ through $x$ and challenging $B$ through $w$. On the other hand it is enough for the opponent to defend $A$ though $x$ for us to need to defend $B$.

Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences

 https://math.mit.edu/~hrm/papers/ss.pdf

Anatta

Anatta was indeed the central philosophy of early, authentic buddhism. But is was not fully understood and rapidly compromised and corrupted. Its recovery, preservation and further exposition and development is found nevertheless in various schools of the East and West, both ancient (and even pre-buddhist) and modern.

Our working hypothesis is that original Buddhism was strictly based on ethics, philosophical insight and meditation (bhavana) and had no religious component whatsoever. Metaphysical questions and questions regarding the afterlife were treated exactly as in Pyrrho or Kant's transcendental dialectics. The concepts of karma and rebirth were originally philosophical teachings whose meaning was lost, being corrupted into literal religious beliefs (wherein the bhikkhu ultimately replaced the brahmin). The original teaching on karma (which involves impersonal polarized psychological energies) did not depend in the least on a literal mythological understanding of rebirth. 

Thus for example we can think of human beings as receptors, generators and transmitters of certain psychological energies, and as being immersed in an environment in which such energies can also exist in a free form.  Negative energies and deeds not only rebound with unfortunate consequences in the life of the individual but also, conceivably, during the process of death the sum-total of such an individual's habitual and accumulated energies may be explosively released into the world and thus influence other beings in a positive, neutral or negative way (there might be an analogy with a supernova in astrophysics). Also during the process of death the individual may experience a kind of final dream in which that individual's life-history and sum-total of energies and habits is lived out in a final synthetic way. 

We hold that Anatta is the essence and culmination of TPC and TPP.  We have discussed and highlighted the importance of Pyrrho, Hume, Nagarjuna - and a certain approach to Hegel.

Let us just say that an entity or structure can be inherently not only fragmentary and incomplete but also inconsistent. A fragmented inconsistent being does not have within itself a representation of its own negation, non-being, of a state where it is not. That is to say, there is no true being without self-transcendence. The best that can happen for an entity is to see clearly the circularity of its own inescapable inner contradiction.

The core of TPP is the development of intentional habits and fundamental energies and forces, virtues of consciousness which counteract from a central source and in all dimensions  the limitations, delusions, inconsistencies and self-postulation and self-proliferation of the fragmentary being.

The categories of consciousness (including space, time, locality) are to be transcended.

We must find strong evidence for the presence and practice of this same TPC and TPP among the most diverse places and people in prehistoric, classical antiquity and subsequent history, all in complete agreement with regards to the only necessary central essential core (which in itself as virtually infinite inexhaustible content).

The primordial tradition is not some kind of primitive human culture, rather it is the ubiquitous and timeless presence of the power of complete transcendence of humanity.

 There is not great difference between logic and argumentation and games. These are the factors involved. One accepts a game  G as a game and holds that the game is interpersonally objective and follows publically agreed upon rules.  One fixes a side, a starting point, a choice X in G. Then X is processed by G (in a way which may involve deliberate choice and participation, or not) and eventually yields an element U from a predefined set of possible outcomes. Logic and argumentation involves furthermore that acceptance of a translation-representation between intentional mental-cognitive content and a choice X as well as the reverse translation from U back to meaningful mental content. Furthermore it is often the case that we postulate a principle of continuity in that the processing steps of G are transparent in their mental interpretability though such do not play any role in the processing itself.

The interpretation of Gödel's dialectica translation in terms of games is one of the most interesting developments in logic in the recent decades (a formula is turned into a game-like formula and proofs become winning strategies codified by closed terms in system T) - and should be compared not only with our reconstruction of the debate game in Aristotle's Topics but also with modern proposals for a logic of dialogue.

We observe that in original Buddhism we meet with many suttas which share a logical-argumentation template and dialectical method which is itself far more important than the contingent details. This method involves a combinatoric bringing to light and articulation of possibilities aiming a bringing about a catharsis and detachment (like the Pyrrhonic epokhe).

Thus Buddhist dialectics and Buddhist phenomenology rest both on a subtle hermeneutics of language, that is, language considered as the vehicle and dwelling place of the proliferation and self-reinforcement of the fragmentary entity. Hermeneutics asks the most fundamental questions which lead to the appearance of cracks and tears in the fabric of constructed reality.