Formal models of consciousness and other matters

From the previous considerations are we able to answer the question: can there be a formal model of consciousness?  This depends on how we understand "consciousness" and how we understand "formal model".  Ordinary human consciousness is just a segment of a vast multi-tiered system of spiritual consciousness of which it is but a small part and emanation - and yet to which it is potentially and essentially identical. Thus if by consciousness we mean the intelligible realm then there can be no direct formal model as we have demonstrated previously.  There can however be a symbolic illustrative mathematical model of the intelligible realm. And also a formal logic and computation based formal model of dialectics which models reversion and the procession of the logoi whereby the human soul reverts and ascends to a higher state.  We do grant that it is conceivable that strictly at the level of soul and its logoi as they unfold and project into the imagination or act upon extension in nature there could indeed be a direct (rather than merely symbolic) mathematical model. Here that our ultrafinitist philosophy of mathematics and theory of analyticity and computability would enter the stage.

We propose the beautiful notion of finitary Turing completeness. Rather than attempt to give a complete formal definition we can give the following preliminary definition for board-games. A board-game is finitary Turing complete if its canonical, most natural or free extension to an infinite (to us indefinite, indeterminate) version of the board is Turing complete. In this sense it has been shown that Chess and Go are finitary Turing complete. Finitary Turing complete systems capture the essence of certain stages of dialectic and the structures and processes of the logical-discursive cognition of the soul. We note that the game of Go has been used traditionally (specially in the Ch'an and Zen tradition) to illustrate profound psychological and philosophical principles. 

If some of the previous considerations were focused on the schema of Hegel's Science and Logic in relationship to the Enneads,  no less important (and indeed more foundational) is the first-person introspective of the phenomenology of consciousness, not only as deployed in Hegel's Phenomenology of Spirit (but in an inverted and distorted way, to be sure), but in the entire first-person, introspective, spiritualist, idealist,  phenomenological and epistemological tradition of philosophy which only achieves its complete validity and goal through spiritual development and transformation (cf. chapters X and XI of Augustine's Confessions and the introduction to Ennead IV.3). 

ζητεῖν τε τὰ ἄλλα καὶ εὑρεῖν βουλόμενοι δικαίως ἂν τὸ ζητοῦν τί ποτ̓ ἐστὶ τοῦτο ζητοῖμεν, τό τε ἐραστὸν ποθοῦν λαβεῖν θεαμάτων. 

Our goal is thus to develop the correct neoplatonic phenomenology of consciousness which expresses the souls' self-knowledge and concomitant re-ascent to the intelligible realm and beyond, and to engage critically with the posterior western phenomenological and proto-phenomenological thought. Mathematics and formal logic can also play an important role here but from an essentially different perspective than in the previous goals. Here we are interested in genesis and constitution, with ultimate foundations and philosophical logic and with the act and effect of logical-dialectical practice.

Note that we must distinguish how intelligible categories unfold and originate sensible categories as it happens in itself and as discursive reason grasps it as a reflection.  But this reflection at the level of the logoi is also part of the dialectical process which leads to the soul's illumination and re-ascent. 

Plotinus starts his investigations in the difficulties related to souls in IV.3-5 with a very non-solipsistic question indeed, regarding the relationship of each soul to others souls and with the world-soul. IV.3.5-6 is of great interest in that the text points to a plurality of noes inseparably united in the total nous. Souls are called the logoi of the nous which represent a stage of evolution/unfolding. Thus each soul has its own particular nous within itself  (IV.3.6 17). Individual souls (note also the remarkable passage in IV.7.14 on the souls of animals) proceed from the world soul at the same time as their proceed from their individual nous while the world soul proceeds from the global nous and is its logos. In order to illustrate unity-and-multiplicity (the soul is the body like light in the air) and the relationship between individual souls and the world-soul, Plotinus states that the same thing can be at different places at once: just as one indivisible soul can be simultaneous present and acting in different parts of the body in different ways so too the world-soul is one and yet completely present in different ways in each individual soul (cf. IV.9). One can use holomorphic functions to symbolize or illustrate Plotinus' theory of how the soul relates to spatial extension or the world-soul to individual souls. If $f$ is holomorphic in a region $U$ then the germ of $f$ at any point $p$ of $U$ completely determines the whole function on $U$ (through the uniqueness of analytic continuation). Thus we can say that f itself (the formative aspect of soul) is virtually entirely present at each point $p$ of $U$ (the body) and at the same time f is spread out through $U$, unfolded into $U$, by taking germs.

In IV.3.9-11 we read that the world-soul is the author of life and mediator of the intelligible realm from which proceeds matter, space and extension and which has received the  logoi spermatikoi from above which she transmits to nature and gives life to all beings and mediates also the soul's re-ascent to the intelligible realm. Later on in the same Ennead the concept of kosmos as a complete and harmonious whole and providence are discussed. The world-soul is also called universal life.  The etheric/astral bodies (called okhêma or vehicle) are mentioned showing that this was not  theory specific to later neoplatonism.

But Plotinus' phenomenological psychology (about sensation, kinds of memory, reflection, reason) can be found for instance in IV.3.25-32 and the following part of IV.4 and IV.6 as well as the key treatise I.1. There in internal textual evidence (in IV.3.25 Plotinus states that memory has been deal with elsewhere and extensively discussed, using the verb θρυλέω)  - as noted by A. H Armstrong in the Loeb Edition of the Enneads, that Plotinus wrote a now lost extensive treatise on memory or perhaps this represented a strictly oral teaching. IV.3.26 30-55 is an important passage wherein Plotinus expounds the purely spiritual active nature of memory and self-consciousness (contrasting to a being in the state of flux), making use of key terms: sunaithesis, parakolouthesis, sunthesis and sunesin. Without negating external sensible objects we can nevertheless consider that Plotinus attains a transcendental phenomenological consciousness in this passage in which consciousness reflects on itself and perceives itself as an independent, free, self-acting, self-representing, self-reflecting, self-conscious being. There follows a discussion on the phantastikon and  antilêpsis and their relation to noesis and memory. In the remarkable section IV.6.3 is developed the refutation of the theory of perceptions and memories as the result of passive imprints and the establishment of the theory that they result from the essentially active role of the soul, the deployment and training of attention (epibolê, prosbolê): directedness, aboutness or intentionality.

The Logic of the Enneads and its mathematical reflections - preliminaries

Plotinus was also a consummate logician in the sense of universal logic, ontological logic or logical ontology.  Buried deep and intricately interwoven within the Enneads we find a sophisticated Science of Logic in the sense of Hegel but expressing a radically different spiritual and philosophical framework and outlook. We will not discuss here the interesting question of a direct influence on Hegel or of Hegel having directly and extensively appropriated neoplatonic doctrines in his own system (we have already given some references concerning this topic). Nor will we discuss the radical rupture in the philosophy and spiritual practices of the neoplatonic school  that occurred after Iamblichus.  We can only stress again the importance of expounding systematically a non-Proclean and indeed anti-Proclean neoplatonism.  

 The project we sketched in our paper "Hegel and Modern Topology" was to given a Hegelian interpretation of key aspects of modern mathematics, logic and computer science and to in turn use these to furnish an interpretation of Hegel (the hermeneutic virtuous circle). We find that most of the insights contained in what we sketched of this project can be transposed from Hegel's Logic to Plotinus's Logic where they seem to find a even more adequate and natural setting. But it must be understood that, as we explained before, in this framework mathematics and formal structures function essentially as an elaborate system of symbolism and hierarchical reflection rather than as candidates for a direct and complete formalization.  The other aspect of mathematics and logic, culminating in dialects to be used as a tool for direct spiritual ascent, has to be considered separately.  

In order to embark on this project we should list the fundamental categories, concepts and processes of Plotinus' Logic which we can organized somewhat along the lines of Hegel's Science of Logic but with fundamental differences which are often the complete inversion of Hegelian conceptions or which collapse what Hegel separated or separate what Hegel joined. Bearing in mind Hegel's Logic of Being we can remark that Plotinus has an elaborate and deeply interesting theory of quantity, extension, number, quality, essence, determination and measure (i.e. the interaction between quantity and quality found in VI.1-3, VI.6 and II. 6) which permeates in the Enneads (and justifies our classification of Plotinus as a finitist) - together with a remarkable theory of relation which when deployed at the intelligible realm fits exactly our considerations of groupoids as models for essential relationality and interconnectedness.  The theory of a self-moving number and relationality of the nous fits well with our previous considerations on Galois groups and Galois extensions. Also the intelligible realm can be symbolized or illustrated as the category of relations.

We note that Proclus's theory of Henads is found in VI.6 but with the difference that for Plotinus they are a proto-determination in Being (as an infinite ouflow) which preceed the intelligibles rather than the One. For a form to exist as a distinct entity it must be bounded, structured, and distinct from other forms. Substantial number is the internal articulation that establishes this multiplicity within unity. In Plotinus’ view, number precedes being: intellect must possess the principle of number to differentiate being into distinct forms. Thus Proclus got his theory of henads from Plotinus' theory of essential number.

 And also a highly developed and articulated theory of infinity (outflow, power) and limit (reversion, theoria)  and a theory of "matter" (both physical and intelligible) which amounts in the physical level to the poorest and emptiest determinations of being (strongly echoing the initial stages in the Logic of Being). Indeed in II.4.9-10 (cf. also VI.6.2-3) Plotinus invites us to perform an abstraction of the quantity and all qualities of a body to arrive a the concept of indetermination. Matter has existence and a disposition to become other things. Intelligible matter is equated to being and intelligible apeiron (the infinity of the One) and physical matter to non-being and to physical apeiron.

Plotinus states explicitly that indeterminate matter will display contradictory dualities - a striking possible interpretation of quantum theory which becomes less paradoxical once we abandon a foundationalist reductionist framework for a top-down emanationist one. 

Plotinus' theory of quantity and quality agrees admirably with the framework of modern topology (he recognizes geometry as a kind of quality in quantity) and differential geometry specially as applied in field theory and differential equations. Indeterminate matter first must receive extension and it is only upon extension that quality can be received. This corresponds exactly to the primordiality of a base space (space-time manifold) relative to its fiber-bundle or sheaf which expresses the local physical-phenomenological properties of the field (Plotinus considers intensive quantities like temperature to be qualities). This lower kind of qualitative structure in turn is only the product of a higher structure and power. For Plotinus space (extension) itself is a product and emanation of higher structure just as a non-commutative $C^\star$-algebra may have within itself various commutative (maximal) subalgebras.  

Thus sensible non-essential quality corresponds closely to Hegel's concept of measure, as quantity determined by something from without and as qualified quantity. The first section on Quality in the Science of Logic corresponds rather to the genesis of quantity in the intelligible realm and soul, where we deal with united multitudes, intelligible forms and their relations and overflow and limit (and we can include indefinite quantity here). Hegel's subsequent treatment of quantity and measure corresponds to sensible quantity and quality in Plotinus and is quite adequate for the mathematical illustrations in Hegel and Modern Topology.

Intelligible quantity exists as pure number and structural order (the archetype of limit and harmony. Intelligible quality exists as the specific power and form (eidos) of an Ideal entity.  At the higher intelligible level, quality and quantity do not conflict or merely sit side-by-side; they are unified aspects of the divine Forms. When projected down into the physical world, they split into extended magnitudes (sensible quantity) and superficial properties (sensible quality). In II.4 we have that pure matter has no magnitude or spatial extension of its own. Extension (megethos) is a form or quality superimposed onto matter. In VI.4-5 dealing with omnipresence we read that the Soul is not spatially spread out or extended. Instead, physical extension is a limitation, whereas incorporeal reality possesses a unified, non-particulate presence that contains spatial extension without becoming spatial itself. In III.7 we read further that just as time is an unrolling or diastasis of the eternal life of intellect into succession, physical extension (megethos) is a dispersion or unfolding of unity into divisible parts.

A fundamental mathematical symbolism or reflection will be that of the limit (let us say, in the category theoretic sense) of a sequence or diagrams expressing an indeterminate outflow or process. This limit is a reversion, completion, perfection and theoria in the Plotinean sense and an aufhebung and being-in-and-for-itself in the Hegelian sense.  We can think of this process as dynamic and limit and reversion as a fixed point (cf. Lawvere's account of Gödel's incompleteness theorems and diagonalization arguments).

The reversion of the physical word and the human soul to the level of Soul and Nous parallels to a certain extent to Hegelian levels of Essence of Concept (for instance the supreme genera in the intelligible realm include identity, difference and in VI.3 rejects that essence or ousia could apply to the physical world) but with many radical differences and fundamental corrections from a neoplatonic point of view. The physical plane can be considered in itself or as it is the result of the activity, influence and emanation of the the plane of soul taking account its act of reversion (i.e. Objective Concept, the theory of life) - and the same goes for the the plane of soul itself. Thus Subjective Concept can correspond to the theory of the soul's logoi, to the theory of the world soul or to such as it relates to the intelligible realm and to the intelligible realm in the aspect of knowledge. Hegel's progression from outwards qualities to essential qualities in Subjective Concept finds its parallel in VI.1.15–22 which distinguishing true intelligible qualities (forms or reason-principles - logoi) from mere sensible qualities (passions, colors, surface shapes).  For Plotinus universal Being (i.e. the Absolute as the perfect emanation of the One) is Being, Intelligence and Life and the perfection of energy/act/power.

When using mathematical structures as neoplatonic symbols and reflections we find that it is best for formulate them and think of them in a strictly constructivist way.  Thus the number of objects in a category must be conceived as being indefinite and defined as one would do in a dependent-type based proof assistant - considering it to be the object object in the definition of an internal category.  The finitist can consider intentionally the sequence $\frac{1}{n}$ as computable process without having to admit the finished extended set of such numbers. And then the finitist can consider $0$ as being its limit.

The considerations presented above are very sketchy and highly insufficient. We point out the the deep exposition of Plotinus' logic can be found in VI.2 and VI.6. The five supreme genera suggest the fundamental importance of the equality type and transformations which preserve structures and express coherence and relational unity or interconnectivity.

Identity is related to stability, difference to movement. Consider the two possible orientations of a vector space. They are indistinguishable, intrinsically identical yet different. Their difference is expressed by a non-trivial automorphism of the underlying vector space which transforms one into the other.  

Plotinus states that number is produces by a combination of movement and rest (as supreme genera) and that extension is posterior to number and is the result of a movement produced by infinity which is halted and limited by rest. In VI.6 we have the theory of multitudes as outflowings and divisions of being. In VI.6.1 we read that if the parts maintain some connection then we have extension which have lost perfection must receive the form of beauty. The negation of negation is the reversion of the outflow through limit.  Negation is related to set-theoretic complementation (for instance it is the interior of the complement for the Heyting algebra of regular open sets of a topology). Thus the dyad, the unlimited, indeterminate outflow, considered in itself independently of its source can be characterized as negation. The negation of negation (a reversion of the outflow) corresponds to a closure operator which encompasses in a limited determined way the object which was initially negated. If negation corresponds to the sun's unlimited rays then double negation corresponds to a halo around the sun's disk.

The remarkable sections  VI.2.4-12, which deduce the supreme categories through an anagogic process of turning inwards, the key concepts being soul, the logoi. one, ousia, life, intellection, reversion, movement and stability, identity and difference, are important for comparison with the Logic of Essence and Logic of Concept and well as for pointing out the radical difference with Hegel. At the same time Plotinus' henological ontology and the deduction of the other genera from the supreme pentad in VI.2.13-21 should be compared to the processes and structures of Hegel's Logic of Being. We note the remarkable fact that VI.2 (which employs many argumentative structures interesting from the point of view of formal logic) can be seen as realizing the idea of dialectic which Plotinus sketched in I.3.  In VI.6.10 we read that Being became Number when it became a multitude, having  had a pre-formation, previous representation,or  allocation for the beings it produced. We note that Augustine copied VI.6.12-13 almost directly. A section in VI.6.3 should be compared to Hegel's absolute negativity in the beginning of the section on Essence: τὸ δὲ μὴ ἀποδρᾶναι ἔχειν, εἴργεσθαι δὲ ἔξωθεν καὶ κύκλῳ καὶ μὴ ἐξεῖναι προχωρεῖν, στάσις ἂν εἴη: ὥστε μὴ μόνον ἐξεῖναι κινεῖσθαι λέγειν.

We note that a fundamental mathematical illustration of the genesis of sensible quality will be that of the unfolding of a singularity (or deformation of a structure). We must bear in the mind that intelligible extension is expressed as the intensive power of a intelligible form which then unfolds and expands spatio-temporally in the world-soul. Or the forms can be symbolized by infinitesimals as we developed in "Hegel in Modern Topology" - and indeed in V.9.6 Plotinus gives us an interesting illustration involving the logoi and a seed.

Note the Hegelian categories of ground and appearance, phenomena, etc. find their treatment in Plotinus theory of sensible substance which is a shadow produced by the Soul casting form onto receptive, featureless matter. Physical substance is a composite (synolon) of matter (hyle) and dynamic immanent form (logos or reason-principle). Physical motion is the life and process of change inherent in time and physical nature. It is the physical shadow of the eternal activity (energia) of Intellect.

Consider an object $A$ in a category. Then $A$ may have internal symmetries in the sense that $hom(A,A)$ may contain more than just $id_A$. Given $f \in hom(A,A)$ we can ask if there exists $g : A \rightarrow B$ and $h : B \rightarrow A$ where $B$ is not isomorphic to $A$ such that $f =   h \circ g$.  This offers an illustration of $B$ being an emanation of $A$ (flowing out in $g$ and reverting by $h$) and proceeding from $A$'s internal structure.

New section in "Hegel and Modern Topology"

The most recent version of "Hegel and Modern Topology" includes a new 4 page section: "Formalization of Hegel's Logic of Concept".

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.

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 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.