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 level, 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. 

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.

Kantian notes

Schematism involves the concepts of  'rule' and  'time',  the realization of a pure concept of the understanding. How can we understand this but as computation ?

The difference between thinking something and knowing something. How can we explain this ?

The ideas of reason as ideal, completed totalities (the unconditioned). There is a largest set, because we can think of the set of all things and thus everything will belong to it. On the other hand if there is a set containing all things then this set itself must belong to it and as such will no longer be the largest set. Thus there is no largest set. Kant's anticipation of Russell's paradox. Or rather Girard's paradox: collapsing universe levels allows one to represent a "totality" that must simultaneously be inside and strictly above itself, producing a contradiction. Kant is finitist and intuitionistic somewhat like Brouwer with his theory of choice-sequences.

The ideas of reason organize and give a direction to philosophy and science but they are also the foundation of practical morality. As we wrote previously: intelligence and morality are one.

In the transcendental dialectic in some cases Kant states that both opposing propositions are false in other cases that they are both true - this recalls the Buddhist tetralemma.

Reason as the intelligible character, the noumenic source of freedom. 

Kant seems to be saying that the phenomenal self and the phenomenal world are mutually dependent and relative. 

Grete Hermann:  quantum mechanics does not disprove causality, but rather clarifies it by separating it from deterministic predictability. She proposed that while quantum predictions are statistical, causal chains can be reconstructed retrospectively after measurement. 

A very important aspect of quantum mechanics is the relationship between the Schrödinger equation and the Hamilton-Jacobi equation (which originally expressed the analogy between mechanics and optics). 

$\frac{\partial S}{\partial t} + H(p, \frac{\partial S }{\partial p} , t) = 0$ 

This relationship was present at the very beginning in multifaceted history of quantum theory.  The Hamilton-Jacobi approach (beyond its use in quasi-classical approximation) is the key to developing a correct pilot-wave type theory (which does not necessarily have anything to do with Bohm's variant or path-integral approaches). Research on droplets bouncing on vibrating fluids (where the "droplet" both causes and is affected by the associated "wave") is of utmost interest and importance. 

We have discussed the mutually implied trinity : logic, computation and arithmetic. But we should also add therein combinatorics and graph-theory...

The Curry-Howard isomorphism expresses to a certain extent the correspondence between logic and the $\lambda$-calculus presentation of computation. Different type-theoretic-logical systems (Gödel's system T, Girard's and Reynold's system F) only capture a fragment of the class of computable functions. Curiously enough there is also a direct correspondence with forms of Peano Arithmetic wherein  provable totality is used to characterize such classes.

And what is a computational object but one which can be reduced, in which a computational process can be carried out ? 

Given a (partial) formal model of computation does it always have some kind of "logical" correspondence? And vice-versa? Note how both proof and computation involve temporality in an essential way...

This is what we call Girard's problem: in the above, what is a priori and what is posteriori? what is analytic and what is synthetic? 

Frege simply defined 'analytic' as that what is derivable in his system of second-order logic (and does not rely on any form of intuition).  Girard seems to view untyped computational objects as analytic and typed-ones as synthetic. One notion of a proposition being analytic is: being true in virtue of its form alone.

Girard's linear logic and proof-nets (and geometry of interaction, transcendental syntax) seeks, so it seems, to delve deeper into the above correspondence, even going beyond the distinction between a proposition and its proof. Girard offers a computational model distinct from the $\lambda$-calculus and still connected to the essence of proof.

Pierre Cartier: A Visionary Mathematician

https://www.researchgate.net/publication/385750025_Pierre_Cartier_A_Visionary_Mathematician

Genius, Intelligence and Sainthood

Schopenhauer expounds a theory of 'genius' in the World as Will and Representation. The analysis and critique of varied historical and present meanings of the terms 'genius', 'intelligence' and 'sainthood' in the west is of utmost cultural and philosophical importance. Part of our philosophical project involves the radical deconstruction of these concepts and unmasking their harmful influences and consequent aberrations. 'Sainthood'  is currently employed in a socio-historical sense or related to organized religions, its true universal ethical and spiritual meaning being lost.  'Intelligence' is a vague and fluid pseudo-concept that has wrought immeasurable harm to human culture and society as well as to the human sciences. The term 'genius' is as a rule completely misapplied.

Our theory of intelligence is that true intelligence is founded on two intimately connected principles. That of knowledge of the universal moral law (perceived as such) and that of attaining at least some elements of transcendental philosophical consciousness (TPC) and transcendental philosophical praxis (TPP).

Extraordinary achievements in TPC and TPP are the mark of genius par excellence. 

The genius of true intelligence in the ethical sphere and TPP is called 'sainthood'. 

The special kind of genius, that while intimately connected implicitly to morality, TPP and TPC (as brilliantly expounded by Schopenhauer),  is that of artistic genius which reveals fundamental important aspects about human nature and the world.  

The quality of intelligence is related directly to the importance of its object and domain. 

Intelligence can be defined as an empathy for reality and moral intelligence has as its main source universal empathy for the suffering of all living beings. 

Intelligence and morality are one. 

The extreme opposite of true intelligence is all that involves a kind of cunning, a skill-set whose only purpose and application lies in achieving personal egotistic wealth and power, everything that involves dominating, exploiting and harming other living beings. There is no 'intelligence' here, no 'genius' and no 'sainthood'. Intelligence is radically incompatible with the will to power or competitive mechanisms of survival (i.e. overpowering others or merely adapting to a contingent environment).

The greatest deception and idol of western society has been attributing 'intelligence' or 'genius' to what amounts to nothing more than gaming, gambling, cunning and calculation. We have to be extremely cautious and insightful about attributing intelligence or genius to work in logic, mathematics, engineering or science,

Genius and intelligence in mathematics is never about the predictable success (theorem proving) in random searches in formal possibility spaces and conceptual engineering (scientific mass production) but rests solely on the relevance of the work to philosophy (more specifically to TPC) and to the philosophical unification and clarification of science.

Playing the game, being lost in the illusion of the game, is very different from the TPC-informed insight and consciousness of the game qua game (which can be compared to the attitude of Alice at the end of Alice in Wonderland...and Schopenhauer uses the metaphor of the chessboard after the game is finished). 

Thus we have such unsurpassed luminaries as Gauss, Grassmann, Riemann, Frege (who wrote that every true mathematician is half a philosopher), Peirce, Hilbert, Turing, Whitehead, Gödel, Brouwer, Russell, Poincaré, Lawvere, Thom, Martin-Löf, Girard, etc. Their writings are never a mere tortuous game with symbols and ad hoc concepts or non-rigorous obfuscation and plagiarism. The light of TPC implicitly shines through. Their work is also a spiritualization of language.

But scientific discoveries with practical applications to the quality and preservation of human life (or of any living being) are certainly meritorious - for indeed they partake this way in the sainthood, intelligence and genius of ethics.  Science aimed at mechanisms for harm and destruction is the ultimate immorality and stupidity. There is nothing brilliant or intelligent about faulty simplistic models of human society such as 'game theory'.

Transcendental Syntax I : deterministic case - Jean-Yves Girard

https://girard.perso.math.cnrs.fr/trsy1.pdf

We study logic in the light of the Kantian distinction between analytic (untyped, meaningless, locative) answers and synthetic (typed, meaningful, spiritual) questions. Which is specially relevant to proof-theory: in a proof-net, the upper part is locative, whereas the lower part is spititual: a posteriori (explicit) as far as correctness is concerned, a priori (implicit) for questions dealing with consequence, typically cut-elimination.

Proof Nets

Elementary Topos Theory

Topos theory is a messy area with various rival factions and a problematic community. However we believe that Lawvere's original theory of elementary toposes and its subsequent development is important and interesting. The most important object in an elementary topos is the subobject classifier $\Omega$. If we think of the topos as a cell then $\Omega$ is a kind of nucleus. An elementary topos is simply a model of mathematics which as a foundation is vastly superior and more cogent than ZFC. To us the most important concept in topos theory is that of the Lawvere-Tierney "topology", which is just a morphism $j : \Omega \rightarrow \Omega$ satisfying three simple "modal" or "closure-operator" type axioms. A very important instance of $j$ is given by double-negation $\neg\neg : \Omega \rightarrow \Omega$, in which we consider the (internal) Heyting algebra structure on $\Omega$. The morphism $j$ then determines a localization of the topos, a new subtopos of the original topos called the topos of $j$-sheaves. 

A central philosophical problem of topos theory is understanding the meaning of the Lawvere-Tierney "topology" and its associated topos of $j$-sheaves as well as the special role of the $\neg\neg$-topology (why is it not abuse to call $j$ in this case a "topology"?).  A key to this is to see how the theory above abstracts the concrete case of presheaves and sheaves. A "sieve" is a curious concept. Think of a set $S$ of open sets (conceived as "cover") in some space $X$. Then take the minimal extension of $S'$ of $S$ under the condition that $U \in S$ and $V\subset U $ implies $V \in S $. Then we have a sieve (generated by $S$). We could rephrase the condition as $W \cap U \in S$ for any $U \in S$ and open set $W$. That is $S'$ is the $\wedge$-ideal generated by $S$. The we have the obvious notion of a principal idea generated by $O$ (called a principal sieve).

 For the presheaf topos on a topological space $X$ we have that the presheaf $\Omega$ associates to each open set $U$ the set of all sieves on $U$. So $\Omega$ is a kind parametric version of local truth values. On the presheaf topos an important example of $j$ is the functor that associates to each sieve $S$ (on a $U$) the principal sieve determined by "what $S$ covers". In the words of Moerdijk and Maclane "What counts is what gets covered". Thus in this case $j$ is nothing more than a kind of parametric generalized union. Logically it is expressing "if something is locally true then it is globally true". That is the subjobject of $\Omega$ determined by $j$ consists in those sieves which are invariant under generalized union, situations in which if something is locally true then it is globally true. This fails for instance for the presheaf of constant functions. This $j$ (which we should call the union topology) seems to be intuitively clear but we still need to understand better why in the topos of continuous functions on  topological space $X$ the internal logic of the topos proves that all functions are continuous.

Thus for the topos of presheaves on a topological space the subobject classifier on an open set $U$ yields all the sieves on $U$ while for the topos of sheaves it yields all the open sets of $U$ (or equivalently the set of principal sieves on $U$. 

But a central problem of topos theory is understanding other $j$s such as the double-negation topology (and its "$j$-sheaves"), in particular as an abstraction of the topological sheaf case. We have written something about this in our "Hegel and Modern Topology". The double negation topology is all about "density" while the union topology is about local-global coherence. Could we associate the union topology with exponentials in linear logic ?

What does the double-negation topology look like for the presheaf topos on a category $C$ ? And for the topos of sheaves over a topological space ? In this case it appears to be simple: an open set $V$ in $\Omega(U)$ is taken by $j = \neg\neg$ to $int(\overline{V})$. Thus the topology subobject $J$ is just the set of regular open sets in $\Omega(U)$. What are the $\neg\neg$-sheaves in this case ? The idea is like the passage from holomorphic to meromorphic functions (or the identification of measurable functions different on measure zero sets). The resulting sheaves can be considered as defined up to empty interior closed subsets. In a way, the $\neg\neg$-topology introduces closed boundaries, qualitative differences between different regions.

The fundamental definition of $j$-sheaf, an object satisfying $hom(E,A) \cong hom(B,A)$ for a $j$-dense subobject $E$ of a an object $B$ and considering the restriction morphism, can be understood as follows. The associated closure operator is a kind of transcendence, expansion beyond itself, saturation, perfection. The condition then picks objects for which their effect on a subobject already contains their effect on that subobject's transcendent expansion (closure). This means that the object satisfying the condition already contains within itself its own self-transcendence.

Put in another way, sheaves for the dense topology are ideal cohesive structures which on are object (open set) are determined by coverings whose union is not equal but only dense in that object. These sheaves allow a form of completion, limit, coherent transcendence. They allow one to construct an object based on coherent data which is yet still incomplete.