The problems of quantum theory

Quantum physics represents a very imperfect, incomplete and highly unsatisfactory theory.  Its interpretations and approaches are multi-faceted and complex. Quantum physics took a drastically wrong turn with the Hilbert space operator  based Von Neumann axiomatization (and the subsequent more sophisticated, but likewise inadequate, $C^\ast$-algebra approach) - abandoning the much more interesting initial historical connections to the 'dualistic theory of radiation', statistical mechanics,  the photo-electic effect, geometric optics, the Hamilton-Jacobi equations, 'wave mechanics', 'matrix mechanics' - and above all the insights of Paul Dirac who was one of the giants of 20th-century physics (Penrose's Twistor Theory is in some sense a continuation of Dirac).  The investigations carried out during the initial development of quantum physics gave rise to meaningful and interesting mathematics and physics which did not depend in any way on probabilistic interpretations (or the collapse) of the wave function - for example the study of the Lorenz-invariant Klein-Gordon and Dirac equations. If the Hilbert-space and and $C^\star$-algebra based quantum theory was at least mathematically rigorous and interesting in its own right (it ultimately gave rise to Alain Connes' Non-Commutative Geometry) this is not the case of Feynman's and Schwinger's approach to  (perturbative) quantum field theory.  Cf.  G.B. Folland's Quantum Field Theory: A Tourist Guide for Mathematicians (2008) where it is stated that once we leave the free field (itself requiring a staggering amount of functional analysis and distribution theory) we have left the realm of a correct mathematical formulation of physics. The problem is that QFT is not only bad mathematics it is also bad experimental science.

Something is rotten in the state of QED (Oliver Consa)

Consa says the much-touted precision of QED is based on measurements of the electron g-factor, but that “this value was obtained using illegitimate mathematical traps, manipulations and tricks”.Theoreticians come up with a calculation that exactly matches an experiment. Then a later experiment shows that the earlier experiment wasn’t quite correct. Then the theoreticians change their calculation to match the new experiment. And so on (...) Consa quotes Dyson from 2006: “As one of the inventors of QED, I remember that we thought of QED in 1949 as a temporary and jerry-built structure, with mathematical inconsistencies and renormalized infinities swept under the rug. We did not expect it to last more than 10 years before some more solidly built theory would replace it. Now, 57 years have gone by and that ramshackle structure still stands”. It still stands because it’s been propped up by scientific fraud. Here we are fourteen years later, and it’s still the same, and physics is still going nowhere. How much longer can this carry on? Not much longer, because now we have the internet.

https://physicsdetective.com/something-is-rotten-in-the-state-of-qed/

Maybe a clue to improving this situation involves a critique and reform of distribution theory - for example along the lines of Sato's theory of hyperfunctions. This has of course already been suggested in the context of the divergent infinite sums of $\delta$-functions appearing in QFT.

We postulate axiomatically that the 'position', 'momentum', even 'energy' of a 'particle' are given by distributions over space-time (we still have our standard PDEs for distributions).  Thus a 'particle' does not necessarily have a definite position at a given moment of time. Nor is a 'particle' a wave or field defined aver space-time. It is a completely different kind of entity which subsumes as particular cases or approximations the aspect of wave or particle (for the wave-like aspect we have regular distributions, for the particle aspect $\delta$-functions, or something similar). There must be an "interactive" (we must carefully re-evaluate the controversies surrounding the interactive interpretation of the collapse of the wave-function as well as the hidden-variable approaches) or alternative way of explaining the collapse of the wave function and the probabilistic aspect based on this perspective (cf. the work of A. Hobson (2012)).  Since experimentally we can only prepare 'test functions' with a limited degree of precision it is not surprising that the output of the distribution should also exhibit a corresponding degree of uncertainty. But of course we need to ask what is the physical nature of the test functions? Do not they have to be (regular) distributions as well?  In practice the test functions will not be exactly regular distributions but only approximately so (determined by some boundary conditions). Thus observations - which correspond to evaluating the test functions - or interactions of localized distributions along a boundary - will have uncertainty corresponding to the non-regular components of the approximate test function.  Note that a distribution is essentially non-local (cf. Hobson's analogy to a bursting balloon) although they can be restricted. It would be interesting to explore how this approach looks like from the point of view of Sato's hyperfunction theory - and sheaf cohomology (Penrose would endorse this !). And maybe Penrose's Twistor theory has an even greater significance in a completely different philosophical context than the one adopted by Penrose himself (who still adheres to the 'collapse of the wave-function' dogma). Consider the double-slit experiment. We need a concept of boundary and interaction for distributions. And to be able to deduce probabilistic information from distributions, boundaries and completely deterministic equations. But we must not forget that the plate used in the double-slit experiment is only approximately a plane - in reality it has a highly irregular surface and there will always be one local region which is the first to "touch" the wave-front proceeding from the slits.

Note that a regular distribution may be localized according to the support of its associated function $f$ in $L_{loc}(\Omega)$.  That is its value for  is equal to the value of its restriction to an open set containing the support of $f$ . Or the support can be disconnected, so we have two disjoint localized centers. A distribution may be regular or localized according to its restriction to a certain time interval but evolve into a different situation - this can be used for a solution to paradoxes similar to the EPR paradox.  The photon ceases to be a localized wave-packet (a regular distribution) becoming a non-regular one (i.e. having a non-local holistic character) (we have a continuous path in the space of Radon measures for instance) thus explaining why an observation (i.e. an interaction) at location A can determine the outcome of an observation at a distant location B.

The whole proposal above is obviously highly sketchy and unsatisfactory.  We need not only the non-locality (which transcends both the field and particle approach) using distribution theory but also the fundamental postulate that the linearity of the theories and equations is only an approximation of the fundamentally non-linear or even chaotic (but deterministic) physics at a finer scale (cf. the Casimir effect which QFT interprets as 'fluctuations of the vacuum'). It is this framework that could explain that in reality observations are interactions with a non-linear component - in general expressing what happens when a non-localized (non-regular) distribution interacts with a regular localized one (recall that there is no satisfactory definition of a product of distributions in general). Maybe we must extend physics to account for an equivalence between energy and information (in observations, the measurement process) perhaps embodied in Psi-phenomena. The wave-function is like the continuous holistic coordinated movement of juggling (or swimming).  If the mind stops and focuses on a localized part (i.e. local interaction energy is exchanged) the system implodes and its non-linear dynamics leads to only apparently random final outcomes or crashes.

A real 'theory of everything' would be a theory which allows one to solve all (differential) equations. 

Boris Eng. A gentle introduction to Girard’s Transcendental Syntax for the linear logician. 2022.

https://hal.science/hal-02977750v7/document

Girard’s philosophical motivation for the transcendental syntax is to establish a whole new architecture for logic which would be free of any logical preconception but also explain the whole logical activity. In this idea, logic is a formatting of computation: everything starts from a chosen very general, simple and natural model of computation. This is what Girard calls the analytics, in reference to Kant. We require that the model includes reducible objects which can be evaluated (what Girard calls performance) to an irreducible object (Girard’s constat). Both are separated by indecidability: the fact that an reducible object cannot always be reduced to its potential results (infinite loop may appear). 

Important papers on Relativity by Lorenz and Poincaré (1899, 1904 and July 1905)

H.A. Lorentz, Electromagnetic phenomena in a system moving with any velocity smaller than that of
light, in: KNAW, Proceedings, 6, 1903-1904, Amsterdam, 1904, pp. 809-831

- Simplified Theory of Electrical and Optical Phenomena in Moving Systems (1899) in Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1899 

Poincaré, H. (1906) Rendiconti del Circolo Matematico di Palermo, 21, 129-176 (paper dated 23rd of July 1905) 

Amnesia, the Zeitgeist and the philosophy of the World Wide Web

What happens in history is not always easy to grasp philosophically at once. There are 'prophets' (like, to a certain extent, the French postmodernists) but only later can we begin to understand what happened or the paradox of what did not happen yet could have. The emergence of the World Wide Web was a monumental happening in the history of mankind. Theoretically this meant that advanced knowledge and higher education (in all humanistic and theoretical disciplines as well as many technological and applied disciplines) was potentially made accessible to every human being having the leisure, patience, inclination and dedication for study and learning. This implied a potential complete overhaul of traditional social, educational and knowledge-guarding institutions, professions and hierarchies (both in their worthless, plutocratic, nepotic, dogmatic aspect but also in their good humanistic aspects). But this is not what happened. This potential universal diffusion of knowledge had apparently to wait for a 'dumbed down' version  based on LLMs for such questions to be even seriously asked.  And yet the World Wide Web became ultimately a tool not of liberation, reform and revolution but a horrific tool for deception and global control by traditional power structures. Language ceased to be means of communication and conveying knowledge morphing instead into a global power tool for psychological and cultural formatting,  uniformization, manipulation and cognitive-semantic degeneration. This monstrous pseudo-culture, pseudo-communication, pseudo-knowledge (engineered and directed and financed by traditional powers) is collected together into vast curated datasets which are then processed to produce bland blends of empty linguistic mimicry.  Thus the semblance of language and knowledge have become the enemy of mankind Homo sapiens sapiens sapiens - and the irony of the fulfillment of the French postmodernists'  prophecies. People nowadays (we could coin the term digitanttes) have all the arrogance and conceit and mindset of the worst kind of 'intellectual'  while having rarely any authentic knowledge or expertise. The salvation of mankind can only come through 

i) systematic, thorough and profound doubt, questioning and skepticism of all alleged knowledge and information and most specially of the alleged knowledge and information offered by LLMs, internet encyclopedias and government, religion and corporate controlled media. The 'dialectic' method of Socrates, Pyrrho and Sextus.

-Develop a stable minimal and optimal software for the logical representation of knowledge and the automatic checking of proofs (such as employing  minimal subsystems of Coq or Agda as in the links on the right).  Continue the work of Vladimir Voevodsky and his original approach to computer assisted proof verification.

Restore the scientific method and the original meaning of positivism and empiricism. 

A book is just a book. You can hold it in your hand and discard it.  But reading a screen is plugging yourself into the matrix (running not only on electric energy but collective focused mental energy).

Owning and reading a book - or any information storing and displaying device not connected to the World Wide Web - is a higher form of revolution and freedom. Put together vast offline libraries (digital or analog).  

ii) a radical reform of language, communication and social interaction (and of course the complete elimination of any engagement with shallow, short, mindless, addictive, memetic, unreliable social media) - perhaps drawing inspiration from the Ch'an tradition, certain literary and poetic traditions as well as the best and most beautiful counter-cultural artifacts of the 20th century. Develop a self-referential, self-transcending  style of expression and thought that is totally beyond the reach of LLMs and internet Encyclopedias.  There are surely more philosophically interesting and advanced forms of cryptography which work at deeper syntactic or semantic levels.

The idea of 'linguistic relativity' can be found already for instance in von Humboldt, Herder and Hamann.  This is something which merits a deep philosophical analysis. We note only that LLMs are almost exclusively trained on datasets in contemporary English with its far-reaching linguistic-cognitive-cultural bias and format. Thus the cultivation of multilingualism (and a rejection of Anglo-supremacy and Anglo-centrism, the proposal of a new lingua franca),  with a special emphasis on ancient languages, is a powerful weapon.

iii) unwavering universal dedication to international law and justice,  human rights and animal rights, demilitarization and the protection of the environment. 

iv) unite science and art, in particular science and literature such as in the best science fiction. 

We must become again truly ourselves (Attadīpā viharatha, attasaraṇā anaññasaraṇā) so we can truly be with others.  

NSF Funding Search Suggests Neuroscientists May Scarcely Believe in Engrams

https://headtruth.blogspot.com/2025/07/nsf-funding-search-suggests.html

Science Literature Is Full of Misleading Claims About Brain Waves

https://headtruth.blogspot.com/2022/11/science-literature-is-full-of.html

Papers

To be published: 

Aristotle's Second-Order Logic and Natural Deduction 

On the Various Translations between Classical, Intuitionistic and Linear Logic

On the nature of Kant's Logic in the Critique of Pure Reason  

Almost ready:

Natural Term Logic

Preprints (work in progress)

Computability, the analytic a priori and differential models of nature

Hegel and Modern Topology

And of course a book about TPC and TPP as well as a book about philosophical anthropology.

Another view of TPC

Perhaps it to attempt to attain passadhi (samatha) through vipassana is putting the cart before the horse. Or rather a different kind of insight is called for as a foundation. Yoga citta vrtti nirodha. Understanding the relationship between consciousness and the body - and the existence of a middle subtle body-consciousness field which carries the feedback interaction between both (the neuro-muscular aspect is important). The unified field has some analogy the solutions of Klein-Gordon and Dirac equations (Dirac was perhaps the greatest physicist of the 20th-century).The models of René Thom come very close to the idea of a field of harmonic oscillators over space-time.  The goal of the fundamental stage of TPP is to attain the pax profunda, possibly using psycho-somatic feedback as a support (to dampen or muffle the spectrum of mutually exciting harmonic oscillators). What we ordinarily call 'body' and ordinarily call 'mind' are two complementary modes of the same underlying consciousness-field. It is only in the deep clarity and stillness of the mind-body field that authentic TPC can blossom 孰能濁以靜之徐清. Thus the initial TPC involves perceiving consciousness as an excitation and self-interaction (producing the illusory perception of the individual self) of an underlying psycho-somatic field, and understanding the effective dynamics (and functional stratifications ) and feedback mechanisms to attain the desired goal. This initial TPC is indeed the pure impersonal perceiving of the flux of consciousness as thus, but also the inner first-person experience of the body (i.e. we have nama-rupa): it is also the perception of how the inner body generates a kind of frame of reference or proto-space (proto-topology) for the total sphere of consciousness. Temporality is on one hand transcendent and a condition for thought and consciousness rather than being generated by consciousness, on the other hand it is generated as an illusory excitation. We must find the lost original deeper meaning of Pyrrhonism, regarding belief, thought and agitation - the same underlying TPC insight.

Some things to explore: how can the role of symmetry in physics be transposed to understand the central role of symmetry in consciousness ? And in the symmetries in logic ? Philosophically, how can our theory of a priori computability relate to the obviously rich computational nature implicit in the study of solutions of PDEs ? Radiation, diffusion, harmonic equilibrium. Geometric optics. Singularities of the solutions of PDEs such as the Hamilton-Jacobi equation. Meaning is perhaps described as symmetry invariance in the dynamical system of consciousness.

What are sense and reference ? It seems that we can reconcile psychologism and objectivity by a theory analogous to that of the invariance under group actions used in physics. The conscious content for two different people can differ, but the contents must be related to each other through a well-defined law, a continuous symmetry group, a deformation. Galilean relativity has perhaps an immense overlooked philosophical significance.  It shows that objectivity is relational (all inertial frames of reference are equivalent, there is no absolute frame of rest), but not less objective for that. Once the idea of group action and group invariance had been brought to light - in physics as well as geometry - it is the most natural thing in the world to investigate hyperbolic geometry (Gauss, Lobachevsky) in this light. Why is the connection of Minkowski space (the pseudo-sphere) to hyperbolic geometry obscured (obtained by the analogue of the stereographic projection of a sphere)?

In order to develop these ideas we recall a previous note on computational linguistics. Large Language Models such as ChatGPT-3 use high dimensional ($dim V$ = 12,288) vector-space representations of meanings of certain textual units ('tokens'). These are generated from context in large data sets. The idea of having certain semantic 'atoms' (sememes) from which are combinatorically constructed possible meanings can be found for instance in Greimas (cf. Osgood's semantic differential for studying the variation of connotation across different cultures). Some (such as René Thom) have claimed that the idea that meaning should have a continuous, geometric aspect is found in Aristotle. Leibniz' characteristica used 'primitive terms' but it is not clear if they are combined in a simple algebraic, combinatorical or mereological way, or if complex logical expessions must be involved (or associated semantic networks). But in embedding matrices we have what would seem to be a quantification of meaning, each 'sememe' is given a 'weight' which determines its geometric relation to other meaning-vectors in a crucial way (the weights cannot be dismissed as probabilistic or 'fuzzy' aspects). To us this would correspond to the 'more-or-less' aspect of species in Aristotle. A very interesting aspect of embedding matrices is how they capture analogy through simple vector operations. This suggests another possible formalization of Aristotelian 'difference' , the same difference operating on two different genera. We get a notion of semantic distance and semantic relatedness. This also revindicates Thom's perception of geometry and dynamics in the spaces of genera.

Some questions to ask: are these token-meaning-vectors linearly independent ? If not can we work with a chosen basis ? If the token is ambiguous is the corresponding vector a kind of superposition of possible meanings, as in quantum theory ? How are we to understand the idea of the meaning of complex expressions being linear combinations of the meaning representations of the tokens occuring in the expression ? It would of course be interesting to analyze these questions relative to the other fundamental components of LLMs (attention in transformers, multi-layer perceptrons) - even if these are more practically oriented rather than reflecting actual linguistic and cognitive reality.

Suppose we are given a large text $T$ generated by a set of words $W$ and a context window $S$ of size $n$. Suppose we wished to represent the elements of $W$ as vectors of some vector space $V$ in such a way that given $v,w \in W$ the modulus of the inner product $|\langle v,w\rangle|$ gives the probability of the two words being co-occurrent in contexts S. Consider the situation: it is very rare for words $s_1$ and $s_2$ to co-occur but words $s_1$ and $s_3$ co-occur sometimes as do $s_2$ and $s_3$. But there is also a word $s_4$ which never co-occurs with $s_3$ but has the same co-occurrence frequencies with $s_1$ and $s_2$ as does $s_3$. Then it is easy to see that there is no way to represent $s_1$,$s_2$,$s_3$,$s_4$ in the same plane in such a way that these properties are expressed by the inner product. Thus the dimension must go up by one value. We can define the geometric $n$-co-occurence dimension as the minimal dimension of a vector space adequate to represent co-occurrence frequencies by an inner product. We can ask what happens as $n$ increases, does the geometric dimension also increase (and in what manner) or does it stabilize after a certain value ?

Thus we can think of different people as having semantic vector spaces which must be related in a well-defined way and in such a way that the semantic information remains coherent. Thus the mental content of the term 'horse' for Alice and Bob may be quite different, but each is related to the other through a kind of continuous deformation related to some structure contrasting the background of Alice and Bob. Thus we need to define a kind of relation space for contrasting and comparing different subjects - and in such a way that we have a representation of the algebraic structure of this space in terms of continuous deformations of mental content.

René Thom proposed that concepts were analogous to living beings and that mathematical models of the regulation structures of living beings could be applied to concepts themselves. This is kind of obvious for natural kinds and not very clear for other kinds of concepts. We need a very different approach.  We need to understand representation, the subject's mental and yet objectified representation of the world. The question: what is a world ? Software engineering and the structure of Object Oriented software aiming at creating virtual worlds (such as Unreal Engine 5, Unity or in general RPG games - we are thinking here only of the classical ones such as the Ocarina of Time which were also works of art besides sophisticated puzzles) including automated agents are of some interest though with great limitations. Generative AI is likely to be followed by more sophisticated models which can train in real-time. The run-time process structure of operating systems is also important. The irony here is that these approaches become more interesting once we discard neuro-reductionism - once we abandon the pointless attempt to view the brain as the hardware of the mind. The central hardware of the mind is to be sought elsewhere, the brain itself is a kind of auxiliary cache.

There is much analogy between the structure of a computer program and that of a novel. 

To obtain a mathematical understanding of consciousness we must first bridge the gap between mathematical models of nature and computer systems.

Also we need to take into account altered or higher states and modes of conscious experience (once harmful and falsified approaches to the spiritual life have been discarded - those that hide the truth that the royal path to spiritual realization consists in a pure love for a real person). 

Tanto gentile e tanto onesta pare la donna mia, quand'ella altrui saluta, ch'ogne lingua deven tremando muta, e li occhi no l'ardiscon di guardare. Ella si va, sentendosi laudare, benignamente d'umiltà vestuta; e par che sia una cosa venuta da cielo in terra a miracol mostrare. Mòstrasi sì piacente a chi la mira, che dà per li occhi una dolcezza al core, che 'ntender no la può chi non la prova: e par che de la sua labbia si mova un spirito soave pien d'amore, che va dicendo a l'anima: «Sospira!»

Do these higher states of consciousness possess a geometry, a topology, a semantics ? It is curious, how many Henads are there is Proclus' system ? Or does cardinality itself not apply to them ? 

Also the entire discipline of lexicology needs to be reformed. Indeed what was the ancient project of the classification and division into genera and species but a lexicological program ? We need to greatly clarify the insight involved in defining a term by its context. It is not only that we need to know the meaning of words to understand a narrative but also narratives themselves give meaning to words.  Being multilingual and practicing translation offers unique insight into the pure semantic universe.

Maybe natural language is a kind of super-mathematics which contains ordinary mathematics as a special case. It is presumptuous to ridicule the concept of an 'ideal language'. Learning other languages and in particular ancient languages is surely on the path of wisdom. In natural language we cannot in general define lexemes in the way we define mathematical or scientific concepts (and the ancient theory of genera and species must have been derived from Euclidean mathematics, law and medicine).  This is polymorphism. Meaning is in an inseparable feedback loop with life and experience, depending on whether we are engaging in solitary discourse or on which person we are conversing with.

Naive dictionaries with obvious circularity in definitions should not be despised as non-scientific. Rather they  express something profound about polymorphism, the circulation, the flow, the dynamics of meaning. Instead of a oriented tree we have a directed graph with cycles. There is an analogy with commutativity and non-commutativity. Meaning circulates like a living current or flow through the whole web or tapestry of language. The name generates a story, the story a name.

Even for mathematical concepts we gain a deeper understanding or apprehension of them through practice, through exercises, through studying proofs in which they are applied. Are these degrees of apprehension - or degrees of meaning?  Formal logic acts as an ultimate arbiter which rarely needs to act, mathematicians with distinct intuitions and apprehensions of a given mathematical concept generally can agree that their concepts are 'the same'.

Thurston's On Proof and Progress in mathematics (1994)

Schopenhauer offers a strikingly alternative theory of consciousness, concepts, intuition and representation as well as super-consciousness. So does Hume. Even Sextus. The problems discussed above are not some kind of puzzle of which one needs to find a solution. Rather they are all the result of delusion and deception, consciousness pulling itself down as an illusion over its own eyes. Only vipassana, only TPC and TPP can break through this illusion. It is foolish to ask about meaning and language without first asking about consciousness and experience. Meaning and language are within consciousness. There is a higher form of non-linguistic cognition. Thus the so-called philosophy of language is not fundamental and does not represent a radical or critical approach to philosophy, rather a dogmatic one (and its arguments against psychologism, against empiricism, against the a priori vs. a posteriori distinction, against the analytic vs. synthetic distinction, all fail). And there is the fact that consciousness can calm itself.

Light on religious anthropology

This post can be seen a continuation of our historical and archaeological posts which focused on ethics and spiritual traditions. It is clear that we do not espouse either any form of  'theism' nor on the other hand any form of physicalism. We made ethics and the philosophical-scientific spirit the corner-stone of what we could call a 'higher culture'. We argued that some of the best aspects of 'modernity' where in fact very ancient. Here we give a further account of what we are to think of  'gods'.  To view 'gods' as a symptoms of a degeneration and decay of human culture, or as embodying a crude preliminary pre-scientific explanation of nature, it too simplistic and shallow.  It is true that in certain cultures 'gods' often express the lowest and most grotesque aspects of that culture itself, its word-view and its values. But it is not so much the question of 'bad gods' but of the badness of the very concept of god. Tear the mask from a god and you find (no matter how monstrous the mask) a human being and human weakness, fear, prejudice and greed. There is no reason why historical male gods are any less crude and absurd than female gods - not any reason to think that the cultures and peoples that made them exhibit any essential moral and spiritual difference. Be this as it may, let us now turn to a vastly different theory of an underlying original unity of theology, mythology, poetry, literature and art. The true god, and true gods, are found in the spiritual consciousness and experience we have with other human beings and living forms in this world. That is to say, in the divinity and numinosity perceived in persons, in the experience and relationships of each human being. This experience is then transfigured and universalized by art and becomes poetry and literature. The spiritual path can be the subject of inspired narrations and inspired narrations can help along the spiritual path. To put it simply: there are no true gods except truly wonderful characters in literature.  There is no genuine sophia or divine feminine except in the ideal yet humanized female characters of inspired literature and art. Art is the aether, the pleroma, the true 'mother of the gods' which exists only in consciousness. Poets  can preserve irrefutable traces of higher more ancient states of culture (just as they do linguistically) in which women were treated as equals. The authentic Sophia, the divine feminine, is simply a term for a process in which the artist (which can of course also be a woman) being acquainted with a noble, refined, cultivated, accomplished, heroic or otherwise admirable woman, gives her an idealized, timeless, universal life. We find this in Homer and Sophocles - and doublessly so many real-life Hypatias, Antigones and Diotimas have tragically been forgotten.   The same goes for ancient Persia (Aredvi Sura Anahita is obviously a reflection of the cultured high-ranking woman in ancient Persian society - and the same was probably true originally for Ishtar and Isis - a figure clearly appropriated by the authors of the 'wisdom literature' of the Old Testament, not ruling out a massive later Hellenistic influence as well, specially with regards to the Stoic world soul), ancient Ireland and Wales (Étaín is at once the most real and the most rich and interesting of mythological characters, the Mabinogion treats the gods as historical figures), ancient Japan (the fact that there is both a Amaterasu Omikami - recalling the Vedic Ushas -  and Sei Shōnagon is no coincidence) even the stories of woman followers of the Buddha in the Pali suttas (cf. the Therigata) - and so much more could be said were our knowledge of the ancient world not so scant and fragmentary.  In summary the true gods and goddesses are not pre-scientific anthropomorphized 'cosmic principles'  but idealized representations of real men and women who attained higher states of virtue, creativity and knowledge of reality - like the bodhisattva characters in the Avatamsaka Sutra. The inspired ideal work of art can of course be corrupted (if not in its artistic merit, were we have a kind of soap-opera Olympus) and put in the service of priestcraft and the religious concept of god, the gods of fertility, war, conquest and punishment we are all too familiar with. Thus gods are not themselves explanations but a celebrations of real persons who sought for (and often found) knowledge. In particular the true goddesses were the celebrations of the remarkable minds and hearts of real women, not pre-scientific anthropomorphic superstitions (which in the worse cases reflect a reduction of women to sexual and reproductive roles: this tragically continues in the bias of modern scholarship which should abandon the fixation on the 'earth mother' and pay more attention to the sun goddess, the sky goddess, the goddess of wisdom and power,  which are older and more authentic, because the reflect higher forms of actual human experience, such as Dante's). There are some interesting points that could be made about the Eddas, the Icelandic Sagas, about Snorri Sturluson, Asbjørnsen and the brothers Grimm which we cannot go into here. Also about the role of the 'good story' (appropriated from previous literature) in the propagation of Christianity.

Some important points.  We should drop the term 'monotheism'. A 'religion' with eight million gods (yaoyorozu no kami) which unconditionally upholds human rights and animal rights - compassion and reverence for all life - is infinitely superior to any murderous, cruel, superstitious, racist, sexist, oppressive 'monotheism' (which historically is just a crude anthropomorphic projection of the tribal war-lord and priest, often with a male appropriation of the 'creative' traits of previous goddesses). We note that the idea that there are beings inhabiting plants (in particular trees and flowers) can be seen as a confused or poetical perception of elevated moral principles.

We have seen thus that 'theology' in its oldest incorrupt form is simply the inspired artistic representation of the qualities and achievements of great men and women, of people of special significance to the experience of the artist.  These heroes and heroines are not themselves principles of knowledge but people who sought and gained the principles of knowledge. The loss of this distinction is the beginning of corruption. But there is another important complementary perspective.

The harm and absurdity of ordinary religion consists in particular in 'worshiping', 'fearing' or 'sacrificing' to what could legitimately could only be philosophical principles and laws of reality, and specially anthropomorphizing and gendering them. Think of the absurdity of what would be the ordinary religious worship of  the transcendental faculties, the pure a priori concepts and principles in Kant's Critique of Pure Reason (closely resembling the ancient Samkhya system), of the pure mathematical structures upon which modern theoretical physics is based (a Pythagorean development), the henadic system of Proclus' Elements of Theology, or logic itself in its fullest Platonic dialectic conception.  Originally the 'gods' considered from this angle can be traced back in high antiquity to pure mathematical and philosophical concepts and principles (the works of Fabre d'Olivet are an historical curiosity with some relevance to this) - and these can indeed by considered as organic and living in the life of consciousness (this is mathematical proof, the scientific method, the philosophical dialectic). These can be the object of, dare we say, quasi-religious silent awe, reverence and love. A proof of original sophistication is furnished by the mathematical structure of ancient languages such as Paninian Sanskrit or what can be gleaned of proto-Semitic through Akkadian, Aramaic and classical Arabic. While in themselves symbolism, analogy and metaphor are powerful, enriching and valid (hence capable of a legitimate mathematical transposition) - anthropomorphic symbolism is dangerously confusing and misleading, for all the reasons expounded above.

We of course need to discuss the alleged vital social-cohesive function of religion, the relation between religion and nature, and the role of the solar mythos, the cycle of the seasons, the great year. The solar year, the passage of the seasons, can be seen as the universal type of the story, the narrative, the epic poem - indeed the archetype of human experience, human life itself. The unity of artistic creation, community celebration and ritual and the process of nature: social-cohesive natural-yearly mythic festivals represent 'living inside the story'.

There is one important caveat. We should be open to the possibility of other intelligent, sentient beings in other domains of our universe or on other planes of reality. There is no reason to suppose that there could not be positive enriching communication and relationship with such beings. But a 'religious' attitude to such contacts would be the worst thing imaginable. For they are either life-forms with some analogy to ourselves or else they are mistaken perceptions of universal scientific, mathematical or philosophical principles.

From what we have seen above, we leave it to the reader to understand why Christian gnosticism, 'neopaganism' and modern psychoanalysis and transpersonal psychology do not offer a valid spiritual-philosophical path or proper remedy for the harms and errors of traditional religions.   

New Writings

We plan to organize a portion of the past posts of this blog (from the previous two years) into several separate texts according to topic. 

We can say that there are two complementary perspectives in transcendental philosophy (at least at an initial methodological stage). The first is that of TPC and TPP - which is focused on philosophical introspective psychology. The second one we will call 'transcendental dialectics' - which is a special approach to the philosophy of logic, philosophy of language and cognitive science. It is the full development of the insights of ancient Skepticism and Madhyamaka but refined and clarified by a computationalist account of the a priori and key aspects of Kant and Hilbert. 

Then there is the application to the philosophy of mathematics and science - with special attention to determinism, computability and the foundations of calculus.

The culmination of philosophy - as mentioned before - is the exposition of ethics. More specifically,  an ethically enlightened philosophical anthropology is massively important for philosophy,  science and culture. This however is beyond the present scope.

We should also address potential criticisms and misunderstandings of TPC. For instance TPC has nothing to do, and indeed is at the antipodes of, the notion of the 'dualistic detached onlooker' of certain feminist (and Heideggerian) theories seen as a vehicle for the devaluation and destruction of nature.  

Also we should point out that we do not recommend or support any particular form  of psychotherapy or spiritual retreat or any kind of religious movement or organization ('Buddhist' or otherwise). And we point out that to us a correct and healthy spiritual path can have nothing to do with anything resembling a 'dark night of the soul'.  And if psychological anguish and troubles are part of life, we make no claim that TPC or the philosophy put forward here constitutes any kind of miracle cure. It is nevertheless our firm conviction that philosophical insight and the cultivation of philosophy can be psychologically beneficial.

If we were a 'guru' or 'spiritual teacher', we would have our students sit on their meditation cushions and read through Sextus' Outlines of Pyrrhonism, Hume's A Treatise of Human Nature and Kant's Critique of Pure Reason. They would actually be learning Abhidharma, Madhyamaka and Yogacara without knowing it !

Philosophical Psychology

Na kiñci loke upâdiyati. Total disinterestedness cannot be the disinterestedness of any finite self. Letting go of all, detachment. Short expressions, easy to apprehend superficially. But in their deep meaning they hold the keys to the heart of the world. To let go of the past, of all persons, to let be, let pass. This means relinquishing all desire and identity. This means freedom. It is utterly foolish to make relinquishment something external and physical. It is a transcendental act in the secret depths of the heart. We come to be, come to be what we are, through what we are attached to and desire. And every person is radically different people throughout the stream of life - where lies this identification of different masks ?

The royal road to TPC is i) through the perception of the universal a priori condition of temporality and transience. This reveals alleged objectified being as but a (modalized) moment in the flux of consciousness - and ii) through the perception that the self is determined through the delusion of the sense of mine, through appropriation and identification. A process and tendency, not a substance - and iii) that thought is often like a screen that consciousness uses to hide from its own nature - the world is born when consciousness wraps itself in the veil of its own thought.

Seeing through thoughts and thought. Analyzing posited 'units' of thought and showing their inner contradictions, essential dependencies and implicit containment of other thoughts.  Like characters in a story that only really have meaning relative to the story as a whole. The excitations of a self-interacting quantum field.

How to transcend the world. There is another aspect of TPC based on an intellectual analysis of the ordinary world, ordinary human life and psychology, which nevertheless has a transcendental philosophical goal of great import. This is ability to clearly see and virtually live various possibilities (for self and others, empathetically taking the perspective of others) and to extrapolate and accelerate a given life-possibility in time. In this way all delusional desires and attachments dissolve once the heap of inevitable sufferings, limitations and anguish entailed by what was taken to be their desired fulfillment are clearly understood. The feeling of possession of something desired is in reality you being possessed by it, its allure, its delusion and its desintegration and downfall.

TPC, self and temporality

We have described TPC as being involved with the transcendental awareness of the total continuum or process of thought considered purely as such, as merely a process of thought.  But TPC has an important aspect: transcendental philosophical consciousness involves the transcendental awareness of those preconditions and structures upon which ordinary consciousness entirely depends and at the same time of which ordinary consciousness as a rule is oblivious of. It is almost as if ordinary consciousness only exists, can only exist, under the condition of it carrying a forgetfulness of its own transcendental preconditions and predetermining structures. Transcendental consciousness is thus also the consciousness of what non-transcendental consciousness must necessary forget.   If TPC in turns does not have full knowledge of such transcendental preconditions and structures, it falls back into a false dualism, limitation and illusory reification.  Two of the most important of such preconditions are selfhood and temporality.  That which claims to be a being which we carry along as a huge burden - all possible transcendental questioning, untangling and analysis being forgotten - and that which we must forget in the first place in order for the very being-ness of being to arise.  Only then we arise to the selfless liberated insight into the pure universal sphere and flux of pure thought processes: only thus is further progress of TPP possible. The ordinary self is the protoype of illusory unquestioned, posited, composite being. The original prototype of being in transcendental temporal oblivion. One task is to study the formative tendency which constructs this prototype employing among other things certain classes of conscious phenomena.This prototype is the the scaffold upon which world-positing and world-directedness and the mutual feedback of the identity construct takes place. Being-in-the world is not a transcendental condition or principle but a non-transcendental illusion conditioned by transcendental ignorance and forgetfulness. Following Plato a philosophical consciousness must include knowledge concerning the phenomenology of love and beauty.  This is a transcendental illusion which yet in its purest form  participates of the some of the modalities of TPP-liberated consciousness.  It paradoxically offers a glimpse of the bliss of beinglessness falsely conditioned and limited by being, by the illusory directed network of the self-prototype and its world. When we are born we are forced to build our identity and learn the world; when 'born into the spiritual life' our duty is to unbuild this identity and learn to see through the illusion and construct of the world. It is as if the first phase were like being plunged into the water, a downwards journey into illusion and forgetfulness. The second phase is when we begin to rise again and make our way back towards the surface and the air of light and truth. TPP is not about an 'individual' detaching itself from the 'world' and still less about an 'individual' transferring its feelings, desires, volitions and tendencies to alternative 'imaginary friends' or any kind of ontological construct - rather it is about seeing through both the 'individual' and the 'world' and their dependent interplay and mutual constitution. There is no contradiction between Yogacara and Madhyamaka. Two of the main tasks of TPC-based philosophy are: i) the untangling of all concepts and categories and the showing forth that they all exhibit the mark of the alleged being of the prototype (self), ii) exhibing the absoluteness of the moral law,  that the absolute can be characterized as morality (dharmakaya). The profound affinity between the consciousness and act of TPC and TPP and that of the essence of morality.  In all of the above the great interest of Kant is patent.  A model of the universal  flux of pure consciousness (Indra's net): neural nets, cellular automata, music theory, formal grammar, tessellations, Wang tiles, knots (and our own theory related to this) - unfolding of the concept of computation (paticcasamupada) and its concurrent parallel interconnected forms. And most importantly the theory of (meta) reflection and representation.  And the logical and mathematical theory of programs, specially the hierarchy of function definitions. The way forward in physics, the way to transcend and rectify quantum field theory, seems to us to involve a radical new foundation for physics based not on the formalism of functional analysis and operator algebras but rather on finitary combinatorics, theoretical computer science, graph theory and algebra. Maybe something following Birkhoff's lattice approach. 

A mathematical model of transcendental pure consciousness (which can only be a limited exterior projection, a partial mirroring...) will involve a spatialization of concepts (structures) wherein their interconnection and mutual (meta) reflection takes place - in a temporal dimension. A general theory of networks which generalizes neural nets, cellular automata, tessellations, crystallography, (higher)category theory and proof theory. This will include the spatial, topological aspects of music, and the architecture of a semantic universe (semantic space) or lexicology. But we can take also an atemporal space-time approach. Has the effects of special relativity on the physical realization of Turing machines been studied? Could special relativity be the key to how physical systems could compute beyond the Turing limit? Perhaps the quantum effects on physical computational systems can be studied in a way different from the current work in 'quantum computing'. In Hegel's  Logic we can discern a marked spatial, geometric way of thinking of concepts, their unfolding, mirroring and interaction.

The knot, the tangle, the projection, the multiplicity is not real, what is real is the formative energy which ties and then unties, tangles and then untangles, projects and then dissolves, multiplies and then unites. Is this not the ultimate meaning of the Hegelian Idea, the holomovement of skepsis which though necessarily dealing in formalism is not itself entirely formalizable ? Each thought (unit of consciousness) containing implicitly all other thoughts and each thought only existing through its relationship to them? That by trying to isolate a thought 'in itself' it inevitably looses the essence of what it really is.

Another approach is to view space-time as characterizing non-transcendental consciousness and being characterized by its algebraic and logical properties. Thus pure transcendental consciousness should be represented by a structure in which some of these logical or algebraic properties are different, the most famous being Heyting rather than Boolean (Brouwer), non-commutativity (Connes) or non-distributivity (Birkhoff).  Birkhoff's explaining away of the uncertainty principle as simply an expression of non-distributivity. However this approach does not go beyond the previous proposal, it is still a priori and phenomenologically a spatialized or space-time based 'collection' of components, parts, connected to each other by some kind of relation or network. And the essence of representation theory is bringing this spatial substrate further into the foreground.

The philosophical elephant in the living room is that we have not explored something which should occupy a fundamental place in TCP. That of symmetry, quasi-symmetry and analogy. Like temporality and the prototype of the self, it is so pervasive as to be habitually consigned to an implicit oblivion. Symmetry and its allied modes pervade biology, psychology, linguistics, chemistry, physics, mathematics, logic, computer science, engineering, the structure of philosophical systems, poetry, music,...so deeply, in such a far-reaching manner, that like temporality or the construction of self, we simply forget and cannot fathom its central, universal transcendental a priori role (symmetries abound in the architecture of Kant's and Hegel's systems, also in the Pali suttas, and linear logic brought forth the implicit symmetries and modalities in classical logic). We end with the following preliminary (Hegelian) descriptions: symmetry is the quality of a being which allows it to change and yet remain the same. Symmetry is also that by which a being may produce itself (notice than in symmetric tiling we can transport any tile to any other tile while preserving the identity of the whole). Symmetry is the foundation for reflection-into-another and reflection-into-self. Recall how spinors capture the 'memory' (homotopy class) of a rotation in SO(3), much like the dialectic in Hegel. An object may be brought back to itself and yet have gained something permanent.  Symmetry is that by which an inherently dynamic process can produce the illusion of stability and being. Symmetry allows us to fathom essentially interrelated holistic systems (illustration of 'maps of consciousness': dictionaries, grammars, thesauri, manuals of stylistics and translation, encyclopedias, biographies, novels).

A radical new look at science

Kantian question: what scientific and mathematical theories are strictly required for the design, engineering and effective implementation and production of modern technological and industrial artifacts (a computer, a drone, a car, a MRI machine, a satellite, a vaccine, a nuclear reactor). This is philosophically similar to the simpler question Aristotle posed about products of art (technê). It is seems plausible that there is a huge amount of published theoretical material which is neither necessary nor useful, strictly speaking.

A disturbing question: there are presently technological artifacts which are not fully understood (or even capable of being practically engineered) by a single human being. For instance current versions of the most common operating systems. Could technological artifacts come eventually to automate their own design, production and maintenance (and even security and defense) outside the possibility of human understanding, design and control ?

This approach favors the view that the differential mathematical apparatus in science is never meant to be nor should be read as a model strictly speaking but as a computational device synthesizing modelling information for concrete more realistic and complex models. In other words, the Navier-Stokes equations do not model fluids, we are in the presence of a mathematical device for compactly representing the potential computational information of actual approximate models of reality. Abstract devices which are used to extract finitary computational content (like the convenient fictions of Leibniz but without taking any philosophical position on mathematical platonism).

There are interesting arguments for the Lebesgue integral being useless and meaningless in the context of actual concrete engineering practice (limited by measurement and scale). 

Philosophical, historical and spiritual inquiries

There are many difficulties with the interpretation and practice of Buddhism as well as with trying to recover original Buddhism. This is well-known. It would be wonderful if we could find an ancient, prehistoric and living tradition which is identical to the highest essence and goal of philosophy - with TPC and TPP, that fully developed leads to the fulfillment of philosophy and vice-versa. It would allow one to consolidate and integrate different philosophical systems and perspectives and to disclose their highest meaning and value. And also it will furnish a basis to discern the scientific, psychological and ethical value of certain systems despite certain fundamental philosophical errors. It is Ariadne's thread that allows us to clarify, diagnose and resolve all confusions and errors in the history of Buddhism. But such a holy grail is not easy to find. What we have are a different distinct parallel ancient traditions and most importantly situations in which pure teachings are misappropriated and integrated into fundamentally antagonistic cultural life-forms. The Upanishads do not represent any kind of single spiritual-philosophical doctrine and practice but a heterogenous plurality of them (a thicket and jungle indeed). Moreover in the Upanishads we find what appears to be an adulteration and appropriation of pure TPC/TPP-oriented teaching in the direction of  proto-Hinduism with its caste system (a complex subject wherein we must be aware of British colonial influence in current perceptions; some argue that the caste system does not correspond to the fluid social division of labor based on qualities rather than birth said to have been present in the Vedic period), ritualism, the cult of procreation, family and material wealth, anthropomorphic and naturalistic religion and a spiritual legitimizing of warfare (notably in the Gîtâ - with parallels in the Kyoto school).  Nowhere is this more evident than in the doctrine of the four ashramas. However this does not rule out that there are ancient portions of both the Upanishads and the Gîtâ which furnish valuable material and information about ancient higher spiritual practices.  We must not overlook the extreme antiquity and importance of texts in Old Avestan.

There can be no doubt that in original Pali Buddhism as well its flowering into the great philosophical Mahâyâna systems of the Yocara and Madhyamaka, represents, together with the parallel traditions of the Samkhya (Burley has vigorously argued for its close affinity to Kant and Husserl and not to naive realist cosmology),  the sister schools of Nyâya and the Kanada's original Vaisheshika,  Râja-Yoga and Jainism, the most pure, profound and beautiful root and fountain that arose in Indian soil and its supreme expression of TPC and TPP. The best aspects of the Vedanta are completely indebted to it, though Advaita Vedanta represents an orthodox Hindu appropriation  of Buddhism (an artificially idealized version of the Vedanta has been presented to Western audiences). Just compare the Pali suttas concerned with caste with what over a thousand years later Shankara writes in his commentary on the Brahmasutras.

These supreme flowering of TPC and TPP  provide the clue to understand what is best in the Western spiritual and philosophical tradition, for instance the Platonic, Pyrrhonic and Plotinean schools, the texts of the Corpus Hermeticum, the ultimate meaning of Hume, Kant, Schopenhauer and Hegel.

It also presents us with a significant challenge and project: the enigmatic gnostics and gnostic texts, for instance the Pistis Sophia and the Nag Hammadi library. One view is that the gnostics and the gnostic texts represent a rare and fortunate survival of very ancient traditions cast in a 'Christian' form. This is the view of G.R.S. Mead in the introduction to his translation of the Pistis Sophia: gnostic texts represent a rare and unique survival of very ancient higher spiritual traditions pertaining to Egypt, Syria (and Phoenicia) as well as Persia and Babylonia. It is also important to study the 'Chaldean Oracles' and their commentaries. A very important place must be occupied by the study of Manicheism and Mandaeism.  Also the investigation of how far the medieval Sufi texts (including those of the Iranian esoteric traditions) represent a a transmission or more-or-less veiled continuation of much earlier neoplatonic, hermetic, manichean, mandaean, zoroastrian, gnostic traditions (the gnostic traditions seen in turn as Egyptian, Syrian, Phoenician and Chaldean). It would also seem that certain more philosophical Sufi texts contain already all the essential ideas of Leibniz and Spinoza (a fact which is historically not that surprising if we consider the possible influences on both these philosophers). It seems plausible that Sufism had a huge overlooked role and influence on all other medieval spiritual traditions (not only Christian but also the medieval Zohar).

However it is also possible that we might arrive at the conclusion that at least some 'gnosticism' does not represent a TCP/TPP directed spiritual-philosophical traditions or its reflection in the higher currents of western philosophy (recall Plotinus's critique of the gnostics). It could be that so-called gnosticism is un-philosophical and does not align with the spiritual orientation and goals of TCP and TPP. Rather so-called 'gnosticism' may represeny an essentially ritual-magical tradition, like certain aspect of later neoplatonism and very much akin to the Tantric schools of India and Tibet. Perhaps there is a kind of myth-making, magic and ritualism - for instance the presence of astrology - that is not only a perversion of science and mathematics but also of art. Maybe this is why TCP and TPP oriented India and philosophical Greece - both guided by TPC and TPP - produced such works of genius in logic, linguistics, mathematics, science and art. Does this assessment apply to Manicheism gnosis (which is of interest for its link to Buddhism) ? Perhaps such texts as the Pistis Sophia heavily indebted to Manicheism (the 'Virgin of Light' is the Manichean Kanig Roshn).

TPC unveils, skeptically analyzes and circulates among fundamental principles and structures of consciousness in absolute pristine given-ness and TPP is concerned with the path of liberation.  There could not be a greater opposition and contrast between TPC and TPP on one  hand and religious anthropomorphic mythologies and cosmologies on the other - like those of the gnostics and Manicheans. It is as if someone took the concepts, categories and principles in Kant's Critique of Pure Reason and turned them into a cosmic mythology of anthroporphic eons and archons mixed with astrology and magic.  It is as if we took Kant's transcendental critical subjective idealism and turned it into the most crude dogmatic realistic anthropomorphic naturalism: for instance associating the twelve categories of the understanding with the twelve signs of the Zodiac.  It seems as if Mani did precisely this with the TPC-doctrines of  Samkhya and Buddhism and the gnostics did this with the Hermetic texts and Plato. This represents a dangerous and unfortunate human cultural-spiritual tendency and in light of this we can appreciate even better the original teaching of the Buddha as well as Pyrrho.  We add: would it be possible to take certain theories of pure mathematics and convert them into a gnostic-like mythology ? 

But we note that certain texts in the Nag Hammadi like the Gospel of Thomas, Gospel of Phillip and the Gospel of Truth do appear to have a certain affinity to TPC and TPP and little in common with typical gnostic cosmologies and mythologies. Or at least they have a more philosophical, critical even Socratic-ironic bent that recalls the Zen koans.  Also we do not know how precisely the gnostic texts were read and interpreted, or who wrote them.

One of the greatest enigma of history is the relationship between such texts and Judaism and the  Old Testament.  Nothing could be more different, unrelated and even directly antagonistic.  What could the Gospel of Thomas have in common with the Pentateuch and book of Joshua ? A vegetarian, animal-rights upholding pacifist community that took the Exodus and Leviticus as holy texts ?  The same absurdities are patent in those who would wish to make the Therapeutae described by Philo  some kind of Buddhists. It seems clear what we have here is the historically pattern of borrowing (in fact plagiarism because the sources are not acknowledged) and (mis)appropriation of certain higher spiritual traditions and materials (probably taking place in Alexandria, but also among the Jewish Stoics who wrote the Pirkei Avot). The world of the Nag Hammadi scriptures is a kind of alternative reality where the only cultural-historical context is the mythology of the Old Testament: the rest of the world and history is almost entirely ignored and cancelled. What about the countless other Middle Eastern peoples (Chaldeans, Syrians, Phoenicians, etc) and Egyptians (whose traditions the Pistis Sophia appropriates) ? We must never loose sight of the pre-eminence and antiquity of the Egyptian tradition and the Graeco-Egyptian gnosis. This is what McBride writes in his book The Egyptian Foundations of Gnostic Thought:

It is in the area of Hellenistic Gnosis that the Egyptian foundations of Gnostic thought attained their greatest synthesis with the diverse strata of metaphysical thought in Hellenistic and Roman times. Here we have the beguiling and obscure phenomenon of literate Jews who were no longer Jews in any real sense of the term, Greeks who were no longer Greeks in their religious affiliations and bloodlines, Egyptians who were no longer “pure” Egyptians, and proto-Gnostic magicians and priests in the period from 100 B.C.E. to 100 C.E., all of who contributed to the evolution of Gnostic thought. Above all, there remains the essential enigma of the literate and bilingual, if not multi-lingual, “Graeco-Egyptian”. In a sense, this group represents both and neither of the scholastic categories of “Greek” and “Egyptian”, so removed were they from traditional modes of thought and even clear ethnic divisions.

 A somewhat similar situation is found in the Pali suttas where other ancient Indian philosophical and spiritual practices is are strangely ignored (except the Jains). It seems pretty clear that the authors of the Nag Hammadi library are supposed to be Jews (or descendents of Jews) who converted to a radically 'heretical' (allegorical, gnostic) form of Judaism.  

Already in the 1st century CE Epictetus was preaching a philosophical, spiritual and ethical doctrine which equals if not far surpasses anything we find in the Bible or the Gospel of Thomas. Along with the strong presence of Pythagorean, Middle Platonic, Hermetic,  Neoplatonic, Stoic and Pyrrhonic schools there are strong arguments for a pervasive transmission of the Buddhist, Jain, Samkhyan and Yogic doctrines in the ancient world. TPC and TPP were both ancient and widespread. Thus there is no spiritual, philosophical or cultural-historical excuse or justification for the Old Testament centric world-view implicit in the New Testament or part of the Nag Hammadi scriptures. Other parts of the Nag Hammadi library furnish radically distinct 'evil demiurge' interpretations of the Old Testament which are even more bizarre (the Old Testament remains central but now for a different reason: offering unique intel into the doings of a bad deity). Docetic Christology (such as found in Manicheism, Marcion, the Acts of John) is also bizarre. It would be interesting the compare this with mythicist views regarding the historical Jesus (which for some reason, which has nothing to do with new evidence, have fallen out of favor). In the Docetic-Manichean view Jesus is a cosmic being and principle (much like the Mahâyâna Bodhisattva) who manifested via a spiritual body in ancient Palestine. It us curious that many aspects of Manichean Christology were adopted by Wagner in the libretto of Parsifal. 

The Aramaic dialects: one of the oldest and most important of the languages (both spoken and written) of ancient Palestine (including Syria).This ancient language was adopted by the inhabitants of Palestine (and Syria) before and during the Roman era (one might call it the semitic counterpart of koinê Greek). A common word for God was Alaha ܐܲܠܵܗܵܐ .  A version of the (cursive) Syriac script was used by the Nestorians and is very similar to the Arabic script. Ancient authors sometimes referred to Aramaic as 'Syrian'. The term derives from the place-name 'Aram' (supposedly in Northern Syria). A common view is that the Aramaic square script, also called Ktav Ashuri (i.e. Assyrian writing), basically the modern Hebrew alphabet, was adopted by Jews after the Babylonian exile (c. 6th century BCE) from Imperial Aramaic, a cursive script used during the Achaemenid Empire.There followed a decline of Aramaic as the spoken language of the Jews which parallels the subsequent domination of Arabic in Egypt and Palestine.There were many important Aramaic speaking currents of early Christianity and the Pesshita is widely used in NT studies. It is possible that some of the neoplatonist philosophers like Porphyry spoke Aramaic. Some Manichean scriptures are in Aramaic.

Finally we address the fundamental question: what criterion do we have that allows us to judge whether a given person, spiritual practice, spiritual school, etc.  has attained TPC and genuine spiritual realization? The answer is simple: we have one perfect criterion which is perfect in the negative sense, it allows us to eliminate false claims of enlightenment. This perfect criterion involves testing the alignment of the person with universal moral principles applicable unconditionally to all human beings (human rights) as well as universal moral principles regarding the treatment of animals and indeed all life forms. For example any philosopher and spiritual teacher who accepted or tolerated things like a caste system, slavery, torture, the mistreatment of animals, cannot categorically be considered as having achieved complete or a higher TPC-enlightenment. The universal moral law is a revelation of the absolute and a path to the absolute. Another infallible criterion is that TPC is inseparable from the scientific spirit and the spirit of criticism and free inquiry.

Elementary Combinatorics

 

Consider the ordered sequence $(1,2,3,4,5,6,7,8)$ organized as in the figure above. We are interested in potential generators of the permutation group $S_8$.  The generators in question are the following (which also explains the diagram):

$F = (1 2 3 4),  B = (5 6 7 8),  L = (1 4 8 5),  R = ( 2 6 7 3), U = (1 5 6 2), D = (3 7 8 4)$.

These generators shift the numbers on their corresponding boundaries  in a clock-wise direction.  These generators $4$-cycles and thus of order 4 and $F^{-1} = F^3$ and so forth for the other generators. For convenience we list the inverses:

$F^{-1} = (1 4 3 2),  B^{-1} = (5 8 7 6),  L^{-1} = (1 5 8 4),  R^{-1} = ( 2 3 7 6), U^{-1} = (1 2 6 5), D^{-1} = (3 4 8 7)$.

We have the following elementary facts about permutations and transpositions:  $(a b c) = (a c) (a b)$ and  $(c d) = (a c b) (a b c d)$ (for $a,b,c,d$ distinct).  We use the notation $[X,Y] =     X^{-1}Y^{-1}XY$. Note that $[X,Y]^{-1} = [Y,X]$ and that if $[X,Y]$ can be written as a composition of disjoint transpositions then $[X,Y]^2 = I$ and $[X,Y] = [Y,X]$.

Let us calculate:

$[D^{-1},F^{-1}] = (14)(37)    \quad [D,R^{-1}]= (37)(42)$

Hence  $T = [D^{-1},F^{-1}][D,R^{-1}] = (14)(37)(37)(42) = (4 2 1)$.

Then $TF = DFD^{-1}F^{-1}D^{-1}RDR^{-1}F =  (4 2 1)(1 2 3 4) = (2 3)$.

It is now easy to see that $F,B,L,R,U,D$ generate $S_{8}$. 

Let $\Sigma = \{id,\sigma,\sigma^2\}$ be the cyclic permutation group on $(1,2,3)$ (of order 3).  

Consider 8 copies of $\Sigma$ denoted by $\Sigma_1,...,\Sigma_8$ and let $W = \Pi^8_{i =1} \Sigma_i$.

Suppose  we had a map $S^i$ acting on $W$ which for $i = (1,2,3)$ multiplies components $1$,$2$ and $3$ of $W$ by $\sigma$. And likewise we have $S^i$ for $i = (2,3,4), (3,4,1),(4,1,2)$ and also $i = (1,2,5)$, etc.

Then $Y = (S^{(2,3,4)})^2 S^{(1,2,3)}$ is equivalent to applying $\sigma$ to the first component and $\sigma^2$ to the fourth component. For instance for $w = (id,id,id,id,id,id,id,id) \in W$ we have $Y(w) = (\sigma,id,id,\sigma^2,id,id,id,id)$. 

 

We now consider the subgroup (of $S_{12}$) generated by certain permutations, where the ordered set $(1,2,3,4,5,6,7,8,9,10,11,12)$ is organized as in the figure above.

The generators are:

$f = (2 \,7\, 3\, 6), b = (1\, 8 \,4\, 5), u =(1 \,10\, 2\, 9), d = (3\, 11\, 4\, 12), l = (5\, 9\, 6 \,12)$ and $r = (7 \,10\, 8 \,11)$ together with

$X = (5\,6\,7\,8), Y = (1\,2\,3\,4)$ and $Z = (9\,10\,11\,12)$

Then $Y^{-1} = (4\,3\,2\,1)$ and $f^2 = (2 3)(67)$. Therefore

$f^2Y^{-1}f^2Y =  (1\,3\,2)$ and hence $f^2Y^{-1}f^2Y^2 = ( 34)$. Now it is easy to see that our generators generate $S_{12}$.

We will define a group morphism via a correspondence of our generators for $S_{12}$ to the generators for $S_{8}$ defined above: $f \longrightarrow F$, etc. 

(to be continued...)

Note on TPC and TPP

Yoga citta vritti nirodhah

. How can we speak of the thought of being, the thought of existence and forget that before all else thought itself is, thought is there, for us, in us, in its convoluted unceasing stream and primordial being. But to be aware of, conscious of, thought as thought, that thought is itself the great a priori screen and window wherein plays out the play of the world, this is already a path to TPC. How does this transcendental awareness of the totality of the thought-process work ? How can thought then think itself and be beyond itself ? How does TPC work ? How can being be a thought and yet thought be a being? How is thought related to language ? How is thought related to the impression or structure of 'self'? How is thought tinctured by sense-images and other psychological material ? How does temporality work in the diverse modes of thought, memory, recollection, anticipation, habit ? We should be very careful about dissections of thought: for instance Brentano's intentionality, Husserl's noesis vs. noema, the concept of 'content', etc. And it is astounding that those who disparage conceptual realism accept at the same time completely arbitrary, dogmatic, unproven and implausible functionalist theories. Neuroreductionism, eliminationism, functionalism and neothomist theories of embodied consciousness should be considered dogmatism and philosophical naivety par excellence. What is the transcendental meaning of the construction or bringing to light of formal systems which are supposed to act as masks, or exterior projections, doubles, of thought ? How does translation and multilingual consciousness work, what do we mean when we say that we think in a given language ? Sextus, Hume and Kant can be read as an absolute untangling and allaying of thought by thought itself transcendentally aware of itself. The true self, pure consciousness (bodhicitta), is revealed to itself and at the same time the deep producer and controller of thought is unveiled.

So TPC and TPP aim at lifting and untangling thought or rather consciousness-thought into a super-consciousness and super-thought.  But ordinary consciousness does not consist of thought alone, it includes feelings, volition, desire, longing,  beauty, love, eros, intimacy, habits and dispositions like pride and humility and so forth. In ordinary consciousness these spheres stand apparently apart from pure thought (to a certain extent). But Kant wrote more than one Critique and the author of the Parmenides and Theaetetus was also the author of the Phaedrus and the Banquet.  Our description of TPC and TPP must be completed and perfected with a philosophical and practical account of how these sister spheres are also untangled and rise above themselves and become united to super-consciousness and super-thought.

The constituted constructed self stands at the center of the division of these two regions and is a common hindrance, obstacle, to the untying and transcending of both.

The significance of temporality and in particular of one's own concrete temporality, memory and identity is immense for TPC and TPP.  We cannot go into here the immense meaning and power of abgeschiedenheit, of 'letting go', of the process whereby we let the past go, let of memory-attachment-persons, intuiting that the past was the present of its own past and turn,  and  learn to embrace a new and better past and identity. We understand the process of coming to be and passing away and gain transcendental acceptance and freedom (freedom from guilt, ought, destiny).

We could also discuss the relationship between TPC/ TPP and the scientific spirit, the scientific method, the engagement in a serious, patient, philosophically aware, critical, detailed study of the sciences.

TCP/TPP is a priori radically incompatible with any form of religion while capable of recognizing positive ethical, philosophical and scientific elements in different spiritual traditions. TCP/TPP is also ideally an effective psychotherapy against the delusions and pathological effects of religion and its misguided spiritual practices. 

The central philosophical theory of TCP/TPP is the exhibition of its essential unity with universal morality. 

Mathematical projections or representations of consciousness. They can have a field-like spatial-like aspect $\phi: S \rightarrow U$ ($S$ expressing certain concomitant psychological material) and a non-local aspect akin to distributions $\Phi(S,U) \rightarrow V$. Distributions can be interpreted in a way similar to the collapse of the wave-function in quantum theory. In quantum theory we derive a probability relative to each eigenvector, for distributions we a have a real or complex value for each test function. Because of freedom all temporal transition $\phi_t$ must be described probabilistically.  There is no reason to suppose that a thought at a given moment has a single 'content' (noema). The extraction of the content can be likened to evaluation at a test function or quantum measurement. The evolution of thought generalizes computations and proofs (arguments). Denotational semantics and content: the (teleological) question what is the program computing ? 

The great project: uniting theoretical computer science and theoretical physics (and the general theory of dynamical systems, differential equations, etc.).  How can we compare the executing of a program and a physical process satisfying a given law ? Put another way: what is a running program, specifically a concurrency of different systems, from the point of view of dynamical systems and theoretical physics ? For a given solution of a partial differential equation we can say that the process represented by the solution is virtually and approximately computing the numerical integration algorithms which could approximately derive that very solution.

Question: are PDEs Turing complete ? That is, given a Turing machine M can we write a partial differential equation having a numerical integration algorithm A such that M can be simulated by the running of A ? 

Prehistoric philosophy and spiritual culture in Europe

 

The close connection of the grammatical structure of Old Church Slavonic to Sanskrit (and Indo-European in general) is fascinating. The Glagolithic alphabet is curious and more interesting are pre-Christian Slavic scripts such as the Alekanovo inscription.  There is nothing 'crude' or non-sophisticated about having a writing system of the Ogham type (Hrabar's чръты и рѣзы, črŭty i rězy).  On the the contrary, it indicates a high level of arithmetical symbolic (or cryptograhic) thinking, which Hrabar probably was not even able to understand.  Take a really good look at the 'Sun ship' from Bronze-age Poland.

 The  Byzantine historical attestation of the Bandura (Ukrainian: бандура) among ancient Bulgarians  points to a survival of an ancient musical tradition (the counterpart of the Celtic Bards).

We entertain the idea that certain peoples among the ancient Slavs may have possessed a form of Buddhism (we see, as attested in the Pali suttas themselves,  Gotama's doctrine and practice as a restoration of something even more ancient) or a similar system of yoga or spiritual cultivation. A common view would be that ancient Scythians and Sarmatians may have played a role (also with later manichean connections: consider the Bogomils and the story of Barlam and Josaphat, this last name being an adaptation of a middle Persian rendition of Bodhisattva). However we obviouly reject arbitrary occultist fantasies (without any serious scientific basis) about Slavic prehistory - which are often unfortunately aligned to questionable political ideologies. But we note also that much of 'mainstream' theories of history and prehistory are equally tainted by ideological agendas and prejudice (just as the account of the pre-Christian Slavs by early Christian writers is not reliable).  Can we find a correlation between the elegance and sophistication of a language (its closeness to Sanskrit or some other measure) and the cultural and spiritual level of its speakers ? And by cultural and spiritual level, we mean consciousness and implementation of universal ethical principles regarding human beings and animals, philosophical consciousness (that is to say, TPC and TPC) and spiritual cultivation based on it, the development of formal logic and analysis, the template of the scientific method, the refined development of the arts, etc. 

From an historical and archaeological perspective we argue strongly against and completely reject the opposition between so-called 'polytheism' and 'monotheism' and all the ideologies and connotation such an opposition or distinction implies. The theological doctrines (cf. kathenotheism, henotheism, etc.)  of Proclus' Elements of Theology are the same as that of the Vedas and the Avesta. Christianity in most of its historical manifestations was never  quite 'monotheistic' in the crude sense and the same goes for the esoteric teachings and practices in Iran.  For indeed the absolute (the good) is that which is beyond all gods which are in the highest sense are its emanations (angels) and mediators (henads). And the absolute as well as the highest gods are essentially to be sought within.

Historically we find high levels of idolatry, anthropomorphism, materialism, violence and barbarism among so-called 'monotheistic' cultures. This so-called monotheism is just a cultural expression of political supremacist ambitions of certain peoples wherein a tribal deity is given traditional attributes of a military conqueror and emperor.

What is far more important is how the concept of divinity is essentially united to that of universal moral principles and philosophically to that of TPC and TPP. 

Returning to the Slavs (the so-called Slavic deities alignment to the natural cycle of the seasons and to the experience of love and beauty  may have been a sign of an elevated philosophical spirituality akin to the Vedic one),  we note that the ancient Slavic woodcraft was quite sophisticated and must have made use of considerable knowledge of geometry and physical engineering. The archaeological site at Biskupin in Poland (c. 8th century BCE) could be proposed as an example, but of course we cannot know how precise and sophisticated this structure originally was. The artifacts related to this site are of exquisite beauty. 

We discussed briefly the prehistory of the Slavs (the Slavic or proto-Slavic speakers) and the thesis of the existence of a higher spiritual literate culture akin to Vedic culture which included a complete system of spiritual realization as well as certain types of science (like Paninian grammar). Furthermore we have hinted elsewhere that the western Slavic  (and Baltic) substrate in the territories of what is now Germany played a central role in higher philosophical and cultural manifestations (for instance Bach, Leibniz and Kant and Frege, mother's name Bialloblotzky). Bach (like Eckhart) was born in Thuringia. 

Thuringia has significant historical Slavic influence, primarily from the Sorbs (Wends) who settled the region in the Early Middle Ages. This influence is most evident in place names and the genetic ancestry of modern inhabitants

A similar and very strong argument could be carried out for a certain substratum of Celtic-speakers, specially in Ireland, Wales and the regions of what now are Belgium and France. This substratum expresses both the persistence of an original higher spiritual culture (and here we cannot go into the question of how proto-Celtic relates to proto-Slavonic) and specially a propensity and affinity for TPC. The testimony of classical authors leaves no doubt as to the high and central place that was given to philosophy (Druids), science (Ovates) and art (Bards). Also a high place accorded to women (cf. the Druidesses Veleda and Aurinia).

Just consider the extraordinary TPC luminaries that arose from northern Celtic soil: 9th-century neoplatonist Scotus Erigena, Berkeley and above all Hume - who can be considered a kind of Celtic Vasubandhu or Nagarjuna. William James' grandfather was from County Cavan, Ireland. British neo-Hegelian McTaggart had (as evidenced by the family name) Scottish roots.

For Celtic France (including the territories of what now is Belgium) we have of course the spiritual culture of  the Troubadours/Trovères, the Cathars (and Gleizes may be a distant heir), the Grail literature of which much could be said regarding its expressing a continuation of an older traditions (Manichean, Buddhist, Hermetic, Avestan, Druidic, etc.). Noteworthy are the philosophical-spiritual writings of Marguerite Porete (a 13th-century Beguine from Hainault, now Belgium) who may have been a source for her more famous contemporary Eckhart.

Beyond Ireland and Scotland, in England there has always been a tension and duality between opposing tendencies, a great number of things high, noble and beautiful flowing forth from the Welsh-Brythonic substrate. And much could be said about the history of the Welsh Quakers.

As for France (whose first great philosopher was Jean Buridan),  there is an evident presence of TPC in the schools of Descartes and Malebranche with a strong connection to Leibniz. Of particular interest is the later merging with the Kantian tradition in such figures as Maine de Biran, Renouvier and Lachelier, showing the essential affinity of the French philosophical genius for TPC-oriented philosophy. Indeed if Husserl's phenomenology certainly has some affinity to TPC then nowhere has this philosophy been expounded with such meticulous care as by Gaston Berger or René Schérer. Whether French existentialism, phenomenology and psychoanalysis had some positive and valuable elements from the perspective of TPC and TPP - following the suggestion of Paul Demiéville in a preface to a book by Walpola Rahula - is something to be investigated.

There were also TPP-oriented poets like Mallarmé. Very noteworthy were the Belgian scholars Louis de La Vallée Poussin and Étienne Lamotte  who were the first to study and expound the Madhyamaka based on the original sources (and already in 1849 H. Chavée produced an indo-european lexicology). While Belgian philosophy (which does not compare to Belgian mathematics and linguistics), beyond the unfortunate influence of neothomism (even Maréchal's more interesting Kantian version),  has been dominated by ad hoc anglo-american neuro-reductionist/functionalist and  'analytic'  influences (Gochet, Devaux, Barzin), the poet François Jacqmin (Le Livre de la Neige) represented a pure and unique expression of the wedding of absolute philosophical insight (TPC) and personal spiritual realization (TPP). Jacqmin was likely a reader of de Waelhens' books on Heideggerian phenomenology (Husserlian phenomenology is represented rather by Marc Richir). 

This 1960 paper by Jean Ladrière is interesting (formal systems as 'doubles' or projections of thought).

We could also investigate the existence of a prehistoric higher spiritual culture among certain branches of Germanic-speakers in certain regions (for instance Frisia, Brabant, the study of the Beguine mystics like Beatrice of Nazareth and what might be authentic in the Oera Linda boek - or certain traditions in Scandinavia related to Baldur). This is a complex topic that we cannot enter into here. 

Hegel and Metaphysics

We can trace much of Hegel back to Spinoza, Leibniz and Kant. But what if Spinoza and Leibniz themselves represented but an adaptation or even simplification of little known medieval philosophical  texts (much earlier than Böhme) ? And that such texts gave a firm foundation for the synthesis of Hegel with Schopenhauer? What if the unfolding of Hegel's Logic or Phenomenology of Spirit can be interpreted as representing the process of spiritual liberation, totalization  and harmonization of all states of being?

In the Zusätze to 573 of the Encyclopedia, we read:

If we want to see the consciousness of the One (...) in its finest purity and sublimity, we must consult the Mohammedans. If, e.g., in the excellent Jelaleddin−Rumi in particular, we find the unity of the soul with the One set forth, and that unity described as love, this spiritual unity is an exaltation above the finite and vulgar, a transfiguration of the natural and the spiritual, in which the externalism and transitoriness of immediate nature, and of empirical secular spirit, is discarded and absorbed.

Essence becoming actuality in the Science of Logic: beyond different models for a theory, different frames of reference for physics, representations for an algebraic structure, etc. Do we not here have the phenomenological reflection which neither attempts to have empty thought grasp directly its own structure nor is thought loosing itself in the objectified engagement in its action, but rather is the self-reflected awareness of thought in its thinking, a shift of perspective which knows itself in its process?  This goes beyond causality, computation and formal logic to inner and infinite spiritual knowledge. Spiritual knowledge is knowledge that what is in consciousness is taken and proceed from its inmost center, a revelation, veiling-unveiling, which must pass to reveal the process: the revelation is re-velation.

Hegel's treatment of modality in the section of Actuality is very relevant to contemporary discussions on possible worlds and arguments against the world being a sum of independent contingent facts. There are also connections that could be made to modal logic (i.e. the connection between possibility, actuality and necessity and the axiom $\lozenge P \rightarrow \square \lozenge P$). 

The considerations on necessity and substance (and indeed the long digressions on Newtonian forces already suggest this) can be interpreted in terms of the solutions of differential equations (or in general smooth vector fields on a manifold). The local existence and uniqueness of a solution for a given initial conditions expresses determinism which extends to a global determinism. 

The first part of Schopenhauer's World as Will and Representation is divided into four books. It would be interesting to establish a correspondence with the Science of Logic.  Clearly books 1 and 2 correspond to Being and Essence while books 3 and 4 correspond to Concept. However books 1 and 2 are written from the perspective 'for us', from the point of view of transcendental reflection. From this perspective books 1 and 2 belong to Essence and their philosophical knowledge expresses with great rigor and detail Hegel's theory of appearance, existence, substance, necessity, causality and actuality.  Or maybe Hegelian substance is much like the spontaneous production and manifestation of the will in Schopenhauer. It goes beyond phenomena and appearance because the will is immanent in its manifestation, totally in each one and beyond any particular one. The will has to manifest. Transcendental reflection, artistic contemplation and spiritual development represent the will's progressive self-knowledge, return to self, the Hegelian Spirit in the form of the concept.

Category Theory: a category expresses essence. Each object is a different mode whose determination is inseparable from its relations (morphisms) to the whole (all possible determinations). An object is an expression of the category and yet not the category (for there are other objects). At the same time this circumstance of the object not being the category is itself internalized and expressed as the object being itself the sum-total of its relations (morphisms) with all other objects (i.e. with the totality of the category). This expressed also that the object is the category. 

The Concept can  be seen as arising when we have the cyclic transcendental consciousness of metatheory. That is to say: we realize that transcendental presuppositions are in cyclic dependency: to do logic we need computation, to do computation we need arithmetic and combinatorics, to these we in turn we need computation and logic. The germ of this in the reciprocal causality at the end of Essence (which we have also associated with higher groupoids). And also, as we have discussed in more detail everywhere, the partial (meta)reflection of one formal system by another (or the formal system by itself) is clearly relevant here: it is the idea of the whole being implicit to some extent in the part. For instance in ZF set theory itself we can prove that there are non-equivalent models of ZF.

Logic, Computability and Grammar

Given a natural language, we can ask what the logical and computational prerequisites for generating and checking and analyzing valid expressions of that language ? And can we define the minimal, optimal way in which expressions are generated and checked and analyzed ? We are looking here at meta-grammar,  the formal language of grammar itself.  How rich are formal languages in themselves compared to standard algebraic systems ! How much remains to be said about the relationship between (specially imperative) programing languages and logic, between logic and recursion theory, between recursion theory and combinatorics and formal languages. These are all topics of our Analyticity and the A Priori.

We view a descriptive grammar as a function which takes a series of choices, semantic categories, and generates the corresponding expression. We should formalize grammar using functional programming. 

Take the rules of Sandhi in Sanskrit. To express them we need the same logical framework as in Aristotle's Topics, in particular as Galen's Relational Syllogism. 

But we can also think of an abstract mathematical framework for Sandhi.  Suppose we have for instance finite sets $A,B,...,G$ and let $P \subset A \times B \times...\times G$.  Then we can define an operation $\otimes: P \times P \rightarrow P \times P$ such that for instance a certain component is right-dominant:

if for $p \in P$ we have $p_C = c_1$ then $q\otimes p = (q', p)$ with $q'_C = c_1$ and $q'_X = q_X$ for $X \neq C$.

For binary sets $A = \{a,b\}$ we can define transposition $p^A$ as $(p^A)_A = a$ if $p_A = b$ and $(p^A)_A = b$ if $p_A = a$.  Then the above example becomes $q \otimes p = (q,p)$ if $q_C = p_C$ and $q \otimes p = (q^C, p)$ otherwise.

In order to represent Sanskrit sounds more precisely we can include in each finite set a 'null' element $0$ expressing that the classification of this set does not apply. For instance $p_N \neq 0$ means $p$ is a nasal and then $p_A = 0$ means for instance that the aspiration  Boolean does not apply.

The algebraic aspect of Sandhi can be illustrated by the existence of left and right (quasi-) absorbing elements or identity elements. $\dot{n}$ is a left identity element. 

While external Sandhi can be given relatively simple well-defined rules, even Max Müller shies away from giving a complete list of rules and exceptions for internal Sandhi.

We will also give  representation of external Sandhi using permutation groups (inspired by the Rubik cube).