Pâninian Linguistics

https://web.stanford.edu/class/linguist289/encyclopaedia001.pdf

Is Pânini's formal system (which appears to be in the form of a term-rewriting system and is the true source of generative grammar) Turing complete ?  That is to say, could the same formal mechanism Pânini used to specify the structure of Sanskrit be used to generated the expressions of any recursive set of strings ? This would make Pânini of supreme logical and philosophical interest. There might be a connection to Vedic rituals: the development of a formal language (a programing language) to codify rituals.

https://ijirt.org/publishedpaper/IJIRT187010_PAPER.pdf

We note that the syntactic transformation of our natural term logic can be expressed more easily as a term-rewriting system than standard quantifier logics.

In fact Pânini is of immense interest to our positions expounded in our paper on Analyticity and the A Priori (and also our paper on Ancient Quantifier Logic).   One view is that true logic (let us say in the form of second-order logic) comprises only universal quantification and implication and their natural deduction rules (cf. Prawitz, Natural Deduction, p.67, Dover edition). This is the logic which can be transcendentally examined to be the pre-condition to understand, check and apply rules, formal term-rewriting systems (or similar systems), such as that of Pânini. We must also examine Pânini as a logic of relations.

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)  points to a survival of an ancient musical tradition.

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. 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.

Claire Ortiz Hill (1951-2025)

https://www.riverafamilyfuneralhome.com/obituaries/claire-ortizhill

Claire Ortiz Hill was an American philosopher (who spent the later part of her life in France) who in the last quarter of the 20th-century authored some lucid and meticulously written books on Frege, Husserl and intensionality as well as offering a cogent critique of some central trends of 20th-century analytic philosophy. In recent years she dedicated herself to Cantor and little known aspects of Hilbert's role in philosophy. Her magnificent article Hilbert's Fight for Philosophy (2021) has many important connections to our paper on Analyticity and the A Priori.

The unknown Dharma of the West

Can we trace a continuous tradition from original Buddhism to Socrates to Plato and the Megarian school to Pyrrho to Sextus to Plotinus to Damascius to Pseudo-Dionysius ? As strange as this may seem to some, there are some striking arguments that can be made to support many links in this chain. In fact we are pleased to note that Sara Ahbel-Rappe in her edition of Damascius' Problems and Solutions concerning First Principles, makes a direct reference to Sextus. The Buddhism-Pyrrho link has been studied for instance by C.I. Beckwith's Greek Buddha (2015). It has also been argued (by C.M. Mazzucchi)  that the author of Pseudo-Dionysius was in fact Damascius who managed to inject core doctrines into the heart of the 'mystical'  tradition of Christianity (which has nothing to do with the literal sense of dogmas): and here we mean its better (predominantly medieval) manifestations such as Hadewijch, Porete, Eckhart, the author of the Cloud of Unknowing, the Viae Sion of Hugh of Balma (which we dedicated a small note to).  It is in this light that we would reinterpret Plato and the close connection to Socrates, Pyrrho and the Megarians.

'Spiritual realization' (we use this term temporarily) is not the same thing as psychotherapy (etymology aside). Psychotherapy aims at helping people adjust and balance personality traits so as to be at peace with themselves and the world around them.  It is about the constitution of an individual 'personality' and 'ego', often seen as the result of an integration of different tendencies or the product of past development and adaptation strategies.  Spiritual realization such as we outlined in our previous account of TPC and TPP more or less presupposes physical and mental health as a condition and its aim involves shining a light through self-hood and overcoming its underlying impulse, motives, determinations and limitations. But this does not rule out that it can also be highly beneficial in a psychotherapeutic context.

We are conscious of the present and the past in the form of recollected or anticipated mental content in the present. Past and future are mediated by the present and are indeed in the present. TPC involves awareness of the presentness of past and future directed or colored mental content. The theory of the 'present moment' and 'letting go of the past and future' and finding 'the timeless present moment' taught in many approaches to Buddhist meditation conceals something of huge psychological and philosophical depth and importance. Temporality is linked to the conditioning of the mind by thought, the constitution of the world by thought, and the falling away from TPC. Time is the magical stage of thought, the impermanence which is made permanent mixing perception and forgetfulness. We might say that this proper philosophical consciousness of psychological (inner) temporality is at the heart of TPC and TPP.  The same goes for self-identification, self-projection,  craving, repulsion.  But the aforementioned practices require the good fortune of being able to retreat to a secluded, quiet, safe and solitary place, the best option  being the countryside.

If we leave this world to attain another state of being we can at least control what we bring along to the next plane.  Let us not bring with us the least trace of anger, hatred, resentment, desire, attachment, clinging - or ignorance and delusion, specially delusion regarding identity and identification or the magic show of our own consciousness.

It is important to note that by 'buddhism' we mean the philosophy, psychology, ethics and spiritual practices that we interpret to be arguably and quite patently present in old Pali and Sanskrit texts (independently of the presence of elements of a different nature, textual history and corruption, etc.). Our meaning of this term does not depend in any way on any particular historical or cultural embodiment of buddhism and as such is not open to the common criticism that we are ignoring its cultural roots or merely picking certain elements which fit a pre-conceived agenda (we are not claiming to represent or interpret any concrete cultural buddhist tradition).  Also we are well aware that modern distorted abstractions and (mis)applications of buddhism can from a psychiatric point of view have harmful effects (cf. negative experiences with so-called 'mindfulness', vipassana and Goenka retreats, experiences which according to our interpretation have nothing to do with the buddhist spiritual path).

We would be quite happy to debate anyone on this topic. Also the focus on buddhism does not imply that we attach less value to the parallel traditions of the Upanishads, Yoga, Samkhya, Vedanta and so forth. 

It is quite possible that the philosophy of Schopenhauer (the philosophy of the principium rationis, the principium individuationis, the central role of voluntas, the theory of representatio, the theory of  supra-individual timeless, ubiquitous cognition,  etc.) offers some of the deepest and most important components of TPC and TPP. Schopenhauer claimed as ultimate source the Upanishads. In Schopenhauer all the sphere of manifest consciousness, individuality, identity, spatial and temporal limitation, conditioned by the Kantian intuitive and conceptual categories, is the product of a primordial impulse and energy.  The only way out of this prison is through a total reversal and repolarization of this fundamental impulse and energy which results in the 'involution' of all the modes, conditions and determinations of individualized consciousness and the consciousness-immanent world.

One might entertain the idea that there is a far greater unity and affinity between Kant, Hegel and Schopenhauer than is generally believed.  Shapshay has argued this with regards to the ethics of Kant and Schopenhauer. And beyond the personal animosity between Hegel and Schopenhauer it is quite possible that Hegel might furnish some valuable structural and interpretative enrichment of Schopenhauer and that Schopenhauer could provide the essential clarification, correction, deeper meaning and completion so lacking in Hegel.

Philosophy of Awakening

 We mentioned before that the is a difference between a true spiritual path (leading beyond anguish and suffering) and a false spiritual path. A true spiritual path involves essentially a habit and state of mind which  we call philosophical awareness or consciousness. The false spiritual path completely lacks it. This state of mind we could attempt to describe as transcendental philosophical consciousness (i.e. awareness, analysis, inquiry)  of the totality of consciousness itself (the immanent world). We use the abbreviation TPC.  One of its oldest and most magnificent attestations is found in the Pali nikayas (with the Socratic dialogues and Pyrrho) and more recently to a certain degree in Hume and Kant (and there was an influence of the late Academy and Pyrrhonism via Augustine and Descartes). This is what Schopenhauer said concerning Kant's Critique of Pure Reason (the analogies of experience): never has the world been looked at from a greater distance. At the present we have a scholar, by name of Dennis Schulting, who has written several books dedicated to defending and expounding the true meaning and spirit of first edition of Kant's Critique, that is, to bringing out the centrality of a form of TPC in the entire deductive and argumentative scheme of the aforesaid work.   Our project is not only to say something about what TPC involves but also to discuss its epistemic and logical aspects, a subject of uttermost difficulty. And of central concern is of course its close relationship to introspective psychology and so-called psychologism (as well as the ambiguous hydra of the term 'idealism'). And discussing the often subtle yet ultimately radical difference with Hegel and Husserl.  There is always a danger that TPC may lapse back into the modes of non-TPC consciousness. That is, a false objectification of TPC insight and knowledge. Such an objectification involves forgetfulness (of constitution, condition and presupposition) and loss of insight and transcendental progress.   TPC crucially involves a way of seeing and questioning (bracketing, suspending) the totality of (conceptual, intuitive, affective, volitional, egological, spatial-temporal, somatic, etc) experience. The total awareness of the total sphere and process of thought and what is beyond all thought and what thought is actually behind and the fact that it is behind it. TPC is a gradual affair,  an ideal which is worked towards by practice and training. TPC traces back and unveils the unified common source of all branches of philosophy. Poetical language achieves its highest function in the expression of TPC. All this we have written about in our compiled Philosophical Monologues collection.

There are no mathematical models of consciousness, and a fortiori for the boundless sphere of TPC,  and no formal axiomatic-deductive systems either. Rather these all proceed from the knowledge and insight offered by TPC. But there could by mini-models, projections, symbols and for every false philosophy or false scientific theory  a certain and valid formal refutation. TPC itself cannot be part of academia (but more concrete, specialized subjects, guided by TPC, can) and it is something that one must find for oneself. TPC can be compared to the state in which when dreaming one knows one is dreaming, or at least, seriously questioning if the totality of one's experience be not in this situation a mere dream. And also importantly, asking: what is a dream?

It is TPC that is about 'saving the phenomena' and involves noticing how there is a vast domain of phenomena, of 'realities' as real as anything else, which includes all forms of sensation, perception, feeling, volition, thought, cognition, meta-cognition, memory, imagination, learning, habit, attachment, sense of self, sense of identity  and so forth which is a central and important part of  the 'world', as important if not much more than anything other alleged aspects (specially those pertaining to so-called 'natural' science). This truth is powerfully present in the Pali suttas which predate Socrates and, as argued by Mikel Burley (2007), in the ancient system of Samkhya.  The terms 'consciousness' and 'the psychological sphere'  are often used in a deflated, trivialized, distorted and illegitimate way. Understanding, such as it is and presents itself, the vast and central role of consciousness and meta-consciousness in all that is, in all the 'world',  is a first step from waking up from  the dream, and it follows the shining wake of Socrates and Pyrrho and there is much to be said about certain aspects of Democritus (atomism is in itself a formal template of logical analysis, as in the abhidharma, and not an hypothesis in natural science or positing of a reductionism between different ontological domains), Carneades, Sextus and Augustine's Contra Academicos: 'I call whatever appears to me the world'. Another powerful force leading to this direction is the theory of knowledge (the earliest known text being the Theaetetus), the questioning of how we come to know things, of how we know that we know (the criterion of truth),  of what knowledge is. The theory of knowledge draws us inevitably back towards the great forgotten pristinely primordial sphere of consciousness. Becoming aware of consciousness as consciousness (and experience as essentially consciousness) and its true position in the scheme of things is truly waking up from a dream and drawing close to TPC.

How does TPC relate to our other distinct considerations on the computable axiomatic-deductive condition and ideal of knowledge (which evidently does not involve any claim to exclusivity or to exhaust valid cognition) which we expounded in our paper on analyticity and the a priori ? The conclusions of that paper can be seen as flowing directly from TPC. Our formal verification principle is epistemically modest. To come to this conclusion relative to formal axiomatic-deductive systems,  TPC must be presupposed. Hence formal axiomatic-deductive systems cannot in themselves be sufficient to unveil their own transcendental determinations and conditions. And formalized transcendental deduction itself cannot subsume TPC but again presupposes it. TPC must be self-deducing, self-illuminating, self-sufficient, self-founding, self-justifying (somewhat like for Kant the 'I think' is the condition and source of the categories and hence of logic itself). And yet does it make sense to speak of a 'deduction' here without circularity ? TPC cannot depend on logic anymore than anything in a dream can in itself awaken one from the dream. And yet this does not decrease the value and importance of the formal verification principle. If a philosopher or anyone else engages in linguistic activity, linguistic expression, then it is legitimate to subject the output to the formal verification principle, to the ideal of a formal game. All logos is bound to logic, including philosophy and any discourse about logic and reason. Philosophical logic is like two mirrors reflecting each other. Yet via TPC one plays the game knowing it is a game (dreams the dream knowing it is a dream) but without trivializing it for being a game. The game is much like a children's game with the purpose of teaching.  Logic is a crutch to help us learn to see (i.e. to develop TPC). But TPC is not deduced or justified from logic anymore than swimming is justified or deduced from water wings. But there is another aspect beyond the simile of a helping crutch. Logic is the laws of thought but not the justification of thought itself. We think, or rather, we are thought, but we seldom see thought itself as it is, from a distance. The ability to see through the game, not only to see the game as a game, but to see immediately non-discursively the game, the playing of the game, the player of the game,  as something that is not necessary and yet at the same time the direct cause of many unpleasant things. Exactly like Alice's final attitude to a 'pack of cards' or Schopenhauer's metaphor about the chessboard after the game is over.  This is related to transcendental philosophical praxis (TPP). TPP involves the coming to awareness that the things that bother us are not really things in themselves and but only images in our own minds. And that we have actually the capacity to exert an enormous power over the entire content of our own mind, conditioned by habit and practice. And the most importance practice and habit is that of viraga, nirodha and patinissagga. Thus TPP (the sister of TPC) gives the true philosophical meaning to the often abused or trivialized assertion: we create our own reality. 

Notes

The embedding matrices used for instance in ChatGPT-3 are a vector space representation of co-occurrence frequency matrices for a given context-window size. This matrix can also be seen as a complete graph with edges labelled by probability values (we can assign values also to polyhedra).  It is important to study the properties of these matrices.

Locally integrable functions (fundamental to distributions and weak solutions to PDEs) are a kind of sheaf-theoretic completion of the $L^p$ spaces. On the other hand from a finitist point of view simple functions are the basic kind of function and these are 'dense' in many fundamental spaces in an appropriate sense. What is the meaning of a PDE in a scientific context ? A positing of certain recursive numerical algorithms and approximation ideals.  How curious that numerical conditions for stability involve the pairing of space and time.  And fascinating is that smooth initial conditions for PDEs as simple as $u_t - g(u)u_x = 0$ may generate shock waves. And what is, from a computational finitist point of view, a weak solution ? A good philosophical goal: to gain a deeper understanding of distributions (cf. Sato's hyperfunctions). Distributions arise from the practical situation of measurement. We do not measure a field at a point but only an integral average weighted with the peculiarity of the instrument employed.

Some of the most interesting concepts in mathematics and applied mathematics: absolutely continuous functions and functions of bounded variation. What does a finitist and computationalist perspective say ? Both the Weierstrass and Cantor step functions are constructed step-by-step. There must be a property expressable in terms of re-scaling.

How could we prove that a physical system is computing beyond the Turing limit ? We could of course produce experimental evidence to the contrary, producing a certain algorithm that agrees with know observations. There are irrational numbers whose sequence of digits are not computable. How are we to view scientific theories about such numbers (which could represent measurements of some fundamental physical constant), calculations and approximations of such numbers, and confrontation with experimental evidence ? Obviously such a question is only interesting from a non-finitist perspective.

The interest in solving the P=NP problem hinges on the complexity of the algorithm for transforming a NP-machine into a P-machine. 

Uncountable infinities (must check Bolzano regarding this subject) are highly questionable (they express a false objectivism). They imply indiscernibles which should be rejected in the light of subjectivism (in terms of computationalism or more generally definability). There is also the problem of the identity and determination of mathematical objects (Benacerraf, etc.).  A foundation for calculus like Lawvere is called for (in which intervals are primitive). The standard definition is that a set A has cardinality greater than a set B if there is no injective function from A to B. But once we restrict the possible functions on philosophical grounds to being recursive or some more general type, then this situation can happen without increase in 'Cantorian' cardinality. Indeed 'infinity' is always present in the concept of recursively enumerable but not recursive and the same goes for the rest of the arithmetical hierarchy (the true hierarchy of 'infinities') ? This makes perfect sense also for ordering. What is the point of ordering something if it is not in a computable way ? In sheaf models (a partial improvement and clarification of the forcing techniques) there is an acknowledgment of the essential relativity of the whole Cantorian framework of monos and cardinalities (already patent in the Löwenheim-Skolem theorems).

A problem, given an inconsistent set of sentence in some language L  is there a canonical disambiguifying language L' (which associates to certain symbols s of L a set of possible clarifications s1,s2,...) such that with a choice of assignment for occurrences of symbols of L to symbols of L' the set becomes consistent ?

Non-classical logic vs. classical logic. What is important is the question: what is the logic required to be able to understand, carry out and check any system of rules ? Kant's rule-based philosophy is a precursor of our computational a priorism. 

Completed papers

Quantifier Reasoning and Multiple Generality in Aristotle and Ancient Logic

Aristotle's Topics and Extended Second-Order Logic

On Analyticity and the A Priori 

Natural Term Logic (in preparation)