Is a mathematical model of consciousness possible?

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

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

 

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

 

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

 

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

The path to peace

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

Philosophical updates

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

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

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

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

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

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

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

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

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

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

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

When was modern computer architecture invented and by whom?

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

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

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

Gödel showed that proofs are strategies in a game

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

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

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

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

Anatta

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

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

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

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

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

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

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

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

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

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

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

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

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

Kantian notes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Pierre Cartier: A Visionary Mathematician

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

Genius, Intelligence and Sainthood

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

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

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

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

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

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

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

Intelligence and morality are one. 

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

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

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

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

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

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

Transcendental Syntax I : deterministic case - Jean-Yves Girard

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

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

Proof Nets

Elementary Topos Theory

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

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

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

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

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

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

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

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

Claire Ortiz Hill on Hilbert's philosophy (excerpts)

Upon several occasions, using almost exactly the same words, Hilbert described what he called the basic philosophical position that he considered necessary for all scientific thinking, understanding, and communicating, and without which no intellectual activity is possible. He stressed that this basic philosophical position was the very least thing that had to be assumed, that it was something that no scientific thinker could dispense with, and that everyone had to adopt, whether consciously or not. According to this approach, as a precondition for the use of logical inferences and the performance of logical operations, certain extra-logical, concrete objects had to be already given in our faculty of presentation and be intuitively present as immediate experience prior to all thought. For logical inferences to be reliable, those objects had to be completely surveyable in all their parts, and the fact that they occurred, differed from one another, followed one another, or were concatenated also had to be immediately given intuitively with them as something neither reducible to anything else, nor requiring reduction. Hilbert labelled this concern for concrete content the finite approach.

 In recognizing that such conditions necessary for the use of “contentual” (inhaltlich) logical inference existed and had to be respected, Hilbert saw himself as being in agreement with philosophers. Specifically, he considered this basic philosophical position to be part and parcel of the teachings of Immanuel Kant who had maintained that extra-logical concrete objects intuitively present as immediate experience prior to all thought had to be given, that, in particular, mathematics could never be provided with a foundation by means of logic alone, that it has at its disposal a content secured independently of all logic. In contrast, he repeatedly stressed that Frege’s and Dedekind’s attempts to provide arithmetic with foundations independent of all intuition and experience and to derive arithmetic by means of logic alone were bound to fail, because logic alone could not suffice, and certain intuitive conceptions and insights were indispensable for scientific knowledge to be possible.

 In the case of mathematics, Hilbert explained, the extra-logical, concrete objects intuitively present prior to all thought were the concrete signs themselves, whose shape was immediately clear and recognizable. As perceptually recognizable, objective and displayable numerals, the numbers of concrete-intuitive number theory met Hilbert’s requirements. They and the proofs of theorems about numbers fell into the domain of the thinkable. Formalized proofs were concrete, surveyable objects communicable from beginning to end. He defined proofs as arrays, that is, objects composed of primitive signs given as such to perceptual intuition and consisting of inferences where each premise is an axiom, directly results from an axiom by substitution, or coincides with the end formula of an inference occurring earlier in the proof or results from it by substitution. The axioms and provable propositions resulting from this procedure were, Hilbert contended, “copies of the thoughts constituting customary mathematics as it has developed till now”.

In his proof theory, only real propositions can be directly verified. “The formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that at the same time the interconnections between the individual propositions and facts become clear. To make it a universal requirement that each individual formula be interpretable by itself is by no means reasonable; on the contrary, a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument”.

Some points about TPC and TPP

Ordinary language discourse about TPC and TPP is always a very a subtle and illusive affair. It does not seem clear that ordinary language can capture with the right degree of precision what should be said and understood. There is always a margin for ambiguity and vagueness and hence an element of error which can be exploited dialectically to lead to apparent contradictions and amphibolies. This is the case for terms and propositions relating to spiritual development and realization. If propositions can appear contradictory (cf. the tension between detachment and integration) on the surface level this is in reality a manifestation of ordinary language attempting to capture something very precise.
An important point is that if there are a plurality of ordinary consciousnesses these manifestly exhibit their own particular characters and qualities and their own particular socio-historical relational network. This specificity and difference can be considered a major guiding principle in the development of TPC and TPP - and specially in the light of considering TPC and TPP as involving a kind of essential integration in communion with insight, calm and detachment.
Insightful detachment is very different from naive detachment. Insightful detachment performs a kind of regression and analysis to find the positive spiritual powers at work in the attachment which have become erroneously monolithically fixated with certain posterior determinations.

Another aspect of transcendental amnesia is forgetfulness that half of our total consciousness experience is spent in a state of dreaming during sleep. Why should one state be downgraded relative to another ? What does wakefulness mean from the point of view of dreaming ? Can lucid dreaming be considered a substantial step in the direction of peace, liberation and freedom ?

In TPC we have a situation in which reasoning, deduction, proof can be likened to how we teach or show somebody a dance or draw a figure in the air with our finger. The proof (or dialectic) is a self-revealing self-unfolding process which dissolves and is lost once any fixed proposition is settled upon independently, in itself. Cf. Sextus' statement "the skeptic is still inquiring" or the famous simile in a letter of Plato. Thus in treating of logic and computability and seeking for their ultimate source we exhibit a cyclic process.

Often what is wrong and harmful with many western spiritual traditions is not certain spiritual states or exercises involved, considered in their purity, but the harmful and erroneous conceptualizations and beliefs arbitrarily associated to them or establishing a feedback loop with them.  As for western spiritual traditions, besides those that stand out such as the pythagorean, neo-pythagorean, middle platonic and neoplatonic schools and other schools of ancient Hellenic and Hellenistic philosophy (such as the Stoics and in particular Epictetus), and the Hellenic adaptation of Egyptian traditions in the Corpus Hermeticum, it a great tragedy that we no longer have access to the first-hand, authentic, original Western sources for much of the better elements of historical Christianity (for instance the Druidical, Chaldean, Syrian, Phoenician traditions or the secret teachings of the Eleusianian and Orphic mysteries). However there is still a wealth of information that can be gathered from Homer, the Homeric Hymns, Greek Theater and references found in Plato. A promising avenue of investigation (that we have already written about) seems to be to focus on ancient Iranian traditions as well a plausible widespread influence of (proto-)Mahayana Buddhism in the ancient world.

But let us recapitulate some principles of TPC and TPP.  

Transcendental awareness of the present moment qua truly present. This requires "detachment" and "letting go of past and future " but such a  present moment is a transcendental and revealed to be the "inner".

Transcendental awareness that this present moment being forgotten and unknown is a condition for ordinary consciousness and experience. 

Transcendental awareness of temporality as a universal conditioner of experience and also transcendental awareness that  forgetfulness of this aspect of temporarlity is itself a condition for ordinary consciousness and experience.

Transcendental awareness  that thought is a distorting illusory screen which hides and distorts transcendental awareness of the pure present moment and awareness that forgetfulness of thought as it is in itself is a condition of ordinary experience.

Transcendental awareness of nature of the self, of the illusion of the self, perceived is such and that forgetfulness of this awareness of the self and its nature is a condition for ordinary experience.

Transcendental philosophical praxis involves overcoming, by skillful means, the above transcendental forgetfulness and the allaying of all illusions and distortions.

Consider the sobriety and circumspection and acumen of Hume and Kant as contrasted with subsequent German idealism (Fichte, Krause, Schelling).  One error of such German idealism is taking the illusory individual self as a philosophical starting point rather than the super-individual transcendent self (which might be called a spiritual space - the 'we ourselves' of Plotinus - alhough there seems to have been some confused awareness of this state on the part of the said philosophers). In a sense the jhânas (which involve illumination) are a foundation of philosophy, or rather, they constitute a higher stage of TPC and TPP. 

Thus TPC and TPP involve a preliminary ascent through transcendental awareness to the jhanas. The jhanas then are a foundation for further transcendental awareness and philosophical development as well as a foundation for even higher forms of consciousness. However there is a great danger of partial or imperfect realizations of the jhanas leading to faulty, incomplete or distorted philosophical views and systems. But it can be said safely that the jhanas allow great insight into the nature of reality and the correct view and assessment of philosophical concepts and systems. 

Thus TPC and TPP can be seen as involving a cyclic process of ascent (through transcendent logical and phenomenological analysis) and then positing and deduction (conceptual unfolding and expression, even axiomatic-deductive expression, flowing from the illumination of achieved higher states of consciousness) which may be repeated many times (perhaps likened to a spiral).  This can be compared to Plotinus' account of Platonic dialectic. We can say, broadly speaking, that the anagogic aspect o TPC is negative, analytic,  "dissolving" in the style of Pyrrho or Nagarjuna or even Hegelian logic - ultimately a turning inward and detachment - while the second component is positive, constructive, deductive, outflowing. 

It is curious and rather uncanny that I just found this passage from Krause:

The analytic-ascending part of science starts with those immediately certain recognitions, which we find in every consciousness, and through self-observation it proceeds steadily to higher recognitions until intellectual intuition is obtained, which has to happen according to this procedure if genuine insight into ultimate reality is possible for the human mind.

 All that we have written about TPC and TPP is incomplete and highly distorted without understanding the fundamental truths about transcendence and selfhood. 

 

A selection of interesting references on Darwinism and evolutionary psychology

References marked with a star (*) are of particular interest and merit. We also included some additional references for anthropology. From the list below we can see clearly that most of these anti-Darwinist authors were i) top experts and deeply knowledgeable in their fields ii) not connected in any way to religion or the occult iii) of manifest intellectual integrity iv) held humanist values.

*Agnes Arber, The Natural Philosophy of Plant Form, Cambridge, 1950.
*David Stove, Darwinian Fairytales - Selfish Genes, Errors of Heredity, and Other Fables of Evolution (1995)
Evolutionary psychology as maladapted psychology, by Robert C. Richardson. Cambridge, MA: MIT Press, 2007
Is Evolutionary Psychology Possible?
J. Bergman, The Darwin Effect (2014)
*Fodor and Piattelli-Palmerini, What Darwin Got Wrong, 2010.
Giving Up Darwin
*D'Arcy Thompson, On Growth and Form.
*Jacques Costagliola, Faut-il Brûler Darwin ? ou l'imposture darwinienne, L'Harmattan, 1995.
*Pierre-Paul Grassé, The Evolution of Living Organisms
Michael Denton, Evolution, Still a Theory in Crisis, 2016.
Eugene K. Balon, Evolution by Epigenesis: Farewell to Darwinism, Neo- and Otherwise
*Søren Løvtrop, Darwinism : The Refutation of a Myth
*Rosine Chandebois, Pour en finir avec le darwinisme
Eugene V. Koonin, The Logic of Chance: The Nature and Origin of Biological Evolution
James A. Shapiro, Evolution, a view from the 21st century
Rémy Chauvin, Le darwinisme ou La fin d'un mythe
Jonathan Wells, Icons of Evolution
Dissecting Darwin
*René Thom, Darwin cent ans après
Bertrand Louart, Aux origines idéologiques du darwinisme, 2010.

https://sniadecki.wordpress.com/2012/05/23/origines-ideologiques-darwinisme/

The homunculus is wrong
Louis Vialleton, L'Origine des Êtres Vivantes, L' Illusion Transformiste
*J. von Uexküll, Theoretical Biology
Michael A. Cremo and Richard L. Thompson, Forbidden Archeology: The Hidden History of the Human Race
https://www.pnas.org/doi/pdf/10.1073/pnas.1200567109 (no human-Neanderthal interbreeding)
Debunking OOA
Finn and Dewar, The making of species (1909)
*P. Wintrebert, Le Vivant créateur de son évolution (1962)
Initial Bipedalism as a realistic model for verterbrate evolution (François de Sarre).
Glazko, G., Veeramachaneni, V., Nei, M., & Makałowski, W. (2005). Eighty percent of proteins are different between humans and chimpanzees. Gene, 346, 215–219.doi:10.1016/j.gene.2004.11.003Bertrand Louart, Aux origines idéologiques du darwinisme (2010):
https://sniadecki.wordpress.com/2012/05/23/origines-ideologiques-darwinisme/

In Ethics, Ridiculed Spirit Theorists Often Outshone the Stars of Darwinism
https://futureandcosmos.blogspot.com/2024/04/in-ethics-ridiculed-spirit-theorists.html

Here are a some key points which we plan to discuss regarding anthropology:

1. Modern (contemporary) humans represent the primitive type of human and have existed at least for a million years. Their primitive homeland was probably in some continent in the Artic regions as already defended by Olaus Rudbeck, Jean Sylvain Bailly, enlightenment anthropology and William Fairfield Warren. Wsiewołod Iwanow, Borys Olszańskij, Konstantin Vasiliev, Andriej Gusielnikow, Alexander Uglanov, Natalya Romanovna Guseva, S.V. Zharnikova, Gimbutas, Joan Marler, B.G. Tilak, R. Sankritiyayana, A.V. Bykov, A.G. Vinogradov, O.N. Trubachev.

2. There is no 'Neanderthal DNA' in modern humans. Cambridge geneticist William Amos debunks Neanderthal interbreeding myth https://www.zoo.cam.ac.uk/directory/william-amos

(...) no one has yet found either a Neanderthal Y chromosome or a Neanderthal mitochondrial sequence in modern humans, despite the huge numbers from across the world who have had their DNA tested. By implication, the Neanderthal versions of both the Y chromosome and mitochondrial DNA must, to some extent, be toxic in humans.

Affecting Up to 216,000 Studies – Popular Genetic Method Found To Be Deeply Flawed

https://scitechdaily.com/affecting-up-to-216000-studies-popular-genetic-method-found-to-be-deeply-flawed/

How Sir William Rowan Hamilton derived the Hamilton-Jacobi equation in 1835

 https://www.diva-portal.org/smash/get/diva2:532943/FULLTEXT01.pdf

The problems of quantum theory

Update: should discuss the original work of de Broglie and the interesting work of Grete Hermann. Also the hydrodynamic analogues involving bouncing droplets on vibrating fluid surfaces.

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.

In another place we have proposed that the fundamental issue at stake regarding quantum theory is that it is necessary to abandon the postulate of differentiability and continuity in our mathematical models of nature - but not necessary computability and determinism. This entails immediately that we have no longer in general any criterion or concept of 'identity' or 'individuality' which in classical (and relativistic) physics is entailed by the temporal continuity of (particle, field) solutions of differential equations. It is has not been proven that $\pi$ is a normal number. Its expansion is computable and deterministic yet it is conjectured that this sequence  is probabilistically indistinguishable from a random sequence (i.e. any finite subsequence occurs with equal probability). We also raised questions about the ordinal (order type) of time. The fundamental problem is studying how this measurable framework can approximate the continuous and differential framework (and this was Boltzmann's problem) - were we need to go beyond the clumsy chimera of the continuous wave-function codifying discrete random information.

It is not because nature is random or non-deterministic that we are driven to use smooth probability distribution equations but rather because nature is not-continuous or smooth and this is the most convenient and logical way of doing modelling bearing in mind the history of physics. However discrete dynamical systems and computability theory may open up new possibilities. And it may be that the non-smooth dynamics are generated by an underlying smooth structure and we return to the beginning.

Our approach will involve considering a measurable function or field (an atomized field) over space-time (or a generalized space-time).  For instance having nowhere dense support or a condition of being discrete up to the Planck scale. A postulate: for every line we have one point (or Planck region) of non-empty intersection. Think of sand on the surface of a drum.

We postulated that a truly free particle (atomized field or light-dust) can be with equal probability anywhere in space at a give moment. This is the analogue of Boltzmann's postulate that for a molecule of a perfect gas each velocity vector has equal probability.

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. The analytic-scientific spirit awakened by correct study of physics, chemistry and biology. This may seem paradoxical but the philosophical and scientific attitude of the Nyâya and Vaisheshika are actually the secret core of correct and successful Buddhist meditation. Investigate the work of Ludwig Bolzmann.

What is true genius in mathematics ?  It it is faithfulness at once to 

i) logical-axiomatic-conceptual  clarity and rigor

ii) algorithmic-combinatorial-geometric-dynamic intuitive clarity 

iii) its simplifying, unifying and clarifying role in the whole of mathematics

iv) its substantial philosophical relevance and significance 

v) its essential connection to science

vi) its artistic  connections and human value

 vii) the clarity and adequacy of its teaching and presentation

This is what defines a true genius in mathematics such as Abel, Gauss, Grassmann, Riemann, Poincáre or Hilbert - or in logic such as Frege, Gentzen, Gödel, Brouwer, Russell, Skolem, Turing, Lawvere, Girard and Martin-Löf 

What a confusion and tragedy in the history of the foundations of mathematics.  With the theory of types of Russell and Church, Gödel's system T and its extensions, Maclane and Lawvere's Categorical Foundations  (Elementary Topoi) and above all the Constructive Type Theory of Martin-Löf.   how can the miserable ZFC foundations (or Cantorian set theory) still be considered more than an embarrassing historical oddity ? Consider how ordered pairs are defined $(a,b) = \{\{a\}, \{a,b\}\}$. What sane mathematician would accept that $\{a\} \in (a,b)$ ?

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 and the analytic a priori

Computability 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 a royal path to spiritual realization can consist in a pure love for a real person). 

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.