Some notes on Augustine's Contra Academicos

This is considered the earliest of Augustine's writings (composed circa 386 CE in Cassiciacum). It is of great historical interest in relationship to the study of the Platonic Academy, Stoicism, Cicero and Ancient Skepticism in general as well as offering some hints regarding Pythagoreanism and Augustine's relationship to Plato and Plotinus (mentioned explicitly as being almost an reincarnation of Plato), not to mention the strange oblique references to Christianity (described in the language of the mysteries - Augustine was supposedly undergoing the preparations for the "initiation" of baptism which he was to receive a year later).  The Christian references may be later interpolations just as the historical-biographical setting of the dialogue could well be a purely rhetorical and literary device (serving a political goal).

In Book III Augustine articulates what can be described as a theory of analyticity.  Concrete instantiations of logical propositions are held up as certain and evident truths.  And these "logical" propositions are equated to truths of "dialectics" and they appear to be Stoic in form (conditionals and disjunctives). In the same book he likewise articulates the theory of self-certifying evidences in the context of  phenomenological epokhê. The perceptive experiences of consciousness - provided they are considered merely as such by suspension or exclusion of further judgment (assent) - are indubitable certain. Augustine thus genuinely effected the transcendental reduction whereby the primordial certainty and truth of the world is that of a system of phenomena (for a given subject) indistinguishable in this respect from dreams. In III.11.25 we read quidquid tale mihi videtur, mundum appello. We could also compare Clitomachus' distinction between assent and approval (cf. Ac.2.104) with key aspects of later phenomenology. The dialogue has a close connection to Cicero's (fragmentary) work Academica and the sections on the ultimate aims and nature of  the later Academy's skeptical stance are of supreme interest and importance. Note that Augustine had access to the full versions of this work.  It suggests that the function of the dialectics of Arcesilaus and Carneades was as a "skillful means" directed at a materialist versions of Stoicism having as ultimate aim the preparation for receiving the Platonic philosophy -  and this is in  perfect agreement with the function of Nagârjuna's Madhyamaka or its later Vedantic version. There is however not a hint of this interpretation in Charles Brittain's introduction to his translation of the Academica (2006). Brittain expounds a humdrum interpretation of Philo and Antiochus in terms of fallibilism and defeasibility. However it seems plausible that it is in the so-called Megarian school (alongside Sextus) that the purest form of Western Madhyamaka (i.e. the highest articulation of dialectics) is to be sought (cf. the description of Arcesilaus in Diogenes Laertius: Plato in front, Pyrrho behind and Diodorus in the middle). We can find much phenomenological psychology in the later Academy and Stoics. 

The major questions which we should answer are the following. What are the historical sources for Augustine's anti-skeptical argument involving the appeal to the certainty of validities of Stoic propositional logic? While such an argument permeates the Platonic dialogues it is to us unthinkable that the great logician Chrysippus would not have used such arguments in his innumerable debates with the later Academy.  The "phenomenological" argument can be traced back to the Pyrrhonist epokhê,  Epicurus' pasa aisthesis estin alethes, and to a certain extent to Protagoras -  but differs significantly from the Stoic theory of kataleptike phantasia. Note that we have argued that this kind of experiential epokhê is found clearly attested in the Pali Nikayas (and the sophisticated development in the Yogãcãra school). A curious fact about the Contra Academicos is that Augustine's famous argument si fallor, sum (Civ. Dei XI. 26, De Trin. X.10.14 ) is entirely absent despite there being ample opportunity to introduce it. While it is highly plausible that Augustine drew this argument from Cicero's Hortensius or other material (cf. Plotinus' treatment of second-order awareness or parakolouthēsis) ,  we note that this argument also appears in Shankara's commentary on the Brahma Sutras as well as several other earlier and later texts in Indian philosophical schools.

Note on the quantum plane

 Let $k$ be a field  of characteristic $0$ and $F(x,y)$ be the free $k$-algebra generated by $x$ and $y$. Then elements of $F(x,y)$ are finite sums of expressions of the form $cx^{n_1}y^{n_2}....x^{n_w}$ where $c\in k$, $n_i \geq 0$ where we consider that only $n_1$ ad $n_w$ can be zero. If $cm$ is such an expression then we consider $m$ to be the non-commutative monomial determined by $s$ and can also consider the 1-dimensional $k$-linear subspace  generated by $m$ (whose elements are of the form $cm$ for $c \in k$).

Now consider $M(x,y) = F(x,y)/ (xy -yx - h)$ for some non-zero $h \in k$. 

Notice that given a non-commutative monomial $m$ then $m$ determines a unique polynomial in $x$ and $y$ (with monomials all ordered in the form $x^ay^b$) with coeficients in $\mathbb{Z}[h]$ and such that the leading coeficient is $1$.  

Then we obtain a map 

$ Q_s:  Mon \rightarrow \mathbb{Z}[h][x,y] \rightarrow k[x,y]$

where $Mon$ are the non-commutative monomials in $M(x,y)$ determined by some $s$. We note that the coeficients in $Q_s(m)$ are uniquely determined by $s$.

Example: $yx^2y = yxxy = (xy - h)xy = xyxy - hxy = x(xy - h)y - hxy = x^2y^2 - hxy - hxy = x^2y^2 -2hxy$. 

This can be  extended to a $k$-linear map $Q$ on $M(x,y)$ but it is not clear which is the best way or what the range should be. Perhaps we define $Q : M(x,y) \rightarrow k[x,y]$ via the condition that its restriction to $Mon$ admits a factorization through $\mathbb{Z}[h][x,y]$ as for $Q_s$ above.

We can think of the $\mathbb{Z}[h]$ coeficients as generalizing energy eigenvalues, particle numbers, or spin and $M(x,y)$ as a kind of Fock space. The field $k$ itself is the "vacuum" and $h = xy - yx$ might be given a process interpretation (but this is not clear at the moment).

The prototype for $M(x,y)$ is the operator algebra generated by the operators $x$ and $\frac{d}{dx}$ on some suitable function  (or distribution) space. Since these are the operators that classical position and momentum are "promoted" to, it is natural to view $M(x,\frac{d}{dx})$ as a deformation of the Euclidean plane (phase-space) $(x,p)$ corresponding to the coordinate ring $k[x,p]$. 

How do we interpret our map $Q$? It suggests $M(x,y)$ is in a sense a more fundamental structure and that its projection (or emanation) onto classical space determines an intrinsic set of "quantum" integers (a kind of symbol).

Generative AI and the kenosis of humanity

We begin with some diachronic considerations concerning thought and language. For brevity's sake we will only distinguish between external language, linguistic utterances externalized in the context of communication or record-keeping and internal language, the inner spiritual substrate (including habitual memory and inner discourse) from which external language proceeds. And let us consider the following simplistic model. A given time $t$ the is a population $P(t)$ of $n$ speakers, for every $p,q \in P(t)$ there is $L(p,q)(t)$, the set of linguistic utterances uttered by $p$ to $q$ and a set of texts $L(t)$ (each authored by some element in $P(t)$) which are public and accessible.  We note that normally, $L(p,q)(t)$ is tied to context, dialogue, questioning, spiritual-mental states, relationships, goals, to the specific community to which $p$ and $q$ may belong - all of which we can consider as taking place in a microlocal time the moments of which are not distinguished by the global time $t$. 

For a given macro-time t, the total possible linguistic content $TPLC$ of $P(t)$ has to determined in terms of the collection of the externalizations and recordings in all possible hypothetical communicative scenarios between elements of $P(t)$.  For instance there are questions which could have been posed to individual a in time t which would have given rise to utterances of great value, deemed worthy of recording, but this scenario historically never in fact happened. We can very roughly think of $TPLC$ as being parametrized by different graphs over $P(t)$ with marked vertex "$a$" representing the social-context for the recorded utterance of $a$. We can think of elements of $P(t)$ as containers which as a rule only pour parts of their contents when interacting with other containers or group of containers. 

Now we can describe the result of the socio-technological transformation which lead to the world wide web and from the world wide web to generative AI (and from generative AI to future totalitarian agentic monitorization and simulation). Generative AI seeks to reach a state P(s) and which $TPLC$ has already been recorded and in which the possible communicative contexts themselves have been automated and made public. Generative AI seeks to effect a universal kenosis (emptying) of the human inner-spiritual linguistic substrate by rendering such a spiritual substrate and all the human social connections and contexts irrelevant and unnecessary.  Returning to the vessel metaphor, generative AI seeks to empty all vessels as to how they empty in every one of their possible interactions and mash together the content into a public monstrous idolatric simulacrum. The essential function of the world wide web was precisely to get people to "empty" themselves onto the great heap of the internet so that such a heap could be then "mashed" together by generative AI. 

Such a kenosis is however illusory and doomed to failure for the following reasons. Behind this thinking is the fallacy that meaning of a linguistic utterance can be precisely determined and exhausted by the extensional sum of all its possible occurrence windows and communicative contexts. This ignores the inexhaustible nuanced connotational dynamic multi-leveled intensional nature of meaning and its ultimate inseparability from the inner spiritual life, past experience and personal relationships of the subject. The contexts in question cannot be captured or formalized on the superficial level in which generative AI operates (the same goes for the earlier shallow encyclopedias and question-forums of the world wide web). Furthermore, the public availability of the generative AI version of $TPLC$ will itself contaminate its own training process in a way recalling diagonalization and self-reference arguments. We can consider prompts like: what would person $X$ think of subject $Y$ if person $X$ interacted with LLM $Z$ in way $W$ for length of time $T$? If LLMs are treated as oracles then they must answer questions which involve oracles. 

Generative AI will effect a kenosis of all the spiritual value and meaning of human language for human spiritual life and culture, not to mention its communicative value and formal objectivity. Generative AI will succeed in making language itself the enemy of mankind robbed of its spiritual, expressive, liberational and higher epistemic function - and historical-cultural continuity with the past.  Equally drastic counter-measures are called for, the development of radically alternative languages (including formal languages and logics) and symbolic modes (maybe including those based on silence and gestures) of thinking, communication and recording which serve and preserve intact the infinite spiritual content of human meaning, experience and social communion.  We note that there are aspects of human experience which can carry profound spiritual and semantic content and connection related to olfactory, gustatory, tactile, proprio-perceptive senses and inner modes of vision and illumination which will remain forever beyond the realm of generative AI. And the experience of a romantic connection, of falling in love, though it can occur in the context of audio-visual data,  cannot be reduced to it. Make philosophical logic and the philosophy of language together with the theory of literature and literary stylistic analysis the core of human education. Encourage multilingualism and metaphoric poetic writing and the appreciation of literary genius. Increase the subtlety, nuances, subtext, connotations of text and metaphoric symbolic expression. Revive symbolist poetry and ancient languages. Develop irreducibly personal, private or intimate languages as forms of resistance (there are some interesting historical precursors).  Demolish arguments against a private language and demolish behaviorism. Use pure mathematical theories as a form of symbolic metaphoric personal private communication. Develop entirely new mathematical theories, puzzles, formal systems and games which have not yet been "extensionalized" in the web to be used by LLMs.  Develop a new system of encryption in which all communications and records can only be decrypted by small groups of people sharing the "private group key". Develop context-dependent programming languages which are intractable by LLMs.

Light on the Philosophy of Consciousness

We now describe briefly our work-in-progress in the philosophy of consciousness. By philosophy of consciousness we mean the science of the soul (psychology) which is inseparable from the science of the nous (noology) which in turn in intimately connected to the science of the one (henology). We will conveniently refer to these aspects as noetic psychology and in this way avoid a terminological confusion with modern psychology.  The great light and genius of noetic psychology was Plotinus (c. 204/5 – 270 CE).  In order to make progress in noetic psychology one needs to be well-versed in analytical dialectics, in intuitive psycho-noetic introspection (phaneroscopy, to borrow a nice neologism) and familiarity with some stages of spiritual realization - all three being present in Ammonius Saccas' pupil Plotinus in a high degree.  Yet Plotinus' noetic psychology is - despite a richness of philosophical concepts, arguments and insights scattered through the Enneads anticipating Kant, German idealism and phenomenology -  not perfectly and completely unfolded in the formal logical plane; nor are the foundations set out clearly accompanied by a detailed manual of inner spiritual phaneroscopy and illumination which serves both as philosophical proof and a guide to spiritual transformation. In other words, a perfect unfolding of noetic psychology must comprise a theory of knowledge imbued system of formal dialectics and a total systematic anagogic and illuminative phaneroscopy progressing from psychology to noology and then back again, establishing metaphysical truths with absolute certainty. In the language of Mahâyâna buddhism the perfect unfolding must encompass both a Madhyamaka and a Yogâcâra including an Abhidharma. Furthermore the special charisma of the Platonic tradition culminating in Plotinus requires that a certain kind of mathematics and formal logic (and there were sophisticated systems of ancient formal logic far beyond the official narrative) be give a special place - and this was done by Porphyry and Iamblichus reviving older mathematical work. While valuable elements for this crucial but lacking unfolding of noetic psychology could be gleaned both from older Middle Platonism (cf. Origen and Saint Clement of Alexandria) and from contemporary Peripatetic, Pyrrhonic and Stoic schools, it fell to Augustine of Hippo (354 - 430 AD) to lay down, as a philosopher  doubtlessly illumined by a Manichean or Christian charisma and theurgy,  a systematic foundation for such a knowledge theoretic, skeptical-dialectical and illuminative phaneroscopic unfolding and articulation. And providentially this occurred in the 5th century as the Neoplatonic School of Alexandria and Athens had reached, under the influence of Syrianus and Proclus, a stage of total corruption and inversion of the original philosophical teaching and spiritual practice of Plotinus and his immediate disciples. Also in the 5th or 6th century it appears that there was a genuine (or "rogue") neoplatonist not connected to the school of Proclus who wrote the foundational texts of the Corpus Dionysiacum. Augustine starts is unfolding of noetic psychology at its proper beginning: the theory of knowledge, the direct and total confrontation with the schools of skepticism, Pyrrhonic and Academic (Contra Academicos), with radical, total and absolute doubt (equal to that of any modern transcendental idealism or phenomenology)  and the recherche de la vérité in the form of luminous indubitable evidence. While within the Islamic Golden Age the true tradition of Plato and Plotinus was continued or revived, it falls to us to construct a systematic, rigorous and formal philosophy of consciousness based primarily on what is called the Platonic-Augustinian tradition: this comprises Augustinian scholasticism which thrived from the early medieval ages in unbroken continuity until the 17th-century (without the direct influence of either Cartesianism or Malebranche), an example being the Milanese Capuchin Friar Valeriano Magni, following the shining wake of Cusa, and then the wonderful school of Ontologism among whose founding fathers are counted Descartes, Leibniz, Malebranche and Fénélon and which thrived as a superior form of Christian philosophy until the second half of the 19th century (Ubaghs, Fabre d'Envieu)  when it was persecuted and destroyed by empiricism-leaning Neothomist fanatics backed by ecclesiastic power, the death-blow being dealt by the encyclical Aeterni Patris (1879)  (see Answer To the Letter of An Empiricist Against Ontologism by Fr. Jules Fabre D'Envieu (1864), for an account of the intellectual, social and religious climate). The second part in particular of Fenélon's work Traité de l'existence et des attributs de Dieu which is entitled Démonstration de l'existence et des attributs de Dieu tirée des idées intelectuelles on pages 82-162 offers a particularly clear, concise and systematic presentation of Ontologism as contained in Descartes and Malebranche but in such a way that the Augustinian origins are clearly manifest. C.G. Ubaghs in his Essai d'Idéologie Ontologique (Louvain,1866) on page 40 writes that the Traité is admirably brief, clear and complete and shows directly the influence of Saint Augustine's De libero arbitrio. We also attach great importance to various "spiritualist" schools which had much in common with Ontologism. What was essentially lacking in Augustinian scholasticism and Ontologism - which is demanded by its Platonic roots and many indications of Augustine himself - was a more systematic development and deployment of formal logic and a special finitist and constructivist philosophy of mathematics (it was only Gödel who developed directly this aspect of Leibniz' work).  Essential spiritual infinity exists, extensive quantitative infinity does not.  A philosophy of consciousness must be essentially a theory of the intelligible light and indeed a theology if by "God" we mean the "place of eternal truths".

We have sketched an outline of the philosophy of Augustine and expounded some core noetic psychological principles regarding Aristotle, Plotinus and Augustine and their anticipation of German idealism and modern phenomenology.

A new critique of pure reason

We present here very briefly some aspects of our larger philosophical project (in continuation of our previous posts).  Here are the main points:

1. Finitism and constructivism are not modern developments but represent the main current of the ancient philosophy of mathematics.  And in the modern age its fathers should be considered to be Leibniz, Kant, Krause, Schopenhauer and Gauss (among others).  Proclus' philosophy of mathematics as expounded in his commentary on the first book of Euclid (and its accompanying epistemology and philosophy of mind) are by and large not original but a transmission of much older material in the finitist and constructivist tradition, for instance Iamblichus, Porphyry and much earlier, Geminus and Eudemus. 

2. Proclus and 5th-century neoplatonism represent a fundamental corruption and distortion not only of the original school of Ammonius Saccas, Plotinus and Porphyry (and to a certain extent Jamblichus) but the original teachings of Platonism, the Peripatetics and the Stoics. Proclus's dogmas ontologically and epistemically downgrade the human soul, imprison it to deny it access to higher modes of being and knowledge and thus betray the  fundamental truth regarding the ultimate essence and possibility of the human soul, clearly articulated in Plato and the Enneads. The flawed destiny of neoplatonism is plausible linked to its failure to take up and incorporate the higher developments of ancient logic which were indubitably attested in Chrysippus, Galen and Boethius.  Proclus was trained as a lawyer and rhetorician, not a logician. The Elements of Theology are, with regards to their pretense of exhibiting a logical-deductive form, are a farce, much like the Ethics of Spinoza.

3.  Our most radical and doubtlessly disturbing claim - which nevertheless finds a definite echo in Plato, Plotinus, Kant, Hegel and to a larger degree in Schopenhauer - involves the inseparability of theoretical and practical reason.  We question if the term "intelligence" denotes anything precise and objective and if much that is encompassed by this term should not really be considered humanly worthless and even contemptible.  That is to say, human so-called "rational" activity based on claims of being independent or abstracted from a moral and spiritual dimension and goal should be considered as lacking any value whatsoever. In particular its very epistemic, semantic and ontological claims should be radically challenged. And this is precisely the highest function of philosophical logic, finitism and constructivism and one of the key aspects of the philosophy of Schopenhauer. 

4.  A child that grows up in a war-torn area, experiences abuse, imprisonment, hunger, famine, sickness, the death of parents, friends, relatives and animals,  is a million times more "intelligent", a million times a greater knower of what the reality of life-experience is, a million times nobler and a greater partaker of being, a million times more capable of compassion and of directly knowing the fundamental aspects of reality, than an arrogant, heartless, shallow, selfish, bigoted, insignificant,  humanly ignorant, "professor", grown up in the privileged lap of material luxury, self-delusion, flattery and conceit, that monkeys around with symbols and language games without any metaphysical understanding or philosophical-logical insight. One cannot divorce true intelligence, true knowledge, from morality, moral knowledge and spiritual transformation.  Profane immoral reason is a deception and delusion which believes in nonsense like completed infinities or non-constructive and non-intuitively verifiable entities and inferences or constructs inconsistent theoretical physics without any connection to concrete experimental data.   Language is a social engineering tool riddled with vague, ambiguous and ultimately meaningless terms. Brouwer and a certain linguistic circle to which he belonged were keenly aware of the immoral and socially manipulative aspects of language. This is our reading of Kant and Schopenhauer: intelligence and morality are one. Or as we read in the ancient Pali texts: by one's deeds is one a Brahmin. Plotinus says the same thing in Ennead I,3,6 15-18: one cannot be wise or a dialectician without the previous or simultaneous cultivation of virtue.

5.  Legitimate reason, good reason, inseparable from morality and illumined by the higher light of transcendental knowledge, has two aspects. That which leads to progress in engineering and medicine and is thus an incarnation of compassion,  and the sword of philosophical logic -  effecting a universal ultrafinitist and constructivist critique of all mathematics, science and language. It is in this way that we should trace Pyrrhonism (and possibly Hume if we believe Gopnik) to their Buddhist roots. But philosophical logic also has a complementary positive philosophical role in unveiling the genetic constitution of logic, computation and legitimate finitary constructive mathematics as well as Platonically unveiling their connection to inner and higher levels of being and cognition. Philosophy should aim at a transcendental phenomenological exposition of Plotinus.

Regarding the first aspect there is a question which nobody wants to answer: what is the minimal essential mathematical apparatus strictly required for all the positive beneficient achievements in the world in engineering and medicine? And can we give numerical analysis a finitist computationalist foundation?  

Regarding the second aspect it is important to point out that this universal critique is based on the possibility of a finitist and computational interpretation or transformation of the mathematics in question (as in the work of Kohlenbach), not a superficial rejection in the style of Kronecker.  

Hegel and Neoplatonism

Platonismus und Idealismus by Werner Beierwaltes

Hegel und der spätantike Neuplatonismus by Jens Halfwassen (2005)

L'illusion Labyrinthiques: L'Hegélianisme et Neoplatonisme by Jean-Louis Vieillard-Baron (1979)

Hegel's Hermeneutics by Paul Redding (2007) / Continental Idealism: Leibniz to Nietzsche

"Hegel’s Programmatic Recourse to the Ancient Philosophy of Intellect" by Jens Halfwassen

"Thinking the One: Studies in Neoplatonic Philosophy and its Reception" by Werner Beierwaltes

"Hegel on Neoplatonism and Proclus" (Lectures on the History of Philosophy) 

We can also remark that Proclus plausibly got most of his philosophy of mathematics from Iamblichus, Porphyry, Geminus and Eudemus and provides us with a fascinating glimpse of much more ancient material that has not come down to us.

Theory of Meaning

Meaning, that most illusive of philosophical concepts, is without doubt a ternary relation M(A,B,C): A means B relative to/in/according to C. To us "meaning" is simply a finitary algorithmic transformation that takes finite structured data of sort 1 into finite structured data of sort 2 (which can be the same sort 1, for instance in normalization of types). The canonical example is a compiler.  The Tarskian "meaning" used in model theory is, as usually conceived, misleading. In reality it is nothing more than a finitary recursive transformation of the expressions of a formal system into expressions in some version of formal set theory.  This transformation is interpretation. And interpretations can themselves be interpreted or transformed just as compilers are themselves programs which must be compiled.

Our natural term logic NTL gives three examples of meaning, of finitary recursive transformation or interpretations. The normalization of NTL terms to reveal the canonical term which conveys core logical "meaning", the tranformation of NTL terms into terms in the more coarse-grained Bealer Logic and the transformation of Bealer Logic terms into normal NTL terms. 

Another interpretation is the one that follows from a priori meta-theoretic postulates whereby deductive data in one system is taken as sufficient grounds for meta-theorems about another systems (meta-theoretic mirroring).

Feedback for queries of data bases, for questions or commands, are also interpretations.  

Theories of "meaning-as-use" are false and circular and do not explain "meaning" and cannot count as a theory of meaning (this is explored elsewhere). 

We can think of interpretations as functors. Natural transformations are themselves particular cases of functors.

The human mind is equipped with a finite series of interpretations which are successively or concurrently applied to symbolic data.  But this process has to stop, it has to have a fixed point. Thus there are finite, bounded, combinatoric, recursive-algorithmic, rule-constituting, self-reflecting  "semantic primes" or "intuitive primes" which have no interpretation beyond themselves (up to isomorphism?). The task of semantics and the philosophy of language is to find these generators and in particular universal components which must be assumed for any adequate possible logic or programing language. It may be the ultrafinitism may shed light on these problems. 

This parallels the synthetic a priori principles of the human understanding where the finite combinatorics of one system has to be assumed to be sufficient epistemic grounds (security, certification, checking) relating to the combinatorics of another system (this is achieved through mirroring and meta-interpretation). Thus we can deduce a priori (using a certain formal system) the number of steps for a given algorithm on a certain system to terminate in function of the input length  and then we can verify this empirically on the system.  Proof mining does this but still in an ideal non-finitary framework. Note a certain Pythagoreanism in category theory. The dyad can be embodies as the category with two objects and no non-identity arrows, as a category with two objects and one non-identity arrow or a category with two objects and two non-identity arrows. These diagrams suffice to calculate all limits and to define natural transformations as functors!

This suggests that  reality corresponds to a hierarchy of different levels of being and on each level finite structures mirror each other in a finite multiplicity of ways, mirror those beneath and mirror those above them in determinate ways. They are woven together like beads on a string.  All these correspondences must be accepted as a priori conditions for the possibility of intelligibility and cognizability and meaning. Thus also our semantic primitives or categories themselves must be considered according to the level they are implemented/manifest on (or participated by). 

Study finite embodiments of the absolute Galois group, the algebraic closure of $\mathbb{Q}$ and of countable models of set theory. Study approximate categories with only finitely  many objects and morphisms.  Study finite versions of quantum field theory. Study finite versions of the calculus based on the approximate implementation of real numbers in standard hardware. Study logical systems with constraints on expressions and rule applications (finite versions of exponentials, etc.).  How much ultrafinitist mathematics, logic, algorithmic theory and philosophy is not hidden deep inside the Pari and Singular algebra software?

Lattices play a central role in algebraic number theory. Why? Because the ring of integers of a number field is finitely generated as a $\mathbb{Z}$-module. This mean that algebraic integers occupy the points in a $n$-dimensional lattice generated by a certain basis $\omega_1,....,\omega_n$ of algebraic integers. The discriminant is nothing more than the covolume of the lattice (measuring how sparse the elements are distributed). From an ultrafinitist perspective it would be very interesting to consider how Pari implements finite fragments and approximations of such lattices.

Finite version of the rational numbers can come in a diversity of forms. Codified by pairs of natural numbers with a maximum bound,  fixed or floating decimal representations, programs of bounded length for computing the decimal expansion, etc. In this sense the ultrafinitist can accept irrational numbers such as $\pi$ whose decimal expansions are defined intensionally and computationally, for instance using Ramanujan's $\pi$ formulae. In a sense there is no $\pi$ as an object, rather there are finite collections of algorithms (intensional descriptions) for generating digits and a formal proof, witness, of their effective equivalence.

It is quantum field theory, not quantum mechanics that is fundamental. Quantum mechanics is at most an incoherent abstraction of quantum field theory (it is chosen as an introduction due to an alleged similarity to classical physics, but such a similarity is based on ad hoc magical postulates about "promoting" physical quantities to operators). Indeed the quantum mechanical position and momentum operators are imbued with particle ontology and do not make sense at all from the point of view of quantum field theory (what is the momentum operator measuring the momentum of ?). And neither does the uncertainty principle. Perhaps the fact that the quantum harmonic oscillator has discrete enemy levels is a consequence of the fundamental finiteness of the universe. We could study if discrete energy levels arise in any classical equations such as a the sine-Gordon equation. Surely there is something "topological" going on here. The uncertainty principle may be tied to the properties of physics assuming the fundamental discreteness of space and time and energy. And quantum field theory be elucidated by the classical finite approximations (vibrating lattice phonons).

A major aspect of ultrafinitist philosophy is a radical critique of mathematics and of the social and cultural value of mathematical activity itself. We call it the ultrafinitist fork. "Good" mathematics pertains either to the domain of radically fundamental and critical logical-philosophical investigations or else to the essential optimal computational foundations of medicine and engineering (but conducted according to an intuitionist and logico-philosophical methodology).  Our task is to develop an ultrafinistic foundations for the calculus which also serves a computational foundation for numerical analysis. In between these "extremes"  should be promoting the clarification and improvement of the formulation and proof of known results as well as a thorough critical re-evaluation of the history of mathematics itself.

And concerning linguistics in this ultrafinistist perspective we propose that the formal logical algebraic structures present in the phonetics and morphology  of natural language be studied with greater care and their philosophical significance be better appreciated. 

The Recursive Finitist Mathematics Program

Pauca sed matura. (Gauss)

It is curious how the following philosophical position of strict or bounded recursive finitism also called ultrafinitism (whose roots can be traced back to Gauss, Kant, Leibniz, Hume,  Proclus, Euclid, Aristotle, Plato and perhaps Chrysippus) has received little attention (apart from Bishop, Maddy and Alexander Yessenin-Volpin) within the broader context of finitism, constructivism and intuitionism. It is wrong to associate Kronecker's philosophically dogmatic and naive empiricist and materialist view of mathematics (never actually worked out, nor the concept of algorithm and computation elucidated) with philosophically and logically reflected finitism or ultrafinitism. Kronecker's views from the end of the 19th century - which had no relationship to the ideas of formal logic - were a crude perversion of the ideal of computability and finitude (which were, due to Kronecker's power and influence, historically harmful to the progress of mathematics) and they cannot be invoked as an anti-finitist argument. They also produced a harmful misconception that ultrafinitism is somehow connected to empiricism and naturalism. Also Kronecker's legitimate and correct focus on effective procedures in number theory is not original but was already, for instance,  at the heart of Gauss' methodology and philosophy, which in turn can be traced back to Euclid. The great names of finitism and the ideal of computability include (besides Maddy, Bishop and Yessenin-Volpin) Skolem, Hilbert, Brouwer, Church, Turing and Martin-Löf among many others. But far more historical research is called for.

We need to recognize the co-implicitness between logic, arithmetic and algorithms and that all mathematics must be subject to bounded finite and computable criteria and rules but at the same time recognize with Leibniz that the fictive imaginary non-finite entities in bounded finite theories, definitions and proofs can be reduced to actual bounded finite numerical computations. But this must not be confused with a pragmatic viewpoint such as as Weyl's in physics.  Because such a reduction with its associated computable elimination of fictive entity symbols,  has to be effective and transparent. The situation is quite different for physics in which even an experimental confirmation of theoretically predicated outcomes does not in the least rule out the radical revision or improvement of the theory in question as well as a healthy methodological skepticism.

The fictive freedom and realism of the birds adds to the perfection of their own finite bounded realization.

 

 1. All legitimate mathematical objects must be bounded, both "extensionally" and "intensionally" (i.e. with regards to their definition). Thus there are intensionally defined natural numbers which are too large to actually exist.

2. All logical expressions, proofs, rules, axioms and instantiations of rules and axioms must be have precise bounds on size.

3. All meaningful quantification must be finitely bounded and interpreted constructively. Constructivism gives us a meaningful way to interpret quantification. But linear logic and other subtstructual logics as well as paraconsistent logic also gives vital contributions. We must replace the exponentials !,? with their bounded finite versions. And the distinction between distributive and non-distributive (intensional) versions of linear version of the quantifiers (corresponding to additive and multiplicative connectives) is of great importance.

4. All valid mathematics must be given either by finite enumeration or finite (possible recursive) specification. Mathematical objects are algorithms (programs) specifiable within a finite bound (this must not be forgotten). One program can be seen as a type of another program. The finding of proofs or construction of proof-terms (with controlled resources) is analogous to assembling the traditional Chinese Luban puzzles.

5. All finite specifications and proofs must be able to be checked by a finite program (proof assistant and proof checker).

6. Valid mathematics corresponds to what can be formalized and checked by a (necessarily finite) proof checker. All valid mathematics must be able to be given direct, concrete, intuitive, operational-combinatorial justification and presentation.

7. The finite and recursive is boostrapping, meta-reflexive, self-referential and self-transcendent guided by regulative ideas (convenient fictions) and synthetic pure a priori principles. We can automatically check proof checkers themselves. But this hierarchy can itself only have finitely many levels. 

Even if a system is inconsistent in the usual sense, may it not have a finite fragment (conditioned by finite rule applications) which derives something coherent and meaningful? 

Finitism may solve the traditional problems of the foundations of analysis (cf. the undecidability of equality for the reals). We need a pure formal algebraic treatment of so-called "approximation",  numerical analysis and implementations of numerical computation. 

Cf. Ulrich Kohlenbach's treatment of the  Krasnoselskii–Mann Iterations.

The above 7 propositions give rise to countless (but hopefully not exceeding the corresponding finite bounds!) philosophical, logical and mathematical problems, and possibilities of radical criticism of contemporary practices and ideas (we can question the usefulness of traditional complexity classes) - all of which we can only  begin to understand. 

What about inconsistent systems whose contradictions have proofs too large to be able known or represented? Systems which within the constraints of finite bounds do not produce contradictions? Or are locally indistinguishable from consistent systems? 

Many-valued logics can be exemplified by a statistical or probabilistic logic based on finite data-sets. 

The 7 propositions permit an intimate fusion between computer science, logic, linguistics, mathematics, cognitive psychology and philosophy as well as art (is there a big difference between mathematical or computational elegance and efficiency and human aesthetic value?). 

"All men are mortal" means that there is an accepted finite process by which from the finite concept "man"  we can extract the finite concept "mortal".  Extensional, distributive readings are untenable. But what about the proposition: "If all A is B and all B is C then all A is C". The quantifiers can be interpreted as above, but what about this whole proposition which is quantified over "monadic predicates" A,B and C ? This gives us another legitimate interpretation of bounded universal quantification: as a postulated rule, an algorithm within finite bounds, i.e., a logical rule, a rule of inference.

It would be interesting the study finite versions of the absolute Galois group.  What is a computer algebra program like Pari or Singular actually doing but implementing the ultrafinitist program?

What if it had been the case that some of the finite simple groups required tables larger than the universe to specify? That is, they could not even be recursively, intensionally specified like the number $10^{10^{10^{10}}}$?

What are modern computers but finite versions of Gödel codings of formal systems? Calculemus! 

Note that Proclus was a finitist and he uses the rejection of even a countable completed infinity in the reductio ad absurdum argument in the proof of the first proposition of the Elements of Theology. His argument is that completed infinities must be an infinity of infinities apeira aperakis, just as the natural numbers contain a union of infinitely many disjoint infinite sets (cf. the standard bijection $\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$).

Mathematics and AI

Referring to certain software and hardware as "AI" and to such alleged "AI" as an agent "solving" problems in mathematics is profoundly misleading. The programs in question are not "intelligent" and are not "solving" mathematical problems as agents in any philosophically meaningful sense. And these programs will certainly never "replace" human scientific endeavor in any conceivable way, rather they are, and will ever remain, mere tools.

Let us be clear. Human beings can only process, check and produce data within definite finite bounds based on symbol systems with definite finite bounds and bounded fragments of finitarily determined rules.

Thus it is a triviality and a truism that all human external symbolic activity and productions could in principle - given a massive enough set of data is made available - be mimicked and processed by brute-force. We can imagine a supercomputer in space with processing power and storage a billion times surpassing any of the human bounds of symbol processing and text processing, checking and production. We can also give our supercomputer some kind of super-luminal processing velocity. This supercomputer through brute-force and crude machine learning algorithms would beat and outperform every man-made "AI" in every possible domain in an instant. There is nothing surprising here and there is nothing here that has anything remotely to do with "intelligence".

Intelligence is rather reflected in doing much with little external, material, processing power. Chess programs are just cheating machines. Our hypothetical supercomputer would beat any current Go or Chess AI and that would not make it "intelligent" in any meaningful way.

As for mathematics, let us take a proof assistant such as Agda or Idris 2. The mathematician develops a theory (or formalizes a previous theory) by means of type definitions, records and type declarations for terms. The problem is to find explicit proof terms. This could be attempted by using brute-force, heuristics or any kind of machine learning method. And this search could fail. Or the search algorithm could be refined and altered. There is nothing unusual going on here. The algorithm is not "doing mathematics", it is not constructing theories, refining definitions, improving its own heuristics or being telologically oriented towards a certain architectural vision. It is the mathematician's legitimate tool.

Speaking of "AI" "solving" mathematical problems or "doing" mathematics or potentially "replacing" mathematicians is sheer and utter nonsense.

If even published journal papers can contain errors, I certainly would not trust any "mathematical" output produced by generative AI that was not formalized and checked in all its details by current proof assistants and proof checkers.

I.R. Shafarevich on the definition of mathematics

In reply to the question 'What does mathematics study?', it is hardly acceptable to answer 'structures' or 'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve
the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion.  

- I. R. Shafarevich  

We must never forget the call for a radical critique and reform of mathematics in the spirit of Hilbert, Brouwer and the constructivist and finistic schools and most of all Voevodsky's wake-up call to the potential errors of published papers and the necessity of a formal mathematics project based on dependent type theory.

Both the formal rigor and certainty of proof assistants and the clarity of pure computational and combinatorial intuition are called for.  

What role does category play here? How can we define and clarify the opposition between good and bad abstraction and construction? 

And the philosophically deep question: what part of mathematics is strictly necessary - including for efficiency and reliability -  for the most important accomplishments in modern technology, medicine and engineering? What tangible beneficial progress in these domains is the direct result of mathematics?

We need a radical (philosophically, logically and  humanistically enlightened) reform of our valuation of  mathematical productivity, mathematical theories, mathematical methodologies and practice, mathematical certainty claims and mathematical foundational frameworks. 

Mathematical Foundations for a Compositional Distributional Model of Meaning (2010)

Mathematical Foundations for a Compositional Distributional Model of Meaning (2010)

Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark

We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are `lifted' to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model' which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.

https://arxiv.org/abs/1003.4394

An enriched category theory of language: from syntax to semantics (2026)

An enriched category theory of language: from syntax to semantics (2026)

Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos

State of the art language models return a natural language text continuation from any piece of input text. This ability to generate coherent text extensions implies significant sophistication, including a knowledge of grammar and semantics. In this paper, we propose a mathematical framework for passing from probability distributions on extensions of given texts, such as the ones learned by today's large language models, to an enriched category containing semantic information. Roughly speaking, we model probability distributions on texts as a category enriched over the unit interval. Objects of this category are expressions in language, and hom objects are conditional probabilities that one expression is an extension of another. This category is syntactical -- it describes what goes with what. Then, via the Yoneda embedding, we pass to the enriched category of unit interval-valued copresheaves on this syntactical category. This category of enriched copresheaves is semantic -- it is where we find meaning, logical operations such as entailment, and the building blocks for more elaborate semantic concepts.  

https://arxiv.org/abs/2106.07890

Emergent Analogical Reasoning in Transformers (2026)

Emergent Analogical Reasoning in Transformers (2026)

by Gouki Minegishi, Jingyuan Feng, Hiroki Furuta, Takeshi Kojima, Yusuke Iwasawa, Yutaka Matsuo

Analogy is a central faculty of human intelligence, enabling abstract patterns discovered in one domain to be applied to another. Despite its central role in cognition, the mechanisms by which Transformers acquire and implement analogical reasoning remain poorly understood. In this work, inspired by the notion of functors in category theory, we formalize analogical reasoning as the inference of correspondences between entities across categories. Based on this formulation, we introduce synthetic tasks that evaluate the emergence of analogical reasoning under controlled settings. We find that the emergence of analogical reasoning is highly sensitive to data characteristics, optimization choices, and model scale. Through mechanistic analysis, we show that analogical reasoning in Transformers decomposes into two key components: (1) geometric alignment of relational structure in the embedding space, and (2) the application of a functor within the Transformer. These mechanisms enable models to transfer relational structure from one category to another, realizing analogy. Finally, we quantify these effects and find that the same trends are observed in pretrained LLMs. In doing so, we move analogy from an abstract cognitive notion to a concrete, mechanistically grounded phenomenon in modern neural networks.

https://arxiv.org/abs/2602.01992v5

See also:  https://www.youtube.com/watch?v=HC_WTCErPcI

On Various Translations Between Classical, Intuitionistic, and Linear Logic

Ferreira, G., Oliva, P. & Protin, C.L. On Various Translations Between Classical, Intuitionistic, and Linear Logic. Stud Logica (2026). https://doi.org/10.1007/s11225-026-10251-y

Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are: (1) to consider extensions of intuitionistic linear logic corresponding to each of these systems, and (2) using this common logical basis, to develop a uniform approach to devising and simplifying proof translations. Through this process of “simplification” we recover most of the well-known translations in the literature.

Investigations into the Idris2 proof assistant and some formalizations

Is there a proof assistant which combines the best aspects of Coq/Rocq and Agda? We would like the construction of proof-terms be done in a natural deduction style, close to actual mathematical practice (at least for the proofs of category theory). This is different at once from the top-down method of constructing proofs in Coq/Rocq and from the emacs interface of Agda and its system of refinement, filling holes, etc.

Dependent type theory - extensions of type theories such as used in Ocaml and Haskell - seems to us philosophically, computationally, logically and practically the best foundation for mathematics. The elegant Idris 2 proof assistant - which we could call an idealized version of Agda (but with linear types!) - seems to be most promising. It is simple and versatile and can work with emacs, vim or nano. Could we use it as a basis to develop our natural deduction style interactive theorem proving?

Idris 2 with its REPL in fact feel like a enhanced version of Ocaml or Haskell (upon which it is based) - but it is not so easy to install (we are running it on an Ubuntu virtual machine in whih we need rlwrap to get command history).

We can seemlessly formalize category theory in Idris 2 (it seems this can be done in a simpler and more direct way than in Coq/Rocq) - a fact which is already of logico-philosophical significance.

https://github.com/owl77/Idris2_formalizations/blob/main/ct.idr

In this module we define general categories, opposite categories, the category of "sets", the empty, singleton and canonical category with two elements, product categories terminal objects, functors and natural transformations, point out the necessity of extensionality to define the category of sets and the need to postulate identity conditions for natural transformations. We prove that functors between categories A and B and their natural transformations form a category. As a result we can define the category of presheaves over a given category A and the yoneda embedding. We define ´diagonal functors, cones and limits. And the composition of functors, the identity functors, the category of categories and the "whiskering operations" (Godement product) and use this to define adjunctions in terms of the triangle identities. There are several interesting points regarding the coherence of natural transformation equality relative to functor identity (cf. transport in Hott). We believe that Iris 2 is the best (and most efficient) dependent-type based proof assistant for this task, once one understands how to use rewrite and Refl.

Consider how we prove that any two terminal objects in a category are isomorphic. To do this we show first that given a terminal object T its canonical morphism to itself must be id_T (this is the lemma1 term). This is how in our natural deduction extension of Idris we would construct the lemma1 term in the way a mathematician would prove the result. 

1 c : Cat Hyp
2 a : obj C Hyp 
3 t : terminal c a Hyp
4 t : forall (x: obj c), exists (g : hom c (x,a)), forall (h: hom c (x,a)), h = g Expand 3
5 t a : exists (g : hom c(a,a)), forall (h : hom c(a,a)), h = g Inst 4,2
6 snd (ta) : forall (h : hom c (a,a)), h = fst (ta) Sigma type stuff
7 id c a : hom c (a,a) By def
8 (snd (ta)) (id c a) : id c a = fst (ta) Inst 6,7
9 (c : Cat)(a : obj c)(t : terminal ca), (snd (ta)) (id c a) : (c: Cat) -> (a : obj c) -> (t : terminal c a) -> h = fst (ta) QED

Critique of Quantum Mechanics and Quantum Field Theory

Here we will present some short logical and philosophical critiques and questions concerning quantum mechanics and quantum field theory with the ultimate goal of going beyond these theories and constructing more satisfactory ones. We will also attempt to bridge the gap between some of our previous speculations and the actual mathematical structure of quantum theory.

We focus on the collapse of the wave-function. But the root of the problem can already be found in the singular situation of observables corresponding to Hermitian operators while the temporal evolution of the state of a system corresponds to unitary operators. The collapse is essentially a projection onto an eigenvector. In classical physics measurement and observations are external to the system. In quantum mechanics they become internalized. The observation process, the applications of hermitian operators, suggest a highly sensitive non-linear response of a system whose nature is still not understood.

What if we viewed space as discrete (like the lattice structures used to explain free fields in QFT)? Then the wave function becomes just a finite vector, the space points mere indexes - just like time is a mere index. We loose the topology.  It is curious that  position can be promoted to an operator but time cannot. A time operator in quantum theory seems to be a notion derived from momentum and as such recalls the classical Aristotelian notion of time being a "measure of change".  

Understanding what the vacuum state $|0\rangle$ is in QFT is not easy - and many different accounts are given in the textbook of why the Green function takes on a particular interpretation of a particle being created at point $x$ and destroyed a point $y$.  Understanding what the spatial-temporally indexed operators are - and what measurements they correspond to - is also not easy. The creation and annihilation operators are not Hermitian and do not correspond to observations and measurements and yet Hermitian operators (or operator-valued distributions) can be written in terms of them.  These supposedly correspond to measurements.  We must first of all understand the Vacuum Expectancy Value, what $\langle 0 | \phi(y) \phi(x) | 0 \rangle$ means (which is obviously similar to the expected value in quantum mechanics $\langle \phi | A | \phi \rangle$). And understand it experimentally. The first $\phi(x)$ corresponds to the high-energy localized injection of a particle (perhaps the product of decay) while the second $\phi(y)$ could correspond to collision with a detector (which thus destroys the particle).

We think that the statistical mechanics motivations of second quantization and QFT is particularly interesting. Our "wave function" gives a probabilistic distribution over occupation numbers $(N_1,...,N_n)$. The interpretation of the creation and annihilation operators in this context is subtle. How are we really to understand

$a_i \Phi (N_1,...,N_i,...,N_k,...) = \sqrt{N_i} \Phi(N_1,...,N_i -1,...,N_k,...)$?

Too often we see physical metaphors applied rather vaguely to mathematical constructs. To us these operators only really make sense in the context of the rigorous formulation of Fock spaces (tensor products, symmetric and anti-symmetric products of Hilbert space $H$). Given an orthonormal basis $\epsilon_i$ of $H$ then the operator $\Sigma_{i=1}^\infty a^\dagger(\epsilon_i)a (\epsilon_i)$ has very clear properties ($Sv = nv$ for $v$ in the $n$th component of the Fock space), far more illuminating than speaking of "particle number". But of course Fock space pertains only to the theory of free fields. The general case, the perturbative case, seems intimately connected to the general concept of formal deformation of an object by formal power series (cf. quantum groups or Hopf algebras in general), but there is no rigorous mathematical structure for perturbative QFT.

 A central flaw in the theoretical treatment of QFT is that it systematically ignores the central unspoken role of human agency, the experimenter and experimental setup, behind the mathematical formalism. Operators (operator valued distributions) are treated as independent active entities in their own right which somehow decide to "act" on fields. The glaring shortcoming is that there is no clear criteria of demarcation between the human observer the what is observed - nature.  Can the measurement process itself be considered a natural process when considered and observed externally by a third observer ? Are Hermitian operators really then descriptions of something objective happening to the system, the external observed observer "zapping" or collapsing the system? And should not this whole process itself be capable of superposition?

This most curious situation: observables must correspond to Hermitian operators - this can be deduced by the requirements of measurement in a probabilistic framework. Also, the unperturbed evolution of a system corresponds to unitary operators (or in the Heisenberg picture conjugation of observables by unitary operators). But what a coincidence that - for a finite discrete approximation - the lie algebra of the unitary group is the space of anti-hermitian operators - itself isomorphic (via multiplication by $i$) to that of hermitian operators. That is, every unitary matrix is can be written $e^J$ for $J$ anti-hermitian or $e^{iH}$ for $H$ hermitian. So measurements correspond to objects in the tangent space of the group of unitary operators - but they are also applied in their own right to the wave function.

Why could not the total energy of the vacuum be infinite? Why could not the total energy of the universe be infinite and yet there be still, in some sense, conservation of energy?  For instance we could have a countable infinite collection of non-communicating finite universes with finite energy and in which the conservation of energy held. Then the total set of universes would be a universe with infinite energy in which there was still conservation of energy in a meaningful sense.  We could even allow communication between the finite components satisfying a conservation law.

It is curious how the discreteness of the energy states is connected to the finitude of the system. 

The vacuum state in QFT is similar to consciousness (to alayavijñana) - and we can argue that classical field theory could never be an adequate representation.

We have discussed before the interest of considering material and energetic constraints in information - and this is precisely the idea behind linear logic (Girard's first papers explicitly mention chemistry), as well as partially already present in the memory management aspect of programming languages. QFT scattering is a lot like communication with material constraints.  The "answer" of the system has to be materially destroyed to be received.

This is certainly a bizarre idea, but could it be that the experiments with LHCs are not teaching us about nature but rather a kind of training process analogous to AI? Nature is changing her habits in conformity to what physicists want to observe? And finally consider how we attempted previously to characterize the massive datasets which allowed LLMs to be trained and perform as they do.  The qualities of the massive data sets in question could they not correspond to fundamental properties of the vacuum or quantum fields in general? Of course the analogy is limited because of the apparent inescapable indeterminism in QFT while LLMs are just deterministic automata.

Type theory / QTF analogy.   Type = Operator, Term inhabiting a type = Vector and application of operator to the vector,  Simply Typed Lambda-Calculus = Quantum Mechanics, Dependent Type Theory = Quantum Field Theory, operators depend on values. Loops, non-termination = infinities. 

Foundationalism and Logic

The point in question concerns the problems of foundationalism involving type theory or term-rewriting systems. In the final part of my Kantian-oriented paper "Analyticity, Computability and the A Priori" I attempt to tackle with this problem (as well as presenting in greater detail my analysis of term-rewriting systems, Turing completeness, the Curry-Howard correspondence and the limits of logic-like formal systems in representing all computable functions and extracts on Hilbert's philosophy taken from a paper by Claire Ortiz Hill). I propose in the end a methodology inspired Piaget's genetic epistemology in which one must effect a sort of regression to relive in a conscious way the stages since early childhood whereby one progressively gained computational competency - centered around the ability to understand, carry out and check the following of rules - and to compare the cognitive structures involved to formal systems and computational models at our disposal. While this cannot lead to the enthroning of any single formal system or model it can, I believe, nevertheless bring to light groups of specific systems and models which are "structurally akin" and "cognitively natural" to consciousness itself. While we cannot enthrone a single system, I believe we can use the language of a given system to express what are in the Kantian terms synthetic a priori principles of the human understanding. For instance when we find a finite derivation in one formal system (the metasystem) and conclude that this derivation "shows" that a certain goal cannot be derived in another system (the object system), which is something that we cannot directly show in the object system because it would require an infinite amount of time. Gödel's incompleteness theorem is (and has to be) formalizable and corresponds to a finite derivation in a system M. But it must be assumed that this finite derivation is sufficient epistemic grounds to conclude something about the infinite set of derivations in Peano Arithmetic, that the sentence G cannot be derived.  

A thesis on generative AI

We present to following thesis on generative AI.  That to understand fully the essence of generative AI we must first understand the socio-cultural and philosophical essence of the advent of the internet.  Generative AI is only possible not merely because of raw hardware capabilities and certain neural net architectures and training processes, but on a more fundamental level because of the nature of the large data sets employed. Generative AI is only possible because of the previous availability of a very special kind of massive data set - precisely the kind of data sets provided by the internet. Generative AI is simply the natural further development of the same factors at work in the genesis of the internet.  So our main task is to describe the nature of internet data as well as its social, psychological, linguistic and cultural correlates.  This is a difficult task. One should consider and evaluate the following aspects: the democratization of knowledge and expertise (what certain authors once called "the cult of amateurism"), the lessening of the both the physical and semantic volume of content - flattening of layers of semantic depth, connotation, subtext and extra-textual reference, a  massive atomization and repetition with small variants of certain templates, memes and "virality", propagation and reproduction of persistent psycho-linguistic patterns throughout the whole body of the data - all which structurally contribute to forming the kind of atomized, grid-like, decentralized, self-entangled and self-connected  massive data set which is precisely what allows the generative AI technology to produce LLMs which perform as they do. While traditional cultural-linguistic data involved relatively few authors with massive linguistic content, the internet data involves a massive amount of authors with relatively scarce linguistic content each.  Humanity is caught itself in the textual universe of the internet.  The internet data set can be described as an enmeshed chaos of micro cognitive-linguistic units - which we can compare to the set P of partial information used in forcing. If we wish to combat generative AI we need to get at the root of the whole historical-cultural process involved in the genesis of the internet. We need to instigate a revolution regarding authorship, epistemic authority and the nature of the written text. The semantic web project should be revived. Curiously enough the Pali canon is an very early example of large textual data which exhibits some of the characteristics of internet data.

If the internet era was the era of communication, personal development through sharing and community, but now not only has the "internet" as a medium become content itself but it has become dead content, i.e., generative AI. Thus we are in the heroic age in which the human person must again find development and peace in isolation again - so that again there is something to share. The highest form of human knowledge must be global and synthetic, partial analysis has as its ultimate aim furnishing the support for such a knowledge. Global and synthetic knowledge is the only knowledge that frees one from Plato's Cave. 

Can we reverse engineering generative AI using generative AI? Could we train an AI model M on the data D = (T,L) consisting of training data T and a LLM L trained on T, to solve the following problem:

Given a prompt P fed into L and response R, identify the portions of T which can be considered as having (in some mode or another) the largest weight in the generation of R from P.

Can we define the concept of essential dependence of response R to P via L on a portion S of T? Could we train M to return, for a given P, portions of T which have a certain structural similarity to R?  

The bridge between logic and geometry - motivating the work of Olivia Caramello

There are several beautiful and profound foundational papers (many from the 1970s) on inituitionistic higher-order logic within the framework of topos theory. Papers by Lawvere, D. Scott, M. Fourman, G. E. Reyes and others which focus on the philosophical aspects of formal presentations of higher order intuitionistic logic inspired by the structure of elementary topoi: partial domains, sorts and types, definite descriptions, definability, non-classical truth values, the interpretation of bounded quantifiers, the axiom of infinity (NNOs), the Russell-Prawitz translation (Scott claims to have discovered this in the 50s), etc.
There is an elegant Hilbert-style formal presentation of intuitionistic higher-order logic employed by Fourman in his paper in the Handbook of Mathematical Logic.

In our paper "Hegel and Modern Topology" we proposed an interpretation of the the Logic of Concept wherein the stages of Subjective Concept and Objective Concept and their integration are interpreted in terms of the duality between theories and models (understood as a geometrically inspired category theoretic framework for general systems theory) and the problem of finding a bridge between the two - something which partakes of the essence of both, perhaps in the style of algebraic logic.

In the paper by Fourman he uses only the most rudimentary notions of category theory. The main result is that any (elementary) topos is equivalent (via a logical morphism) to the topos of a definitionally complete theory E(T) and that any theory T gives rise to a topos E(T). Thus the objects E(T) -  bearing in mind their universal property (theorem 8.9 in the paper by Fourman) - would seem to be a candidate for a bridge between theories and models. E(T) classifies models of T: there is a correspondence between models M of T and logical morphisms E(T) -> M.

But Fourman remarks (he terms it an "embarassement") that this still does not constitute a bridge between (higher-order) logic and geometry (i.e. Objective Concept), between theories and Grothendieck topoi and their geometric morphisms. Fourman mentions a paper by Reyes in which Grothendieck topoi are described in terms of adding to Set a "generic model" for a possibly infinitary first-order theory. 

As I see it some of the main problems with topos theory are the following:

1) Some of the toposes that I find most interesting, such as the Hyland's Effective Topos (a model of higher-order computable mathematics) are not Grothendieck toposes.
2) Even the Effective Topos still depends crucially on the category of sets (it is an enrichment of Set) - so it does not seem to provide in itself an alternative foundations for mathematics. However Fourman and Scott's theory of Omega-sets may provide a different perspective...
3) This needs to be checked, but I think that for a general topos with a Natural Number Object we cannot prove that there is no mono m: P(N) -> N. But for a Grothendieck topos the situation may be different...

A nice application of Grothendieck toposes is provided by Moerdijk and MacLane's treatment of forcing models in Sheaves in Geometry and Logic. I like to think of the underlying poset P as a kind of training data (partial information, not always consistent) and the construction of the sheaf model Sh(P) for the dense topology as analogous to models constructed by machine learning (a kind of passage to the limit, extrapolation and smoothing out of the data of P).

We have written in Ocaml a minimal proof assistant based on the logical system presented by Fourman in his paper The Logic of Topoi:

https://github.com/owl77/Intuitionistic-Higher-Order-Logic-and-Topos-Theory

Note on Large Language Models

LLMs have a certain analogy to compression. From the training data D we obtain a LLM T(D) which is supposed to contain (or "extract") the essential "information" or "statistical patterns" present in D. T(D) is much smaller than D. It is speculated that Claude models are trained on D of the size of a petabyte and that the models themselves range from 150 to 500 GB. The response to a given prompt is analogous to decompression. Supposedly T(D) can "generate" an approximation of all the information originally contained in D. Some questions:

1. Is it not true that the passage D -> T(D) is not lossless, that important information present in D is lost in T(D) and cannot be recovered by it?
2. Is there any way to study T(D) as a mathematical object, detect its structure and geometry? And to study likewise the correspondence between D and T(D)? If there are limitations to doing this are they practical or theoretical?
3. There is an analogy between passing from D to T(D) and passing from general to countable models of ZF set theory (which exist by the downward Löwenheim-Skolem theorems)?
4. Is there not some analogy between forcing using countable models and generic sets and the process of training to generate T(D)? In both cases there is pattern generalization from fragmentary data.
5. Is there any structural correspondence between the structure of T(D) and structures found in the world (not counting neurological analogues of MLPs)?
6. Can we construct toy universes, toy languages and toy training data and study how D -> T(D) works in this simplified idealized scenario to gain more insight regarding real world LLMs?
7. Do LLMs express an essentially emergent phenomenon in which hardware capabilities are a crucial factor? Can we formalize rigorously such a concept of emergent phenomenon or capability?
8. But most importantly LLMs are linear statistical predictors (next token predictors) and they are trained as such. We need to formalize clearly what LLMs are supposed to do in the first place. Suppose we have a (first-order) model M that represents the world. We want our LLM T to be able to deal with a good degree of approximation with the theory of M, Th(M). We are given a finite large set L of first-order formulae with probabilities of their belonging to Th(M). A transformation is applied to L to obtain the object T which is able to include the reliable part of L in Th(M) and to extrapolate to other elements of Th(M). Is this to be understood as both logical and statistical inference?
9. A LLM is just a finite state automaton. But recursively axiomatizable theories are in general not recursive. Can we can construct a theory T such that for any finite subset L of T all LLMs trained on L will err to an arbitrarily with regards to infinitely many sentences of T. We define metrics on expressions, that's the key.
10. And most importantly: are LLMs analogous to syntactic (and algebraic) models used in logic and category theory?  Or the training data is like the a poset P with the dense topology and the LLM is like the topos of sheaves over this site?

11. Do LLMs function essentially by analogy, metaphor, induction and extrapolation? This is an old idea in AI. 

New section in "Hegel and Modern Topology"

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

The topological dynamics of plant physiology

We have mentioned Goethe's method in the context of the history of phenomenology as well as its connection to the modern topological and geometric theory of differential equations and dynamical systems. Anyone who has studied biology, or plant biology in particular, learnt about the systems of nutrition and transport in vascular plants, the two main types of vascular tissues being xylem and phloem. The first system is easy to understand enough (and its analogy with animal systems obvious) and not incompatible with a general Goethean conception of a plant with its vertically rising gradient. But phloem is a different matter.  Topologically phloem tubes - transporting sugars and amino acids synthesized in the leaves - are spread across the entire plant in a structure analogous to animal circulatory systems - a perplexing situation which begs for deep topological and dynamical questions and explanations. For if we consider the the system of phloem tubes to be topologically equivalent to a closed ball $B^3$, then the circulation of nutrients will define a vector field on $B^3$ which is tangent to its boundary $\partial B^3 = S^2$.  So there must be sources and sinks. We can ask: where are the sources and sinks in the plant? Is there an analogy with animal digestive systems - or even neural pathways? And even more importantly: does this vector field change in time (undergo a continuous bifurcation)? Modern plant biology of course allows us to answer these questions. The point here is the great philosophical and methodological importance of asking precisely these kinds of question and the role of topological-dynamical thinking.

A point about subjective idealism

Consider the opening sentence of Schopenhauer's famous book: "Die Welt ist meine Vorstellung/The world is my representation".  How can subjective idealism seem at once so utterly certain and so dubious? The answer is quite simple and lies in the definite article "Die/the" in Schopenhauer's dictum. The world in the sense of the world in which each one of us is acquainted with through first-hand experience - my world that I take immediately as the world - this cannot be in reality any more than a modification of the subject, of consciousness, a self-projection, self-occultation of consciousness inside itself. Even from neuro-reductionist physicalist assumptions there is no way of avoiding this conclusion without dogmatism (the ruse of invoking the objectivity or public nature of logic or language) or non-physicalist explanations and we can turn Benaceraff's overrated objection against Platonism against the physicalist who wishes to refute subjective idealism:  basic neuro-physiology and gestalt psychology show that a direct causal connection between nerve impulses and the "causality" of the phenomenal subjectively constituted world - is sheer nonsense. There are physical impacts on receptors which do not produce effects on consciousness and there are effects on consciousness which are not the direct result of impacts on receptors.

The world in the sense of that world with which I am acquainted with through first-person experience, is entirely phenomenal and exists in consciousness only, despite its objectifying externalizing intentions and pretenses.  However from this it does not follow in the least that a world, any world, is my representation. It does not follow that outside my world there is not another world. Nor can we rule out that there could be some sort of cognizable connection between our world and such a world, the world which is not our world.  In Kantian terms there does not follow that there is no thing-in-itself and it does not follow that we cannot know anything about this thing-in-itself (we already know that we know that it does not necessary not exist).  Despite Kant's own arguments, we may hold that such a world-in-itself is perhaps a grey, timeless, purely mathematically structured world but in which other world-creating consciousness subsist. We can even discern such a conception in Hume's Treatise. None of our causal scientific theories or explanation depend on the phenomenal intuition or concept of time, only on a vicarious geometric abstraction, a flattened timeless time.

Consider the stereographic projection of a sphere into a plane. The sphere is the subject,  the tangent bundle of the sphere is the modifications of consciousness,  the plane is the postulated world outside consciousness.  

But is this not logical dogmatism where we endow logic, a product (or condition?) of consciousness, with the right to judge that which is beyond consciousness? If logic corresponds to the ability to understand, carry out and check rules and to meta-interpret rule-systems,  where resource constraints can be taken into account,  then perhaps continuous processes - as infinitary rule systems - transcend logic. But logic functions through necessary principles of meta-logical agreement - thus it is reasonable to endow logic with transcendental epistemic scope. If a derivation in one formal system has to be considered sufficient for the cognition of a non-finitary property of non-derivability in another formal system, then surely the belief in the extensive epistemic range of formal systems - of logic - is reasonable.

If the above considerations do not seem convincing, we can point out that the validity and transcendental scope of logic (the highest a priori condition) is an essential assumption in Hume (as we discussed in detail before), Kant, Schopenhauer and Husserl.  We can always ask Kant: what gives reason the right to judge itself and its own limits? Does not transcendental logic have to follow the rules of general logic as well (even if it includes important differences)?  And another important approach involves the formal verification principle: challenge the meaning-as-use ordinary language philosophers and neoscholastic dogmatists  to set up their theory of knowledge and mind in a formal system and then proceed to critique those systems using their own rules. The weak conclusion remains the same: either physicalism is false or (qualified) subjective idealism is true. The stronger conclusion is that if consciousness is essentially non-physical and non-local (or even trans-temporal) then this fact tends to strengthen rather than weaken the case for qualified subjective idealism. And if the dogmatists invoke a direct intuitive "given" or direct intuitive acquaintance with allegedly real exterior things then they have no right to discard the direct intuitions of consciousness being the fundamental reality that can be experienced during spiritual development and altered states of consciousness.

If we are certain that there are universal moral laws known a priori then it would seem to follow that we should treat phenomena which appear to us as being sentient beings as if they really were sentient beings (treat them according to the moral law),  regardless of the non-reality of their first-person phenomenal presentations. This is because we do not have nor could ever have certain knowledge that they do not exist (either entirely or in part) and secondarily because regardless of the situation the practice of moral deeds can only strengthen character and morality itself.  If a child is nice to its toys it will probably be nice to people and animals as well. The similarity of this argument to Pascal's wager is noted.

But subjective idealism is a misleading term because it suggest than an individual ego-subject, finite personality or self-in-the-world is at the same time some kind of substrate of the world. But in reality cosmo-ontology and egology are illusions to be dissolved together. This is the task of transcendental dissolving and liberating insight,  true philosophical phenomenism/positivism/empiricism and skeptical dialectics.

Also considerations on temporality, finitism, mereology, the mysterious nature of concepts and meaning - and in general intrinsic intentionality, synthesis, constitution and constructivity - applied to both the self and experience  - are quite sufficient to establish the above liberating insight with regard to the egological and cosmo-ontological illusion.

How do we interpret quantification from a finitist perspective? How can we develop intentionalist finitism? What does alleged arithmetic knowledge mean from a finitist perspective? Was Hilbert really defeated? Was Leibniz actually a finitist?  Can finitism offer new perspectives on the theory of knowledge? Can we solve the problem of the stubborn non-finitism of category theory? Do we need a new interpretation of Carnap and Russell?

Can statements about numbers involve an infinite number of relations to actual numbers? What is the significance of Ockham here? Can psychologism be revindicated?

The new synthetic a priori principle: finite derivations must ratify and represent an indefinite but bounded set of truths. This is applied also to specification and verification of programs.

And yet, what are logical, semantic, phenomenal, physical atoms? What could they be? Can paraconsistent and infinitary logic be justified in the framework of our "Analyticity, Computability and the A Priori"?

Very short note on formal concepts in science

From whence do we get the impression that some mathematical models are closer to physical and spatio-temporal intuition,  more down to earth and intimately tied to concrete applications, while others hover close to the heights of allegedly less useful  'abstract-nonsense' ? The real numbers and differential equations, these are seen as tied to dynamic-geometric intuition and of vast applicability and interest in engineering and science.  But abstract algebra and category theory appear to have no direct relevance to applicable mathematical models or to their kind of concrete geometric-dynamic intuition. We present here a few short speculations.

Perhaps the abstract models do capture fundamental levels of reality (maybe even more fundamental than the so-called concrete spatio-temporal ones) and what is required is first of all the methodic development of a special kind of intuition or cognition to grasp the planes of reality which they model (maybe something like Goethe's method is called for?). And then further models are required which can effect a mediation or transfer between these two levels, or at least allow a continuity or gradual deformation between concrete spatio-temporal reality and higher levels of reality.

A beautiful illustration of such a meditation and construction is furnished by the double fibration in Penrose's twistor theory which allows a mediation and transfer between physically significant objects on Minkowski space and abstract cohomological objects on the complex algebraic variety $\mathbb{P}^3$.This operation allows solutions to conformally invariant differential equations on spacetime (e.g., Maxwell's equations, Yang-Mills, or linearized gravity) to be identified exactly with Cech or Dolbeault cohomology classes on specific regions of twistor space. This appears to have been Penrose's attempt to give a geometric (topological) interpretation of quantum non-locality. 

Another illustration is the theory of $\mathcal{D}$-modules which presents a mediation between derived categories (and monoidal categories) and concrete models of systems of partial differential equations.

The idea of Kant's schematism of the pure concepts of the understanding can be interpreted as finding the mediation between logic and geometry (including mathematical physics). This is of immense contemporary significance as we find versions of Kantian schematism in topos theory, homotopy type theory and in areas involving monoidal categories in algebra, geometry and physics and in linear logic and computer science. 

There is also apparently a mediation and transition between the theory of bifurcations of smooth vector fields, fundamental  and ubiquitous in concrete applied mathematics,  and monoidal category and operad theory. See also Physics, Topology, Logic and Computation: A Rosetta Stone by John C. Baez and Mike Stay.

An objection can be raised that our concrete allegedly intuitive mathematical models are themselves quite abstract but are only perceived as immediate and intuitive in our present cultural context by a deliberate forgetting of the complexity of their cognitive-historical past and genesis. For instance real numbers are constructed from the rationals by quite abstract, cardinality increasing, procedures involving equivalence classes and identification. And the same goes for the concepts of continuity, differentiability and so forth. Surely the ancient Greeks had other forms of intuitive perception and concreteness in their geometry, physics and engineering. We answer this objection by our arguments that the fundamental dynamic and geometric-topological intuitions of modern science were in fact identical to those of ancient Greece. It is the modern foundations in terms of Dedekind cuts and $\epsilon$s and $\delta$s that can be critiqued from a philosophical and logical point of view and alternative foundations (locales, realizability topoi, synthetic differential geometry, homotopy type theory) can be defended which are also more aligned to the theory of space and change found for instance in Aristotle's Physics.

Meditation, phenomenology and obfuscation

There is an important missing link in the history of philosophy which has significant consequences for the history of phenomenology. This is the work of the young Rudolf Steiner. It is not fair to judge this work by his later abhorrent and manifestly false views as an occultist and cult leader any more than it would be fair to judge Heidegger's Sein und Zeit by his later similar political and cultural-anthropological views. The most important work of the young Steiner (who attended lectures by Brentano) is a work which usually goes under the title of Philosophy of Freedom (1894) but was published and revised numerous times under different titles.  Along with this work there is also much value in Steiner's earlier work on Goethe's scientific method and worldview. 

There can be no doubt that The Philosophy of Freedom, like Sein und Zeit, is a very important work and one of the masterpieces of phenomenological philosophy. In this work (and in portions of his later books such as the one on "Initiation") we find something similar to an exposition of the approach and methodology involving the transcendent self-transparency and self-illumination of thought and consciousness which we have repeatedly attempted to expound, the correct combination of "epokhê" and "satipatthâna".  

Equally important is Steiner's Goethean method of variation and "imagination". This seems to us to be the original and correct form of what later was called the method of "eidetic variation" (and the affinity for Ruskin's approach in Modern Painters seems to have gone unnoticed).  But the consequences are enormous. The combination between the philosophical-spiritual insight offered by Steiner's exposition of the Goethean method on one hand and on the other the development of modern geometric and topological theory of differential equations and dynamical systems would have been a "marriage made in heaven", two complementary aspects of the same essence,  whose union would have furnished the key to a powerful new science.

But the harmful influence of 20th-century materialist cults and their pseudoscientific dogmas (Freudian psychoanalysis, evolutionary psychology) hindered the emergence of such a science. This is the tragedy of René Thom whose work flawed by Freudian and materialist-evolutionary dogma did indeed point to such a qualitative topological science with a method based on developing higher dimensional dynamic and geometric intuition. The fact that Thom never mentioned Theodor Schwenk's Sensitive Chaos: The creation of flowing forms in water and air (preface by Jacques Cousteau) which predates Structural Stability and Morphogenesis can only be seen as an example of intellectual dishonesty. As far as I know, Thom never referred to the wonderful book by A.T. Winfree, The Geometry of Biological Time (1980) which shows how advanced topological concepts in applied science can be presented in an accessible, rigorous and clear way (cf. the explanation of phase singularities on p.26).

A major fault of René Thom was also ironically to be found on the mathematical plane. Thom, like V.I. Arnold, engaged in obfuscation and sloppy presentation of mathematical ideas.

An expert in a field has all the right to develop a conceptual shorthand for their own use or that of fellow experts. In particular, a shorthand for writing down ideas to be developed and unfolded in lectures. Such texts can also employ beautiful drawings and diagrams which may impress even the non-expert. However all this has nothing to do with writing a book whose aims is to open the gates of a knowledge to those who, while not having such a knowledge, seek earnestly for it. A book that is a genuine introduction (i.e. initiation) in the full sense of the word, and which does not only "bring in" the neophyte but presents the beauty, clarity and order of the temple in all its details and adornments.  This is the kind of book, the kind of spirit, style and approach, which every student deserves, and certainly the kind of spirit. style and approach which should be employed when presenting important philosophical, scientific and mathematical ideas to the world. To present the young student with any other kind of book can only be considered a pernicious and immoral act of deliberate obfuscation and perversion. On another occasion we hope to develop the topic of cryptography, deliberate obfuscation and "secrets of the trade". Let it just be said that this has traditionally been a not infrequent vice of academia, the deliberately ponderous, opaque, pedantic jargon which not only hinders understanding of comparatively simple concepts and arguments but also masks the author's debt (or even plagiarism) to their contemporaries or predecessors. The aim is also to appear outwardly as outstandingly intelligent, original and sophisticated and to close the doors of science to anyone outside a select few.

Besides in phenomenological philosophy and Goethean science, a tragedy occurred regarding "meditation".  There is no such thing as "meditation" in the popular sense, there is no secret powerful technique of self-help where one sits down, closes their eyes, breathes in and out, focuses their mind or attempts to detach themselves from troubling thoughts, and thus finds a cure for mental anguish or a way to gain paranormal faculties or blissful states. This goes in particular for everything connected to the term "mindfulness".  And there is evidently no easy quick philosophical-phenomenological method for the self-observation and self-transparency of consciousness.

This is not what "bhavana" or "katharsis" as in neoplatonism and the ancient mysteries was about. Rather bhavana can be likened to the building of a beautiful high-towered palace or temple, including the preparation and purification of all the materials needed as well as the ground and foundations. The "mindfulness" at work here cannot be separated from action. The popular concept of "meditation" actually only corresponds to the final royal stage in which one enters the newly built palace and climbs up many flights of stairs to survey the beauty of all the sparkling rooms and luminous windows until reaching the most wonderful of all views at the top. Before that it is the long toil and active work of the artisan and soldier - which can be be interwoven with periods of peaceful surveillance and contemplation of yet unfinished work. We do not meditate as some kind of easy, all-sufficient, way to overcome our desires, rather we first diligently and actively uproot desire-images-complexes in order to attain the peace and illumination of "meditation".  Using a software engineering metaphor, we must start with cultivation at the abstract object-oriented level before proceeding to reverse engineering at the assembly level. Thus taught both the Pali suttas and Plotinus.