How can philosophy and science be a source of joy?

Philosophical and scientific work are a source of joy as long as one remains conscious of its ultimate goal and applications relating to the improvement of human culture and society leading to it being based on the omnipresence and supreme value of ethics and justice - the equal dignity and inviolable rights of all human beings and animals - and  to the promotion of a spiritual culture in which each individual does not separate their happiness from that of all other sentient beings,  and last but not least,  the unveiling of the truth regarding human history and prehistory. 

A Python tool for testing normalization in Natural Term Logic

https://github.com/owl77/natlog/blob/main/ntltest.txt

This tool allows you to interactively reduce NTL terms and to ultimately check that the normal form is independent of the path chosen. In the example above we start from 

$\Pi^{(1,1,2)} \Upsilon^{\{1,4\}\{2\}\{3\}} \Pi^{(1,3)} \Pi^{(1,1)} C D D D D C B C$

and we derive the normal form 

$\Upsilon^{\{1,5\}\{2\}\{3\}\{4\}} \Pi^{(1,4)} C \Pi^{(1,1,1,1,1)} D \Pi^{(1,1,1,1,1)} D C I I I I I I I I \Pi^{(4,1,1,1,1)} D \Pi^{(1,2,1,1,1)} D B C C I I I I I I$

by following two different paths.  Here the primitive terms $B,C,D$ have arities $1,2,5$ respectively.

The program is run from the command line : python -i reductions.py and in general one needs to specify the primitive terms and their arities through the command AddPrimitive(name, value). But $B,C$ and $D$ above come by default.

Formal models of consciousness and other matters

From the previous considerations are we able to answer the question: can there be a formal model of consciousness?  This depends on how we understand "consciousness" and how we understand "formal model".  Ordinary human consciousness is just a segment of a vast multi-tiered system of spiritual consciousness of which it is but a small part and emanation - and yet to which it is potentially and essentially identical. Thus if by consciousness we mean the intelligible realm then there can be no direct formal model as we have demonstrated previously.  There can however be a symbolic illustrative mathematical model of the intelligible realm. And also a formal logic and computation based formal model of dialectics which models reversion and the procession of the logoi whereby the human soul reverts and ascends to a higher state.  We do grant that it is conceivable that strictly at the level of soul and its logoi as they unfold and project into the imagination or act upon extension in nature there could indeed be a direct (rather than merely symbolic) mathematical model. Here that our ultrafinitist philosophy of mathematics and theory of analyticity and computability would enter the stage.

We propose the beautiful notion of finitary Turing completeness. Rather than attempt to give a complete formal definition we can give the following preliminary definition for board-games. A board-game is finitary Turing complete if its canonical, most natural or free extension to an infinite (to us indefinite, indeterminate) version of the board is Turing complete. In this sense it has been shown that Chess and Go are finitary Turing complete. Finitary Turing complete systems capture the essence of certain stages of dialectic and the structures and processes of the logical-discursive cognition of the soul. We note that the game of Go has been used traditionally (specially in the Ch'an and Zen tradition) to illustrate profound psychological and philosophical principles. 

If some of the previous considerations were focused on the schema of Hegel's Science and Logic in relationship to the Enneads,  no less important (and indeed more foundational) is the first-person introspective of the phenomenology of consciousness, not only as deployed in Hegel's Phenomenology of Spirit (but in an inverted and distorted way, to be sure), but in the entire first-person, introspective, spiritualist, idealist,  phenomenological and epistemological tradition of philosophy which only achieves its complete validity and goal through spiritual development and transformation (cf. chapters X and XI of Augustine's Confessions and the introduction to Ennead IV.3). 

ζητεῖν τε τὰ ἄλλα καὶ εὑρεῖν βουλόμενοι δικαίως ἂν τὸ ζητοῦν τί ποτ̓ ἐστὶ τοῦτο ζητοῖμεν, τό τε ἐραστὸν ποθοῦν λαβεῖν θεαμάτων. 

Our goal is thus to develop the correct neoplatonic phenomenology of consciousness which expresses the souls' self-knowledge and concomitant re-ascent to the intelligible realm and beyond, and to engage critically with the posterior western phenomenological and proto-phenomenological thought. Mathematics and formal logic can also play an important role here but from an essentially different perspective than in the previous goals. Here we are interested in genesis and constitution, with ultimate foundations and philosophical logic and with the act and effect of logical-dialectical practice.

Note that we must distinguish how intelligible categories unfold and originate sensible categories as it happens in itself and as discursive reason grasps it as a reflection.  But this reflection at the level of the logoi is also part of the dialectical process which leads to the soul's illumination and re-ascent. 

Plotinus starts his investigations in the difficulties related to souls in IV.3-5 with a very non-solipsistic question indeed, regarding the relationship of each soul to others souls and with the world-soul. IV.3.5-6 is of great interest in that the text points to a plurality of noes inseparably united in the total nous. Souls are called the logoi of the nous which represent a stage of evolution/unfolding. Thus each soul has its own particular nous within itself  (IV.3.6 17). Individual souls (note also the remarkable passage in IV.7.14 on the souls of animals) proceed from the world soul at the same time as their proceed from their individual nous while the world soul proceeds from the global nous and is its logos. In order to illustrate unity-and-multiplicity (the soul is the body like light in the air) and the relationship between individual souls and the world-soul, Plotinus states that the same thing can be at different places at once: just as one indivisible soul can be simultaneous present and acting in different parts of the body in different ways so too the world-soul is one and yet completely present in different ways in each individual soul (cf. IV.9). One can use holomorphic functions to symbolize or illustrate Plotinus' theory of how the soul relates to spatial extension or the world-soul to individual souls. If $f$ is holomorphic in a region $U$ then the germ of $f$ at any point $p$ of $U$ completely determines the whole function on $U$ (through the uniqueness of analytic continuation). Thus we can say that f itself (the formative aspect of soul) is virtually entirely present at each point $p$ of $U$ (the body) and at the same time f is spread out through $U$, unfolded into $U$, by taking germs.

In IV.3.9-11 we read that the world-soul is the author of life and mediator of the intelligible realm from which proceeds matter, space and extension and which has received the  logoi spermatikoi from above which she transmits to nature and gives life to all beings and mediates also the soul's re-ascent to the intelligible realm. Later on in the same Ennead the concept of kosmos as a complete and harmonious whole and providence are discussed. The world-soul is also called universal life.  The etheric/astral bodies (called okhêma or vehicle) are mentioned showing that this was not  theory specific to later neoplatonism.

But Plotinus' phenomenological psychology (about sensation, kinds of memory, reflection, reason) can be found for instance in IV.3.25-32 and the following part of IV.4 and IV.6 as well as the key treatise I.1. There in internal textual evidence (in IV.3.25 Plotinus states that memory has been deal with elsewhere and extensively discussed, using the verb θρυλέω)  - as noted by A. H Armstrong in the Loeb Edition of the Enneads, that Plotinus wrote a now lost extensive treatise on memory or perhaps this represented a strictly oral teaching. IV.3.26 30-55 is an important passage wherein Plotinus expounds the purely spiritual active nature of memory and self-consciousness (contrasting to a being in the state of flux), making use of key terms: sunaithesis, parakolouthesis, sunthesis and sunesin. Without negating external sensible objects we can nevertheless consider that Plotinus attains a transcendental phenomenological consciousness in this passage in which consciousness reflects on itself and perceives itself as an independent, free, self-acting, self-representing, self-reflecting, self-conscious being. There follows a discussion on the phantastikon and  antilêpsis and their relation to noesis and memory. In the remarkable section IV.6.3 is developed the refutation of the theory of perceptions and memories as the result of passive imprints and the establishment of the theory that they result from the essentially active role of the soul, the deployment and training of attention (epibolê, prosbolê): directedness, aboutness or intentionality.

From the Lankavatara Sutra

Lankavatara Sutra, Chapter 3, XVI (Tr. Suzuki)

What is meant by an eternally-abiding reality? The ancient road of reality, Mahamati, has been here all the time, like gold, silver, or pearl preserved in the mine, Mahamati; the Dharmadhatu abides foreover, whether the Tathagata appears in the world or not; as the Tathagata eternally abides so does the reason (dharmata) of all things; reality foreover abides, reality keeps its order, like the roads in an ancient city. For instance, Mahamati, a man who is walking in a forest and discovering an ancient city with its orderly streets may enter into the city, and having entered into it, he may have a rest, conduct himself like a citizen, and enjoy all the pleasures accruing therefrom. What do you think, Mahamati? Did this man make the road along which he enters into the city, and also the various things in the city? Mahamati said: No, Blessed One. The Blessed One said: Just so, Mahamati, what has been realised by myself and other Tathagatas is this reality, the eternally-abiding reality (sthitita), the self-regulating reality (niyamata), the suchness of things (tathata), the realness of things (bhutata), the truth itself (satyata).

The Logic of the Enneads and its mathematical reflections - preliminaries

Plotinus was also a consummate logician in the sense of universal logic, ontological logic or logical ontology.  Buried deep and intricately interwoven within the Enneads we find a sophisticated Science of Logic in the sense of Hegel but expressing a radically different spiritual and philosophical framework and outlook. We will not discuss here the interesting question of a direct influence on Hegel or of Hegel having directly and extensively appropriated neoplatonic doctrines in his own system (we have already given some references concerning this topic). Nor will we discuss the radical rupture in the philosophy and spiritual practices of the neoplatonic school  that occurred after Iamblichus.  We can only stress again the importance of expounding systematically a non-Proclean and indeed anti-Proclean neoplatonism.  

 The project we sketched in our paper "Hegel and Modern Topology" was to given a Hegelian interpretation of key aspects of modern mathematics, logic and computer science and to in turn use these to furnish an interpretation of Hegel (the hermeneutic virtuous circle). We find that most of the insights contained in what we sketched of this project can be transposed from Hegel's Logic to Plotinus's Logic where they seem to find a even more adequate and natural setting. But it must be understood that, as we explained before, in this framework mathematics and formal structures function essentially as an elaborate system of symbolism and hierarchical reflection rather than as candidates for a direct and complete formalization.  The other aspect of mathematics and logic, culminating in dialects to be used as a tool for direct spiritual ascent, has to be considered separately.  

In order to embark on this project we should list the fundamental categories, concepts and processes of Plotinus' Logic which we can organized somewhat along the lines of Hegel's Science of Logic but with fundamental differences which are often the complete inversion of Hegelian conceptions or which collapse what Hegel separated or separate what Hegel joined. Bearing in mind Hegel's Logic of Being we can remark that Plotinus has an elaborate and deeply interesting theory of quantity, extension, number, quality, essence, determination and measure (i.e. the interaction between quantity and quality found in VI.1-3, VI.6 and II. 6) which permeates in the Enneads (and justifies our classification of Plotinus as a finitist) - together with a remarkable theory of relation which when deployed at the intelligible realm fits exactly our considerations of groupoids as models for essential relationality and interconnectedness.  The theory of a self-moving number and relationality of the nous fits well with our previous considerations on Galois groups and Galois extensions. Also the intelligible realm can be symbolized or illustrated as the category of relations.

We note that Proclus's theory of Henads is found in VI.6 but with the difference that for Plotinus they are a proto-determination in Being (as an infinite ouflow) which preceed the intelligibles rather than the One. For a form to exist as a distinct entity it must be bounded, structured, and distinct from other forms. Substantial number is the internal articulation that establishes this multiplicity within unity. In Plotinus’ view, number precedes being: intellect must possess the principle of number to differentiate being into distinct forms. Thus Proclus got his theory of henads from Plotinus' theory of essential number.

 And also a highly developed and articulated theory of infinity (outflow, power) and limit (reversion, theoria)  and a theory of "matter" (both physical and intelligible) which amounts in the physical level to the poorest and emptiest determinations of being (strongly echoing the initial stages in the Logic of Being). Indeed in II.4.9-10 (cf. also VI.6.2-3) Plotinus invites us to perform an abstraction of the quantity and all qualities of a body to arrive a the concept of indetermination. Matter has existence and a disposition to become other things. Intelligible matter is equated to being and intelligible apeiron (the infinity of the One) and physical matter to non-being and to physical apeiron.

Plotinus states explicitly that indeterminate matter will display contradictory dualities - a striking possible interpretation of quantum theory which becomes less paradoxical once we abandon a foundationalist reductionist framework for a top-down emanationist one. 

Plotinus' theory of quantity and quality agrees admirably with the framework of modern topology (he recognizes geometry as a kind of quality in quantity) and differential geometry specially as applied in field theory and differential equations. Indeterminate matter first must receive extension and it is only upon extension that quality can be received. This corresponds exactly to the primordiality of a base space (space-time manifold) relative to its fiber-bundle or sheaf which expresses the local physical-phenomenological properties of the field (Plotinus considers intensive quantities like temperature to be qualities). This lower kind of qualitative structure in turn is only the product of a higher structure and power. For Plotinus space (extension) itself is a product and emanation of higher structure just as a non-commutative $C^\star$-algebra may have within itself various commutative (maximal) subalgebras.  

Thus sensible non-essential quality corresponds closely to Hegel's concept of measure, as quantity determined by something from without and as qualified quantity. The first section on Quality in the Science of Logic corresponds rather to the genesis of quantity in the intelligible realm and soul, where we deal with united multitudes, intelligible forms and their relations and overflow and limit (and we can include indefinite quantity here). Hegel's subsequent treatment of quantity and measure corresponds to sensible quantity and quality in Plotinus and is quite adequate for the mathematical illustrations in Hegel and Modern Topology.

Intelligible quantity exists as pure number and structural order (the archetype of limit and harmony. Intelligible quality exists as the specific power and form (eidos) of an Ideal entity.  At the higher intelligible level, quality and quantity do not conflict or merely sit side-by-side; they are unified aspects of the divine Forms. When projected down into the physical world, they split into extended magnitudes (sensible quantity) and superficial properties (sensible quality). In II.4 we have that pure matter has no magnitude or spatial extension of its own. Extension (megethos) is a form or quality superimposed onto matter. In VI.4-5 dealing with omnipresence we read that the Soul is not spatially spread out or extended. Instead, physical extension is a limitation, whereas incorporeal reality possesses a unified, non-particulate presence that contains spatial extension without becoming spatial itself. In III.7 we read further that just as time is an unrolling or diastasis of the eternal life of intellect into succession, physical extension (megethos) is a dispersion or unfolding of unity into divisible parts.

A fundamental mathematical symbolism or reflection will be that of the limit (let us say, in the category theoretic sense) of a sequence or diagrams expressing an indeterminate outflow or process. This limit is a reversion, completion, perfection and theoria in the Plotinean sense and an aufhebung and being-in-and-for-itself in the Hegelian sense.  We can think of this process as dynamic and limit and reversion as a fixed point (cf. Lawvere's account of Gödel's incompleteness theorems and diagonalization arguments).

The reversion of the physical word and the human soul to the level of Soul and Nous parallels to a certain extent to Hegelian levels of Essence of Concept (for instance the supreme genera in the intelligible realm include identity, difference and in VI.3 rejects that essence or ousia could apply to the physical world) but with many radical differences and fundamental corrections from a neoplatonic point of view. The physical plane can be considered in itself or as it is the result of the activity, influence and emanation of the the plane of soul taking account its act of reversion (i.e. Objective Concept, the theory of life) - and the same goes for the the plane of soul itself. Thus Subjective Concept can correspond to the theory of the soul's logoi, to the theory of the world soul or to such as it relates to the intelligible realm and to the intelligible realm in the aspect of knowledge. Hegel's progression from outwards qualities to essential qualities in Subjective Concept finds its parallel in VI.1.15–22 which distinguishing true intelligible qualities (forms or reason-principles - logoi) from mere sensible qualities (passions, colors, surface shapes).  For Plotinus universal Being (i.e. the Absolute as the perfect emanation of the One) is Being, Intelligence and Life and the perfection of energy/act/power.

When using mathematical structures as neoplatonic symbols and reflections we find that it is best for formulate them and think of them in a strictly constructivist way.  Thus the number of objects in a category must be conceived as being indefinite and defined as one would do in a dependent-type based proof assistant - considering it to be the object object in the definition of an internal category.  The finitist can consider intentionally the sequence $\frac{1}{n}$ as computable process without having to admit the finished extended set of such numbers. And then the finitist can consider $0$ as being its limit.

The considerations presented above are very sketchy and highly insufficient. We point out the the deep exposition of Plotinus' logic can be found in VI.2 and VI.6. The five supreme genera suggest the fundamental importance of the equality type and transformations which preserve structures and express coherence and relational unity or interconnectivity.

Identity is related to stability, difference to movement. Consider the two possible orientations of a vector space. They are indistinguishable, intrinsically identical yet different. Their difference is expressed by a non-trivial automorphism of the underlying vector space which transforms one into the other.  

Plotinus states that number is produces by a combination of movement and rest (as supreme genera) and that extension is posterior to number and is the result of a movement produced by infinity which is halted and limited by rest. In VI.6 we have the theory of multitudes as outflowings and divisions of being. In VI.6.1 we read that if the parts maintain some connection then we have extension which have lost perfection must receive the form of beauty. The negation of negation is the reversion of the outflow through limit.  Negation is related to set-theoretic complementation (for instance it is the interior of the complement for the Heyting algebra of regular open sets of a topology). Thus the dyad, the unlimited, indeterminate outflow, considered in itself independently of its source can be characterized as negation. The negation of negation (a reversion of the outflow) corresponds to a closure operator which encompasses in a limited determined way the object which was initially negated. If negation corresponds to the sun's unlimited rays then double negation corresponds to a halo around the sun's disk.

The remarkable sections  VI.2.4-12, which deduce the supreme categories through an anagogic process of turning inwards, the key concepts being soul, the logoi. one, ousia, life, intellection, reversion, movement and stability, identity and difference, are important for comparison with the Logic of Essence and Logic of Concept and well as for pointing out the radical difference with Hegel. At the same time Plotinus' henological ontology and the deduction of the other genera from the supreme pentad in VI.2.13-21 should be compared to the processes and structures of Hegel's Logic of Being. We note the remarkable fact that VI.2 (which employs many argumentative structures interesting from the point of view of formal logic) can be seen as realizing the idea of dialectic which Plotinus sketched in I.3.  In VI.6.10 we read that Being became Number when it became a multitude, having  had a pre-formation, previous representation,or  allocation for the beings it produced. We note that Augustine copied VI.6.12-13 almost directly. A section in VI.6.3 should be compared to Hegel's absolute negativity in the beginning of the section on Essence: τὸ δὲ μὴ ἀποδρᾶναι ἔχειν, εἴργεσθαι δὲ ἔξωθεν καὶ κύκλῳ καὶ μὴ ἐξεῖναι προχωρεῖν, στάσις ἂν εἴη: ὥστε μὴ μόνον ἐξεῖναι κινεῖσθαι λέγειν.

We note that a fundamental mathematical illustration of the genesis of sensible quality will be that of the unfolding of a singularity (or deformation of a structure). We must bear in the mind that intelligible extension is expressed as the intensive power of a intelligible form which then unfolds and expands spatio-temporally in the world-soul. Or the forms can be symbolized by infinitesimals as we developed in "Hegel in Modern Topology" - and indeed in V.9.6 Plotinus gives us an interesting illustration involving the logoi and a seed.

Note the Hegelian categories of ground and appearance, phenomena, etc. find their treatment in Plotinus theory of sensible substance which is a shadow produced by the Soul casting form onto receptive, featureless matter. Physical substance is a composite (synolon) of matter (hyle) and dynamic immanent form (logos or reason-principle). Physical motion is the life and process of change inherent in time and physical nature. It is the physical shadow of the eternal activity (energia) of Intellect.

Consider an object $A$ in a category. Then $A$ may have internal symmetries in the sense that $hom(A,A)$ may contain more than just $id_A$. Given $f \in hom(A,A)$ we can ask if there exists $g : A \rightarrow B$ and $h : B \rightarrow A$ where $B$ is not isomorphic to $A$ such that $f =   h \circ g$.  This offers an illustration of $B$ being an emanation of $A$ (flowing out in $g$ and reverting by $h$) and proceeding from $A$'s internal structure.

Svetla Slaveva-Griffin, Plotinus on Number (2009)

https://bmcr.brynmawr.edu/2010/2010.02.17/

Ennead VI,6, which deals with Plotinus' philosophy of number, is a very difficult text to understand.  However the text offers prima facie strong evidence for enlisting Plotinus squarely in the ultrafinistic and constructivist camp (not only for the essential qualitative numbers but for their monadic quantitative emanations as well).  We need to investigate how the second kind of number may be traced back to the first kind. We suggest this be effected by viewing the sequence of natural numbers as embodying increasingly rich qualitative and relational properties and use this as a basis for a spiritual exercise. That is, we view the sequence of positive integers as embodied in the corresponding sequence of finite groups and their group rings over finite fields,  finite fields (for powers of primes), their irreducible polynomial and primitive elements and their matrix rings and finite projective planes (for numbers of the form $n^2 + n + 1$) which has a connection to the game Spot It!, general finite rings and other finite algebraic and combinatorial structures, different kinds of automata, algorithms, games, graphs, formal grammars of classical languages etc., and visualize the unfolding of their structure and relations. Note that the finite necessarily algebraic extensions of finite fields have cyclic Galois groups.The metareflection principle (which includes the principle of induction) expresses an ascent and return and conversion. Of great interest are the finite-field techniques used by the Singular algebra software, including Rational Reconstruction via the Extended Euclidean Algorithm (also known as the Farey Map). So, contrary to certain opinions, the finite fields $\mathbb{F}_q$ can indeed be considered as good approximations of $\mathbb{Q}$. Note that Book X of Euclid is a structural study of towers of quadratic field extensions over the rational numbers $\mathbb{Q}$.

Let the following postulates be assumed in common for the totality of the objects of mathematics: that their mode of existence is incorporeal and selfsubsistent, holding an intermediate rank between indivisible beings and those that are divided about bodies, both as forms and as reason- principles, being assigned the middle status between the simple and the divisible, being purer than the latter, more variegated than the former; that they make use of composition and division, but oversee the synthesized and the divided without coming to be and eternally; they are inferior to intelligible beings but prior to natural ones; they are in beauty, in order, and in exactitude superior to things visible but inferior to the intelligible, and similarly they have an intermediate symmetry and compatibility; they have the power to transport and lead over to the indivisible forms, since they are akin to them, and they lead away from corporeal concerns those who have become accustomed to them, converting them to the divine beings, as if on a ladder leading up to the heights. - Iamblichus, De Com. Math. Sci.

But we must emphasize just how profoundly obscure, opaque, incomplete, vague and metaphorical all the above neoplatonic and neopythagorean accounts of mathematics are, cosmo-ontologically and with regards to spiritual realization - and even to basic epistemology and psychology.

While mathematics is placed at an intermediate rank, we can certainly understand their theory of how mathematical objects operate in nature and at the same tie subsist in the human mind (but then they must be finite in number). We can also to some extent understand  how, in this framework, the study of mathematics is spiritually beneficial, helping to energize the soul and turn her away from lower passions, sense-impressions and opinions. However it remains totally unknown and incomprehensible how mathematical objects or theories could be considered lower-level emanations of the intelligible realm and what their higher principles and prototypes could be.  And beyond the basic understanding of the cathartic and anagogic function of studying mathematics outlined above it remains totally incomprehensible just how mathematics (or what branch or version of mathematics, ancient or modern) is to be studied as part of a spiritual path leading to the intelligible realm and exactly how mathematics could effect such an ascent and reversion. Our authors speak in hints and riddles and vague but attractive metaphoric suggestion. 

Here is our proposal for a solution. 

1. The neoplatonists and neopythagoreans did not grasp Plato's genuine thought concerning the cathartic and anagogic function of mathematics (such as expounded in Book VII of the Republic). Plato is quite clear that the key role here is not to be played by mathematics but by dialectic, by pure logic. The key spiritual role of mathematics is as a starting point from the exercise of pure logic or dialectics. Just as the visible triangle is a tool for grasping the idea of triangle so too are mathematical objects and theories the tools for the self-revelation of pure logic. The fatal flaw of neoplatonism, as mentioned before, is not taking up the most sophisticated developments of formal logic and dialectics inherited from the Stoics and Megarians (and attested by Galen and Boethius). That is, they did not comprehend or develop or deploy Plato's hymn of dialectic. This only is the anthos tês psychês which can lead the soul back to the unity of the intelligible realm. Dialectics is both the most sophisticated  formal logic (which realizes logicism by extracting theories from pure thought and analyses hypothesis while being beyond them or extracting them directly from itself) and pure power and energy (pure general fluid intelligence and concentration) which creates and dissolves, separates and combines - and we may interpret the cryptic account given by Plotinus himself in Ennead I,3. Dialectic is the infinite power which dissolves all concepts, reveals their hidden layers, tensions and suppressed genesis, reveals their mutual dependency with other concepts and then builds them again on a higher level by combination and fusion. Its goal is to obtain a vision of their perfect holistic circularity, co-dependence and mirroring of all by each.  This is why it is important to understand the deep spiritual meaning and function of the dialectics of the later Academy, the Megarians, the Stoics, Pyrrho and Sextus - in light of the correct spiritual understanding of Nagârjuna, Yogâcãra and other dialectical Buddhist traditions as well as their Vedantic counterparts. The finitism and constructivism above must be understood as furnishing the clearest, cleanest and richest ground for dialectics to reveal itself and blossom. But note that the range of dialectic transcends mathematics and encompasses all concepts and theories. 

2. The relation between mathematical objects and mathematical theories and the intelligible realm must essentially be one involving symbolism, analogy, reflection, and the same goes on a higher level for the processes of dialectic itself. But like all symbolism the correspondence is always limited and restricted to certain aspects.  Plotinus in Ennead VI,6 speaks of the infinite as the incomplete, indefinite, which is bounded by limit. So completion, passing to the limit, can be seen as a finitization (compactification), a reflection, a return - an deepening and internalization of the previous spontaneous outflow. Nous is not created through a single, static event, but rather exists as a dynamic two-stage emanation from the One: an unformed "outward" movement (the Unlimited) followed by a reflective "inward" turning back (the Limit). Of importance is the contrast between a static extensional and a dynamic relational global representation of a mathematical object (for instance symmetry groups or in particular Galois groups which measure the algebraic relationality between the roots of a minimal polynomial defining a finite normal field extension).

Finite projective plane with $13$ points and $13$ lines, $13 = 3^2 + 3 +1$

 

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.

On p.xliii of the Introduction Brittain remarks that Cicero plausibly deliberately mistranslated the original Greek technical terms so as to conflate katalêptikê and katalêptos, the phenomenon as such and its intentional objective correlate. This gave rise to a misunderstanding which would lead naturally to Augustine's reaction in the Contra Academicos.  

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)

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) 

Hegel the Consummate Neoplatonist (Philip Stanfield, Marxist orientation)

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.