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.

The Recursive Finite 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! 

Mathematics and AI

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

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

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

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

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

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

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

I.R. Shafarevich on the definition of mathematics

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

- I. R. Shafarevich  

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

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

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

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

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

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

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

Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark

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

https://arxiv.org/abs/1003.4394

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

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

Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos

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

https://arxiv.org/abs/2106.07890

Emergent Analogical Reasoning in Transformers (2026)

Emergent Analogical Reasoning in Transformers (2026)

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

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

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

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

On Various Translations Between Classical, Intuitionistic, and Linear Logic

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

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

Investigations into the Idris2 proof assistant and some formalizations

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

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

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

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

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

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

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

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

Critique of Quantum Mechanics and Quantum Field Theory

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Foundationalism and Logic

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

A thesis on generative AI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note on Large Language Models

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

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

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

New section in "Hegel and Modern Topology"

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

The topological dynamics of plant physiology

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

A point about subjective idealism

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

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

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

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

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

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

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

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

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

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

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

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

Very short note on formal concepts in science

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

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

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

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

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

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

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

Meditation, phenomenology and obfuscation

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

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

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

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

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

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

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

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

Unwanted popup on this blog

Unfortunately the theme of this blog employed an outdated and suspicious external script from polyfill.io. This script has been removed. 

From phenomenology to a philosophy of consciousness to spiritual science

Alas there is no clear definition of what phenomenology is (which should not to be confused with the specific philosophical work of Edmund Husserl) or even what phenomenism, empiricism, positivism and the theory of knowledge exactly are.  There is no clear definition of what the philosophy of consciousness is or how it could differ from the philosophy of mind. And there is no clear idea of what a spiritual science could be or a science of consciousness. We are in possession however of the bodies of knowledge encompassed by the various branches of psychology and cognitive science - and there is surely much of value there.

In our writings we have emphasized a few important points relevant to the entire interwoven enterprise of all these subjects. The centrality of first-person experience, its descriptive insight and analysis and most importantly:  there can be no genuine progress without a radical and thorough form of personal transformation aimed at at self-transparency and self-transcendence of consciousness. We attach importance to the first-person experience of the body and the various feedback mechanisms involved, specially those related to sensory systems and the peripheral nervous system.

Is there any relevance or legitimacy in the idea of mathematical (or formal) models of consciousness? 

This question, which we will not attempt to give any definite answer for now, seems related to the circumstance of there being two distinct paths (but not necessarily non-complementary or non-convergent)  passing through all the subjects mentioned above, whose ultimate aim is not only spiritual science but the attaining of ultimate spiritual knowledge. We might characterize these two paths as pertaining to the Buddhist and the (neo)Platonic philosophical and spiritual sciences - it being understood that these terms should be read in a very universal sense encompassing in particular a huge range of philosophical literature from antiquity to the present, both from the east and the west. Both their differences and their common core (we have written on the connection between the Platonic-Socratic dialogues, Pyrrhonism, Damaskios and the Pseudo-Dionysian literature) are important.

The differences manifest in the kind of mathematical or formal model of consciousness we could associated with each one of them. The (neo)Platonic path would seem to point to geometric and topological models of consciousness (due to its synthetic, global, anagogic methods) while the Buddhist path (analytic and actively illuminating through insight) would prefer event and process-oriented computational models. Its western embodiment has great luminaries such as Sextus Empiricus, Hume, Kant and Brentano. Dialectics is found in both paths but with different modes of employment and emphasis. For both thorough psycho-somatic peace, calm and relaxation is a precondition for any progress.The dichotomy between the paths is not meant to be universal and disjoint as it is easy to find traditions that sit somewhere in the middle or incorporate core elements of both approaches (Yoga, Samkhya, various schools in the Upanishads and Bhagavad-Gita).

About the geometric-topological model of consciousness. This resolves around the ancient notion of topos (cf. Aristotle's Physics Book IV), the modern concepts of stratified space, and the granting legitimacy to traditional cartographical, architectural, horticultural and other metaphorical language regarding consciousness and the diverse faculties and regions of the soul (cf. the ars memoriae).  Its foundation is  the work of Plotinus (which integrates the best elements in the Platonic and Aristotelian texts). There are also interesting elements from neuroscience and psychology that can be integrated into such an account.

We can cautiously suggest that Plotinus' process of spiritual simplification, illumination and reversion (we would say rather 'involution' and 'inversion')  involves a radical shift in the self-perception of consciousness and at the same time a radical shift of perspective regarding what is perceived and conceived as spacetime and the laws of physics. We propose that this transformation, this shift of perspective, involves the Penrose transform of complexified (and compactified) Minkowski spacetime. The new reality is twistor space $\mathbb{P}^3$ and the previous Minkowski spacetime is something secondary and generated (its points correspond to lines in $\mathbb{P}^3$). As regard to the hyperboloid hypersurface which appears naturally in twistor theory (it corresponds to the image of real Minkowski space and stratifies twistor space), we can recall this verse by Alan Turing:

Hyperboloids of wondrous Light
Rolling for aye through Space and Time
Harbour those Waves which somehow might
Play out God's wondrous pantomime. 

 

The initial process of complexification and compactification can be likened to unveiling a more fundamental ontological domain, the intentions, meanings or processes of the world-soul at work in physics - which corrresponds to a parallel process in consciousness. Fiber bundles are perhaps a natural mode for the human mind to grasp high-dimensional spaces. The isomorphism of the fibers at different points represent their ubiquity and unity in the Plotinean sense.

 

Spiritual science will involve a geometry and cartography of  consciousness employing a stratification into spaces with their own dynamics and deformative processes which become aligned within a global anagogic process.  However we must begin with the most humble processes related to the psychosomatic domain and their geometry and dynamics. The geometry of biophysics furnishes the clues and can be transposed - through a process of variation, complexification and compactification -  into the domain of a geometry of consciousness. The branches of mathematics which offer a unification, fusion, of logic and geometry, are surely of great relevance here. 

 

There are several interrelated geometries of the mind. The geometry of the architecture of our cognitive-semantic spaces. The geometry of our personal narrative (our biography) and of narratives in general, specially the initiatic journey. The geometric physiology of the soul's powers and functions. An important aspect of such geometries is not only their stratification, multidimensionality, dynamic nature, relation to projective space,  but also their foliated nature, all forms manifest as a simultaneous united continuum of many different levels. An opposing view would see all this as a cultural inheritance of forms of illusion (prolongations of a standard naive pre-biochemical understanding of biology), of aggregation and identification and put in its stead Hume's sublime concurrent process vision of consciousness. We distinguish between objects and beliefs constituted by function (mistaken for essence) and true essence revealed in the reality of the underlying concurrent processes which do not reveal themselves necessarily as they are when they construct belief-objects subordinate to function.

 

But perhaps indeed the vision between the two paths is not so significant in comparison to what they have in common - the science of the subtle and spiritual bodies. Also recall also how Hegel and Husserl would hold that, as strange as it may seem, Plato, Sextus Empiricus and Hume  present complementary aspects of the one truth and are capable of unification at a higher level. But also postulating unity can be harmful at the level of spiritual paths, contrary to a common trend. Rather one must be lucky to find a path that is 'beautiful in the beginning, beautiful in the middle and beautiful in the end'.

 

Before the question "how can we gain higher knowledge of consciousness itself?" we should ask "what is the best knowledge for consciousness?"  It is the knowledge that lets it see through itself and liberate itself from itself. A central object of the best knowledge is the difference and unity between perception, feeling, volition, thought and the sense of self and awareness. Also of temporality and temporal intentions and the construction of objects and concepts. And is not logic the science of consciousness par excellence? And even more importantly: the best knowledge is not just knowledge of the highest object but the best knowledge is the knowledge of the best knowledge itself in its illuminating and transcending function. Consciousness must be unbounded light and its knowledge be knowledge of its own knowledge and of its own knowledge's illumination. Recursion and reflection-into-self permeate the Pali Canon and its path of spiritual development. Consciousness lives usually passively immersed in and hypnotized by its own self, forgetting its own power not only of freeing insight but its ability to act directly upon itself, dissolving the mistaken belief in the necessity of certain mental content.

 

And we can give a symbolic-anagogic formalization of consciousness properly understood, or at least partially mirror its self-clarifying, self-transcending, self-subsuming process. We have sketched some of this in our "Hegel and Modern Topology".  We could add thereto some considerations on monoidal categories (which furnish a common ground for linear logic,  concurrent processes, algebraic geometry, PDEs and quantum theory). What is important here is the internalization of their own "boxes" into wires (input and output) allowing the modelling of dynamic rewiring (closed monoidal categories).

 

About the coherence conditions in monoidal categories. The coherence condition can be viewed as the necessary amnesia which forgets the history of a process to look only at the final state and  conditions the connection isomorphism only on the final result (cf. Kant's example of surveying a house in the CPR). This is what allows objectivity to emerge: objectivity is the postulated forgetfulness of the plurality of different presentations and the postulate of their essential equivalence. As Aristotle wrote: the road from Athens to Thebes is the same road as that from Thebes to Athens. Coherence expresses the emergence of objectivity (hence order) by a voluntary forgetfulness of a system's memory via focusing only on the final state. The condition of (shallow) objectivity is that something is what it is independently of its process of coming to be.  If you start with a certain composition of $\otimes$-products X and switch them around using tensorings of ids and associators to obtain a new composition of $\otimes$-products Y then the resulting composed isomorphism linking X and Y will always be the same, no matter the intermediate route. Without a partial repulsion of memory one will be constantly becoming something different - this act is also that of the Understanding in Hegel's Logic that forgets, attempts to cancel,  its own process of coming to be.

Short note on philosophical methodology

Cultivating what we should call not transcendental consciousness, but transcended consciousness, the true epokhê in the school of Pyrrho and Hume, we realize the following.

That the formal mathematics and formal philosophy projects and the philosophy developed in our paper Analyticity, Computability and the A Priori are themselves perfectly legitimate forms of transcendental philosophy and neutral superpositivism.

That the purpose of life is to die well, finis vitae bene mori. 

Transcended consciousness is not only the power of being beyond time and its manifold constructions, but to have the liberty and freedom to concretely imagine and to virtually live consequences and alternative possibilities and to change perspective regarding a given contingent time.

Thus desire is dissolved by the simple transcendental consideration of: what if I had that which I am now desiring? Thus the taking-for-granted, the masking, of the good and blessings we have in the present and the twisted mirage of having a better past are dissolved by the recognizing that we will almost always live in our past, in the golden age, relative to the total temporality of our life. 

Thus we may long to live again in the world before the advent of LLMs.  But if we had the total perspective of how the world will change in our lifetime, we would have greater appreciation of living in the precise present that is present to us.

Yet more short considerations on AI

The philosophical significance of LLMs and a potentially powerful philosophically based critique of LLMs have not yet been developed and perhaps their importance has not even been recognized yet.

Could LLMs be tied to a certain philosophical view regarding the mind, language and the world? And if such a view is manifestly erroneous could not this furnish a sound basis for acknowledging - alongside numerous other reasons - the social and cultural harm of LLMs?

For instance, could we not explore the relationship between LLMs and Quinean extensionalism and meaning-as-use theories? Are not LLMs based on the rejection of the irreducible intensionality of human symbolic activity? Behind every symbol there is an intension. And a formal theory of intensionality must itself acknowledge the intensions of its meta-symbols.  But there is no intensionality in LLMs beyond that of the humans involved in their creation. Searle's Chinese Room ignores the deeper philosophical meaning of computation - or a potential associated geometric theory of meaning - but may be of interest for a critique of LLMs.

Do LLMs compute? Computation is a primordial intensional human activity (see our paper Analyticity, Computability and the A Priori). 

Perhaps there are other machine learning models of language of a more geometric or even combinatorial-algebraic nature which would have interested Riemann who wanted to relate meaning to a kind of cognitive geometry.   Perhaps statistical regularities should be traced back to geometry.

What are LLMs really, formally? Can we formalize the critical values wherein they become 'adequate' for their proposed task? What exactly in 'large'? How can this be formalized rigorously? Can LLM techniques be used for formal axiomatic-deductive systems and automated theorem proving? Or can we prove certain fundamental limiting theorems about the powers of LLMs akin to the unsolvability of the halting problem and Gödel's incompleteness theorems?

LLMs only exist because of the Internet. The Internet and LLMs are part of the same historical-cultural-technical process.  This ontological process might be described as the datafication of humanity. Language ceases to be a tool of human thought, communication and culture-building but rather a tool for the reduction, degrading, emptying, perversion and commodification of humanity itself. The Internet and LLMs are the anti-Gutenberg. Man has become text, sound, image, data, statistics. LLMs are a counterfeit reality, a monstrosity, the world becomes one big corporate controlled screen.

 If we compare the training data and the resulting LLMs is there or not a loss of information? Would it not be more worthwhile to develop sophisticated search algorithms and querying language to access the training data directly?

 LLM culture is the culture superficiality, atomization, banality, cosmetics and deception (a LLM is almost a trained deceiver in the biological sense - it detects and mimics human patterns). There is a loss of the multiple layers of meaning behind every symbol which cannot be reduced to statistical correlations with other symbols. Meaning is replaced with arbitrary social-statistical emergent correlative patterns.

The realm of pure mathematics - and that of pure logic, combinatorics and computability - is a pure realm which LLMs cannot touch or corrupt.  So the formal mathematics and formal philosophy projects, contrary to popular misconception likely resulting from deliberate propaganda and deception - are the antithesis of and antidote for LLM culture. There is a pure universal computational-mathematics-akin language (far beyond the natural language or the audio-visual data  that can be perverted and imitated by LLMs) and a pure logic and a pure thought and mankind may indeed hope to attain them. 

LLMs are not intelligent and statistics cannot solve formal computational problems nor can they encompass the pure a priori synthetic principles of formal computational systems (i.e. the cognitive certainly of the foundational principles for metatheoretic knowledge) as detailed in our paper mentioned above.

No statistical pattern analysis of the shadows are sufficient to lead to knowledge of the object. Pixel injection in image recognizers demonstrates this fact, and similarly for formal reasoning.  Besides lacking intensionality, LLMs lack reference and context (despite the misleading terminology of context windows). 

And most important of all since the true intellect is inseparable from morality, empathy and compassion, completely beyond the reach of LLMs. LLMs do not have the bondage to an illusion of a self.

A major task of philosophy is to destroy the evil empire of LLMs and a good starting point is deconstructing and refuting the worldviews (extensionalism, meaning-as-use) which LLMs embody. 

Is a lawyer someone trained in a specific system of laws or someone who has developed the skill to study, interpret and apply any given system of laws or perhaps be able to cope with significance changes in present laws?  Such a metalawyer is the analogue of a universal Turing Machine. A LLM could never be a judge or a metalawyer - it could not grasp the spirit of legal institutions or the deeper meaning of a given legal context.

Billions use light-bulbs without understanding the underlying physics. Billions could use LLMs thinking they are conscious.

LLMs will become more interesting in the measure in which the multilayer perceptrons models are replaced by geometrically and mathematically more sophisticated models (KANs are a step in the right direction).

Short philosophical considerations on AI

Hegel and Heidegger were thinkers about their own time, thinkers about historical events and happenings. Few have the insight and courage to fathom the full depth of the meaning of an historical event, the coming to be (coming of age?) of an historical process. Tragically,  it is only some time after the event (the time of monsters?) has hit humanity with full force that Hegel's famous owl can spread her wings. Is it not true that some of what the prophetic author of Sein und Zeit wrote about technology only makes full sense at the present?

This seems to us particularly true of the emergence of the age of the Internet and the age of generative AI which is its logical development.

And yet nothing could be further from our own philosophizing than any form of historicism or historical philosophy. As such both Hegel and Heidegger, for all their interest and insight, must be considered as having crafted systems based on an incorrigible error. 

Social progress is not a law of nature but a legitimate hope - even if at present it seems a distant one - and it is our moral duty to work towards it in the midst of uncertain outcomes.

The advent of the internet was the advent of connection between people. This connection carried rhetorically moral undertones and echoed enlightenment ideals about the desirability of sharing and making knowledge available.  In the present age of the generative AI based Internet powered and controlled by corporations and governments aligned to anti-enlightenment ideals, it may be that it is morally called upon us to practice instead the process of disconnection and the purification and preservation of knowledge(not obviously in the sense of the 'great simplification' of the Canticle of Leibowitz).   

The most basic step is ensuring locality of core information. That is, to be in possession of machines onto which have been downloaded significant portions of Internet Encyclopedias (despite their serious shortcomings) as well as some decently performing LLM. To this we add, it needs not be said, massive of digital preservation of human cultural artifacts, notably libraries.

One can use Kiwix and download for offline viewing the most recent English Wikipedia (50GB text-only 150GB with pictures).  With a AMD Ryzen 7 5825U processor with 16GB RAM and 2GB Radeon Graphics one can use Ollama and download and use some decently performing LLMs (gemma4 comes in E2B, E4B, 31B and 26B A4B). 

Most living beings alternative between states of being awake and of sleep. Can would we design a dynamic LLM which similarly alternates between states of user interaction and re-training based on this interaction? The most important being the correction and/or updating of knowledge or perhaps the removal of harmful and biased content and "thought patterns". If LLMs can improve then it is not only  a question of having the number of parameters equal to the number of neurons of the human brain.

Computers can enhance and aid human cognition as well as hinder and destroy it (there is a growing body of evidence concerning the disastrous effect of excessive or inappropriate generative AI use for individual mental health and cognitive development, not to mention for society as a whole). 

But the harms of generative AI have little to do with lesser-known extremely powerful and beneficial aspects of the computer for human cognition. We cannot go into this in detail here. Let it just be said that it involves using adequate software for the rigorous formalization of human scientific theories and concepts (specially logic and mathematics and formal methods in the sciences) and the vital feedback-loop between human thought and the software interface (IDE) which results in the simultaneous enhancement of human understanding and production and the quality of the software-based formalization and implementation. 

The software in question includes not only Rocq (formerly Coq), Agda and functional programming language but such languages as Python, Javascript and C/C++. Python is a multi-paradigm and highly versatile language with an elegant syntax. Python comes close to achieving the ideal of a universal language in the Leibnizian sense and is a wonderful tool for formalization, implementation,  verification and exploration in a variety of areas in mathematical logic and finite mathematics. 

We note also the importance of minimalism (using as few dependencies as possible) and building things from the ground up - this goes for scientific and philosophical projects, not of course for commercial and industrial ones.  We will address in the future the question of the possible role of machine learning in this process.

The geometry of cognitive biases and fallacies

 

Categorizations of objects are certainly a useful, even necessary, part of human cognitive development. We cannot underestimate the interest of so-called Venn diagrams which while going back to Leibniz arguably are strongly yet somewhat implicitly present in Aristotle's Organon. Venn diagrams can describe human categorical-cognitive architecture but with the caveat that we should be aware of the extreme geometric complexity of the sets involved as well as how a vast array of psychological tendencies and schemata are at work in shaping our perception and reasoning about such sets. 

Here is one example of cognitive fallacy (linked to geometric inadequacy) which doubtlessly occurs in political propaganda and statistically flavored pseudoscience. We have two categories of human beings A and B (the universe U can represent a given human population). A can be a desirable or desirable trait while B can represent a self-identified human group. The situation in the diagram above represents a situation in which for a standard notion of probability we can say both that 'most As are Bs' and yet  'most B's are not A's. A typical fallacy (perhaps linked to a tendency for the mind to generate a symmetrical geometric scenario) is to assume that if  'most As are Bs' then most 'Bs are As'.   We can however say that B is over-represented in category A and seek a sociological explanation. In some instance a category may appear to be defined precisely by its intersection geometry with other categories.

The above considerations show that while categories are useful cognitive constructs one must be aware of  fallacies and biases - some linked to a poor grasp of geometry - and never lose sight of what is cognitively of far greater importance: the individual.