Light on the Philosophy of Consciousness

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

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

A new critique of pure reason

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

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

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

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

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

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

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

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

Hegel and Neoplatonism

Platonismus und Idealismus by Werner Beierwaltes

Hegel und der spätantike Neuplatonismus by Jens Halfwassen (2005)

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

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

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

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

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

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

Theory of Meaning

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Recursive Finitist Mathematics Program

Pauca sed matura. (Gauss)

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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