Natural Term Logic

https://www.academia.edu/143539685/Natural_Term_Logic

https://www.researchgate.net/publication/394776125_Natural_Term_Logic

Finitism and resource consciousness

In formal logic and theoretical computer science not enough attention has been payed to the explicit study of resource limitations and their implications (cf. the study of the length of proofs).  We should reform these disciplines by explicitly postulating finite bounds in every aspect of the object of study and attach importance to the quantitative study of the interdependence of these bounds.  For example: study the length of proofs in function of the formula to be proved in system based on a finite language. Study resource limitation in the grammar of natural language. Study the computational capacities of automata seen as finite approximations of Turing machines. This may be a way a developing abstract theories with a higher degree of fundamenta in re.  Study the bounds and limitations of encodings of information and attempt to understand the question: can there be provable mathematical statements which have proofs which are too big to be written down or processed by a computer given our physical limitations ? Are there numbers to big to be represented by any means in the physical universe ? Finitism means not only finitely many basic or 'atomic' elements but finitely many higher-order relations between them.  This all is related to the mysterious problem of qualitative changes in scale (and also the relativity of scale as illustrated in Gulliver's travels).  Remaining in the finite domain by increase the quantity of a certain parameter (size, speed, etc.) suddenly beyond a certain limit the behavior of the system can change radically.  This is usually studied using the infinitary models of mathematical analysis and geometry, but surely this occurs in a finitary  context at a basic level.  The question of finitism is related to some of the most fundamental questions both of phenomenology and the very concept of 'analysis' and 'scientific consciousness'.  An authentic phenomenology must give a central place to the concepts of 'illusion' and 'delusion' as well as the two-pronged nature of the 'social' and the 'cultural'.  

To understand the synthesis of phenomenology, logic and analytic scientific consciousness (in a playful symbolic way we could say: the synthesis between Hume, Sextus and Democritus - and we would add Epictetus for moral theory)  it is good to look at the structure and dynamics of the theories of biology and biochemistry and their surprising correspondence with certain aspects of contemporary mathematics.  The analogy between a tissue and its cells and the definition of algebraic variety (or better sub-analytic objects or general stratified objects) which has both a 'smooth' and 'extensive' aspect and a local, discrete, algebraic 'intensive' aspect.  The algebraic locality is much like the discrete biochemistry of a cell and homology theory a kind of genome or centrifuge of the algebraic structure (so too are graded ring constructions).  What is lacking is of course the dynamic, transformative aspect.  The connection between biochemistry and logic (or between processes and inferences) is profound and Girard explicitly acknowledges this in his foundational papers on linear logic. We propose that the basic local-global (sheaf theoretic) concepts of geometry be applied to logic as well.  We have an organism made up from local logics considered as individual cells. For each cell hypothesis are what are given from without and what is deduced (asserted) is what exits the cell. The difference from standard sheaf (and topos) theory is that we have a dynamic global aspect related to (rapid) transport and distant interactions (cf. the $\pi$-calculus).  This TeX package is interesting because it bridges the gap between logical (and linguistic) syntax and chemical syntax by developing a linear system to represent two and three dimensional chemical diagrams (just as logical expressions codify syntactic trees).

If reality is represented by $U$ then a 'perspective' or 'approximation' or 'abstraction' or 'construction' is a system of relational logical atomism $A \rightarrow U$.  These perspectives can be organized by a partial orders $A < A'$ signifying that $A'$ extends $A$ or that $A$ abstracts from details of $A'$. A very interesting aspect of this abstraction are the convenient fictions of mathematical analysis, the 'passage to the infinite', seen also in statistical mechanics and the kinetic theory of gases. Some comparisons might be made with Jain logic. The theory of scale is fundamental here.

All formal systems involve the characteristic of proliferation, generativity (cf. our previous discussion on semi-Thue systems).  This is also omnipresent in algebra and geometry. The philosophical and scientific ideal of a 'system'.  This reproduces fundamental characteristics of mental processes (note how Chatterjee's book on the Yogacara makes a connection to the broadly Hegelian subjective idealism of Gentile).  But this generativity is limited by resources and is also error prone. Pyrrho and Sextus have a radically distinct approach and goal. 

Addendum: Boethius and the medieval tradition support out interpretation of a topic as a 'maximal proposition', i.e. as an axiom.  There is an interesting notion of definition as an 'unfolding'. We should mention Boethius' 'hypothetical syllogisms' in our note on Kant's logic.  Can we find where Boethius is more explicit about a universally quantified conditionals ? Note that his indefinite  propositions is our indefinite article selector and that singular propositions are introduced. Disjunction seems to have been studied from a profoundly mereological perspective (see also Kant and Hegel) which echoes our treatment. 

Another view of philosophy

Philosophy is divided into pure grammar, ethics and liberation practices (eleutherology).  Pure grammar holds that the main legitimate domain and goal of philosophy is the description and direct awareness of the deep structures and processes of natural language (and this includes exploration and classification of the total semantic universe).  It is called 'pure' because this is to be done without any extraneous reductionism and ideological, theoretical presuppositions or interpretation, anything beyond the domain of the analytic a priori computationalist  'core logic' discussed in our previous work. Pure grammar criticizes standard philosophy (from its historical roots onwards) for its shaky or questionably linguistic confusions and simplifications (rather than taking the formalist logical-dialectical approach we have discussed in the past). There is also the problem of ideological distortions and arbitrary dogmas both in their views and applications of language, specially the most recent physicalist and behaviorist ones (and this applies equally to many contemporary linguistic theories with physicalist, behaviorist and social-pragmatic premises, or the dubious theories which reject meaning altogether.). Though of course historically many philosophers in some of their works have approached to realm of pure grammar (Aristotle's Topics, the Stoics, Buridan, Leibniz, Peirce, Frege, Montague, etc.).  Some of the great linguists are revealed to have been great philosophers (Pânini,  the (nava)Nyâya school, Saussure, Anna Wierzbicka). It is also 'pure' because it claims to be (or shows itself to be) cognitively and epistemically a priori and more fundamental than all the sciences: pure grammar is 'first psychology' and 'first cognitive science' (of course we distinguish between surface and depth grammar and bring to the foreground the plurality of natural languages and the study of universals). Pure grammar shows the limits of the world (without making claims about the world) and its shows what must and can be transcended in order to go beyond the world. Pure grammar challenges not so much the idea of formal logic in itself but the faulty paradigms which dominated the extensionalist logic of the 20th century (in particular its inability to distinguish between distributive and merely intensional quantification). Pure grammar  challenges both dialectics (Madhyamaka, Hegel) and idealism (Yogâcâra, Plotinus) - while at the same time ascribing a certain value to both of these (and more can be said in relation to eleutherology).  Perhaps Ch'an (Zen) - which we might call linguistic buddhism (the practice of the Koan) -  can be seen as related closely to pure grammar in its eleutheric aspect. Pure grammar should not be confused with the ideas of Mauthner, Wittgenstein and 'ordinary language philosophy'  - certain superficial similarities mask a profound, radical and irreconcilable difference and opposition (there is no question of pure grammar attempting to reduce language and meaning to any alleged social act, game or toolset: also a language need not be in perfect working order): a task of pure grammar is critical analysis and refutation of such schools. The same goes for structuralism (including the Prague School),  'grammatology' and postmodernist linguistic philosophy.  Comparison with Husserl's Logical Investigations (and to Bolzano, Lotze, Brentano, Meinong,  Twardoswski, Mally and specially Marty)  is an interesting and involved subject that is well worth exploring.  An important work for pure grammar is Bernard Pottier's Linguistique Générale (1974). Also Jean-Louis Gardies. Of interest is also medieval scholastic logic and grammar.

Wilhelm Wundt - Psychologismus und Logizismus (1910)

Some of our preliminary studies for our project of a 'pure grammar': Bealer's Intensional Logic, Aristotle's Second Order Logic,  On Analyticity and the A Priori.  It is important to compare the theory of definition in the Topics with modern 'analytic semantics' and its sememes. Furthermore, our papers  Differential Models, Computability and Beyond and Hegel and Modern Topology can be considered essays on regional ontologies and the pure categories of the understanding.

Pure grammar: an extended universal philosophical semantics and grammar as transcendental knowledge in the Kantian sense. Pure grammar is about seeing, about coming to awareness. It has no method. And it has nothing to do with social acts, behaviors or games. Learning language is a linguistic activity and is not a game or an action (for games presuppose language).  We cannot act without language (or logic). Sneezing is not an act.  Philosophy does not result from linguistic confusions and misapplication: rather the vice of philosophy is simply the ignorance of the importance and complexity of the pure linguistic a priori.

Pure grammar puts forward linguistic (arti)facts which being surveyed automatically  lead to pure knowledge.  That is, the artifacts are presented which disclose immediately cognitive-linguistic-semantic structural symmetries and dualities and triads - as well as proto-spatial and proto-temporal a priori categories. Thus pure grammar contains a kind of pure  a priori immediately evident geometry.  Pure grammar has not as primary aim the foundation of science but rather eleutherology. Kant anticipated much of contemporary linguistics.

The recursion of subordinate clauses is limited in natural language (maybe maximum three levels before intelligibility is lost ?).   We should investigate how subordinate clauses can be interpreted or transformed away.  This is of course intimately connected to intensional logic.  John believes that Mary believes that John believes something about her.  Can we transform this multiply embedded subordinate clause sentence into a sentence without subordinate clauses ?  John believes something about Mary. That is something Mary could be considered to believe.  And John believes that Mary believes that. This is of course just a way of disguising nominalization of sentences through backwards pointing demonstrative pronouns.

So, there is the triad of psychological (phenomenological) a priorism, logical a priorism and grammatical-semantic a priorism.  A particular kind of psychological a priorism is psycho-somatic a priorism with profound eleutheric consequences. As Wundt discusses in the text linked above, there is a deeply confusing entanglement, dance  and even contest between all the members of this triad. Which one, we can wonder, holds the key to a synthetic illumination of human existence ? To pañña and the overcoming of dukkha ?

The correct 'phenomenism': we note  that the naive, default world-view is actually 'idealist' in the sense that it is the result of an unconscious projection and constitution by an ego, a mine-making force and determination. Correct phenomenism  looks at things as they really are, looks at what really is in fact there and discerns the fundamental properties shared by the  things* that are there, not what is made from, by, in, or with them by the ego and mine-making force and its constitutive conceptual proliferation. False idealism attempts to logically and causally derive the world from the ordinary ego or subject (or will).  True phenomenism lets what really is shine forth as it really is and liberates from the world and from the world-tending ego and its superimposition, distortion  and positing.  Seeing psycho-physicality from a neutral third-person perspective. It is the coming to intimate yet detached awareness of the process of one's own thoughts qua such and merely as such as well as the perception of the world (and the mundane ego) as immanent and constituted in this thought-current (into which is woven sense-imagination). yogascitta vritti nirodhah. Correct phenomenism unveils the right reference point which synthesizes and makes a compendium of the sphere of core semantic categories of human existence (and hence thought and language and life). This is the critical 'phenomenology' we have discussed extensively in this blog.

*we do not mean 'things' in the ontological sense, but in the sense of mind-stuff manifestations.

Philosophy of quantifiers

Are quantifiers convenient fictions with fundamenta in re ? What does constructivism and dependent type theory have to say about this ? And functional interpretations ? 

Universal quantification is either an abbreviation for a expression of finite conjunctive knowledge or else something about concepts and not pluralities or extensions.

Universal quantification is determined by a computable function.

A universal Turing machine or equivalent machine (we will not discuss finiteness arguments here) is enough to check any proof or run proof-searches. And all machines imply human intentionality.

The first principle of synthetic a priori knowledge: that a specific computation of one machine can be taken as showing a certain (non-finitarily verifiable) property of the computations of another machine (for instance, output X cannot be produced from any input).  But this property itself may on the surface involve quantification in its expression. Again quantification needs to be seen as determined by a computable function or functional.

What is a typing judgment $t:T$ ? It is a statement that a given machine produces a certain output for input $t$ (inference) or $t$ and $T$ (checking).

Is there a particular philosophical interest in considering Boolean circuits or cellular automata (or neural network) models of universal Turing machines ?

It might be possible to elaborate an alternative version of formal logic, closer to the syntax and mechanisms of natural language and without any implicit extensionalismFF.  We call this system natural term logic (NTL).  The building blocks consists of primitive terms (denoted by capital letters) and primitive constructors (denoted by small-case letters).  To each term we associate a tag which can be $I$, $\omega$ or a number $n \geq 0$ and to each constructor we associate an ordered sequence of tags $t_1,...t_n,t$ denoted by $t_1 \times...\times t_n \rightarrow t$.  We form complex terms and associated to each of them a tag in the expected way. We indicate the tag of a term as $M^{(n)}$.  The most basic constructor corresponds to predication and is denoted simply by concatenation in some cases. There is also partial predication. The tag $i$ is for individual substances and $\omega$ is to indicate that a place in a constructor can accept any term. 

 NTL rejects unbounded quantifiers and does not presuppose extensionalism.  Universal quantification in its most basic form is given by a constructor $a$ of tag $1 \times 1 \rightarrow 0$. Thus for $M^{(1)}$ and $N^{(1)}$ we have the $0$ term (i.e. propositional term) $aMN$ which can be read 'all $M$ is $N$'. Likewise we a similar constructor for 'some'. Let $F^{(2)}$ be the primitive relation term of fatherhood. Then how do we form the $1$ term corresponding to having a father ? To we have implicit boundedness here too ? Before discussing this we can mention the we posit a class of constructors for relational terms which permute or diagonalize certain arguments.  But back to 'having a father'. We need a constructor $s^{1 \times 2 \rightarrow 1}$ such that $sMF$ would be read as 'being an M having an F'.  We can also deal with a unique existential quantifier.  

Can we conceive of a logic in which syntax and logic and intertwined (a way of dealing with non-denoting terms) ?

Notice the difference between relations and  (choice) functions.  We can take a binary relation $R$ and obtain a  (partial) function $cR$ which assigns each M to a possible F (if any).  This suggests that we should consider an alternative approach to NTL, closer to Church's intensional logic (but without variables and with bounded quantifier constructors) - or indeed to Stoic logic ! Each term will have a type. The genitive and other natural language constructions are clearly functional.  Thus for $M,N: \iota \rightarrow p$ we obtain $\Pi_1 MN : p$ and for $R: \iota\rightarrow \iota\rightarrow p$ we obtain $\Pi_2 MR : \iota \rightarrow p$, 'being the R of all M'.   We have a family of choice functions. For instance $cR: \iota \rightarrow \iota$  which corresponds to 'an R of something'  which is not the same as 'being the R of something'.  To define paternal grandfather we still need a combinatoric version of lambda terms (see our paper on Bealer's logic).

Being the father of someone's father, that is, $\lambda xy Fx\iota z Fzy$, which then is to be converted according to our formulation of Bealer's logic. The mechanisms of natural language for expressing multiple generality and logical constructions in general are wondrous and intricate.

Project: find a variable-free combinatoric version of our system ASOL  The metasyntactic operators can be expressed directly in the logic through the 'second-order' functions. 

Did the Stoics anticipate dependent type theory (DTT) ? If one argues that 'all men are mortal' would be implicatively expressed by the Stoics as: if something is a man then it is the case that that man is mortal - this corresponds to there being a term of type $\Pi$(x: man) mortal(x). This term is a 'function' which takes every man m to a proof of the proposition "m is mortal" (but this can also be another kind of type such as what is called a 'set' and even one that does not depend on $x$).  Of course there are other readings in dependent type theory. Anyhow so much remains to be understood about quantifications and the philosophical implications of homotopy type theory. 

If we take 'all men are mortal' as a function which takes each man $m$ into a witness or justification or proof or evidence or element in the set $mortal(m)$ which is a kind space for the mortality of each man.  Note that for adjectives such as 'big' this is necessary: a big flea does not use the same standard as a big house. The fibers have only an analogical correspondence. If we want to read a type for individual substances in an extensional way, the types for adjectives or qualities have to be considered otherwise. Notice how complex 'some A are B' is in DTT. 

See also our latest note on 'Natural Term Logic' (available on researchgate and academia.edu)

Commentary on the first book of Aristotle's Topics

In this note we assume the reader is familiar with the system of logic expounded in our paper 'Aristotle's Second-Order Logic'.  We refer to this system now as second-and-a-half order logic as it sits between second and third-order logic. In the above paper we argue that this is the natural logic to formalize Aristotle's philosophy. 

Chapter 1. Aristotle's definition of 'syllogism'  here is quite general (and should not be confused with the 'syllogisms' of the Analytics) and a good translation would be 'inference', the kind of inference represented by a sequent in the sequent calculus (with the cut-rule) with one formula on the right $A_1,A_1,....,A_n \vdash B$.  Aristotle's 'true' or 'primary'  things are sequence of the form $\vdash A$.

Chapter 2. But what is dialectic in the Topics ? Does it investigate the axioms of the particular sciences themselves (which cannot be investigated in those sciences) ? The passage 101b1-3 is mysterious.  Dialectic is a critical path having the 'beginning of all methods'.

Chapter 4.  The protasis. Each protasis indicates (is made up from) either property, genus or accident. Difference is strangely classed as 'pertaining to genus (generic)'. And   'problems' can be constructed from every protasis by changing the 'mode'.  This is very subtle and interesting point regarding intensional logic and, so it seems, a term-formation corresponding to interrogation.  

Chapter 5. This is a very important section.  We have a 'logos' which 'semainein' (signifies).  It is not clear if the logos is meant as a mere signifier or as the sign (signifier + signified).  Here logos is contrasted with onoma.  Apparently this corresponds to the difference between simple and complex terms. We see that there are definitions of complex terms.  An important question is: can we accept a definition consisting of a simple term ? Here Aristotle hesitates but admits that such protasis are at least useful for definition. A fundamental concept is that of 'antikategorein' (to be convertible with).  In 'Aristotle's Second-Order Logic' we argue that the Fregean distinctions between Sinn and Bedeutung  as well as between concept, object (and extension) are all present in the Topics.  A is convertible with B for Aristotle if A and B are predicates with the same extension (but not necessarily with the same meaning).  Aristotle's formula in 102b20-23 is : if (it) is A then (it) is B and if (it) is B then (it) is A.  Clearly definition and property have the same extension but different meaning (they both do not signify essence).  The rest of the discussion is valuable for elucidation of 'accident' and how it overlaps with relative (and temporary) property. The example of the 'only man sitting' (in a group) suggests a connection to definite descriptions of individuals.

In second-and-a-half order logic, all quantifiers should be bound - and how are we to interpret quantification, for instance in chapter 1 of the second book.  How did Aristotle distinguish between: 'all men are animals',  'man is an animal', 'animal is the genus of man', between extensional and intensional predication.  In the above chapter Aristotle seems to give the rule: from $\Gamma \vdash \forall_{x:Ax} B$ we can derive $\Gamma \vdash \exists_{x:Ax} B$.   In the beginning of chapter 6 of the second book we find the Stoic exclusive disjunction.  It is interesting that 'connectives' for Aristotle are just as much term operators as operators on propositions.

We are concerned with finding evidence for the natural deduction rule of existential quantifier introduction. For instance Aristotle explicitly stating that we can infer from Socrates is mortal that some man is mortal. Unfortunately direct evidence is lacking. Rather it would seem that a dependent, alternative, version of existential quantifier introduction is required.  We could reserve universal quantification for explicitly distributional, extensional quantification (such as in the the expression of the topics itself) and in other case use $\gamma$, etc. And $\exists^\gamma_A B$ would be third-order predicate meaning $\exists C\prec B \gamma AC$. Thus our new version of the existential quantifier rule (which is a topic discussed in book 2, 109b) would look like this: from $\gamma AC$ and $C\prec B$ deduce $\exists^\gamma_B A$.

Chapter 7. Peri Tautou (sameness, identity).  There is a kind of homotopy or qualitative sameness considered here and the example of the water drawn from a given spring is noteworthy. The  question of identity is examined in our paper and is crucial with regards to extensionalism.  What is Aristotle's 'arithmetical identity':  the name being many the thing being one (103a9-10).  Here Aristotle may be interpreted as postulating that identity is not a primitive notion but a polysemic and it to be defined in terms of either homonymity, definition or property  or even accident ! This rules out any extensionality (Frege's Law V).

Chapter 8.  Definition consists of genus and differences and these are said to be 'in'  the definition. There is still the question, however, of the precise relationship between difference and accident. 

Chapter 9.  For Aristotle 'kategoria' means 'predicate'.  What is the relationship between onoma, pros, protasis, logos and kategoria?  Category in the ordinary sense is actually 'genus of predicate'.  This chapter is very important and very subtle.   The ten classes seem to be genera both of predicates and of things in general  - an ontology.  In our paper (and in Modern Definition and Ancient Definition) we raise the question of the definition of objects that do not belong to the category of substance. Elsewhere we have inquired about Aristotle's view on statements of the form 'A is A'.  Aristotle is stating here that if the thing and predicate belong to the same class then we have an essential predication, otherwise we do not.  But how can we accept Aristotle's example of predicating 'man' of a given man being a predication according to essence ?  How can 'white is white' signify essence? 

Chapter 10. This chapter offers us the rudiments of a new kind of intensional logic: a doxastic logic, and is of considerable interest.  We can think of a modal operator Dox(P) satisfying certain logical rules. 

Chapter 13.  Differences of meaning of a term and a term qua term can be objects themselves of propositions.  To formalize the Topics we thus may need a third-order semantic identity relation.

Chapter 15.  It would be interesting to investigate formal systems in which each term is assumed to be interpreted as having possibly a set of references (and meanings) rather than one (or none).  This is the kind of polysemic logic that looms large in the Topics. A kind of semantic set theory, perhaps.  The task is to construct expressions which are singletons and to detect them within the formal logical and grammatical rules of the system. Aristotle must accept that there is a notion of semantic identity (which is not the same as that of 'antikategorein' or extensional equivalence).  We tried to formalize this notion in our second-and-a-half order logic. See the previous remark. 

The Legacy of Abel in Algebraic Geometry

https://publications.ias.edu/sites/default/files/legacy.pdf

We hope to share similar papers on Galois, Euler, Cauchy, Lagrange, Legendre, Bolzano, Hamilton, Gauss, Sophus Lie, Dirichlet, Dedekind, Grassmann,  Kummer,  Sylow, Liouville, Wronski, Riemann, Weierstrass,  Schröder,  Hecke,  Sofya Kovalevskaya, Hilbert, Poincaré,  Couturat, Hermite, Picard, Camille Jordan,  Poussin,  Felix Klein, Ramanujan, Hermann Weyl, Teichmüller,  Élie Cartan, Henri Cartan, Oka, Ehresmann, Pontryagin, Siegel, etc.  And also more recent mathematicians such as Smale, Thurston, Hamilton, Milnor, etc.

A philosophy of mathematics:

i) the importance of the rigorous logicist ideal of Leibniz and Frege (not to be confused with formalism)

ii) but equal importance to the training of (higher-dimensional) geometrico-dynamic intuition

iii) and preserving a connection either to philosophy or to applied science (and the unity of mathematics itself)

iv) and the dangers of wrong or faulty abstraction (not to be confused with good, natural or intelligible abstraction)

Thus the wrong direction or trends in mathematics has three aspects: the deviation into unintelligible and exaggerated abstraction, the loss of logical rigor and clarity in concepts and proofs and the loss of the philosophical vision of the unity of mathematics as a whole and its connection both to philosophy and science.  To this we add the lack of a criteria to evaluate progress and quality in mathematical work (and separate it from mere programmed automatic productivity without a unifying synthesis, transparency and purpose). 

Much harm has been done to mathematics (including the teaching of mathematics) through the distortion of core disciplines in number theory and geometry and analysis via wrong and deviated abstractions which obstruct both the logical and intuitive clarity and essence (and dare we say beauty) of the fundamental objects of study.  Also there is a bad habit of naming theorems after people who merely stated them (including when they produced erroneous proofs) rather than the person actually proving them. 

Consider also the arbitrariness in a certain selection of 'great problems'.  And so often the proof of a famous conjecture resembles the building of a cathedral, a collective collaborative endeavor  spanning several generations.  How unfair it is to give the prize merely to the person who happens to put the cherry on the top when the real hard work and deepest insights and ideas belong to other people. And specially if the said 'cherry'  is incomplete or has some dubious elements or 'holes'.

Pedagogically, for undergraduates and basic graduate courses, the focus should be on finite extensions $\mathbb{Q} \subset F  \subset \mathbb{Q}^{ab} \subset \mathbb{C}$ and $\mathbb{F}_q [x]$ wherein all the richness and essential content of basic algebraic number theory (including Galois theory)  happens naturally and in a more illuminating way than with standard abstract approaches.  Algebraic geometry should be concretely focused on complex algebraic curves (and surfaces) using, among other approaches, the excellent philosophy of the book on the same subject by Kirwan - therein the concrete essential algebra of polynomial rings over the complex numbers and complex analysis (the Riemann surface approach - via elliptic functions and the Weierstrass elliptic function; also modular groups) are brought into play. Elliptic functions and elliptic curves (including over the rationals) should also be fundamental. In algebraic topology the foundation should be in concrete topological and combinatorially based homology theory and natural cohomology theories (de Rham and to a certain extent Čech) avoiding abstract cohomology and relative (co)homology - and we need to focus on the right kind of 'space', more concrete and adequate than topological manifolds - spaces like complex varieties and the 'stratified' spaces of René Thom. Sheaf theory should return to its concrete and complex-analytic roots in Leray and Oka and the traditional topological formulation should be revived (based on local homeomorphisms and covering spaces).  Real algebraic geometry must be revived. Discard derived categories and go back to spectral sequences.

How mathematics leads to reversion to the logoi and to the nous

 

The following note sketches some ideas that attempt to make sense of Proclus' theory of mathematics and dialectic in the Commentary of the First Book of Euclid and Commentary on the Parmenides. How does the study or doing of mathematics lead to the unveiling of the system of the essential logoi in the soul and consequently the souls reversion (according to its mode) to the nous ? What mathematics should be studied or done and how should it be approached ? Is there an essential philosophical difference between ancient and modern mathematics ?

To attempt to answer some of these questions we propose the following theory of mathematics.  The structure of mathematics (be it ancient or modern) resembles the structure of living tissues, it is composed of a grid, a tiling, of 'cells' which are also evidently (logically and conceptually) interconnected. But each cell (even if incomplete and fragmentary from a purely formal mathematical point of view, from the point of view of concepts employed and results derives) exhibits a certain essential unity and sufficiency from a higher perspective. 

In the figure above the lowermost layer of cube represents mathematics with its natural division into cells (small cubes), each representing an autonomous intelligible unit of mathematical theory. It is important to be abe to carve out mathematics according to its natural cells or units. Now mathematics is constantly growing (both in scope and in detail) and self-revising.  But this growth should be represented as a horizontal growth represented by the expansion of the lower layer of the cube (adding new cubes). Over each cube in the upper layer is a column of cubes progressing in the upward direction. These represent the progressive unveiling of the logoic and noetic content of that particular mathematical cell: for each mathematical cell is like a microcosm of self-sufficient intellectual and noetical content and potential.  It would be more accurate to represent the cube as converging like a cone in the upward direction, for the ultimate goal of the vertical process of every cell is the same. It is this upwards interpretation which is also a source of synthesis and progress in mathematics.  It is clear that Proclus' anagogic process cannot depend in any way on the further horizontal progress of  mathematical theory (or on the difference between ancient and modern mathematics).  Rather it must be sufficient to consider one (or a few) genuine mathematical cells and use it a starting point for the anagogic process.

Common mathematical practice is concerned almost exclusively with horizontal expansion and the birth of more cells,  a frenzy for finding proofs, defining concepts and producing new results - which justifies in a certain sense some of the censure addressed at mathematics in Hegel's Science of Logic (the proofs are left behind like a ladder). There is not so much of a return-to-self via dwelling on a given cell, or a gradual development and deepening philosophical and spiritual intuition of a given organic unit of mathematical theory.  All genuine units of mathematical theory have at first sight something 'difficult',  'mysterious' , 'non-evident' or 'surprising' about them (and this is the source of the addictive nature of mathematics), even if this be regarded as proceeding from a mere fortuitous combinations of clever tricks. 

Thus for the Proclean anagogic and reversion process based on mathematics our first, vitally important, task is to identify and natural intelligible cells, noetically self-sufficient units, in the great body of mathematical literature and knowledge.

And yet there are so many factors and qualities involved in a portion of mathematical theory that it seems difficult to assign perfection,  completeness and sufficiency to any given theoretical portion (either ancient or modern).  So the corresponding anagogic process will, it seems, always be approximative only, if we consider merely its dependency with  regards to its purely mathematical basis.  Something else will be required to supplement the defect. 

Problems of philosophy

What is the nature of 'dialectics' according to Plato, Plotinus and  (in particular in the context of the Parmenides) ? How did it relate to other forms of ancient logic such as Stoic logic and Aristotle's Topics ?  There is also the following interpretation of some aspects of dialectics. Given a logical system L we can study different axiomatic theories in L and how they relate to each other  (for instance, are they mutually consistent) and whether they are in themselves consistent or incomplete.   A major paradigm is starting from a hypothesis H and arriving at a contradiction or staring from the negation of H and arriving at a contradiction and taking this to be (as in classical logic) a proof of H.

And we can study different logical systems and their relationship as well as the relationship of their theories. However all such logical systems use and epistemically presuppose recursion theory and arithmetic - and along with deduction exhibit some of the order-characteristic of temporality and also it seems cyclic temporality. Also the different  theories can be projected outwards in the form of concrete models, specially geometric models. Such models in turn can lead to other discoveries. And models can be reflected in other formal systems (see our theory of reflection) and concepts such as categoricity come into play, which must not be misunderstood in some kind of absolutist sense. And from a Fregean point of view we can consider the theory of the  informal elucidation of the primitive terms and axioms.  It is not at all clear how considering different theories (hypothesis) in logical systems can lead to disclosure of the primitive terms - but we must consider first of all the problem of the meaning of  meaning, proposition, truth, of logical connectives and quantifiers as well as the concept of deduction and inference (this is already a self-reflection of logic), etc. The quest for primitive terms must involve the theory of definition.  All these primitive terms, definitions, axioms in logical systems and theories concern the foundations of all possible knowledge and thought.

Mathematical logic and in particular formal theories of arithmetic and recursion must be seen as a reflection-into-self of logic, recursion theory and arithmetic itself. Gödel's incompleteness theorems are a unique example of reflection-into-self followed by reversion. Arithmetic projects itself outwards, reflects on the insufficiency of this projection and at the same time mediated by the projection cognizes a truth that leads it back to itself, the fact that the undecidable sentence is in fact true. Gödel's famous result gives us noetic knowledge.

Thus the projection into formal systems bound up absolutely with recursion theory and arithmetic (and hence combinatorics and graph theory and finitary set theory) is part of the cyclic process of investigation of the primitive concepts of thought (which appear to be known and clear but actually are not), a process which unfolds through formal projection and clash and comparison with other projections and hopefully leads to self-reversion. 

What is the relationship between more purely 'logical' primitive terms and others which seem to relate more to ontology, metaphysics, philosophy of mind, physics, etc. ?  In what sense the logical more fundamental (the old question of psychologism, etc) ? These concepts must be treated in the same way as logical and mathematical ones (see the quote from Leibniz under the blog header).

We can suspect a term is primitive if it does not seem to be easily definable. Can we define the logical connective 'and' ? We could group it together with other connectives and specify it by its truth-value properties in inferences, but in doing so we are already making use of it. For example, saying that 'A and B' is true iff A is true and B is true.

The biggest error of western philosophy was abandoning the neoplatonic (and augustinian) concept of the soul as an autonomous immaterial substance with potentially unlimited epistemic and ontological capabilities, and of taking the 'self' to be merely peripheral and mixed aspects of ordinary somatically and sensually conditioned psychological experience (this is the target of the original buddhist theory of anatta) or having a 'depth psychology' and elaborating a theory of the 'unsconscious' or 'subonscious' based merely on inferior aspects of the soul while totally ignoring the true spiritual depth which is both 'within and above' oneself. 

The neoplatonic philosophy of  mind and consciousness (through its theory of analogy and projection and reversion) allows us to reconcile logicism, realism and 'psychologism' and both species relativism and absolutism and both subjective idealism and natural science. Note that category theory besides being a rather specialized theory of relations is at the same time an interesting example of a theory of analogy and this is how it arose in the first place.

With regards to mathematics: how are we to understand why and how complex analysis  and complex analytic geometry (for instance developed by the great geniuses Abel and Riemann)  became so central in 19th century mathematics and beyond ?  How is it connected to problems in number theory and physics (and the significance of the work of Grassmann and Clifford is yet to be fully explored) ?  Hyperbolic geometry seems to be of immense philosophical interest, it perhaps represents the geometry of the soul or nous as opposed to the geometry of nature. The Beltrami surface gives an image of the 'inverted sphere' the return-to-self  which is also the projection to infinity of the soul. Hyperbolic geometry expresses the consistency of infinite different possibilities (a point outside a given line has infinitely many lines going through it which do not intersect the given line).

The cause of the descent of the soul must be some kind of internal disorder and forgetfulness which, by means of the descent, is projected and given external manifestation intimately correlated with the soul's own inner activity, the goal being that the soul will recognize through the world and through this correlation the very internal disorder and forgetfulness it started with, but now known clearly as such and by this insight be lead to a spontaneous and total 'reversion'.  So the descent of the soul is a fundamental 'mistake' and a 'fall' which at the same time is necessary to cure the internal 'mistake' that the soul was carrying within herself before the descent.

Prop. 1 of Proclus' Elements of Theology and Brouwer's intuitionism

The proof of the first proposition of Proclus' Elements of theology is among the most difficult to understand from a formal point of view.  Here is out attempt to make some sense of it using concepts which are also employed in Brouwer's intuitionism (or certain forms of finitism) -  the proof then assumes a structure somewhat like the standard proof of König's lemma.

The proposition reads: every multitude partakes in some way or another of the One.  We take 'multitude' to be represented by a mereological relational system in the form of a tree.

Consider the following interpretation. Proclus assumes that no tree  can have more than a countably infinite number of nodes (and hence branches) because, for him,  there is no infinity greater than countable infinity (the cardinality of the natural numbers). 

Hence there does not exist a tree in which every node has at least one successor and a fortiori in which each node has infinitely many successors- because then the set of branches would be of the cardinality of the continuum.

Proclus' proposition attempts to characterize trees with countable many branches.

Here are at least three types. Type 1 may have finitely many infinitely branching nodes but all branches of finite length.  Thus it participates of unity in a type 1 way (we may think of the terminal node as a 'unity').

Type 2 may have infinite branches but only finitely many nodes with more than one branch passing through them. Would Proclus accept this ? What are we to make of such chains (perhaps they express return-to-self) ? 

Type 3 has finitely branching nodes and finite length branches (what in intuitionism is called a 'barred spread').   Note that in a finitely branching tree if the length of the finite branches is not bounded then - in classical mathematics - König's lemma implies that the tree has an infinite branch.  The contrapositive of this lemma - called the fan theorem - is in fact intuitionistically valid. Thus Type 3 trees must  have bounds for the length of their branches. This is certainly a participating in 'unity' and 'limit' ! 

The following considerations may also have a connection to Proclus' proposition, but we leave this for future study: a basic fact about logic is the need to work with (complete) Boolean algebras or (complete) Heyting algebras as truth values. And that predicates assign such values to each elements of a set (Boolean-valued models, tripos, etc.).  A 'set' is a hierarchical structured tree in which at each level 'membership'  is assigned an algebraic value.  Consider also the authentic meaning of dense and generic subsets of a partially ordered set $P$. If P represents a an infinite 'tree' but in which branches can join,  we can think of P as being linearly ordered representing possible states of (finite, imperfect) information  regarding an object along time. A dense subset is a kind of sequence of bars which guarantees for a given set that as time progresses that set will also progress. A generic set represents a the infinite continuous information trajectory of a possible cohesive object. But a generic set is more than that, it is a kind of amalgamation and transcendent diagonalization of cohesive objects - which belong to the transitive model M in question.  Since it touches all it can be entirely none (when  $f = \cup G$ it is a like a synthesis, integration and transcending of all the limited and possibly contradictory points of view in the ground model M). Cohen's forcing proof (ignoring the for the moment the highly questionably philosophical value of transitive models  of ZF and indeed ZF(C) as a foundation for mathematics) seems to be a rather ad hoc combination of many different ideas and tricks (stemming from Skolem's fundamental insight on countable models) which definitely demands further clarification, simplification and refinement. What about when sheaf semantics comes in. Predicates with generalized truth values correspond to sheaves over the truth values.  In the sheaf semantics approach to forcing the generic object is not required (or is implicitly constructed).  Consider the forcing poset $P$ used to falsify the continuum hypothesis. An alternative would be to take $P$ as a pure poset and to construct a presheaf explicitly which associates to each $p \in P$ a finite function in the form of a finite subset of $B \times N$ satisfying the expected compatibility conditions (i.e. functorial conditions), where $B$ is some large cardinal (and any such finite subset is represented uniquely by some $p$). Then the presheaf topos on $P$ is already an intuitionistic model which falsifies the continuum hypothesis, since we can construct the expected mono in a coherent way ? The double negation topology (which corresponds to the dense topology on $P$) is used only to obtain a Boolean topos ?  It appears that this is not so: the double negation (or dense) topology is required for the local mono condition which makes the total sheaf morphism mono.  There is a very deep philosophy behind this sheaf theoretic construction. An object that cannot exist at once can yet exist spread out through time (or a space of situations) if it does so in a coherent way (the globality of a sheaf is determined essentially by its local qualities: thus if something is locally a mono in a coherent way, the sheaf morphism is mono). A fundamental insight is missing to understand the logical and model theoretic properties of the category of sheaves. Note we are considering presheaves over $P$ which unlike a topological space has no maximal element.   Could we construct a topos in which there is a bounded non-constant holomorphic function ? The construction above is like a completion and like a limit and like the construction of infinitesimals (and internalization of flux into an object). The construction of a topos of sheaves in which every function is continuous is just as interesting and important (if not substantially more) than the forcing methods for transitive ZF models. The same goes for the discovery of the topos theoretic version of forcing. Indeed, consider the sheaf model in which all functions are continuous. Classically, continuous functions have the cardinality of the continuum, while general real functions have that of $P\mathbb{R}$. 

Spiritual development

There is the following ancient and widespread theory regarding consciousness (which is found presented in mythological, philosophical and highly detailed practical form).  That human consciousness normally finds itself is state which is very different from its original state or states which it is ultimately capable. This situation has a cause.  Consciousness is mapped out according to certain domains and powers (without implying that they are not all closely interconnected) and it is found that for each of these domains and powers (which we can come into conscious contact with) there is a certain obstacle or counter-energy, in particular in the form of deeply-ingrained habits and tendencies. All these obstacles work together the ensure consciousness stays in its current state.  If to each of these obstacles and 'illnesses' we apply the right remedy and 'virtue' (in the form of an deeply-ingrained counter-habit and counter-tendency) then this will function like so many keys which will remove the shackles binding consciousness to its unhappy condition.  Be it noted that the transformation involved is total and radical and all-encompassing. This freed and purified consciousness becomes apt to receive higher influences and powers and to be ultimately transformed and transfigured to its original state. Since there are billions of different consciousness it is natural that there are many different kinds of corrupt and fallen conditions which require subtle differences and balances of medicinal virtues and counter-energies.  And as regards to religions and spiritual and esoteric traditions and philosophies, besides the pure universal moral law this is the only important and valid core we should look for - and they should be purified again and again (including through restoration of symbolic and esoteric hermeneutics) until only the pure gold of the core shines forth. Anything beyond morality and yoga or which does not contribute directly to them is to be utterly rejected. 

The synthesis between ancient (neo)platonic philosophy (ultimately deriving from Orphism) and Buddhist philosophy  (together with the traditional darshanas and daoism) offers us a solution to all the problems of modern philosophy (and morality and culture) as well as a reconciliation between ancient and modern philosophy.  The connection between Hegel and neoplatonism is very deep by it is neoplatonism that should be taken as our guide and authority. Also Buddhism appears to have exerted a huge influence in the ancient world. Thus we have the two 'good angels' of the west (who often had to go 'undercover'). 

We must not get lost and drown in the tempest and torrent of our minds,  trying to do introspective psychology and platonic dialectics without preparation.  Rather we must first return to the root uncovering the vast unknown aspects of our bodies, feelings and general aspects of consciousness which have great practical consequences. We need to know what is this 'we' and were it needs to dwell and focus and what it should do.  It is not easy to understand the authentic original meaning of satipatthâna and thus its perfect agreement  with and complementarity to platonism.

There are certain difficulties involved with reconstructing buddhism in its most original authentic form as well as extracting the most relevant philosophical exposition thereof (and we highly recommend Bhikkhu Ñânananda's book Concept and Reality). The Nikayas are a vast and complex collection of texts which demand careful historical-critical analysis. The collection of texts in the abhidhamma division of the Pali Tipitaka is likewise a complex and heterogenous collection of texts clearly reflecting later sectarian dogma but also containing older material of the highest philosophical (and logical) value and interest (and we must not forget the importance of studying parallel Chinese versions of many sections of the Pali canon).

The complete mutual consistency, complementary and even essential identity between platonism and original buddhism may appear to be quite a controversial claim, even if the connection to Pyrrhonism has gained some scholarly acceptance (cf. C.I. Beckwith's 2015 book Greek Buddha). Some important points are the following:

i) The meaning of the Buddha's employment of the term anattâ became lost and confused with a doctrine of the denial of the existence of a 'soul'.  In reality this term is used as part of a practice of dis-identification  (cf. the Atthakavagga) entirely consistent with Plotinean anthropology (for instance Enn. I,1,) and purificatory practices. We also have written about the uncanny correspondence with Aristotle's De Anima. 

ii) The correct methodological and epistemic role of dialectics and cognitive abstention involving undecidable or equipollent pairs of propositions also was ultimately lost, leading to a confusion with logical and conceptual nihilism and relativism.  Thus neither madhyamaka nor Pyrrhonism are consistent with original buddhist dialectics. Rather such dialectics  (see Ñânananda's excellent discussion in Concept and Reality) most closely resembles the anagogic and gradual process of Platonic dialectics (see  Enn. I,3).  

Also (as Jayatilleke holds in his famous book) original buddhism was based on direct evidence (which in modern terms could be described as 'positivism', 'phenomenology' and 'the return to the things themselves') simultaneously with the cultivation of the 'eye'  which is necessary to see things as they really are - and in this again there is perfect agreement with platonism.

iii) The sophisticated formal logic and ontology of Stoicism certainly was known to have played a role in neoplatonism and even middle platonism (specially in the context of the controversy between the stoics and later academy) - and we can inquire into the relationship between the Stoic lekta and Proclus' theory of the logoi (in a proto-Fregean way Platonic ideas at a discursive level can be seen as incomplete lekta). Likewise buried within the Pali abhidhamma literature we find (as already acknowledged in the literature on the Katthâvatthu)  a fairly elaborate deployment of formal logic and a sophisticated theory of types of cause.

We note that in neoplatonism the logoi of the soul and the 'ideas' of the nous are to be understood as living beings in communion with each other in a kind of eternal process of cyclic generation and unification...

iv) Both buddhism and platonism have cultural-political dangers and problems. But note that the passages on race and caste found in the Pali texts are some of the most important in the history of mankind. The philosophical content of original buddhism allows us to reject mythological interpolations regarding kamma, previous existences and  the afterlife - without rejecting such concepts in themselves or an alignment between the ethical and cosmic law. A  problem in the subsequent development of buddhism is the order of bhikkhus itself becoming somewhat like the traditional brahmin caste in all except the requirement of birth: for instance the claim that a layman cannot attain full enlightenment, more emphasis placed on accumulating merit by supporting the monks than personal spiritual development or doing good to others.   A problem with original Platonism is the militarism and totalitarianism (among other troubling aspects) of the Republic as well as many aspects of the Laws.  Militaristic values were deeply ingrained within the fabric of Athenian society (and there were of course natural historical causes for this) and it is noteworthy that the iconoclasm of the famous passage of the Theaetetus which rejects many key values of contemporary Athenian culture does not touch the adulation and idealization of the soldier and warrior (or indeed of the athlete).  The concept of a 'noble lie' is one of the lowest points of the surviving Platonic texts.  We hope to show that we can reject all these problematic elements based on the Platonic philosophy itself.

Parapsychology and the philosophy of science

It is far from clear what exactly is the so-called 'scientific method'  but it is clear that is actually a complex and fluid combination of various different methodologies and attitudes all of which are inextricably genealogically and logically connected to theoretical assumptions and hermeneutic decisions.

The scientific method conceived as the 'experimental method'  pertains principally to a certain limited and partial domain of reality - that of 'matter'  or 'physicality' or the  strictly physical-chemical dimension and aspects of living beings - and as thus the kind of theory associated exclusively with it must be essentially an abstraction of reality (rather than a negation of other aspects of reality).

The experimental method is not logically or theoretically self-contained or self-justifying or self-sufficient (for instance it depends on previous theory, hermeneutics and mathematical theory).    It has no claim to supremacy and exclusivity as far as a source of knowledge in its particular associated domain nor a fortiori claims regarding other domains of reality which it may well be totally inadequate for. 

Also if the ultimate aim of physical science is the construction of machines that serve mankind and the good of the world or the development of treatments in medicine, then  the kind of deep intuition which guides the engineer or medical doctor is just as important as any experimental protocol: for there is no greater proof or validation than the machine actually working or the treatment being actually effective.

Experimental science is not the only not the best or most certain or even most important source of knowledge (for instance there are the more certain, more important and more fundamental epistemic domains of  logic, mathematics and ethics, all of which have nothing to do with physical experimentation). Nor does its particular limited domain of application exhaust the totality of reality. Nor can experimental science justify any kind of reduction or alleged correlation (supervenience) between its domain and other different domains.  In fact the actual experimental results and evidence contradict  such reductionist claims. Experimental science cannot a priori impose its epistemic methodology on other domains of reality - and much less claim that a physicalist philosophy is somehow justified by the experimental method itself or its results (which is factually false).

When natural science and the 'scientific method'  violate basic ethical principles such as  when causing harm, suffering and death to human beings or animals in the course of its  methodology and 'experiments' , it shows itself to be profoundly mistaken and driven by the same kind of blind superstition, dogmatism and fanaticism it often projects onto and decries in others.

Spirit, soul, mind, consciousness - this is an entirely distinct domain of reality which cannot be reduced to and does not necessarily supervene on physical matter (the physical brain and body).  There is no reason why the experimental method should be the best method  (as opposed for instance to an axiomatic-deductive or first-person phenomenological and instrospective method - both of which were developed to high degree in Ancient Greece and India)  for exploring and obtaining knowledge regarding this domain of reality. 

And yet since spirit, soul, mind, consciousness are in a way connected to or associated with the physical brain and body it leaves indirectly its footprint in the legitimate domain of physical science.  Thus it should be possible to additionally 'beat physicalism in its own domain', to exhibit tangible, measurable phenomena which even the most convinced physicalist could not deny.

This brings us to the subject called 'parapsychology'.  On the surface this subject consists in certain experimental protocols which as a rule tend to lead to plausible conclusions or bring to light evidence which is radically at variance with a physicalist worldview, or  to exhibit a class of phenomena that while involving the physical world suggests that there are forces at play which transcend it.  So parapsychology  while wearing the cloak  of experimental science does patently have  philosophical concerns.

There is the following major problem with parapsychological research.  A massive amount of scientific activity has been funded with the goal of proving or finding evidence for physicalism (neural reductionism) or for various other theories which assume neuro-reductionist premises.  A substantial and important part of parapsychological research should be devoted to a critical analysis of such experiments and their methodology and protocols showing how they completely fail to establish physicalist claims but rather strongly suggest opposite conclusions. Also parapsychology should point out that there is a massive amount of direct evidence (which was not obtained in a parapsychological context)  suggesting the untenability of neuro-reductionist physicalism.  There are also powerful theoretical deductions that can be made based on known neuroscientific facts (for instance regarding the impossibility of dendritic spines being involved in memory) which again refute physicalism.  None of this involves 'spooky' phenomena and is perhaps not as 'fun' and 'exciting' as the usual concerns of parapsychology, and yet its importance and value is immense and fundamental.

While we hold that much of the experimental protocols and results in parapsychology are both valuable and interesting (specially the work of Rupert Sheldrake) it is a mistake to make such experiments and results a sole foundation for the rejection of physicalism (for there are much more powerful, extensive and conclusive arguments and evidence to be found elsewhere as discussed briefly above).  Indeed it seems that as the rule the researchers in this field have still at least half-consciously profess a kind of confused semi-neuro-reductionism in which mind, consciousness and brain are too easily confused and conflated. This opens the door to a kind of theoretical  neuroscience in which these phenomena could be explained by speculations  pertaining to theoretical physics (for instance telepathy is compared to quantum entanglement).  It becomes not about refuting neuron-reductionism but about exploring the quantum superpowers of the brain (or the interconnectivity of brains rather than primarily of consciousnessness).

Some other flaws we find in parapsychology are arguments from authority which also suggests a kind of implicit western supremacy and exceptionalism.   For instance:  person A was a great scientist and he or she thought parapsychology was a legitimate field of study therefore this counts as evidence that it is so.  We have also seen it implied that a non-western person who undergoes a western academic education (or is involved in business) is somehow bound to be more intellectually honest (or less liable to deception) about paranormal phenomena than his counterpart who has not undergone such an education or training. 

It also should be mentioned that in the past both in the east and west there was already a systematic science (first-person or axiomatic-deductive) involving the kind of phenomena (or powers) studied in parapsychology but with the caveat that no great spiritual importance was attached to them and they were rather seen as dangerous distractions and potential obstacles.

Finally we find it quite disturbing that the interest and use of parapsychology by government, military and intelligence agencies is mentioned - the military was interested in it and funded research in it, therefore this consists of evidence that there must be something to its claims (regarding, for instance, remote viewing) - all the while completely omitting to mention the terrible crimes and violations of human rights documented among such projects.  This topic should first of all be mentioned as a cautionary tale that parapsychology can also be perverted  and misused in criminal activities and that the aspiring parapsychologist must be wary of government and military funding and involvement.

Studies regarding the ability of directed thought to influence other minds and living bodies can have dangerous implications. For instance if in a given community things are not going well or there seems to be consistent 'bad luck' then would not  the popularization of such studies encourage finding a culprit (somebody who allegedly is a source negative directed thought-energy) and even engaging in 'witch-burning' ?  Also what about government and military applications of these facts ? Or massive activity of social media generating automatically a kind of powerful psychic field influencing  public opinion, a kind of spontaneous 'brain-washing' at a distance ?

Addendum to our note 'Differentiability, Computability and Beyond'.  We wish to add some considerations to this note which also have some connection to experiments with random number generators and microPK. Recall that we postulated that a truly free particle must have a completely random completely discontinuous trajectory in space.  This begets the problems: i) define this rigorously. ii) this trajectory is not unique but there are uncountably infinite many such trajectories and so there is no well-defined free state of a particle.  And for i) we can draw inspiration from random number generators and the mathematical definitions used (this corresponds to the discrete case).  Now in our note we considered that a field would act on this random particle in a certain way introducing a geometric form to its associated density or distribution. The similarity to the results in experiments involving random numbers generators is patent.

Just as physical bodies appear separated in space we can ask if the multiplicity of consciousnesses is 'situated' in some kind of analogous medium (which may have a very different 'metric' or concept of separation which need not coincide with the spatial aspect of their corporeal counterparts).  In neoplatonism this might be the 'soul of the universe'  and physical space would be its emanation.  So the 'geometry'  of the soul of the universe must be distinct from ordinary geometry and yet this last must be able to be derived from it (as a special case or projection). Also (certain levels) of soul might occupy a 'body' in such a space which is more extensive and complex than the physical body in ordinary space.

Addendum: there is the also the following very important point about NDEs.  It is important to distinguish the hypothesis of survival,  that is, the independence of conscious experience and personality from the physical brain,  from the interpretation of the content of such experiences which may in most cases be of no more objective significance (or intrinsic value)  than lucid dreaming or vivid imagination or recollection, their varied content being mostly drawn from memory and experience.  There is very little agreement or correspondence between such 'lucid dreams' beyond certain generic emotions and perceptions (light, warmth). The dubious hypothesis regarding mediums and channeling apply equally to NDEs. NDEs are of less value and do not generally do not have anything near the transcendental cognitive content of higher states of meditation (save in their emotional content).

Logical notes III (Mathesis Universalis)

Many of the problems which concerned western philosophy are just consequences of an a priori rejection (and this rejection also reflects a spiritual, cultural and intellectual regression) of the platonic philosophy (and its sophisticated form found in Plotinus and Proclus and the Proclean philosophy of mathematics).  And indeed it seems that we can do justice all at once to the geniuses of Frege, Gödel, Hilbert,  Russell, Church, Turing, Brouwer, Skolem, Gentzen, Girard, Lawvere, Martin-Löf and to Meinong and his school and to Hegelian phenomenology and dialectics (which has a striking correspondence with Proclus' theory of eternity, time, dianoia, dialectics, the logoi and their projection and the process of reversion to the nous). 

There is nothing wrong with thinking of consciousness as a spiritual substance and as a place wherein are 'located' a system of pure concepts which are independent of and not derived from sensation or imagination.  Our access to these pure concepts is purely objective and yet they are 'subjective' in the sense that they are part of the substance of consciousness and not (directly) outside it.

They are also involved in the morphogenesis and activity of the body.

This system of pure concepts in human consciousnesses is one and the same because it has one and the same cause beyond ordinary human consciousness and this cause is also involved in the explanation of how the system of pure concepts adequately relates to the knowledge of nature (thus the universe is permeated by reflection and analogy). In our ordinary knowledge these pure concepts come into play, there is also a lower rank involved in abstraction from sensation.

Hegel's science of logic gives us an illustration of the Proclean account of dialectics. Furthermore Hegel's science of logic has some deep connections to modern mathematics and mathematical logic and foundations of mathematics (in particular category theory).  Hegel allows one to reconcile  Frege and Brouwer within a larger and more thorough framework (which is to be an typed, intensional, computational-algorithmic-oriented logic and mathematical foundations - which rejects completed infinite cardinalities in the extensional sense).

Modern mathematics (as well as modern physics) needs very much a clarification, improvement and radical reformation of its foundations.  Voevodsky opened up a promising approach. Category theory is not to be seen as universal theory but rather a specialized and partial one suited for the particular turn modern mathematics took in the 20th century.  It is to be replaced with a structure related to dependent type theory or a more universal theory of higher-order relations.

Plotinus and Proclus offer an integral solution to the problems of the theory of knowledge (which in antiquity are associated to the Academics, Pyrrhonism and the debates with the middle Platonists and Stoics - but also found in Augustine).

Neoplatonism also offered a consistent and insightful theory of spiritual yoga within a coherent philosophical and scientific context. And indeed the theory of dialectics gives the genuine  clarification and possible higher meaning of madhyamaka and Pyrrhonism. Also, the apparent discrepancy between Proclus and Plotinus can be explained by a better understanding of procession and emanation  as a kind of instantaneous continuous current between levels: thus there is no difference between the attainment of nous or henosis by the soul and the metaphor of a drop of water merging into the ocean without loosing its individuality.  Or rather, reversion and return is not to be seen as a lower level reflection but as a direct 'plugging in' to a higher current connecting the levels in eternal continuous simultaneity.

Mally : Gegenstandstheoretische Grundlagen der Logik und Logistik (1912)

This work of Meinong disciple E. Mally (tr. Object theoretic foundations of logic and symbolic logic) can be found (in the original German) here:

https://mally.stanford.edu/mally.html

See also this interesting paper by E. Zalta about the relationship to Husserl's Ideen:

https://philpapers.org/rec/ZALMDA

Logical Notes II

Sausurre's Course in General Linguistics (1916) expresses some fundamental ideas of category theory: that an object's 'value' only make sense in the context of a system of other objects to which it is both similar and dissimilar.  His concept ' sign' is furthermore much like a functor from the category of phonetic materials to that of 'concepts'.  Also in Carnap's Aufbau we find an interesting graph-theoretic oriented discussion of the calculus of relations which is category-theoretic in flavor. In fact Carnap's discussion suggests a more general structure than that of category (which can be seen as appropriate mainly to the 'regional ontology' of mathematics).  This structure simply consists in a collection of objects $x,y,z,...$ which are themselves collections of certain elements and for each pair (possibly the same) of objects $x,y$ a  (possibly empty) class of relations subject to the condition that if we have relation $R$ on $x,y$ and one $S$ on $y,z$ then the composite relation $SR$ belongs to the class of relations associated to $x,z$. A special case is in which we consider for each pair of objets all possible relations.  An apparent difference from the concept of category involves the lack of 'arrows', or the polarity of the relations between two objects.   It is curious how relations become important in tripos theory and the effective topos. But here when we have a relation $R(x,y)$ there is always an orientation which we can take as being defined 'from the first argument object to the second argument object' (like the arrows in Sowa's conceptual graphs).

Thus the correct prototype of the concept of category is a collection of objects each pair of which may be subject to a plurality of different relations.  This prototype concept is superior because the relations are implicitly classified - and this is what is often done in practice in category theory. Morphisms (as usually considered) are rather redundant as relations and only special classes of morphisms become in fact relevant (and relations worthy of that name).  Another problem is that 'naive' category theory assumes we have a notion of equality between 'morphisms' (for instance in universal constructions).  But when are two relations 'equal' ? Clearly mere extensional equivalence is not sufficient or at least problematic.

But is not a category a collection of objects that are like different species of the same genus ? And according to our discussion above, species which can have a multiplicity of different relations between them ? But this is precisely the theory in Aristotle's Topics. The immediate species of a given genus are subject to many possible relations (perhaps involving a third factor): opposition, more-or-less, more desirable,  better known, etc.

What are some of the most significant results in philosophy ? The clarification of the pure concept of computability (both in terms of machines and term rewriting: Turing, Church) and the pure general concept of axiomatic-deductive system and in particular the clarification of axiomatic-deductive systems which to some extent mirror actual human reasoning processes (natural deduction, dependent type theory).  The complex interaction between psychological experience and objectivism (if not already thoroughly explored by Hegel) made definite progress with Brouwer's intuitionism and subsequent advances in constructivism (or intuitionism) - specially dependent type theory (Martin-Löf type theory). A major philosophical error: Zermelo-Frankel foundations and the standard proliferation of the concepts of 'topological space' and abstract theory of rings and fields.  Superior to the concept of topological space is J.R. Isbell's concept of 'locale' (1972) which recaptures aspects of Aristotle's, Leibniz's and Kant's (and Hegel's) concept of space - or better still, the notion of Heyting algebra.

Originally there was a profound unity between algebraic geometry, analytic geometry, mathematical analysis and differential equations - between algebra, geometry and the method of infinitesimals and indivisible points.  This 'good' mathematics was neglected or became marginal during the disaster of 20th century mathematics. It is represented by lesser known disciplines of 'real algebraic geometry' (which we can also say is the 'real' algebraic geometry having roots in the work of the Italian algebraic geometers), the study of analytic, semi-analytic and sub-analytic sets, singularity theory and Thom's theory of stratified morphisms: some of this mathematics seems to have been adumbrated by Hegel's long notes on the section on Quantum in the Science of Logic.  The bad mathematics consisted in wrongly abstracted algebraic geometry based on Noetherian rings and fields (A. Weyl, Grothendieck) and the wrong abstraction of analysis based on general topology and 'infinite-dimensional' vector spaces (all ultimately based on the Bourbaki framework and Zermelo-Frankel set theory). The theory of finitely determined germs and universal unfolding is 'true algebraic geometry' which is also the approach to the calculus based on the basis of powers found in Hegel (also we should prefer étale spaces to the abstract definition of sheaf and covering spaces to locally constant sheaves).  The opposition between Thom and Grothendieck (we mean here scheme theory not his later work on topoi, homotopy theory, dessins d'enfants, etc. )  might be seen broadly as the opposition between authentic and false mathematics. A nice project would be to continue Lawvere's work on synthetic differential geometry and Grassmann and also go back to the great algebraic geometers of the past (such as Bonaventura Cavalieri) and give a rigorous foundation to their infinitesimal and 'indivisible'  techniques

It is interesting to read Husserl's Philosophy of Arithmetic in light of Hegel's treatment of Quantity in his Science of Logic. Indeed Hegel's treatment of number, magnitude, infinite progression, ratio,  measure, etc. can be given interesting interpretations in terms of category theory (specially the treatment of the natural number objet and computability in a topos) and modern singularity and bifurcation theory. 

The theory of knowledge involves the analysis of the essence of reason. But it assumed that human consciousness cannot completely abrogate and go beyond reason (and thus ultimately see reason), not in the direction of something 'inferior' to reason (presumably the realm of dangerous lower instincts or to a kind of 'irrationalism' which is a real problem of our times), but something superior to reason, super-rational. The same goes for the 'self' and 'volition'. Indeed Hume's theory of the self does not deny that there is  systematic  'energy' at work causing the impression of 'self' (even if allegedly this 'idea' is unfounded).  Modern western theory of knowledge has the fatal error of ignoring the fundamental principles of ancient philosophy (both western and eastern) relating to the necessity of engaging in systematic practices relating to the purification of consciousness in order to be able to gain access to knowledge (and in particular self-clarity relating to consciousness itself).  But note that these practices themselves already require (partial) philosophical insight. See MacIsaac's thesis on 'The Soul and Discursive Reasoning in the philosophy of Proclus'  for some interesting and overlooked ideas regarding the theory of knowledge. Sausurre's theory of syntagmas and association is an example of truly insightful phenomenology (or cognitive depth-psychology). Interesting also is Peirce's phenomenology.

If to be at home in the world of objectivity much practice and preliminary methodology is required, a cyclic return and refinement,  why should not the same hold for what Peirce called 'phaneroscopy',  direct internal spiritual cognition ? And the 'subjective' and 'objective' being subtly connected, when easy with the objective world is attained some subjective intuitive clarity is automatically attained. And indeed introspection involves an object, consciousness itself or aspects thereof become objects themselves.

We do not understand what consciousness is nor how consciousness can directly perceive itself nor what in that case would be perceived and how it would perceive and how this perception relates to objectivity. What is clear is that we never perceive atomic sensations.  The concepts of 'subject' and 'object' are liable to criticism.  Maybe consciousness can only at first look at itself indirectly when engaged in some cognitive activity.

Speak not of subject or object but of ceasing to look without and starting to look within (Plotinus). And look within in a way in which you are looking at what is really there without distortions and projections. But what is this 'looking without' and 'looking within' and first of all 'looking' ?  (Husserl himself quotes Augustine's noli foras ire, in te ipsum redi, in interiore homini habitat veritas.).

If we need logic to grasp and operate an axiomatic-deductive system, what sense is there is trying to capture logic itself as an axiomatic-deductive system ? (it is a kind of reflection-into-self). Cf. also our previous criticism of Sextus. The net of logic is very vast yet there is no easy escape from it.

The situation of ordinary consciousness: it cannot just immediately gaze inward on itself, see itself through inner perception and perceive the truth (it will just loose itself in a pool of shadows and mirages and fleeting dreams).  It can only manifest itself to itself gradually in and through its activity and specially clear logical cognition. However there is also the possibility of making a constant progressive effort at 'conversion' or 'turning inward'  its aim and target (its intention). 

Stoic logic was not a kind of 'propositional logic', rather it was closer to quantifier free many-sorted first order logic.

Just as different physical phenomena can be described and studied by the same mathematical structure, so too different mathematical theories can contain the same unity and energy of reversion arising from the same source.

See this excellent paper by Butorac on Proclus' theory of dialectic in his commentary on the Parmenides.

Head Truth: Reasons for Doubting Thought Comes from the Fronta...

Head Truth: Reasons for Doubting Thought Comes from the Fronta...: Scientists lack any coherent explanation for how a brain could generate thought or intellect. Thoughts are immaterial things, so how could ...

Logical notes

In this post we place brief sketches of some ideas to be developed.  The criticism of the concept of 'possible world'. Our knowledge of the 'world' is approximative, incomplete, local and relative.  Thus it makes little sense to speak of an 'alternative world' or of a certainty that such is even 'possible'.  Rather there are local restricted domains and aspects of the world, coming from different spatio-temporal regions of the actual world, which we loosen and group together and inappropriately call 'worlds'.  For instance possible gardens are based on the collection of actual (present or historical) gardens in the actual world (which depend, in this case, on human agency and choice), and these may or not contain roses. It is absurd to speak of an alternative global world in which the red roses in this garden are white.  Possible worlds are an undue reification of aspects of imagination and consciousness. Criticism of statistical and probabilistic concepts: cannot judgments involving such concepts all be transformed into judgments without them, based on spatial-temporal events ? Is not the statistical and probabilistic inherently eliminable and reducible to mere arithmetic judgments regarding a collection of evidence from a set of spatio-temporal regions  ? And are not such concepts based on partial, artificial, approximative abstractions which totally ignore the underlying epistemically open-ended structure of reality ?  One cannot enter into a relation to something without entering into a relation with the relation and so forth. Logic is an a posteriori useful descriptive tool, not a foundation. There is a large amount of evidence that consciousness can subsist independently from the physical brain and that consciousness is not generated from the brain nor in particular are different psychological faculties determined by specialized functional regions of the brain. A philosophy which ignores this evidence is not philosophy but propaganda. Consciousness does not supervene on the brain or physical matter.  

If linguistic utterances were a soup then Wittgenstein would hold that the meaning of 'soup' is reduced to the movements of the spoon (such movements could indeed serve as a non-verbal sign indicating 'soup').  One cannot abstract any symbol from meaning, nor use from mention (which is a misuse of the term 'mention' anyhow).  When attempting to grasp an empty purely syntactic meaning-free entity one is actually trying to grasp an ideal type and structure with an unlimited number of possible perceptual variants and instances, an extension which may indeed by vague (for instance in the case of poor sight when we are not sure of a certain letter). Thus 'the letter 'b'' is an abstract concept with an extension and hence with a meaning.  Frege and Husserl pointed out again and again that the meaning and reference  of terms or propositions certainly cannot be exhausted by the accompanying 'mental images' which need not coincide for different people and cannot provide thus a foundation for a so-called objectivity of meaning. True. Yet digging deeper  and applying the right introspective-descriptive method we may discover that intuitive mental content of a different order can be disclosed to consciousness which does indeed furnish a ground for inter-subjective, objective agreement. Yet this need not be naively taken to be some kind of identical extra-mental ghostly entity, and even less some kind of external sociological structure and dynamics,  rather it is a directly accessible aspect of consciousness which while sharing the same type for the same meaning across different minds also allows room for  individual variation. For instance if the common mental image is like the adornments that may adorn a statue (and there may be none at all), the statue representing a god, say for instance Athena, can be different one for different minds and yet be clearly isomorphic in a suitable sense across different minds for the same term or proposition. The fact that human beings must have hearts with the same structure does not imply that there is a common objective heart outside each body.  The objectivity of meaning can be understood in a way similar to the isomorphic structure of our bodies, rooted in what we are,  in the things themselves.  If incomplete and having errors, few western philosophers have ever probed consciousness like Hume and Kant and laid bare the deep forces underlying it, including the nature and constitutive role of 'ego' and the construction of 'naive realism'.  To these we add the powerful contributions of the fathers of introspective psychology (Brentano, the William James of 'The Principles of Psychology').  And see the work of K.N. Jayatilleke and the early Carnap (and also Rosado Haddock's on Carnap). Carnap uses the suggestive term : autopsychological.

After the autopsychological was chosen as basic domain, thus, the processes of consciousness or experiences of consciousness of the I, it must still be determined which formations of this region are going to serve as basic elements. One could, let us say, consider taking as basic elements the ultimate constituent parts obtained by means of psychological and phenomenological analysis of the experiences of consciousness, thus, let us say, the simplest sensory sensations (as Mach), or more generally: psychic elements of different sorts, from which the experiences of consciousness are formed. On a closer examination, however, we must acknowledge that in this case not the given itself, but abstractions from it, thus something epistemologically secondary, has been taken as basic elements…. Since we, however, wanted also to require from our constitutional system the consideration of the epistemological order of the objects, we shall, thus, start from what is epistemologically primary to everything else, from “[the] given”, and those are the experiences of consciousness themselves in their totality and closed unity…. To the chosen basic elements, those experiences of consciousness of the I as unities … we refer as “elementary experiences of consciousness. [Der logische Aufbau der Welt, pp. 91–2].

Is not the concept of vagueness itself rather vague ? Many natural language predicates admit more-and-less (in the terminology of Aristotle's Topics) but our adverbial resources are often clumsy or insufficient to express the underlying linear order of the corresponding 'semantic space'. Language is discrete but consciousness and the world are continuous.   Thus all difficulties involving vagueness and ambiguity can be resolved by introducing a fine enough (though finite) linear scale: this in practice is what is used in many sorts of questionnaires.  Baldness is a predicate capable of more-or-less hence with an underlying linearly graded semantic space.  It is absurd to believe that you can split a continuous linear segment into a binary classification. 

Language  - together with the orientation found in the Pali texts - is the antechamber and the initial map for entry into philosophical psychology, into the domain of  pure consciousness.  Thus read the Logical Investigations and then Sowa's Conceptual Structures for an a posteriori formal elaboration (but discarding the erroneous neuroscience found in the otherwise phenomenologically and logically interesting book). 

In the Logical Investigations, Husserl is very frank about the intricacies, problems, confusions, pitfalls and puzzles of his project.  The desire to separate mental experience, the actual lived experience in the process of consciousness,  from some kind of ideal 'other' (meaning, the thing itself) which nevertheless, and most paradoxically, is also the object of pure, direct, intuition and hence just as much a part of lived conscious experience as the perception of an apple.  Sowa holds that concepts cannot be directly intuited and pertain to some kind of underlying neural substrate - and yet they can be translated into inner verbal discourse and imagination.  Husserl naturalizes psychology (misrepresents it) too much in order to contrast  it with his method without presuppositions (already expressed in the Logical Investigations) and yet, I think, any philosophical psychology does not engage in this kind of naivety.  The psychologist is concerned with consciousness and how consciousness comes to constitute or construct a representation of the world. Its building blocks are not naturalistic presuppositions but such things as sense icons, percepts, atomic sense data (which Husserl himself brings back in a very Kantian way into his noesis-noema scheme),  or basic categories or whatever. But the fundamental principle (found already in the Pali texts) involves developing a pure detached awareness of the stream of consciousness as it is in itself (as distinguished for what is constructs and the exterior projections taken to be real) and in this way to obtain definite direct knowledge of (and even power over) consciousness itself - but without any necessity of denying an exterior world in itself, only the naive mental imaginative way in which the mind takes itself to be relating to it.  If  'the world is my representation' it does not follow that there is no world, only that - in our mental life - we actually live and interact with a dream and projection immeasurably more than with the world itself in a strict causal sense.  While we radically reject Sowa's neuro-reductionism, there is a similarity in that transcending consciousness there is likewise a physical world which can causally affect consciousness. We need to learn how to observe and see the meaning of words in our own minds.  Rather than speak of mental experience, meaning, intention, meaning-fulfillment, or the complex structure of noema and noesis,  we ask: what is the mental content associated to concepts (for the moment abstracting how such content can differ at different instances of the concept being thought or with context) ? For all their shortcoming, the older philosophers, Locke, Berkeley, Hume and Mill where very much more profound and radical than those influence by rationalism and scholasticism - and we hope to view their theories in a novel profound and consistent way (including the fundamental issue of the relationship between self and consciousness).  The concept 'triangle'  does not correspond to any hidden neural 'conceptual graph' nor social behavioral or pragmatic pattern nor to a mysterious eidetic form, purified meaning, etc. Rather (thanks to the right method of introspection) to a directly perceptible complex dynamic structure  (an open-ended proliferation or papanca) in consciousness which is linked to the life-history of the mind  (and to feelings and emotions, as Sowa discusses) and involves 'genetic epistemology'. All concepts have an 'original baptism' in the mind. Concepts are somewhat like living entities. Perhaps Hegel's logic (which brings to life the genetics and internal life, turmoil, psychology and complexes and politics of concepts) offers us an interesting template which is distinct both from the elusive intuitive transcendence of Husserl and from the crude and contrived decompositions and definitions of modern symbolic logic.  It is the childish simplicity and unabashed naivety as a way to embracing concepts that is the key to wisdom, not artificial sophistication and abstraction.  How do we explain that certain people can have the concept of 'triangle' and yet not know that the heights intersect at a single point ? Or that the sum of the internal angles is two right angles ?  The difference is like someone who has read David Copperfield and someone who only has heard a vague outline of the plot (mathematical proofs are like stories). In fact concepts are learned in a context and as part of a theory and a system (in a loose sense). Like leaves and branches they cannot be totally separated and abstracted from their tree and root. We see this is the way formalizations of mathematical theories are carried out (for instance in Coq or Agda). Concepts are very fine grained and the mode or presentation and the enveloping theory and context are everything, logical equivalence is a posteriori and comes later. The ancients had a good insight into this in their theory of genera and species but their theory was never fully developed and remained mostly simplistic and artificial. But consider just how we possess concepts of extensive complex objects, like a novel. We have a concept of the novel 'David Copperfield': what are the mental contents associated with this expression ? We can survey them as a unified whole at once from a distance, without detail, much like surveying a landscape,  or traverse according to a certain order the actual contents of the book in detail. It is this seeing as a whole, from afar, that needs to be phenomenologically elucidated.  The perception from afar of a complex concept might need to evolve a choice of salient features and markers (cf. Sowa's discussion on the difference between recollection and recognition).  If we can understand this we might be able to understand 'abstract' concepts as well. The concept of 'triangle'  is associate to a spontaneous mental content which is a pair (T,D) in which T is an actual imagined triangle and D is a set of permissible deformations of T, for instance changing the length of the basis and changing the position of the upper vertex. Husserl's eidetic variation as a process is itself the expression of the mental content of the concept. More precisely we are given a space $X$ and a figure $F \subset X$ together with $n$-parametric groups of diffeomorphisms $D_1,D_2,...$ acting on $X$. Also recall our previous discussions about the refinement of concepts.  We can also say that when we consider a concept we are implicitly considering the process whereby that concept comes to be as a species in its proximate genus and at the same time the its own internal process whereby it may be further divided into further species: heteroeidogenesis and autoeidogenesis. Or we may take the cue from object oriented programming (and Sowa) and consider the the mental content corresponding to a concept is like that of a code of a class (or interface).  Another point is that feeling, volition, certain subtle physiological connections to the body and the constitution-energy of a 'self'  (see Schulting's books on Kant) are entangled and bound up with conceptual thinking, though this has been traditionally discarded.

Another overlooked point about Husserl. There is a strong and hidden scholastic strain in Husserl. But what is scholasticism ? Medieval scholasticism is often presented as a continuation of the philosophy of Augustine (which in turn borrowed heavily from Plotinus) and other neoplatonic texts. But in reality this continuation of certain currents of Islamic Aristotelianism is quite different and is not compatible with  original neoplatonism or the more neoplatonic aspects of Augustine (it is only in so-called 'mystical theology' that genuine neoplatonic themes make an appearance). Much of medieval scholasticism looks forward to modern rationalism, empiricism and nominalism and as thus has very little relation to neoplatonism (Scotus, Ockham, Buridan, the 'Oxford Calculators').   With the scholastic revival of the 19th century we have a yet further estrangement from neoplatonism, we have radically altered version of medieval scholasticism (neothomism) which rejects even further core augustinian and neoplatonic epistemological and psychological elements (this is related with the Roman Church's rejection of Ontologism). Thus it appears that Husserl latches on to some of the ideals of neoscholasticism (in particular via Brentano and Bolzano) and many of the problems involved may well stem from a confusion between a theory in its pure neoplatonic (or even Augustinean) form and its scholastic and neoscholastic distortions.  Husserl's theory of pure intuition of eidetic forms and so forth is not original, if found in a certain form in Schopenhauer and Goethe, its true origin is in neoplatonism and its theory of the logoi and the nous. The subtext of the later Husserl is precisely the development of a unusual (supernatural one might say)  mode of spiritual contemplation and intuition which presupposes a profound spiritual transformation of the subject. Recall also our criticism of Husserl in the context of interpretation of the allegory of the cave.  It is the the silencing and emptying of the proliferations of mental thought-content (and its feeling, volition, mine and self - making, etc) which is the condition to obtain pure vision and wisdom. The maze of concepts (and the enigma of meaning) is only solved and overcome by discarding the whole thing (maybe as in Sextus) and gazing at it from the outside. Theory of knowledge aside, it is wonderful to understand how the mind works, how consciousness works - and if this understanding appears itself paradoxically part of the mind (and as yet unfounded) we say that it is made possible by something coming from without and truth is shown in power.

The Logical Investigations are based on there being something beyond psychic experience, something pointed to by signification acts,  an intended objectivity beyond any concrete lived psychic experience. Thus the Logical Investigations might be construed as an essay in neoscholastic epistemology and psychology in sharp contrast to the more philosophical psychologist approaches of Meinong and Twardowski ( On the Content and Object of Presentations: A Psychological Investigation, 1894). Indeed we do not find here yet 'the enigma of transcendence' of the object (how can mental content refer to something beyond itself, just as in the incompleteness theorem a sentence seems to jump out of the system and engage in meta-talk)  or even an awareness of Kantian questioning - all which will appear later in the Cartesian Meditiations.  This intentionality and its transcendent objectivity is yet still something completely immanent in consciousness expressesing fundamental categories of the mind.  The idea of objectivity itself  (or intersubjectivity, it is based on the previous idea of another self) is mere necessary categorical mental content accompanying the more immediate mental experience: for instance when we think and make the judgment '7 + 5 = 12' . And this accompanying idea of objectivity is related to the self.

The Pali abhidhamma literature is of great interest (this type of sophisticated philosophical-psychological literature was found in different schools of early Buddhism, many of which are now extinct).  The abhidhamma concerns the fundamental elements (or factors) of consciousness (the cetasikas which come together to form mental states or cities) and their combinations and dynamic causal structure.  The lucidity and insight of this philosophy is a wholesome antidote to much of western mental conditioning and allows one to know what to look for in a detached neutralized global introspective psychology (or phenomenology).

One must not try to conceptualize the process of pure introspection and insight into consciousness, rather one must let it manifest spontaneously 'by itself'.  The principle for obtaining this insight and following the right method is the awareness of temporality in its universal transcendental sense permeating and constituting the totality of the immanent flux of consciousness. It is from a firm and clear insight into temporality that insight-analysis-decontruction of thought and self arises and thus the necessary 'conversion' or 'revulsion'. The self is a tendency to reify vortices and thought is  lost  in a maze and tangled forest.  Understand anicca, anatta and dukkha (there is also a connection between anatta and a certain tradition of love poetry). Yet one must not hastily conceptualize one's liberating insight such as in the atomism of the later abhidhamma.

And there still remains the open problem regarding how all this is to be integrated with our considerations on the Platonic dialectic. One could for instance put forward the hypothesis: most of ordinary mental life is too dim, weak and vague to effect a self-intuition and self-unveiling of consciousness: but the practice of Platonic dialectic and its cyclic refining concentration of pure concepts and their network energizes and illumines consciousness rendering it vast and powerful, self-transparent and self-luminous, the necessary qualities required to carry out higher phenomenology.

Or rather it should be seen as follows: we must start with concentrating on the highest ideals of objectivity and truth which are also the highest ideals of knowledge and science: the chief position is occupied by logic and mathematics.  Then we must investigate the claims of objectivity and truth phenomenologically, unveiling the essence of their  formalism, ideality, and a priority with the corresponding lived subjective psychological intentionality and meaning acts. By a special act of reflection we cognize the ideas presupposed or which necessarily accompany certain types of mental act, for instance those pertaining to logic, mathematics or the sciences. But what precisely is the nature of this reflection and how does it cognize ?

Consider Masefield's book: Divine Revelation in Pali Buddhism.  There is much to be elucidated in the Pali texts. For instance, the nature of pañña,  what is going on precisely when the dharma is being listened to (cf. 'shravakas'),  what is the function of their structure and repetition, what the dharma itself is and how it is cognized. Is the dharma itself made of the 5 khandas and how could the khandas contain in themselves knowledge leading to their own overcoming ? We may consider the thesis that in the Pali texts there is postulated a process of 'ideation', 'abstraction' and 'luminous evidence'  entirely analogous to the one discussed in the Logical Investigations. Pañña is a kind of evidence that cannot be reduced to the ordinary psychological experience of an ego.   Jayatilleke, p.428:

The Buddhist theory of truth (v. supra, 596) also makes it clear that truth and therefore knowledge is objective, as telling us the nature of'things as they are' (yathâbhûtam). The knowledge of things as they are consists in knowing 'what exists as "existing" and what does not exist as "not existing" ' (santam vâ atthl ti nassati asantam vâ natthi ti nassati, A. V.36). 'Knowing things as they are', it is said, 'wherever they are, is the highest knowledge' (etad anuttariyam . . . nânânam yadidam tattha tattha yathäbhütanänam, A. V.37). What is taught by the Buddha is claimed to be objectively valid: 'Whether the Tathâgata preaches the dhamma to his disciples or does not preach it, the dhamma remains the same' (desento pi Tathâgato sâvakânam dhammam tâdiso va adesento pi hi dhammo tâdiso va, M. L331).

Brentano, Meinong,  Mally,  Marty, Twardowski, Stumpf, von Ehrenfels are clearly closer to buddhist epistemology than Husserl (or Bolzano, Lotze, Frege, Natorp). The delicate and subtitle interaction between the subjective and the objective in Husserl renders his writings an excellent propadeutic to a whole lost content of philosophy (when philosophy took a wrong turn after the first quarter of the 20th-century). We must investigate the relationship between psychological introspection, phenomenological reflection and satipatthana and vipassana. Indeed the true sati and vipassana of the content of consciousness goes beyond the ordinary Erlebnis of sense-perception and inner sensual imagination and feeling - it involves also a clear awareness of the inseparable 'deep structure'  khandas: samkhara and viññana. Also we must investigate the terms close to that of 'object', 'intentionality', 'signs' and the arupa jhanas. It is the Buddhist theory of knowledge that requires further detailed investigation.

For all the richness and historical references found in the Logical Investigations, this work still does not address directly the fundamental problems of the theory of knowledge, and thus remains a kind of elegant (even if Kantian-like) scholasticism combining previous logical-objectivist and psychological-descriptive currents.

Later we shall analyse poetry and the phenomenology of poetry as a means of access to the truth as well as a path of spiritual development.