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 ?

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.

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 units 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.

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' ! 

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.

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.