Holism. Note Quote.


It is a basic tenet of systems theory/holism as well as of theosophy that the whole is greater than the sum of its parts. If, then, our individual minds are subsystems of larger manifestations of mind, how is it that our own minds are self-conscious while the universal mind (on the physical plane) is not? How can a part possess a quality that the whole does not? A logical solution is to regard the material universe as but the outer garment of universal mind. According to theosophy the laws of nature are the wills and energies of higher beings or spiritual intelligences which in their aggregate make up universal mind. It is mind and intelligence which give rise to the order and harmony of the physical universe, and not the patterns of chance, or the decisions of self-organizing matter. Like Capra, the theosophical philosophy rejects the traditional theological idea of a supernatural, extracosmic divine Creator. It would also question Capra’s notion that such an extracosmic God is the self-organizing dynamics of the physical universe. Theosophy, on the other hand, firmly believes in the existence of innumerable superhuman, intracosmic intelligences (or gods), which have already passed through the human stage in past evolutionary cycles, and to which status we shall ourselves one day attain. There are two opposing views of consciousness: the Western scientific view which considers matter as primary and consciousness as a by-product of complex material patterns associated with a certain stage of biological evolution; and the mystical view which sees consciousness as the primary reality and ground of all being. Systems theory accepts the conventional materialist view that consciousness is a manifestation of living systems of a certain complexity, although the biological structures themselves are expressions of “underlying processes that represent the system’s self-organization, and hence its mind. In this sense material structures are no longer considered the primary reality” (Turning Point). This stance reaffirms the dualistic view of mind and matter. Capra clearly believes that matter is primary in the sense that the physical world comes first and life, mind, and consciousness emerge at a later stage. That he chooses to call the self-organizing dynamics of the universe by the name “mind” is beside the point. If consciousness is regarded as the underlying reality, it is impossible to regard it also as a property of matter which emerges at a certain stage of evolution. Systems theory accepts neither the traditional scientific view of evolution as a game of dice, nor the Western religious view of an ordered universe designed by a divine creator. Evolution is presented as basically open and indeterminate, without goal or purpose, yet with a recognizable pattern of development. Chance fluctuations take place, causing a system at a certain moment to become unstable. As it “approaches the critical point, it ‘decides’ itself which way to go, and this decision will determine its evolution”. Capra sees the systems view of the evolutionary process not as a product of blind chance but as an unfolding of order and complexity analogous to a learning process, including both independence from the environment and freedom of choice. However, he fails to explain how supposedly inert matter is able to “decide,” “choose,” and “learn.” This belief that evolution is purposeless and haphazard and yet shows a recognizable pattern is similar to biologist Lyall Watson’s belief that evolution is governed by chance but that chance has “a pattern and a reason of its own”.

While the materialistic and mystical views of mind seem incompatible and irreconcilable, mind/matter dualism may be resolved by seeing spirit and matter as fundamentally one, as different grades of consciousness-life-substance. Science already holds that physical matter and energy are interconvertible, that matter is concentrated energy; and theosophy adds that consciousness is the highest and subtlest form. From this view there is no absolutely dead and unconscious matter in the universe. Everything is a living, evolving, conscious entity, and every entity is composite, consisting of bundles of forces and substances pertaining to different planes, from the astral-physical through the psychomental to divine-spiritual. Obviously the degree of manifested life and consciousness varies widely from one entity to another; but at the heart of every entity is an indwelling spiritual atom or consciousness-center at a particular stage of its evolutionary unfoldment. More complex material forms do not create consciousness, but merely provide a more developed vehicle through which this spiritual monad can express its powers and faculties. Evolution is far from being purposeless and indeterminate: our human monads issued from the divine Source aeons ago as unself-conscious god-sparks and, by taking embodiment and garnering experience in all the kingdoms of nature, we will eventually raise ourselves to the status of self-conscious gods.

Badiou’s Materiality as Incorporeal Ontology. Note Quote.


Badiou criticises the proper form of intuition associated with multiplicities such as space and time. However, his own ’intuitions’ are constrained by set theory. His intuition is therefore as ‘transitory’ as is the ontology in terms of which it is expressed. Following this constrained line of reasoning, however, let me now discuss how Badiou encounters the question of ‘atoms’ and materiality: in terms of the so called ‘atomic’ T-sets.

If topos theory designates the subobject-classifier Ω relationally, the external, set-theoretic T-form reduces the classificatory question again into the incorporeal framework. There is a set-theoretical, explicit order-structure (T,<) contra the more abstract relation 1 → Ω pertinent to categorical topos theory. Atoms then appear in terms of this operator <: the ‘transcendental grading’ that provides the ‘unity through which all the manifold given in an intuition is united in a concept of the object’.

Formally, in terms of an external Heyting algebra this comes down to an entity (A,Id) where A is a set and Id : A → T is a function satisfying specific conditions.

Equaliser: First, there is an ‘equaliser’ to which Badiou refers as the ‘identity’ Id : A × A → T satisfies two conditions:

1) symmetry: Id(x, y) = Id(y, x) and
2) transitivity: Id(x, y) ∧ Id(y, z) ≤ Id(x, z).

They guarantee that the resulting ‘quasi-object’ is objective in the sense of being distinguished from the gaze of the ‘subject’: ‘the differences in degree of appearance are not prescribed by the exteriority of the gaze’.

This analogous ‘identity’-function actually relates to the structural equalization-procedure as appears in category theory. Identities can be structurally understood as equivalence-relations. Given two arrows X ⇒ Y , an equaliser (which always exists in a topos, given the existence of the subobject classifier Ω) is an object Z → X such that both induced maps Z → Y are the same. Given a topos-theoretic object X and U, pairs of elements of X over U can be compared or ‘equivalized’ by a morphism XU × XUeq ΩU structurally ‘internalising’ the synthetic notion of ‘equality’ between two U-elements. Now it is possible to formulate the cumbersome notion of the ‘atom of appearing’.

An atom is a function a : A → T defined on a T -set (A, Id) so that
(A1) a(x) ∧ Id(x, y) ≤ a(y) and
(A2) a(x) ∧ a(y) ≤ Id(x, y).
As expressed in Badiou’s own vocabulary, an atom can be defined as an ‘object-component which, intuitively, has at most one element in the following sense: if there is an element of A about which it can be said that it belongs absolutely to the component, then there is only one. This means that every other element that belongs to the component absolutely is identical, within appearing, to the first’. These two properties in the definition of an atom is highly motivated by the theory of T-sets (or Ω-sets in the standard terminology of topological logic). A map A → T satisfying the first inequality is usually thought as a ‘subobject’ of A, or formally a T-subset of A. The idea is that, given a T-subset B ⊂ A, we can consider the function
IdB(x) := a(x) = Σ{Id(x,y) | y ∈ B}
and it is easy to verify that the first condition is satisfied. In the opposite direction, for a map a satisfying the first condition, the subset
B = {x | a(x) = Ex := Id(x, x)}
is clearly a T-subset of A.
The second condition states that the subobject a : A → T is a singleton. This concept stems from the topos-theoretic internalization of the singleton-function {·} : a → {a} which determines a particular class of T-subsets of A that correspond to the atomic T-subsets. For example, in the case of an ordinary set S and an element s ∈ S the singleton {s} ⊂ S is a particular, atomic type of subset of S.
The question of ‘elements’ incorporated by an object can thus be expressed externally in Badiou’s local theory but ‘internally’ in any elementary topos. For the same reason, there are two ways for an element to be ‘atomic’: in the first sense an ‘element depends solely on the pure (mathematical) thinking of the multiple’, whereas the second sense relates it ‘to its transcendental indexing’. In topos theory, the distinction is slightly more cumbersome. Badiou still requires a further definition in order to state the ‘postulate of materialism’.
An atom a : A → T is real if if there exists an element x ∈ T so that a(y) = Id(x,y) ∀ y ∈ A.
This definition gives rise to the postulate inherent to Badiou’s understanding of ‘democratic materialism’.
Postulate of Materialism: In a T-set (A,Id), every atom of appearance is real.
What the postulate designates is that there really needs to exist s ∈ A for every suitable subset that structurally (read categorically) appears to serve same relations as the singleton {s}. In other words, what ‘appears’ materially, according to the postulate, has to ‘be’ in the set-theoretic, incorporeal sense of ‘ontology’. Topos theoretically this formulation relates to the so called axiom of support generators (SG), which states that the terminal object 1 of the underlying topos is a generator. This means that the so called global elements, elements of the form 1 → X, are enough to determine any particular object X.
Thus, it is this specific condition (support generators) that is assumed by Badiou’s notion of the ‘unity’ or ‘constitution’ of ‘objects’. In particular this makes him cross the line – the one that Kant drew when he asked Quid juris? or ’Haven’t you crossed the limit?’ as Badiou translates.
But even without assuming the postulate itself, that is, when considering a weaker category of T-sets not required to fulfill the postulate of atomism, the category of quasi-T -sets has a functor taking any quasi-T-set A into the corresponding quasi-T-set of singletons SA by x → {x}, where SA ⊂ PA and PA is the quasi-T-set of all quasi-T-subsets, that is, all maps T → A satisfying the first one of the two conditions of an atom designated by Badiou. It can then be shown that, in fact, SA itself is a sheaf whose all atoms are ‘real’ and which then is a proper T-set satisfying the ‘postulate of materialism’. In fact, the category of T-Sets is equivalent to the category of T-sheaves Shvs(T, J). In the language of T-sets, the ‘postulate of materialism’ thus comes down to designating an equality between A and its completed set of singletons SA.

Priest’s Razor: Metaphysics. Note Quote.


The very idea that some mathematical piece employed to develop an empirical theory may furnish us information about unobservable reality requires some care and philosophical reflection. The greatest difficulty for the scientifically minded metaphysician consists in furnishing the means for a “reading off” of ontology from science. What can come in, and what can be left out? Different strategies may provide for different results, and, as we know, science does not wear its metaphysics on its sleeves. The first worry may be making the metaphysical piece compatible with the evidence furnished by the theory.

The strategy adopted by da Costa and de Ronde may be called top-down: investigating higher science and, by judging from the features of the objects described by the theory, one can look for the appropriate logic to endow it with just those features. In this case (quantum mechanics), there is the theory, apparently attributing contradictory properties to entities, so that a logic that does cope with such feature of objects is called forth. Now, even though we believe that this is in great measure the right methodology to pursue metaphysics within scientific theories, there are some further methodological principles that also play an important role in these kind of investigation, principles that seem to lessen the preferability of the paraconsistent approach over alternatives.

To begin with, let us focus on the paraconsistent property attribution principle. According to this principle, the properties corresponding to the vectors in a superposition are all attributable to the system, they are all real. The first problem with this rendering of properties (whether they are taken to be actual or just potential) is that such a superabundance of properties may not be justified: not every bit of a mathematical formulation of a theory needs to be reified. Some of the parts of the theory are just that: mathematics required to make things work, others may correspond to genuine features of reality. The greatest difficulty is to distinguish them, but we should not assume that every bit of it corresponds to an entity in reality. So, on the absence of any justified reason to assume superpositions as a further entity on the realms of properties for quantum systems, we may keep them as not representing actual properties (even if merely possible or potential ones).

That is, when one takes into account other virtues of a metaphysical theory, such as economy and simplicity, the paraconsistent approach seems to inflate too much the population of our world. In the presence of more economical candidates doing the same job and absence of other grounds on which to choose the competing proposals, the more economical approaches take advantage. Furthermore, considering economy and the existence of theories not postulating contradictions in quantum mechanics, it seems reasonable to employ Priest’s razor – the principle according to which one should not assume contradictions beyond necessity – and stick with the consistent approaches. Once again, a useful methodological principle seems to deem the interpretation of superposition as contradiction as unnecessary.

The paraconsistent approach could take advantage over its competitors, even in the face of its disadvantage in order to accommodate such theoretical virtues, if it could endow quantum mechanics with a better understanding of quantum phenomena, or even if it could add some explanatory power to the theory. In the face of some such kind of gain, we could allow for some ontological extravagances: in most cases explanatory power rules over matters of economy. However, it does not seem that the approach is indeed going to achieve some such result.

Besides that lack of additional explanatory power or enlightenment on the theory, there are some additional difficulties here. There is a complete lack of symmetry with the standard case of property attribution in quantum mechanics. As it is usually understood, by adopting the minimal property attribution principle, it is not contentious that when a system is in one eigenstate of an observable, then we may reasonably infer that the system has the property represented by the associated observable, so that the probability of obtaining the eigenvalue associated is 1. In the case of superpositions, if they represented properties of their own, there is a complete disanalogy with that situation: probabilities play a different role, a system has a contradictory property attributed by a superposition irrespective of probability attribution and the role of probabilities in determining measurement outcomes. In a superposition, according to the proposal we are analyzing, probabilities play no role, the system simply has a given contradictory property by the simple fact of being in a (certain) superposition.

For another disanalogy with the usual case, one does not expect to observe a sys- tem in such a contradictory state: every measurement gives us a system in particular state, never in a superposition. If that is a property in the same foot as any other, why can’t we measure it? Obviously, this does not mean that we put measurement as a sign of the real, but when doubt strikes, it may be a good advice not to assume too much on the unobservable side. As we have observed before, a new problem is created by this interpretation, because besides explaining what is it that makes a measurement give a specific result when the system measured is in a superposition (a problem usually addressed by the collapse postulate, which seems to be out of fashion now), one must also explain why and how the contradictory properties that do not get actualized vanish. That is, besides explaining how one particular property gets actual, one must explain how the properties posed by the system that did not get actual vanish.

Furthermore, even if states like 1/√2 (| ↑x ⟩ + | ↓x ⟩) may provide for an example of a  candidate of a contradictory property, because the system seems to have both spin up and down in a given direction, there are some doubts when the distribution of probabilities is different, in cases such as 2/√7 | ↑x ⟩ + √(3/7) | ↓x ⟩. What are we to think about that? Perhaps there is still a contradiction, but it is a little more inclined to | ↓x⟩ than to | ↑x⟩? That is, it is difficult to see how a contradiction arises in such cases. Or should we just ignore the probabilities and take the states composing the superposition as somehow opposed to form a contradiction anyway? That would put metaphysics way too much ahead of science, by leaving the role of probabilities unexplained in quantum mechanics in order to allow a metaphysical view of properties in.

Metempsychosis of the Ancients’ Veritability


The impenetrable veil of arcane secrecy was thrown over the sciences taught in the sanctuary. This is the cause of the modern depreciation of the ancient philosophies. Much of Plato’s public teachings and writings had therefore to consist of blinds or half-truths or allegories, and just as Jesus spoke in parables, so the Mysteries were ever reserved for special groups of neophytes – and, needless to say, they did not reach the Church of the days of Constantine, which never held the keys of the Mysteries, and hence can hardly be said to have lost them.

The ancient philosophers seem to be generally held, even by the least prejudiced of modern critics, to have lacked that profundity and thorough knowledge in the exact sciences of which a couple of last centuries and present are so boastful. It is even questioned whether they understood that basic scientific principle: ex nihilo nihil fit. If they suspected the indestructibility of matter at all – say these commentators – it was not in consequence of a firmly established formula, but only through intuitional reasoning and by analogy. The philosophers themselves had to be initiated into perceptive mysteries, before they could grasp the correct idea of the ancients in relation to this most metaphysical subject. Otherwise – outside such initiation – for every thinker there will be a “Thus far shalt thou go and no further,” mapped out by his intellectual capacity, as clearly and unmistakably as there is for the progress of any nation or race in its cycle by the law of karma. Much of current agnostic speculation on the existence of the “First Cause” is little better than veiled materialism — the terminology alone being different. Even so a thinker as Herbert Spencer speaks of the “Unknowable” occasionally in terms that demonstrate the lethal influence of materialistic thought which, like the deadly sirocco, has withered and blighted most of current ontological speculation. For instance, when he terms the “First Cause” — the Unknowable — a “power manifesting through phenomena,” and an “infinite eternal Energy,” (?) it is clear that he has grasped solely the physical aspect of the mystery of Being — the energies of cosmic substance only. The co-eternal aspect of the ONE REALITY — cosmic ideation — (as to its noumenon, it seems nonexistent in his mind) — is absolutely omitted from consideration.

The doctrine of metempsychosis has been abundantly ridiculed by scientists and rejected by theologians, yet if it had been properly understood in its application to the indestructibility of matter and the immortality of spirit, it would have been perceived that it is a sublime conception. Should we not first regard the subject from the standpoint of the ancients before venturing to disparage its teachers? The solution of the great problem of eternity belongs neither to religious superstition nor to gross materialism. The harmony and mathematical equiformity of the double evolution – spiritual and physical – are elucidated only in the universal numerals of Pythagoras, who built his system entirely upon the so-called metrical speech of the Hindu Vedas.

Symmetry, Cohomology and Homotopy. Note Quote Didactic.

Given a compact Kähler manifold Y, we know that the cohomology groups H(Y,C) have a Hodge decomposition Hp,q. Now because we have Poincaré duality, and the comparisons between singular and De Rham cohomologie, we know that any other space Z that has the same homotopy type as Y will have the same cohomology groups. Consequently they will share the same Hodge diamond, thus its symmetries.


This means that the symmetry of the Hodge diamond is mostly attached to the homotopy type of Y . This is not surprising anymore because it can already be seen from the equivalence of the De Rham cohomology which is analytic, and the Betti cohomology which is something purely simplicial. In fact, it can also be seen from the (smooth) homotopy invariance of De Rham cohomology.

This symmetry can be understood using the Quillen-Segal formalism as follows. Given Y , let’s consider Ytop ∈ Top. We have a Quillen equivalence U : Top → sSetQ, where U = Sing is the singular functor whose left adjoint is the geometric realization. When we consider the comma category sSetQU[Top] = sSet ↓ U, we are literally creating in French a “trait d’union”, between the two categories. And when we consider the subcategory of Quillen-Segal objects then we have a triangle that descends to a triangle of equivalences between the homotopy categories. In fact there is a much better statement.

 IMG_20170419_063100 (1)

It turns out that if we choose the Joyal model structure sSetJ, we get the Homotopy hypothesis.

A fibrant replacement of Y in the model category sSetQU[Top], is a trivial fibration F → U(Y), where F is fibrant in sSetQ, that is a Kan complex. But a Kan complex is exactly an ∞-groupoid. ∞-Groupoids generalize groupoids, and still are category-like. In particular we can take their opposite (or dual), just like we consider the opposite category Cop of a usual category.

Given Y as above, we can think of the mirror of Y as the opposite ∞-groupoid Fop. A good approximation of Fop can be obtained by the schematization functor à la Toën applied to the simplicial set (quasicategory) underlying Fop. We can take as model for F the fundamental ∞-groupoid Π(Y). And depending on the dimension it’s enough to stop at the corresponding n-groupoid. Toën schematization functor can also be obtained from the Quillen-Segal formalism applied to the embedding

U : Sh(Var(C)) ֒→ sPresh(Var(C),

where on the right hand side we consider the model category of simplicial presheaves à la Jardine-Joyal. The representability of the π0 of the schematization has to be determined by descent along the equivalence type.

‘Pranic’ Allostery (Microprocessor + Functional Machine)


Michael J. Denton in Nature’s Destiny (1998) gives an interesting example from biochemistry, that of proteins. Proteins are built of chains of amino acids which mainly consist of carbon, hydrogen, and nitrogen. Proteins have a specific spatial structure which, as we have seen above, is very sensitive – for example to the temperature or acidity of the environment – and which can very easily be changed and restored for specific purposes within a living organism. Proteins are stable, but remain in a delicate balance, ever on the threshold of chaos. They are able to bond themselves to certain chemicals and to release them in other situations. It is this property which enables them to perform a variety of functions, for example catalyzing other chemical reactions in a cell. Proteins have the power to integrate information from various chemical sources, which is determined by the concentration within the cell of the chemicals involved. As we have seen when discussing the eye, proteins enable the processes in the cell to regulate themselves. This self-regulation is called allostery.

Thus proteins have a remarkable two-sided power – firstly, the performance of unique chemical reactions and the integration of the information of diverse chemical components of the cell; and secondly, intelligent reaction to this information by increasing or decreasing their own enzymic activity according to present needs. How this is possible is still regarded as one of the greatest mysteries of life. It means that the functional units which perform the chemical processes are at the same time the regulating units. This property is crucial for the functioning of the cell processes in orderly coherence. It prevents the chaos that would no doubt follow if the enzymic activity were not precisely adjusted to the ever-changing needs of the cell. It is thus the remarkable property of proteins to unite the role of both a microprocessor and a functional machine in one object. Because of this fundamental property, proteins are far more advanced than any man-made instrument. An oven, for example, has a thermostat to regulate temperature, and a functional unit, the burner or electric coil, which produces heat. In a protein these two would be unified.

Blavatsky maintained that every cell in the human body is furnished with its own brain, with a memory of its own, and therefore with the experience and power to discriminate between things. How could she say so within the context of the scientific knowledge of her day? Her knowledge was deduced from occult axioms concerning the functioning of the universe and from analogy, which is applicable on all levels of being. If there is intelligence in the great order of the cosmos, then this is also represented within a cell, and there must be a structure within the cell comparable to the physical brain. This structure must have the power to enable the processes of intelligence on the physical level to take place.

G. de Purucker wrote some seventy years ago about life-atoms, centrosomes, and centrioles. He stated that “In each cell there is a central pranic nucleus which is the life-germ of a life-atom, and all the rest of the cell is merely the carpentry of the cell builded around it by the forces flowing forth from the heart of this life-atom.” A life-atom is a consciousness-point. He explained that

the life-atom works through the two tiny dots or sparks in the centrosome which fall apart at the beginning of cell-division and its energies stream out from these two tiny dots, and each tiny dot, as it were, is already the beginning of a new cell; or, to put it in other words, one remains the central part of the mother-cell, while the other tiny dot becomes the central part of the daughter-cell, etc. All these phenomena of mitosis or cell-division are simply the works of the inner soul of the physical cell . . . The heart of an original nucleolus in a cell is the life-atom, and the two tiny dots or spots [the centrioles] in the centrosome are, as it were, extensions or fingers of its energy. The energy of the original life-atom, which is the heart of a cell, works throughout the entire cellular framework or structure in general, but more particularly through the nucleolus and also through the two tiny dots. — Studies in Occult Philosophy 

Along these lines Blavatsky says that the

inner soul of the physical cell . . . dominates the germinal plasm . . . the key that must open one day the gates of the terra incognita of the Biologist . . . (The Secret Doctrine).

Theosophical Panpsychism


Where does mind individually, and consciousness ultimately, originate? In the cosmos there is only one life, one consciousness, which masquerades under all the different forms of sentient beings. This one consciousness pierces up and down through all the states and planes of being and serves to uphold the memory, whether complete or incomplete, of each state’s experience. This suggests that our self-conscious mind is really a ray of cosmic mind. There is a mysterious vital life essence and force involved in the interaction of spirit or consciousness with matter. The cosmos has its memory and follows general pathways of formation based on previous existences, much as everything else does. Aided by memory, it somehow selects out of the infinite possibilities a new and improved imbodiment. When the first impulse emerges, we have cosmic ideation vibrating the first matter, manifesting in countless hierarchies of beings in endless gradations. Born of the one cosmic parent, monadic centers emerge as vital seeds of consciousness, as germs of its potential. They are little universes in the one universe.

Theosophy does not separate the world into organic and inorganic, for even the atoms are considered god-sparks. All beings are continuously their own creators and recorders, forming more perishable outer veils while retaining the indestructible thread-self that links all their various principles and monads through vast cycles of experience. We are monads or god-sparks currently evolving throughout the human stage. The deathless monad runs through all our imbodiments, for we have repeated many times the processes of birth and death. In fact, birth and death for most of humanity are more or less automatic, unconscious experiences as far as our everyday awareness is concerned. How do we think? We can start, for example, with desire which provides the impulse that causes the mind through will and imagination to project a stream of thoughts, which are living elemental beings. These thoughts take various forms which may result in different kinds of actions or creative results. This is another arena of responsibility, for in the astral light our thoughts circulate through other minds and affect them, but those that belong to us have our stamp and return to us again and again. So through these streams of thought we create habits of mind, which build our character and eventually our self-made destiny. The human mind is an ideator resonating with its past, selecting thoughts and making choices, anticipating and creating a pattern of unfolding. Perhaps we are reflecting in the small the operations of the divine mind which acts as the cosmic creator and architect. Some thoughts or patterns we create are limiting; others are liberating. The soul grows, and thoughts are reused and transformed by the mind, perhaps giving them a superior expression. Plato was right: with spiritual will and worthiness we can recollect the wisdom of the past and unlock the higher mind. We have the capacity of identifying with all beings, experiencing the oneness we share together in our spiritual consciousness, that continuous stream that is the indestructible thread-self. All that it was, is, or is becoming is our karma. Mind and memory are a permanent part of the reincarnating ego or human soul, and of the universe as well.

In the cosmos there are many physical, psychic, mental, and spiritual fields — self-organizing, whole, living systems. Every such field is holographic in that it contains the characteristics of every other field within itself. Rupert Sheldrake’s concepts of morphic fields and morphic resonance, for instance, are in many ways similar to some phenomena attributed to the astral light. All terrestrial entities can be considered fields belonging to our living earth, Gaia, and forming part of her constitution. The higher akasic fields resonate with every part of nature. Various happenings within the earth’s astral light are said to result in physical effects which include all natural and human phenomena, ranging from epidemics and earthquakes to wars and weather patterns. Gaia, again, is part of the fields which form the solar being and its constitution, and so on throughout the cosmos.

Like the earth, human beings each have auric fields and an astral body. The fifty trillion cells in our body, as well as the tissues and organs they form, each have their own identity and memory. Our mental and emotional fields influence every cell and atom of our being for better or worse. How we think and act affects not only humanity but Gaia as well through the astral light, the action of which is guided by active creative intelligences. For example, the automatic action of divine beings restores harmony, balancing the inner with the outer throughout nature.

Symmetry: Mirror of a Manifold is the Opposite of its Fundamental Poincaré ∞-groupoid


Given a set X = {a,b,c,..} such as the natural numbers N = {0,1,…,p,…}, there is a standard procedure that amounts to regard X as a category with only identity morphisms. This is the discrete functor that takes X to the category denoted by Disc(X) where the hom-sets are given by Hom(a,b) = ∅ if a ≠ b, and Hom(a,b) = {Ida} = 1 if a = b. Disc(X) is in fact a groupoid.

But in category theory, there is also a procedure called opposite or dual, that takes a general category C to its opposite Cop. Let us call Cop the reflection of C by the mirror functor (−)op.

Now the problem is that if we restrict this procedure to categories such as Disc(X), there is no way to distinguish Disc(X) from Disc(X)op. And this is what we mean by sets don’t show symmetries. In the program of Voevodsky, we can interpret this by saying that:

The identity type is not good for sets, instead we should use the Equivalence type. But to get this, we need to move to from sets to Kan complexes i.e., ∞-groupoids.

The notion of a Kan complex is an abstraction of the combinatorial structure found in the singular simplicial complex of a topological space. There the existence of retractions of any geometric simplex to any of its horns – simplices missing one face and their interior – means that all horns in the singular complex can be filled with genuine simplices, the Kan filler condition.

At the same time, the notion of a Kan complex is an abstraction of the structure found in the nerve of a groupoid, the Duskin nerve of a 2-groupoid and generally the nerves of n-groupoids ∀ n ≤ ∞ n. In other words, Kan complexes constitute a geometric model for ∞-groupoids/homotopy types which is based on the shape given by the simplex category. Thus Kan complexes serve to support homotopy theory.

So far we’ve used set theory with this lack of symmetries, as foundations for mathematics. Grothendieck has seen this when he moved from sheaves of sets, to sheaves of groupoid (stacks), because he wanted to allow objects to have symmetries (automorphisms). If we look at the Giraud-Grothendieck picture on nonabelian cohomology, then what happens is an extension of coefficients U : Set ֒→ Cat. We should consider first the comma category Cat ↓ U, whose objects are functors C → Disc(X). And then we should consider the full subcategory consisting of functors C → Disc(X) that are equivalences of categories. This will force C to be a groupoid, that looks like a set. And we call such C → Disc(X) a Quillen-Segal U-object.

This category of Quillen-Segal objects should be called the category of sets with symmetries. Following Grothendieck’s point of view, we’ve denoted by CatU[Set] the comma category, and think of it as categories with coefficients or coordinates in sets. This terminology is justified by the fact that the functor U : Set ֒→ Cat is a morphism of (higher) topos, that defines a geometric point in Cat. The category of set with symmetries is like the homotopy neighborhood of this point, similar to a one-point going to a disc or any contractible object. The advantage of the Quillen-Segal formalism is the presence of a Quillen model structure on CatU[Set] such that the fibrant objects are Quillen-Segal objects.

In standard terminology this means that if we embed a set X in Cat as Disc(X), and take an ‘projective resolution’ of it, then we get an equivalence of groupoids P → Disc(X), and P has symmetries. Concretely what happens is just a factorization of the identity (type) Id : Disc(X) → Disc(X) as a cofibration followed by a trivial fibration:

Disc(X)  ֒→ P → Disc(X)

This process of embedding Set ֒→ QS{CatU[Set]} is a minimal homotopy enhancement. The idea is that there is no good notion of homotopy (weak equivalence) in Set, but there are at least two notions in Cat: equivalences of categories and the equivalences of classifying spaces. This last class of weak equivalences is what happens with mirror phenomenons. The mirror of a manifold should be the opposite of its fundamental Poincaré ∞-groupoid.

Badiou, Heyting Algebras cross the Grothendieck Topoi. Note Quote.

Let us commence by introducing the local formalism that constitutes the basis of Badiou’s own, ‘calculated phenomenology’. Badiou is unwilling to give up his thesis that the history of thinking of being (ontology) is the history of mathematics and, as he reads it, that of set theory. It is then no accident that set theory is the regulatory framework under which topos theory is being expressed. He does not refer to topoi explicitly but rather to the so called complete Heyting algebras which are their procedural equivalents. However, he fails to mention that there are both ‘internal’ and ‘external’ Heyting algebras, the latter group of which refers to local topos theory, while it appears that he only discusses the latter—a reduction that guarantees that indeed that the categorical insight may give nothing new.

Indeed, the external complete Heyting algebras T then form a category of the so called T-sets, which are the basic objects in the ‘world’ of the Logics of Worlds. They local topoi or the so called ‘locales’ that are also ‘sets’ in the traditional sense of set theory. This ‘constitution’ of his worlds thus relies only upon Badiou’s own decision to work on this particular regime of objects, even if that regime then becomes pivotal to his argument which seeks to denounce the relevance of category theory.

This problematic is particularly visible in the designation of the world m (mathematically a topos) as a ‘complete’ (presentative) situation of being of ‘universe [which is] the (empty) concept of a being of the Whole’ He recognises the ’impostrous’ nature of such a ‘whole’ in terms of Russell’s paradox, but in actual mathematical practice the ’whole’ m becomes to signify the category of Sets – or any similar topos that localizable in terms of set theory. The vocabulary is somewhat confusing, however, because sometimes T is called the ‘transcendental of the world’, as if m were defined only as a particular locale, while elsewhere m refers to the category of all locales (Loc).

An external Heyting algebra is a set T with a partial order relation <, a minimal element μ ∈ T , a maximal element M ∈ T . It further has a ‘conjunction’ operator ∧ : T × T → T so that p ∧ q ≤ p and p ∧ q = q ∧ p. Furthermore, there is a proposition entailing the equivalence p ≤ q iff p ∧ q = p. Furthermore p ∧ M = p and μ ∧ p = μ for any p ∈ T .

In the ‘diagrammatic’ language that pertains to categorical topoi, by contrast, the minimal and maximal elements of the lattice Ω can only be presented as diagrams, not as sets. The internal order relation ≤ Ω can then be defined as the so called equaliser of the conjunction ∧ and projection-map

≤Ω →e Ω x Ω →π1 L

The symmetry can be expressed diagrammatically by saying that


is a pull-back and commutes. The minimal and maximal elements, in categorical language, refer to the elements evoked by the so-called initial and terminal objects 0 and 1.

In the case of local Grothendieck-topoi – Grothendieck-topoi that support generators – the external Heyting algebra T emerges as a push-forward of the internal algebra Ω, the logic of the external algebra T := γ ∗ (Ω) is an analogous push-forward of the internal logic of Ω but this is not the case in general.

What Badiou further requires of this ‘transcendental algebra’ T is that it is complete as a Heyting algebra.

A complete external Heyting algebra T is an external Heyting algebra together with a function Σ : PT → T (the least upper boundary) which is distributive with respect to ∧. Formally this means that ΣA ∧ b = Σ{a ∧ b | a ∈ A}.

In terms of the subobject classifier Ω, the envelope can be defined as the map Ωt : ΩΩ → Ω1 ≅ Ω, which is internally left adjoint to the map ↓ seg : Ω → ΩΩ that takes p ∈ Ω to the characteristic map of ↓ (p) = {q ∈ Ω | q ≤ p}27.

The importance the external complete Heyting algebra plays in the intuitionist logic relates to the fact that one may now define precisely such an intuitionist logic on the basis of the operations defined above.

The dependence relation ⇒ is an operator satisfying

p ⇒ q = Σ{t | p ∩ t ≤ q}.

(Negation). A negation ¬ : T → T is a function so that

¬p =∑ {q | p ∩ q = μ},

and it then satisfies p ∧ ¬p = μ.

Unlike in what Badiou calls a ‘classical world’ (usually called a Boolean topos, where ¬¬ = 1Ω), the negation ¬ does not have to be reversible in general. In the domain of local topoi, this is only the case when the so called internal axiom of choice is valid, that is, when epimorphisms split – for example in the case of set theory. However, one always has p ≤ ¬¬p. On the other hand, all Grothendieck-topoi – topoi still materially presentable over Sets – are possible to represent as parts of a Boolean topos.

Dark Matter as an Ode to Ma Kali. Note Quote.


Arcane knowledge provides some answers assuming we ask the questions. If Isis is “Infinite Stars, Infinite Space”, then what is Nepthys? Being the opposite side of Isis we have to assume she plays a part in Universe. And, if Kali’s re-creation of Universe is possible, then can we see it in the process? The answer to both of these lies in the Dark Matter. This is very intriguing but resolved in the connection in Isis’s dark twin, Nepthys. She is dark (like Kali) because she is hidden, manifested but unseen. It is speculated that she became dominant when Isis was shedding lunar blood (sacred to Kali), this is when the unfertile seeds are being discarded. For the aspirant this is a time of great power, and danger. Nepthys is the goddess of the night magicks, the red magick of Vamamarg sometimes referred to as the “left hand path”. Hers is the force of re-creation which is so vital to the growth of the aspirant. IAO, Isis-creator, Apophis (Set, husband of Nepthys)-destroyer, and Osiris-re-creator. In Tantra, Kali is all three. She gives birth to Universe, devours it when all life has expended its energy, and re-creates it from the seeds of the old Universe. It’s uncertain whether there is enough Dark Matter to cause the collapse of Universe, but clearly if there is a chance, it is in this manifestation of the Dark Goddess. Her body is the body of matter that lies “between” known spaces and stars, her power is felt in the pull of matter itself, “Love is the law, love under will” is the axion of gravity where all particles seek to unite with all others. Her books are written in the night sky, her rites are the rites of ancient humans awed by the power of the Great Sleep, and equally awed by it’s power of re-creation. If Kali/Nepthys manifests at the end of time, it will be as the mouths of numerous black holes, each larger one devouring the smaller, uniting in one undifferentiated monad of space-time, not only matter sucked in but the net of creation on which it resides as well. In the Dark Matter is the new creation. Dark matter is maddeningly shy. More like a de-terrestrial-centric potency for sure with none of the considerations for earthlings.