Black Hole Entropy in terms of Mass. Note Quote.


If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)


S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Disjointed Regularity in Open Classes of Elementary Topology


Let x, y, … denote first-order structures in St𝜏, x ≈ y will denote isomorphism.

x ∼n,𝜏 y means that there is a sequence 0 ≠ I0 ⊆ …. ⊆ In of sets of 𝜏-partial isomorphism of finite domain so that, for i < j ≤ n, f ∈ Ii and a ∈ x (respectively, b ∈ y), there is g ∈ Ij such that g ⊇ f and a ∈ Dom(g) (respectively, b ∈ Im(g)). The later is called the extension property.

x ∼𝜏 y means the above holds for an infinite chain 0 ≠ I0 ⊆ …. ⊆ In ⊆ …

Fraïssé’s characterization of elementary equivalence says that for finite relational vocabularies: x ≡ y iff x ∼n,𝜏 y. To have it available for vocabularies containing function symbols add the complexity of terms in atomic formulas to the quantifier rank. It is well known that for countable x, y : x ∼𝜏 y implies x ≈ y.

Given a vocabulary 𝜏 let 𝜏 be a disjoint renaming of 𝜏. If x, y ∈ St𝜏 have the same power, let y be an isomorphic copy of y sharing the universe with x and renamed to be of type 𝜏. In this context, (x, y) will denote the 𝜏 ∪ 𝜏-structure that results of expanding x with the relations of y.

Lemma: There is a vocabulary 𝜏+ ⊇ 𝜏 ∪ 𝜏 such that for each finite vocabulary 𝜏0 ⊆ 𝜏 there is a sequence of elementary classes 𝛥1 ⊇ 𝛥2 ⊇ 𝛥3 ⊇ …. in St𝜏+ such that if 𝜋 = 𝜋𝜏+,𝜏∪𝜏 then (1) 𝜋(𝛥𝑛) = {(x,y) : |x| = |y| ≥ 𝜔, x ≡n,𝜏0 y}, (2) 𝜋(⋂n 𝛥n) = {(x, y) : |x| = |y| ≥ 𝜔, x ∼𝜏0 y}. Moreover, ⋂n𝛥n is the reduct of an elementary class.

Proof. Let 𝛥 be the class of structures (x, y, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c ≤ a a family Ic = {I(c, i, −, −)}i∈x of partial 𝜏0-𝜏0–isomorphisms from x into y, such that for c < c’ ≤ a : Ic ⊆ Ic and the extension property holds. Describe this by a first-order sentence 𝜃𝛥 of type 𝜏+ ⊇ 𝜏0 ∪ 𝜏0 and set 𝛥𝑛 = ModL(𝜃𝛥 ∧ ∃≥n x(x ≤ a)}. Then condition (1) in the Lemma is granted by Fraïssé’s characterization and the fact that x being (2) is granted because (x, y, <, a, I) ∈ ⋂n𝛥n iff < contains an infinite increasing 𝜔-chain below a, a ∑11 condition.

A topology on St𝜏 is invariant if its open (closed) classes are closed under isomorphic structures. Of course, it is superfluous if we identify isomorphic structures.

Theorem: Let Γ be a regular compact topology finer than the elementary topology on each class St𝜏 such that the countable structures are dense in St𝜏 and reducts and renamings are continuous for these topologies. Then Γ𝜏 is the elementary topology ∀ 𝜏.

Proof: We show that any pair of disjoint closed classes C1, C2 of Γ𝜏 may be separated by an elementary class. Assume this is not the case since Ci are compact in the topology Γ𝜏 then they are compact for the elementary topology and, by regularity of the latter, ∃ xi ∈ Ci such that x1 ≡ x2 in L𝜔𝜔(𝜏). The xi must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the Ci. By normality of Γ𝜏, there are towers Ui ⊆ Ci ⊆ Ui ⊆ Ci, i = 1,2, separating the Ci with Ui, Ui open and Ci, Ci closed in Γ𝜏 and disjoint. Let I be a first-order sentence of type 𝜏 ⊇ 𝜏 such that (z, ..) |= I ⇔ z is infinite, and let π be the corresponding reduct operation. For fixed n ∈ ω and the finite 𝜏0  ⊆ 𝜏, let t be a first-order sentence describing the common ≡n,𝜏0 – equivalence class of x1, x2. As,

(xi,..) ∈ Mod𝜏(I) ∩ π-1 Mod(t) ∩ π-1Ui, i = 1, 2,..

and this class is open in Γ𝜏‘ by continuity of π, then by the density hypothesis there are countable xi ∈ Ui , i = 1, 2, such that x1n,𝜏 x2. Thus for some expansion of (x1, x2),

(x, x,..) ∈ 𝛥n,𝜏0 ∩ 𝜋1−1(𝐶1) ∩ (𝜌𝜋2)−1(C2) —– (1)

where 𝛥𝑛,𝜏0 is the class of Lemma, 𝜋1, 𝜋2 are reducts, and 𝜌 is a renaming:

𝜋1(x1, x2, …) = x1 𝜋1 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜋2(x1, x2, …) = x2 𝜋2 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜌(x2) = x2 𝜌 : St𝜏 → St𝜏

Since the classes (1) are closed by continuity of the above functors then ⋂n𝛥n,𝜏0 ∩ 𝜋1−1(C1) ∩ (𝜌𝜋2)−1(C2) is non-emtpy by compactness of Γ𝜏+. But ⋂n𝛥n,𝜏0 = 𝜋(V) with V elementary of type 𝜏++ ⊇ 𝜏+. Then

V ∩ π-1π1-1(U1) ∩ π-1(ρπ2)-1 (U2) ≠ 0

is open for ΓL++ and the density condition it must contain a countable structure (x1, x*2, ..). Thus (x1, x*2, ..) ∈ ∩n 𝛥𝑛,𝜏0, with xi ∈ Ui ⊆ Ci. It follows that x1 ~𝜏0 x2 and thus x1 |𝜏0 ≈ x2 |𝜏0. Let δ𝜏0 be a first-order sentence of type 𝜏 ∪ 𝜏* ∪{h} such that (x, y*, h) |= δ𝜏0 ⇔ h : x |𝜏0 ≈ y|𝜏0. By compactness,

(∩𝜏0fin𝜏 Mod𝜏∪𝜏*∪{f}𝜏0)) ∩ π1-1(C1) ∩ (ρπ2)-1(C2) ≠ 0

and we have h : x1 ≈ x2, xi ∈ Ci, contradicting the disjointedness of Ci. Finally, if C is a closed class of Γ𝜏 and x ∉ C, clΓ𝜏{x} is disjoint from C by regularity of Γ𝜏. Then clΓ𝜏{x} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.

Vedic Mathematics: Sixteen Sutras and their Corollaries


The divergence embraces everything other than the fact of intuition itself – the object and field of intuitive vision, the method of working out experience and rendering it to the intellect. The modern method is to get the intuition by suggestion from an appearance in life or nature or from a mental idea and even if the source of the intuition ie the soul, the method at once relates it to a support external to the soul. The ancient Indian method of knowledge had for its business to disclose something of the Self, the Infinite or the Divine to the regard of the soul – the Self through its expressions, the infinite through its finite symbols and the Divine through his powers. The process was one of Integral knowledge and in its subordinate ranges was instrumental in revealing the truths of cosmic phenomena and these truths mere utilised for worldly ends.

Tirthaji_S.B.K. Vedic mathematics or sixteen simple mathematical formulae from the Vedas

Banking? There isn’t much to it than this anyways.

Don’t go by the innocuous sounding title, for this is at its wit alternative. The modus operandi (Oh!, how much I feel like saying modis operandi in honor of the you Indians know who?) for accumulation of wealth to parts of ‘the system’ (which, for historic reasons, we call ‘capitalists’) is banking. The ‘capitalists’ (defined as those that skim the surplus labor of others) accumulate it through the banking system. That is nearly an empty statement, since wealth = money. That is, money is the means of increasing wealth and thus one represents the other. If capitalists skim surplus labor, it means that they skim surplus money. Money is linked to (only!) banks, and thus, accumulation is in the banks.

If interest is charged, borrowers will go bankrupt. This idea can be extended. If interest is charged, all money is accumulated in banks. Or, better to say, a larger and larger fraction of money is accumulated in the banks, and kept in financial institutions. The accumulation of wealth is accumulation of money in and by banks. It can only be interesting to see whom the money belongs to.

By the way, these institutions, the capitalists naturally wanting to part with as little as possible from this money, are often in fiscal paradises. Famous are The Cayman Islands, The Bahamas, The Seychelles, etc. With the accumulated money the physical property is bought. Once again, this is an empty statement. Money represents buying power (to buy more wealth). For instance buying the means-of-production (the Marxian mathematical equation raises its head again, MoP), such as land, factories, people’s houses (which will then be rented to them; more money). Etc.

Also, a tiny fraction of the money is squandered. It is what normally draws most attention. Oil sheiks that drive golden cars, bunga-bunga parties etc. That, however, is rather insignificant, this way of re-injecting money into the system. Mostly money is used to increase capital. That is why it is an obvious truth that “When you are rich, you must be extremely stupid to become poor. When you are poor, you must be extremely talented to become rich”. When you are rich, just let the capital work for you; it will have the tendency to increase, even if it increases slower than that of your more talented neighbor.

To accelerate the effect of skimming, means of production (MoP, ‘capital’), are confiscated from everything – countries and individual people – that cannot pay the loan + interest (which is unavoidable). Or bought for a much-below market value price in a way of “Take it or leave it; either give me my money back, which I know there is no way you can, or give me all your possessions and options for confiscation of possessions of future generations as well, i.e., I’ll give you new loans (which you will also not be able to pay back, I know, but that way I’ll manage to forever take everything you will ever produce in your life and all generations after you. Slaves, obey your masters!)”

Although not essential (Marx analyzed it not like this), the banking system accelerates the condensation of wealth. It is the modus operandi. Money is accumulated. With that money, capital is bought and then the money is re-confiscated with that newly-bought capital, or by means of new loans, etc. It is a feedback system where all money and capital is condensing on a big pile. Money and capital are synonyms. Note that this pile in not necessarily a set of people. It is just ‘the system’. There is no ‘class struggle’ between rich and poor, where the latter are trying to steal/take-back the money (depending on which side of the alleged theft the person analyzing it is). It is a class struggle of people against ‘the system’.

There is only one stable final distribution: all money/capital belonging to one person or institute, one ‘entity’. That is what is called a ‘singularity’ and the only mathematical function that is stable in this case. It is called a delta-function, or Kronecker-delta function: zero everywhere, except in one point, where it is infinite, with the total integral (total money) equal to unity. In this case: all money on one big pile. All other functions are unstable.

Imagine that there are two brothers that wound up with all the money and the rest of the people are destitute and left without anything. These two brothers will then start lending things to each-other. Since they are doing this in the commercial way (having to give back more than borrowed), one of the brothers will confiscate everything from the other.
Note: There is only one way out of it, namely that the brother ‘feels sorry’ for his sibling and gives him things without anything in return, to compensate for the steady unidirectional flow of wealth….

Dialectics of God: Lautman’s Mathematical Ascent to the Absolute. Paper.


Figure and Translation, visit Fractal Ontology

The first of Lautman’s two theses (On the unity of the mathematical sciences) takes as its starting point a distinction that Hermann Weyl made on group theory and quantum mechanics. Weyl distinguished between ‘classical’ mathematics, which found its highest flowering in the theory of functions of complex variables, and the ‘new’ mathematics represented by (for example) the theory of groups and abstract algebras, set theory and topology. For Lautman, the ‘classical’ mathematics of Weyl’s distinction is essentially analysis, that is, the mathematics that depends on some variable tending towards zero: convergent series, limits, continuity, differentiation and integration. It is the mathematics of arbitrarily small neighbourhoods, and it reached maturity in the nineteenth century. On the other hand, the ‘new’ mathematics of Weyl’s distinction is ‘global’; it studies the structures of ‘wholes’. Algebraic topology, for example, considers the properties of an entire surface rather than aggregations of neighbourhoods. Lautman re-draws the distinction:

In contrast to the analysis of the continuous and the infinite, algebraic structures clearly have a finite and discontinuous aspect. Though the elements of a group, field or algebra (in the restricted sense of the word) may be infinite, the methods of modern algebra usually consist in dividing these elements into equivalence classes, the number of which is, in most applications, finite.

In his other major thesis, (Essay on the notions of structure and existence in mathematics), Lautman gives his dialectical thought a more philosophical and polemical expression. His thesis is composed of ‘structural schemas’ and ‘origination schemas’ The three structural schemas are: local/global, intrinsic properties/induced properties and the ‘ascent to the absolute’. The first two of these three schemas close to Lautman’s ‘unity’ thesis. The ‘ascent to the absolute’ is a different sort of pattern; it involves a progress from mathematical objects that are in some sense ‘imperfect’, towards an object that is ‘perfect’ or ‘absolute’. His two mathematical examples of this ‘ascent’ are: class field theory, which ‘ascends’ towards the absolute class field, and the covering surfaces of a given surface, which ‘ascend’ towards a simply-connected universal covering surface. In each case, there is a corresponding sequence of nested subgroups, which induces a ‘stepladder’ structure on the ‘ascent’. This dialectical pattern is rather different to the others. The earlier examples were of pairs of notions (finite/infinite, local/global, etc.) and neither member of any pair was inferior to the other. Lautman argues that on some occasions, finite mathematics offers insight into infinite mathematics. In mathematics, the finite is not a somehow imperfect version of the infinite. Similarly, the ‘local’ mathematics of analysis may depend for its foundations on ‘global’ topology, but the former is not a botched or somehow inadequate version of the latter. Lautman introduces the section on the ‘ascent to the absolute’ by rehearsing Descartes’s argument that his own imperfections lead him to recognise the existence of a perfect being (God). Man (for Descartes) is not the dialectical opposite of or alternative to God; rather, man is an imperfect image of his creator. In a similar movement of thought, according to Lautman, reflection on ‘imperfect’ class fields and covering surfaces leads mathematicians up to ‘perfect’, ‘absolute’ class fields and covering surfaces respectively.

Albert Lautman Dialectics in mathematics

Conjuncted: Occam’s Razor and Nomological Hypothesis. Thought of the Day 51.1.1


Conjuncted here, here and here.

A temporally evolving system must possess a sufficiently rich set of symmetries to allow us to infer general laws from a finite set of empirical observations. But what justifies this hypothesis?

This question is central to the entire scientific enterprise. Why are we justified in assuming that scientific laws are the same in different spatial locations, or that they will be the same from one day to the next? Why should replicability of other scientists’ experimental results be considered the norm, rather than a miraculous exception? Why is it normally safe to assume that the outcomes of experiments will be insensitive to irrelevant details? Why, for that matter, are we justified in the inductive generalizations that are ubiquitous in everyday reasoning?

In effect, we are assuming that the scientific phenomena under investigation are invariant under certain symmetries – both temporal and spatial, including translations, rotations, and so on. But where do we get this assumption from? The answer lies in the principle of Occam’s Razor.

Roughly speaking, this principle says that, if two theories are equally consistent with the empirical data, we should prefer the simpler theory:

Occam’s Razor: Given any body of empirical evidence about a temporally evolving system, always assume that the system has the largest possible set of symmetries consistent with that evidence.

Making it more precise, we begin by explaining what it means for a particular symmetry to be “consistent” with a body of empirical evidence. Formally, our total body of evidence can be represented as a subset E of H, i.e., namely the set of all logically possible histories that are not ruled out by that evidence. Note that we cannot assume that our evidence is a subset of Ω; when we scientifically investigate a system, we do not normally know what Ω is. Hence we can only assume that E is a subset of the larger set H of logically possible histories.

Now let ψ be a transformation of H, and suppose that we are testing the hypothesis that ψ is a symmetry of the system. For any positive integer n, let ψn be the transformation obtained by applying ψ repeatedly, n times in a row. For example, if ψ is a rotation about some axis by angle θ, then ψn is the rotation by the angle nθ. For any such transformation ψn, we write ψ–n(E) to denote the inverse image in H of E under ψn. We say that the transformation ψ is consistent with the evidence E if the intersection

E ∩ ψ–1(E) ∩ ψ–2(E) ∩ ψ–3(E) ∩ …

is non-empty. This means that the available evidence (i.e., E) does not falsify the hypothesis that ψ is a symmetry of the system.

For example, suppose we are interested in whether cosmic microwave background radiation is isotropic, i.e., the same in every direction. Suppose we measure a background radiation level of x1 when we point the telescope in direction d1, and a radiation level of x2 when we point it in direction d2. Call these events E1 and E2. Thus, our experimental evidence is summarized by the event E = E1 ∩ E2. Let ψ be a spatial rotation that rotates d1 to d2. Then, focusing for simplicity just on the first two terms of the infinite intersection above,

E ∩ ψ–1(E) = E1 ∩ E2 ∩ ψ–1(E1) ∩ ψ–1(E2).

If x1 = x2, we have E1 = ψ–1(E2), and the expression for E ∩ ψ–1(E) simplifies to E1 ∩ E2 ∩ ψ–1(E1), which has at least a chance of being non-empty, meaning that the evidence has not (yet) falsified isotropy. But if x1 ≠ x2, then E1 and ψ–1(E2) are disjoint. In that case, the intersection E ∩ ψ–1(E) is empty, and the evidence is inconsistent with isotropy. As it happens, we know from recent astronomy that x1 ≠ x2 in some cases, so cosmic microwave background radiation is not isotropic, and ψ is not a symmetry.

Our version of Occam’s Razor now says that we should postulate as symmetries of our system a maximal monoid of transformations consistent with our evidence. Formally, a monoid Ψ of transformations (where each ψ in Ψ is a function from H into itself) is consistent with evidence E if the intersection

ψ∈Ψ ψ–1(E)

is non-empty. This is the generalization of the infinite intersection that appeared in our definition of an individual transformation’s consistency with the evidence. Further, a monoid Ψ that is consistent with E is maximal if no proper superset of Ψ forms a monoid that is also consistent with E.

Occam’s Razor (formal): Given any body E of empirical evidence about a temporally evolving system, always assume that the set of symmetries of the system is a maximal monoid Ψ consistent with E.

What is the significance of this principle? We define Γ to be the set of all symmetries of our temporally evolving system. In practice, we do not know Γ. A monoid Ψ that passes the test of Occam’s Razor, however, can be viewed as our best guess as to what Γ is.

Furthermore, if Ψ is this monoid, and E is our body of evidence, the intersection

ψ∈Ψ ψ–1(E)

can be viewed as our best guess as to what the set of nomologically possible histories is. It consists of all those histories among the logically possible ones that are not ruled out by the postulated symmetry monoid Ψ and the observed evidence E. We thus call this intersection our nomological hypothesis and label it Ω(Ψ,E).

To see that this construction is not completely far-fetched, note that, under certain conditions, our nomological hypothesis does indeed reflect the truth about nomological possibility. If the hypothesized symmetry monoid Ψ is a subset of the true symmetry monoid Γ of our temporally evolving system – i.e., we have postulated some of the right symmetries – then the true set Ω of all nomologically possible histories will be a subset of Ω(Ψ,E). So, our nomological hypothesis will be consistent with the truth and will, at most, be logically weaker than the truth.

Given the hypothesized symmetry monoid Ψ, we can then assume provisionally (i) that any empirical observation we make, corresponding to some event D, can be generalized to a Ψ-invariant law and (ii) that unconditional and conditional probabilities can be estimated from empirical frequency data using a suitable version of the Ergodic Theorem.

Noneism. Part 2.


Noneism is a very rigourous and original philosophical doctrine, by and large superior to the classical mathematical philosophies. But there are some problems concerning the different ways of characterizing a universe of objects. It is very easy to understand the way a writer characterizes the protagonists of the novels he writes. But what about the characterization of the universe of natural numbers? Since in most kinds of civilizations the natural numbers are characterized the same way, we have the impression that the subject does not intervene in the forging of the characteristics of natural numbers. These numbers appear to be what they are, with total independence of the creative activity of the cognitive subject. There is, of course, the creation of theorems, but the potentially infinite sequence of natural numbers resists any effort to subjectivize its characteristics. It cannot be changed. A noneist might reply that natural numbers are non-existent, that they have no being, and that, in this respect, they are identical with mythological Objects. Moreover, the formal system of natural numbers can be interpreted in many ways: for instance, with respect to a universe of Skolem numbers. This is correct, but it does not explain why the properties of some universes are independent from subjective creation. It is an undeniable fact that there are two kinds of objectual characteristics. On the one hand, we have the characteristics created by subjective imagination or speculative thought; on the other hand, we find some characteristics that are not created by anybody; their corresponding Objects are, in most cases, non-existent but, at the same time, they are not invented. They are just found. The origin of the former characteristics is very easy to understand; the origin of the last ones is, a mystery.

Now, the subject-independence of a universe, suggests that it belongs to a Platonic realm. And as far as transafinite set theory is concerned, the subject-independence of its characteristics is much less evident than the characteristic subject-independence of the natural numbers. In the realm of the finite, both characteristics are subject-independent and can be reduced to combinatorics. The only difference between both is that, according to the classical Platonistic interpretation of mathematics, there can only be a single mathematical universe and that, to deductively study its properties, one can only employ classical logic. But this position is not at all unobjectionable. Once the subject-independence of the natural numbers system’s characteristics is posited, it becomes easy to overstep the classical phobia concerning the possibility of characterizing non-classical objective worlds. Euclidean geometry is incompatible with elliptical and hyperbolic geometries and, nevertheless, the validity of the first one does not invalidate the other ones. And vice versa, the fact that hyperbolic and other kinds of geometry are consistently characterized, does not invalidate the good old Euclidean geometry. And the fact that we have now several kinds of non-Cantorian set theories, does not invalidate the classical Cantorian set theory.

Of course, an universally non-Platonic point of view that includes classical set theory can also be assumed. But concerning natural numbers it would be quite artificial. It is very difficult not to surrender to the famous Kronecker’s dictum: God created natural numbers, men created all the rest. Anyhow, it is not at all absurd to adopt a whole platonistic conception of mathematics. And it is quite licit to adopt a noneist position. But if we do this, the origin of the natural numbers’ characteristics becomes misty. However, forgetting this cloudiness, the leap from noneist universes to the platonistic ones, and vice versa, becomes like a flip-flop connecting objectological with ontological (ideal) universes, like a kind of rabbit-duck Gestalt or a Sherrington staircase. So, the fundamental question with respect to the subject-dependent or subject-independent mathematical theories, is: are they created, or are they found? Regarding some theories, subject-dependency is far more understandable; and concerning other ones, subject-independency is very difficult, if not impossible, to negate.

From an epistemological point of view, the fact of non-subject dependent characteristic traits of a universe would mean that there is something like intellectual intuition. The properties of natural numbers, the finite properties of sets (or combinatorics), some geometric axioms, for instance, in Euclidean geometry, the axioms of betweenness, etc., would be apprehended in a manner, that pretty well coincides with the (nowadays rather discredited) concept of synthetical a priori knowledge. This aspect of mathematical knowledge shows that the old problem concerning the analytic and the a priori synthetical knowledge, in spite of the prevailing Quinean pragmatic conception, must be radically reset.

Organic and the Orgiastic. Cartography of Ground and Groundlessness in Deleuze and Heidegger. Thought of the Day 43.0


In his last hermeneutical Erörterung of Leibniz, The Principle of Ground, Heidegger traces back metaphysics to its epochal destiny in the twofold or duplicity (Zwiefalt) of Being and Thought and thus follows the ground in its self-ungrounding (zugrundegehen). Since the foundation of thought is also the foundation of Being, reason and ground are not equal but belong together (zusammenhören) in the Same as the ungrounded yet historical horizon of the metaphysical destiny of Being: On the one hand we say: Being and ground: the Same. On the other hand we say: Being: the abyss (Ab-Grund). What is important is to think the univocity (Einsinnigkeit) of both Sätze, those Sätze that are no longer Sätze. In Difference and Repetition, similarly, Deleuze tells us that sufficient reason is twisted into the groundless. He confirms that the Fold (Pli) is the differenciator of difference engulfed in groundlessness, always folding, unfolding, refolding: to ground is always to bend, to curve and recurve. He thus concludes:

Sufficient reason or ground is strangely bent: on the one hand, it leans towards what it grounds, towards the forms of representation; on the other hand, it turns and plunges into a groundless beyond the ground which resists all forms and cannot be represented.

Despite the fundamental similarity of their conclusions, however, our short overview of Deleuze’s transformation of the Principle of Sufficient Reason has already indicated that his argumentation is very different from Heideggerian hermeneutics. To ground, Deleuze agrees, is always to ground representation. But we should distinguish between two kinds of representation: organic or finite representation and orgiastic or infinite representation. What unites the classicisms of Kant, Descartes and Aristotle is that representation retains organic form as its principle and the finite as its element. Here the logical principle of identity always precedes ontology, such that the ground as element of difference remains undetermined and in itself. It is only with Hegel and Leibniz that representation discovers the ground as its principle and the infinite as its element. It is precisely the Principle of Sufficient Reason that enables thought to determine difference in itself. The ground is like a single and unique total moment, simultaneously the moment of the evanescence and production of difference, of disappearance and appearance. What the attempts at rendering representation infinite reveal, therefore, is that the ground has not only an Apollinian, orderly side, but also a hidden Dionysian, orgiastic side. Representation discovers within itself the limits of the organized; tumult, restlessness and passion underneath apparent calm. It rediscovers monstrosity.

The question then is how to evaluate this ambiguity that is essential to the ground. For Heidegger, the Zwiefalt is either naively interpreted from the perspective of its concave side, following the path of the history of Western thought as the belonging together of Being and thought in a common ground; or it is meditated from its convex side, excavating it from the history of the forgetting of Being the decline of the Fold (Wegfall der Zwiefalt, Vorenthalt der Zwiefalt) as the pivotal point of the Open in its unfolding and following the path that leads from the ground to the abyss. Instead of this all or nothing approach, Deleuze takes up the question in a Nietzschean, i.e. genealogical fashion. The attempt to represent difference in itself cannot be disconnected from its malediction, i.e. the moral representation of groundlessness as a completely undifferentiated abyss. As Bergson already observed, representational reason poses the problem of the ground in terms of the alternative between order and chaos. This goes in particular for the kind of representational reason that seeks to represent the irrepresentable: Representation, especially when it becomes infinite, is imbued with a presentiment of groundlessness. Because it has become infinite in order to include difference within itself, however, it represents groundlessness as a completely undifferentiated abyss, a universal lack of difference, an indifferent black nothingness. Indeed, if Deleuze is so hostile to Hegel, it is because the latter embodies like no other the ultimate illusion inseparable from the Principle of Sufficient Reason insofar as it grounds representation, namely that groundlessness should lack differences, when in fact it swarms with them.

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The Sibyl’s Prophecy/Nordic Creation. Note Quote.



The Prophecy of the Tenth Sibyl, a medieval best-seller, surviving in over 100 manuscripts from the 11th to the 16th century, predicts, among other things, the reign of evil despots, the return of the Antichrist and the sun turning to blood.

The Tenth or Tiburtine Sibyl was a pagan prophetess perhaps of Etruscan origin. To quote Lactantus in his general account of the ten sibyls in the introduction, ‘The Tiburtine Sibyl, by name Albunea, is worshiped at Tibur as a goddess, near the banks of the Anio in which stream her image is said to have been found, holding a book in her hand’.

The work interprets the Sibyl’s dream in which she foresees the downfall and apocalyptic end of the world; 9 suns appear in the sky, each one more ugly and bloodstained than the last, representing the 9 generations of mankind and ending with Judgment Day. The original Greek version dates from the end of the 4th century and the earliest surviving manuscript in Latin is dated 1047. The Tiburtine Sibyl is often depicted with Emperor Augustus, who asks her if he should be worshipped as a god.

The foremost lay of the Elder Edda is called Voluspa (The Sibyl’s Prophecy). The volva, or sibyl, represents the indelible imprint of the past, wherein lie the seeds of the future. Odin, the Allfather, consults this record to learn of the beginning, life, and end of the world. In her response, she addresses Odin as a plurality of “holy beings,” indicating the omnipresence of the divine principle in all forms of life. This also hints at the growth of awareness gained by all living, learning entities during their evolutionary pilgrimage through spheres of existence.

Hear me, all ye holy beings, greater as lesser sons of Heimdal! You wish me to tell of Allfather’s works, tales of the origin, the oldest I know. Giants I remember, born in the foretime, they who long ago nurtured me. Nine worlds I remember, nine trees of life, before this world tree grew from the ground.

Paraphrased, this could be rendered as:

Learn, all ye living entities, imbued with the divine essence of Odin, ye more and less evolved sons of the solar divinity (Heimdal) who stands as guardian between the manifest worlds of the solar system and the realm of divine consciousness. You wish to learn of what has gone before. I am the record of long ages past (giants), that imprinted their experience on me. I remember nine periods of manifestation that preceded the present system of worlds.

Time being inextricably a phenomenon of manifestation, the giant ages refer to the matter-side of creation. Giants represent ages of such vast duration that, although their extent in space and time is limited, it is of a scope that can only be illustrated as gigantic. Smaller cycles within the greater are referred to in the Norse myths as daughters of their father-giant. Heimdal is the solar deity in the sign of Aries – of beginnings for our system – whose “sons” inhabit, in fact compose, his domain.

Before a new manifestation of a world, whether a cosmos or a lesser system, all its matter is frozen in a state of immobility, referred to in the Edda as a frost giant. The gods – consciousnesses – are withdrawn into their supernal, unimaginable abstraction of Nonbeing, called in Sanskrit “paranirvana.” Without a divine activating principle, space itself – the great container – is a purely theoretical abstraction where, for lack of any organizing energic impulse of consciousness, matter cannot exist.

This was the origin of ages when Ymer built. No soil was there, no sea, no cool waves. Earth was not, nor heaven above; Gaping Void alone, no growth. Until the sons of Bur raised the tables; they who created beautiful Midgard. The sun shone southerly on the stones of the court; then grew green herbs in fertile soil.

To paraphrase again:

Before time began, the frost giant (Ymer) prevailed. No elements existed for there were ‘no waves,’ no motion, hence no organized form nor any temporal events, until the creative divine forces emanated from Space (Bur — a principle, not a locality) and organized latent protosubstance into the celestial bodies (tables at which the gods feast on the mead of life-experience). Among these tables is Middle Court (Midgard), our own beautiful planet. The life-giving sun sheds its radiant energies to activate into life all the kingdoms of nature which compose it.

The Gaping Void (Ginnungagap) holds “no cool waves” throughout illimitable depths during the age of the frost giant. Substance has yet to be created. Utter wavelessness negates it, for all matter is the effect of organized, undulating motion. As the cosmic hour strikes for a new manifestation, the ice of Home of Nebulosity (Niflhem) is melted by the heat from Home of Fire (Muspellshem), resulting in vapor in the void. This is Ymer, protosubstance as yet unformed, the nebulae from which will evolve the matter components of a new universe, as the vital heat of the gods melts and vivifies the formless immobile “ice.”

When the great age of Ymer has run its course, the cow Audhumla, symbol of fertility, “licking the salt from the ice blocks,” uncovers the head of Buri, first divine principle. From this infinite, primal source emanates Bur, whose “sons” are the creative trinity: Divine Allfather, Will, and Sanctity (Odin, Vile, and Vi). This triune power “kills” the frost giant by transforming it into First Sound (Orgalmer), or keynote, whose overtones vibrate through the planes of sleeping space and organize latent protosubstance into the multifarious forms which will be used by all “holy beings” as vehicles for gaining experience in worlds of matter.

Beautiful Midgard, our physical globe earth, is but one of the “tables” raised by the creative trinity, whereat the gods shall feast. The name Middle Court is suggestive, for the ancient traditions place our globe in a central position in the series of spheres that comprise the terrestrial being’s totality. All living entities, man included, comprise besides the visible body a number of principles and characteristics not cognized by the gross physical senses. In the Lay of Grimner (Grimnismal), wherein Odin in the guise of a tormented prisoner on earth instructs a human disciple, he enumerates twelve spheres or worlds, all but one of which are unseen by our organs of sight. As to the formation of Midgard, he relates:

Of Ymer’s flesh was the earth formed, the billows of his blood, the mountains of his bones, bushes of his hair, and of his brainpan heaven. With his eyebrows beneficent powers enclosed Midgard for man; but of his brain were surely all dark skies created.

The trinity of immanent powers organize Ymer into the forms wherein they dwell, shaping the chaos or frost giant into living globes on many planes of being. The “eyebrows” that gird the earth and protect it suggest the Van Allen belts that shield the planet from inimical radiation. The brain of Ymer – material thinking – is surely all too evident in the thought atmosphere wherein man participates.

The formation of the physical globe is described as the creation of “dwarfs” – elemental forces which shape the body of the earth-being and which include the mineral. vegetable, and animal kingdoms.

The mighty drew to their judgment seats, all holy gods to hold counsel: who should create a host of dwarfs from the blood of Brimer and the limbs of the dead. Modsogne there was, mightiest of all the dwarfs, Durin the next; there were created many humanoid dwarfs from the earth, as Durin said.

Brimer is the slain Ymer, a kenning for the waters of space. Modsogne is the Force-sucker, Durin the Sleeper, and later comes Dvalin the Entranced. They are “dwarf”-consciousnesses, beings that are miðr than human – the Icelandic miðr meaning both “smaller” and “less.” By selecting the former meaning, popular concepts have come to regard them as undersized mannikins, rather than as less evolved natural species that have not yet reached the human condition of intelligence and self-consciousness.

During the life period or manifestation of a universe, the governing giant or age is named Sound of Thor (Trudgalmer), the vital force which sustains activity throughout the cycle of existence. At the end of this age the worlds become Sound of Fruition (Bargalmer). This giant is “placed on a boat-keel and saved,” or “ground on the mill.” Either version suggests the karmic end product as the seed of future manifestation, which remains dormant throughout the ensuing frost giant of universal dissolution, when cosmic matter is ground into a formless condition of wavelessness, dissolved in the waters of space.

There is an inescapable duality of gods-giants in all phases of manifestation: gods seek experience in worlds of substance and feast on the mead at stellar and planetary tables; giants, formed into vehicles inspired with the divine impetus, rise through cycles of this association on the ladder of conscious awareness. All states being relative and bipolar, there is in endless evolution an inescapable link between the subjective and objective progress of beings. Odin as the “Opener” is paired with Orgalmer, the keynote on which a cosmos is constructed; Odin as the “Closer” is equally linked with Bargalmer, the fruitage of a life cycle. During the manifesting universe, Odin-Allfather corresponds to Trudgalmer, the sustainer of life.

A creative trinity plays an analogical part in the appearance of humanity. Odin remains the all-permeant divine essence, while on this level his brother-creators are named Honer and Lodur, divine counterparts of water or liquidity, and fire or vital heat and motion. They “find by the shore, of little power” the Ash and the Elm and infuse into these earth-beings their respective characteristics, making a human image or reflection of themselves. These protohumans, miniatures of the world tree, the cosmic Ash, Yggdrasil, in addition to their earth-born qualities of growth force and substance, receive the divine attributes of the gods. By Odin man is endowed with spirit, from Honer comes his mind, while Lodur gives him will and godlike form. The essentially human qualities are thus potentially divine. Man is capable of blending with the earth, whose substances form his body, yet is able to encompass in his consciousness the vision native to his divine source. He is in fact a minor world tree, part of the universal tree of life, Yggdrasil.

Ygg in conjunction with other words has been variously translated as Eternal, Awesome or Terrible, and Old. Sometimes Odin is named Yggjung, meaning the Ever-Young, or Old-Young. Like the biblical “Ancient of Days” it is a concept that mind can grasp only in the wake of intuition. Yggdrasil is the “steed” or the “gallows” of Ygg, whereon Odin is mounted or crucified during any period of manifested life. The world tree is rooted in Nonbeing and ramifies through the planes of space, its branches adorned with globes wherein the gods imbody. The sibyl spoke of ours as the tenth in a series of such world trees, and Odin confirms this in The Song of the High One (Den Hoges Sang):

I know that I hung in the windtorn tree nine whole nights, spear-pierced, given to Odin, my self to my Self above me in the tree, whose root none knows whence it sprang. None brought me bread, none served me drink; I searched the depths, spied runes of wisdom, raised them with song, and fell once more from the tree. Nine powerful songs I learned from the wise son of Boltorn, Bestla’s father; a draught I drank of precious mead ladled from Odrorer. I began to grow, to grow wise, to grow greater and enjoy; for me words from words led to new words, for me deeds from deeds led to new deeds.

Numerous ancient tales relate the divine sacrifice and crucifixion of the Silent Watcher whose realm or protectorate is a world in manifestation. Each tree of life, of whatever scope, constitutes the cross whereon the compassionate deity inherent in that hierarchy remains transfixed for the duration of the cycle of life in matter. The pattern of repeated imbodiments for the purpose of gaining the precious mead is clear, as also the karmic law of cause and effect as words and deeds bring their results in new words and deeds.

Yggdrasil is said to have three roots. One extends into the land of the frost giants, whence flow twelve rivers of lives or twelve classes of beings; another springs from and is watered by the well of Origin (Urd), where the three Norns, or fates, spin the threads of destiny for all lives. “One is named Origin, the second Becoming. These two fashion the third, named Debt.” They represent the inescapable law of cause and effect. Though they have usually been roughly translated as Past, Present, and Future, the dynamic concept in the Edda is more complete and philosophically exact. The third root of the world tree reaches to the well of the “wise giant Mimer,” owner of the well of wisdom. Mimer represents material existence and supplies the wisdom gained from experience of life. Odin forfeited one eye for the privilege of partaking of these waters of life, hence he is represented in manifestation as one-eyed and named Half-Blind. Mimer, the matter-counterpart, at the same time receives partial access to divine vision.

The lays make it very clear that the purpose of existence is for the consciousness-aspect of all beings to gain wisdom through life, while inspiring the substantial side of itself to growth in inward awareness and spirituality. At the human level, self-consciousness and will are aroused, making it possible for man to progress willingly and purposefully toward his divine potential, aided by the gods who have passed that way before him, rather than to drift by slow degrees and many detours along the road of inevitable evolution. Odin’s instructions to a disciple, Loddfafner, the dwarf-nature in man, conclude with:

Now is sung the High One’s song in the High One’s hall. Useful to sons of men, useless to sons of giants. Hail Him who sang! Hail him who kens! Rejoice they who understand! Happy they who heed!

The Womb of Cosmogony. Thought of the Day 30.0

Nowhere and by no people was speculation allowed to range beyond those manifested gods. The boundless and infinite UNITY remained with every nation a virgin forbidden soil, untrodden by man’s thought, untouched by fruitless speculation. The only reference made to it was the brief conception of its diastolic and systolic property, of its periodical expansion or dilatation, and contraction. In the Universe with all its incalculable myriads of systems and worlds disappearing and re-appearing in eternity, the anthropomorphised powers, or gods, their Souls, had to disappear from view with their bodies: — “The breath returning to the eternal bosom which exhales and inhales them,” says our Catechism. . . . In every Cosmogony, behind and higher than the creative deity, there is a superior deity, a planner, an Architect, of whom the Creator is but the executive agent. And still higher, over and around, withinand without, there is the UNKNOWABLE and the unknown, the Source and Cause of all these Emanations. – The Secret Doctrine


Many are the names in the ancient literatures which have been given to the Womb of Being from which all issues, in which all forever is, and into the spiritual and divine reaches of which all ultimately returns, whether infinitesimal entity or macrocosmic spacial unit.

The Tibetans called this ineffable mystery Tong-pa-nnid, the unfathomable Abyss of the spiritual realms. The Buddhists of the Mahayana school describe it as Sunyata or the Emptiness, simply because no human imagination can figurate to itself the incomprehensible Fullness which it is. In the Eddas of ancient Scandinavia the Boundless was called by the suggestive term Ginnungagap – a word meaning yawning or uncircumscribed void. The Hebrew Bible states that the earth was formless and void, and darkness was upon the face of Tehom, the Deep, the Abyss of Waters, and therefore the great Deep of kosmic Space. It has the identical significance of the Womb of Space as envisioned by other peoples. In the Chaldaeo-Jewish Qabbalah the same idea is conveyed by the term ‘Eyn (or Ain) Soph, without bounds. In the Babylonian accounts of Genesis, it is Mummu Tiamatu which stands for the Great Sea or Deep. The archaic Chaldaean cosmology speaks of the Abyss under the name of Ab Soo, the Father or source of knowledge, and in primitive Magianism it was Zervan Akarana — in its original meaning of Boundless Spirit instead of the later connotation of Boundless Time.

In the Chinese cosmogony, Tsi-tsai, the Self-Existent, is the Unknown Darkness, the root of the Wuliang-sheu, Boundless Age. The wu wei of Lao-tse, often mistranslated as passivity and nonaction, imbodies a similar conception. In the sacred scriptures of the Quiches of Guatemala, the Popol Vuh or “Book of the Azure Veil,” reference is made to the “void which was the immensity of the Heavens,” and to the “Great Sea of Space.” The ancient Egyptians spoke of the Endless Deep; the same idea also is imbodied in the Celi-Ced of archaic Druidism, Ced being spoken of as the “Black Virgin” — Chaos — a state of matter prior to manvantaric differentiation.

The Orphic Mysteries taught of the Thrice-Unknown Darkness or Chronos, about which nothing could be predicated except its timeless Duration. With the Gnostic schools, as for instance with Valentinus, it was Bythos, the Deep. In Greece, the school of Democritus and Epicurus postulated To Kenon, the Void; the same idea was later voiced by Leucippus and Diagoras. But the two most common terms in Greek philosophy for the Boundless were Apeiron, as used by Plato, Anaximander and Anaximenes, and Apeiria, as used by Anaxagoras and Aristotle. Both words had the significance of frontierless expansion, that which has no circumscribing bounds.

The earliest conception of Chaos was that almost unthinkable condition of kosmic space or kosmic expanse, which to human minds is infinite and vacant extension of primordial Aether, a stage before the formation of manifested worlds, and out of which everything that later existed was born, including gods and men and all the celestial hosts. We see here a faithful echo of the archaic esoteric philosophy, because among the Greeks Chaos was the kosmic mother of Erebos and Nyx, Darkness and Night — two aspects of the same primordial kosmic stage. Erebos was the spiritual or active side corresponding to Brahman in Hindu philosophy, and Nyx the passive side corresponding to pradhana or mulaprakriti, both meaning root-nature. Then from Erebos and Nyx as dual were born Aether and Hemera, Spirit and Day — Spirit being here again in this succeeding stage the active side, and Day the passive aspect, the substantial or vehicular side. The idea was that just as in the Day of Brahma of Hindu cosmogony things spring into active manifested existence, so in the kosmic Day of the Greeks things spring from elemental substance into manifested light and activity, because of the indwelling urge of the kosmic Spirit.